A Constant Force Bicycle Transmission

Rochester Institute of Technology RIT Scholar Works

Theses

8-1-1983

Thomas Chase

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Recommended Citation Chase, Thomas, "A constant force bicycle transmission" (1983). Thesis. Rochester Institute of Technology. Accessed from

This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. A CONSTANT FORCE BICYCLE TRANSMISSION by Thomas R. Chase

A Thesis Project Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mechanical Engineering

Approved by: Prof. Richard Budynas Thesis Adviser

Prof. Dr. Bhalchandra V. Karlekar Department Head

Prof. '"egible Signature

Prof. Ray C. Johnson

DEPARTMENT OF MECHANICAL ENGINEERING ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NHJ YORK August 1983 A CONSTANT FORCE BICYCLE TRANSMISSION

ABSTRACT

A prototype design for a human powered automatic transmission intended for use on an ordinary touring bicycle is presented. The transmission is intended to automatically adjust the gearing of the bicycle to maintain an optimum pedal force, regardless of the current riding conditions. Therefore, the transmission eliminates the need for the cyclist to manually adjust the bicycle gearing. The entire transmission is a self-contained unit designed to bolt onto the rear wheel of an otherwise unmodified 27-inch bicycle.

The transmission combines a unique adaptation of a commercially popular continuously variable traction drive with a totally mechanical integral feedback controller. The features of the traction drive unique to its application to a bicycle are outlined in detail, along with an analysis of the important traction drive design parameters.

The control system is also uniquely adapted to the requirements of bicycling. A detailed classical analysis of the controller is pre sented to verify its desirable performance characteristics. In addi tion, a numerical simulation of the transmission is included to accurately predict its performance under typical operating condi tions.

A knowledge of typical pedal force profiles for normal touring cycling is prerequisite for the design of the constant force bi cycle transmission. This data has not been previously published. n

An instrumentation package was developed especially to determine this data. A description of the system and a summary of the results are presented. These results are reduced to a set of design criteria for the constant force transmission. Table of Contents

Abstract i

List of Symbols 1

1. Introduction 7

1.1) Scope 7

of 1.2) History the Constant Force Bicycle Transmission . 9

1.3) History of the Pedal Data Instrumentation 12

1.4) Summary 15

2. Overview of the Constant Force Bicycle Transmission ... 16

2.1) Introduction 16

2.2) Continuously Variable Transmission Overview 16

2.3) Control System Overview 19

2.4) Additional Features 28

3. Pedal Data Instrumentation 31

3.1) Introduction 31

3.2) System Overview \ . . 31

3.3) Pedal Force Sensor 35

3.4) Pedal Position Sensor 48

3.5) Frequency to Voltage Converter 53

4. Design Criteria for the Constant Force Bicycle Transmission 60

4.1) Introduction 60

4.2) Overview of the Test Data 60

i i i IV

4.3 Estimated Pedal Crank Velocity. . 63

4.4 Pedal Force Profile 64

4.5 Estimated Ideal Pedal Force . . . 65

4.6 Design Maximum Pedal Force. . . . 68

4.7 Estimated Loading Schedule. . . . 71

The Continuously Variable Transmission 75

5.1 Introduction 75

5.2 Principle of Operation 76

5.3 Unique Characteristics of the Constant Force Bicycle Transmission CVT 80

5.4) CVT Analysis 91

5.4.1) Worst-Case Tractive Force Requirements . . . 91

5.4.2) Derivation of the Traction Ball Normal Force 93

5.4.3) Traction Fluid Performance 99

5.4.4) Maximum Stress in the CVT 102

5.4.5) Transmission Life Estimate 105

The Constant Force Controller 106

6.1 Introduction 106

6.2 Principles of Operation 107

6.3 Basic Modelling of the Constant Force Controller. . 113

6.4 Shift Torque Requirements 118

6.5 Modelling the Control System Input 123

6.6 Numerically Modelling the Constant Force Controller 128

6.7 Simulation of the Control System Performance. . . . 132

6.8 Linear Model of the Constant Force Controller . . . 153 -V-

6.9) Frequency Response Analysis of the Constant Force Controller 164

7. Conclusion 173

Acknowledgments 176

Bibliography 177

Appendix: Constant Force Controller Simulation Program . . . .180 List of Symbols c-j : Chain reduction between the output of the continuously variable transmission and the rear wheel of the bicycle

c- : Total chain reduction between the pedal crank and the input

of the continuously variable transmission

c- : Total geardown between the integrator driven wheel, a,

and the traction ball shift angle, e

c. : Leverage of the floating traction ball lever

f : Frequency of the sinusoidal input force from the pedal crank (hz)

F : A general force (lb)

f : Corner frequency of the constant force controller (hz)

Fp : Force in the chain driving the input sprocket of the continuously variable transmission (lb)

FCR : Normal force of a traction ball against a cradle bearing of the continuously variable transmission (lb)

FD : Peak value of the ideal pedal force (lb)

F. : Normal force between the integrator driving disc and the

integrator driven wheel (lb)

FN : Normal force of a traction ball against the input or output disc of the continuously variable transmission (lb)

Fp : Pedal force (lb)

-1- -2- fpp : Output frequency of the pedal position sensor instrumen tation (hz)

Fg : Sum of the forces in the two springs of the chain force sensor linkage (lb)

F__t : Force in the equivalent spring (with spring constant k )

of the force sensor linkage when the force sensor linkage

displacement, y, is zero (lb)

F-.R : Tractive (frictional) force between the traction ball and the input or output disc of the continuously variable transmission (lb)

F,, : Total wedging force of a traction ball into the input and

output discs of the continuously variable transmission

(measured in the plane parallel to the input and output discs) (lb)

G : Transfer function of the constant force controller

K : Gain of the constant force controller

of parallel chain k . : Actual spring constant of each the two act force sensor linkage springs (lb/in.)

k : Equivalent spring constant of the two parallel chain eq force sensor linkage springs acting through the chain

force sensor linkage (lb/in.)

M.A. : Mechanical advantage of the chain force sensor linkage

balls rb : Spherical radius of the traction (in.) -3-

: Radius of the rdisc input or output disc of the continuously

variable transmission at the point of contact with the traction ball (in.)

ri : Radius of the integrator driven wheel (in.)

r-jn : Radius from the axis of rotation of the traction ball to the point of contact with the input disc (in.)

rout : Radius from the axis of rotation of the traction ball to the point of contact with the output disc (in.)

rpc : Length of the pedal crank (in.)

rSCVT : Pitch radius of the input sprocket of the continuously variable transmission (in.)

s : A complex number

t : Time (sec)

T : Period of the pedal crank rotation (sec)

T : Time constant of the constant force controller (sec)

TDW : Torque developed by the integrator driven wheel (lb-in.)

T. : Input torque to the continuously variable transmission (lb-in.)

: Constant torque value for T T-_v f (lb-in.) -4-

Tout : Output torque from the continuously variable transmission (lb-in.)

Tref : Function describing the peak value of the sinusoidal torque required to drive the rear wheel of the bicycle (lb-in.)

TRW : Torque required to drive the rear wheel of the bicycle (lb-in.)

Tshl.ft : Torque required to change the shift angle of a traction ball of the continuously variable transmission (lb-in.)

T"w : Input torque to a worm gear of the continuously variable transmission (lb-in.)

Tw . : Output torque from a worm gear of the continuously variable transmission (lb-in.)

response of model of constant force Too/ I,---- 2% time the linear the .c/d, ii near controller to a true step input (sec)

x : Input function to the linear model of the constant force

controller (lb)

X : Magnitude of the sinusoidal input function to the linear

model of the constant force controller (lb)

y : Chain force sensor linkage displacement; i.e., the relative

displacement between the center of the integrator driving

disc and the center! ine of the integrator driven wheel (in.) ymax : The maximum allowable displacement of the chain force sensor linkage (in. )

a : Angular displacement of the integrator driven wheel (rad)

y : Angular displacement of the right pedal crank from top dead center (rad)

n : Efficiency of the worm gears of the continuously variable transmission

e : Shift angle of the continuously variable transmission (rad, deg)

: state shift angle of the variable ecc3SS Steady continuously transmission (deg)

y. : Coefficient of friction between the integrator driving

disc and the integrator driven wheel

u : Maximum assumed value of the coefficient of friction of nriax the traction fluid of the continuously variable transmis

sion

traction u . : Minimum permissible coefficient of friction of the mm fluid of the continuously variable transmission

a : Compressive stress (lb/in.2)

function to the linear w : Frequency of the sinusoidal input

model of the constant force controller (rad/sec) -6-

: Corner fc>c frequency of the constant force controller (rad/sec)

to . : Angular -i velocity of the integrator driving disc (rad/sec)

: An9ular pedal velocity of the pedal crank (rad/sec) Chapter 1

1. Introduction

1 . 1 ) Scope

The prototype design for a constant force bicycle transmission

is presented here. The constant force transmission is a human

powered automatic transmission intended for use on an ordinary 27

inch touring bicycle. The transmission is designed to automatical

ly adjust the bicycle gearing to maintain the pedal force at a

rider-prescribed level under all riding conditions (assuming this

is possible within the gearing limits of the transmission). Thus,

the transmission eliminates the need for the cyclist to manually

adjust the bicycle gearing, as on an ordinary 10-speed, making it

simpler for the rider to maintain an optimal power output at all

times.

The transmission is a self-contained combination of a con

tinuously variable speed drive and control system (Figure 1.1).

The unit is intended to bolt on the rear wheel shaft of an other

wise unmodified bicycle frame; it replaces the standard 10-speed

derailleur. The transmission is driven by the chain of a standard

front pedal crank. The unit requires no external power.

A complete description of the prototype design and an

performance is analysis of its mechanical integrity and predicted

presented here. The prototype has not been constructed at the

time of this writing; however, the analysis presented here fully

justifies its actual construction. All parts in the transmission

-7- EZ o T Ul I/) r- E in a ta s-

o >

in c o o

i- o cr> -9-

have been dimensioned with full consideration for adequate sizing to withstand the expected loading (detailed in Chapter 4). All assembly drawings included in this report are based on the actual dimensions.

The actual prototype transmission will have three undesirable characteristics: it will be large, heavy, and expensive. However, the "overdesigned" prototype has been to help insure its proper

initial operation. For example, the control system has been de signed to produce the maximum power that the variable ratio drive could reasonably require while shifting (see Chapter 6), and all shafts subjected to fatigue loading have been designed for infinite life at the maximum expected input load. Therefore, if the proto type performance is encouraging, optimization of the design will substantially reduce its size, cost, and weight. A suitably opti mized design would be intended for marketing as a bolt-on substi- tute for derailleur gearing on an ordinary 10-speed touring frame.

2) History of the Constant Force Bicycle Transmission

Widespread interest in alternatives to the conventional derailleur and multispeed hub gears for use as bicycle transmis sions has existed for many years (reference 21). However, only a few attempts at a transmission that automatically adjusts the gearing to maintain optimal power input from the rider have been published. The known examples are summarized in this section. -10-

r^rj\>_

1*1 \ I tic

Figure 1.2) The BTCA Automatic Bicycle Transmission (Photos extracted from Reference 5)

The only automatic transmission known to have actually been

marketed is the BTCA drive (reference 5) . This transmission is also the most similar of earlier attempts to the constant force bicycle transmission; it combines a continuously variable trans mission with a mechanical control system. However, both of these elements are ^ery different from their counterparts in the constant force bicycle transmission. The continuously variable transmission of the BTCA drive is an unusual geometry consisting of a planetary gear-like system with incrementally-driven planets. The planets

1 transmis The only information available to the author on this sion is an advertising brochure from its developer, the Bicycle Technology Corporation of America (reference 5). All observa tions made here are based on that brochure. The commercial status of the device is unknown. -11-

are driven a by variable eccentric; the degree of eccentricity of

the driver determines the shift ratio. The transmission is adver

tised as torque" "responding to input speed and with an "inertial

device" control and a "centrifical anticipatory control". The

principle of operation of the controller is not clear; it appears

to be a proportional system based on the speed of the chain and

the rear wheel. The unit is housed in a custom rear wheel hub.

The transmission and bicycle frame are apparently sold only as an

integral unit.

A second attempt at an automatic transmission combines a

proportional force-sensitive controller on a variable ratio front

sprocket (reference 15). The variable sprocket itself is a 16-step

version of a Hagen-type geometry (reference 23) . Proportional

force control is obtained by spring-loading the variable pitch

elements of the sprocket.

Several attempts have been made at hydraulic bicycle trans missions. The typical configuration consists of a pedal-powered

hydraulic pump driving a hydraulic motor on the rear wheel (refer

ences 4 and 25); reference (4) mentions the possibility of includ

ing automatic shift features. Another design (reference 19) utilizes a hydraulic pump attached to the ring gear of a planetary gear set to vary the bicycle shift ratio. The practicality of such systems is questionable because of the low efficiencies

Variable pitch sprocket drives have not become popular; they apparently tend to throw the chain. -12-

associated with hydraulics (on the order of 80% at best); the

efficiency of a bicycle transmission must exceed 90% to be com

petitive with conventional positive drives (reference 21).

Several other unusual bicycle transmissions have been proposed.

Most attempt to modify the standard circular pedal crank cycle.

A representative summary is provided in reference (11). Outside

of those already mentioned, none address the problem of automati

cally adjusting the gearing.

The constant force bicycle transmission is unique from all

other automatic bicycle transmissions in providing integral con

trol of the pedal force. The advantages of this approach will be

clarified in Chapter 2. The high-efficiency traction drive uti

lized in the design has also never previously been successfully

adapted to a bicycle.

1. 3) History of the Pedal Data Instrumentation

Knowledge of typical pedal force profiles, describing the

input to the constant force bicycle transmission, is prerequisite

to designing the transmission. Specifically, the pedal crank

speed, the general shape of the pedal force input, the ideal pedal

force (i.e., the pedal force the constant force transmission will

attempt to maintain), and the design maximum pedal force expected

from average cyclists when riding a modern lightweight touring

bicycle under normal conditions are required. Little of this

information has been previously published. Therefore, an -13-

instrumentation package was custom designed to determine these

data; the instrumentation and resulting data are discussed in

Chapters 3 and 4, respectively. However, a great deal of ergo-

nomic data concerning bicycling has been published; the content

of these studies is briefly summarized here.

An early study of the force exerted on a bicycle pedal under

actual riding conditions is described in reference (16). A

mechanical (clockwork-driven) recorder was attached to a spring-

"ordinary" loaded pedal of an bicycle. The data is interesting

in that it shares the approximately sinusoidal pedal force profile

determined in Chapter 4; however, the applied forces differ sub

stantially from that applied to a modern 10-speed frame.

Additional data concerning average cyclists under normal

touring conditions is extremely limited. Reference (1) addresses

this problem; however, the pedal force is not measured. Instead,

the power output of the rider is estimated by measuring the oxygen

consumption of the rider. Reference (7) utilizes a similar tech

nique to evaluate the effect of tire size on bicycling efficiency.

Reference (22) compares the oxygen consumption technique of the

previous references with power estimates based on a knowledge of the expected tractive resistances; once again, the pedal force is not examined. Reference (21) presents an interesting summary of several similar ergonomic studies of bicycling, emphasizing the maximum possible power output of a cyclist. -14-

Three additional references examine the actual pedal force exerted by a cyclist; however, none of these studies have been

"average" conducted under conditions. Reference (18) measures the pedal force exerted by racing cyclists at start-up, in addi tion to examining the output of racing cyclists with stationary dynamometer and photographic studies. Reference (6) studies pedalling efficiency with an instrumented pedal on a stationary dynamometer. The data from these studies are again useful only

in validating the assumption of a roughly sinusoidal pedal force profile estimated in Chapter 4. Reference (12) presents a plot of maximum crank moment measured as a function of crank angle

under static conditions only.

The pedalling studies presented in Chapters 3 and 4 are

unique in examining the pedalling habits of average cyclists

under actual road conditions. The peak levels of the pedal force were found to differ vastly from the "maximum possible power out

"ideal" put" nature of the previous tests. The estimate of an

pedalling force is also unique. The data was collected using

the same bicycle frame to which the prototype constant force

transmission is fitted, and the data was generated under riding conditions closely resembling the conditions to which the proto type transmission will be subjected. Therefore, even though the

and 4 is sample size is limited, the data presented in Chapters 3

expected on the far more representative of typical loading to be -15-

constant force transmission than data extrapolated from the earlier references cited here.

4) Summary

All important information underlying the design of the prototype constant force bicycle transmission is presented here.

Chapter 2 presents a general description of the prototype transmis sion and its operating characteristics. Chapter 3 describes the instrumentation package used to determine the typical input to the transmission. Chapter 4 summarizes the data collected with the instrumentation package described in Chapter 3 and reduces these data to a set of design criteria. Chapter 5 presents a description and an analysis of the continuously variable traction drive used to vary the gearing in the transmission. Chapter 6 presents a de tailed analysis of the constant force controller and predicts the response of the transmission to a set of typical inputs. An Appen dix lists the computer program used to simulate the performance of the transmission. Chapter 2

2. Overview of the Constant Force Bicycle Transmission

2. 1) Introduction

The constant force bicycle transmission is a unique

adaptation of a popular commercial continuously variable traction

drive combined with a totally mechanical feedback control system.

The front assembly drawing of the complete transmission is shown

in Figure 2.1. The entire self-contained system is designed to

bolt on the rear wheel shaft of a standard 10-speed bicycle. The

prototype is fitted to the popular Peugeot UO-8 10-speed touring

frame. This chapter provides a general description of the two

main components of the transmission and concludes with a summary

of the general design features.

2. 2) Continuously Variable Transmission Overview

The gear ratio of the constant force bicycle transmission is

CVT is set via a continuously variable transmission, or CVT. The

frictional a "traction drive"; i.e., it transmits power through

enables contact between smooth steel surfaces. Utilizing a CVT

which is stepless shifting over a wide speed range, particularly

well suited for coupling with a control system.

shares the basic The constant force bicycle transmission CVT

popular Cleveland principle of operation of the commercially

1 Peugeot is a brand name; UO-8 is a model number.

-16- 17-

3 OJ

c o

in m r

to c

o >>

CD O S- o

c re +-> to c o o

3: 10 S- Q

OJ to (0

OJ s- 3 18-

2 Speed Variator. The transmission has a geometry similar to a large ball bearing. Shifting is accomplished by varying the angle of the axis of rotation of three "traction balls" between input and output discs (see Chapter 5). The axis of rotation of each traction ball is positioned by a worm gear. The three worm gear shafts are coupled with a roller chain (the lower chain vis ible in Figure 2.1). The upper right worm shaft is equipped with a second sprocket, which is driven by the control system.

Bicycle gearing is expressed as the diameter of the equiva lent driving wheel; i.e., gear ratio times the diameter of the driving wheel. Stock Peugeot UO-8 derailleur gearing varies from a low gear of 38.6 inches (1.43:1 sprocket ratio) to a high gear of 100 inches (3.71:1 sprocket ratio) in 10 finite steps. The constant force bicycle transmission provides continuous gearing from 32.7 inches (1.21:1 equivalent sprocket ratio) through a high gear of 151 inches (5.58:1 equivalent sprocket ratio).

Therefore, simply replacing the derailleur with the uncontrolled 3 CVT would in itself be desirable.

The chain speed step-up at the input to the CVT and the step-down at the output are necessary to reduce the stresses in the CVT to an operable level. The entire step-up/step-down

Manufactured by: Eaton Yale and Towne, Inc. Cleveland Worm and Gear Division 3262 East 80th Street Cleveland, Ohio 44104

3 Note, however, that additional expenses are accrued in cost and weight. -19-

mounts on freewheel.4 assembly the standard Peugeot UO-8 An end

view is provided in Figure 2.2. The right two sprockets are

rigidly fastened to form the step-up idler; the sprocket pair

rides over the freewheel via two ball bearings. The leftmost

step-down sprocket drives the freewheel directly, as in an ordi

nary bicycle transmission.

The CVT is lubricated with "traction fluid", a synthetic

lubricant combining a relatively high coefficient of friction

with good anti-wear characteristics. A sealed plexiglass case

is provided around the critical CVT components to keep the trac

tion fluid in and contaminants out. Plexiglass provides a light

weight, durable case that allows visual inspection of the CVT

components. The rotary seals on the case are all standard 0-

rings; the sliding seals are packed felt; the coverplate seals

are cork composition gasket material.

2. 3) Control System Overview

The transmission control system is devised to produce a

constant average force at the pedals regardless of riding condi

tions. The controller is totally mechanical, eliminating the

need for external electric, hydraulic, or pneumatic power. The

The freewheel is the base of the stock Simplex (trade name) 10-speed rear chainwheel cluster.

5 Sun Oil Company Automotive Traction Fluid Type TDF 88 -20-

^standard

Jf 20T V2P

chainwheel

sprocket

J \J

Figure 2.2) Assembly Drawing of the Step-Up/Step-Down Idler Assembly. -21-

control system is analyzed in detail in Chapter 6; the geometry

and features of the controller are outlined here.

The controller consists of a chain force sensor, a mechanical

a gear-down to integrator, the traction ball worm gears, and a set

of manual controls (see Figure 2.1). The chain force sensor con

sists of a floating sprocket carried by a Roberts-type straight

line mechanism (see Figure 2.3). The mechanical integrator is

comprised of a floating integrator driving disc, carried by the

chain force sensor linkage and powered by the floating sprocket,

and a fixed integrator driven wheel (see Figure 2.4). The gear-

down system, in addition to the worm gears themselves, consists

of a 1:4 bevel gear set and a 15:36 chain reduction to the worm

gear shafts (see Figure 2.5). Manual controls are provided for

adjusting the desired pedal force, shutting off the automatic

control system, and adjusting the transmission shift ratio (assum

ing the automatic system is disabled). Thus, total manual control of the CVT is available. The nominal peak value of the desired pedal force is approximately 35 pounds (see Chapter 4); it may be set between 25-65 pounds.

Ergonomic data indicate that bicyclists most optimally produce constant power at a constant rate of pedalling of about

60 RPM. Since power is the product of force times speed, this is equivalent to inputting a constant pedal force at a constant

c "optimal" Various pedalling rates between 40-100 RPM are re ported by different investigators; 60 RPM is typical for touring cycling. See References 7, 11, and 21. -22-

d) CJ3 fO

S- o to cz (D CO

CD O S- o Ol

CZ "r- TJ (0 o O __ w C_3

CD x: 4->

4- O

C7)

s CO S- Q

J_ E 0) to to

Csj

OJ i-

Ol -23-

integrator 7 integrator driving / driven disc / wheel

(l): JO

-integrator movable driven integrator wheel driving disc

Figure 2.4) Basic Mechanical Integrator Geometry.

pedal crank rotational speed. Therefore, a control system that will adjust the bicycle gearing to maintain either a constant

pedal force or a constant pedal crank rotational speed will proba

bly maintain optimal gearing.

It has been assumed here that the pedal force is a more dependable indicator of improper gearing than pedal crank rota tional speed. Therefore, the input chain force, which is equiva lent to the component of the total pedal force being transmitted to the rear wheel, is measured in the constant force bicycle transmission. The rider is assumed to be able to maintain the optimal crank rotational speed if the force demanded at the pedal is maintained at the ideal level. 24-

01 to to a:

cD CD

CZ CD >

S- o +-> CO s- D1 *D

-I-) C

cz I 3: CO S- Q

-_ = a> in to <:

o> -25-

As will be shown in Chapter 6, the controller is of integral

type. Earlier attempts at automatic control systems for bicycles

have been of the proportional type, as reviewed in Chapter 1. The

use of integral control produces two major advantages over propor

tional control. First, an integral controller produces zero

steady-state error to a step input, while a proportional control

ler will have finite steady-state error, or offset. Second, the

integral controller acts as a low-pass filter, reducing effects

of high frequency input to the control system.

The physical significance of zero steady-state error is well

illustrated by envisioning the behavior of bicycles equipped with

the two types of control systems while riding the bicycles up a

hill. In order for the bicycle with the proportional controller to remain downshifted, the force on the pedal must remain too

high. In contrast, the integral controller will detect the high pedal force, downshift the transmission until the ideal pedal force is resumed, and then leave the transmission in the down shifted state until another deviation from the ideal pedal force

is detected. In simpler terms, the rider of the bicycle with the proportional controller will have to pedal too hard all the way

integral up the hill, while the rider of the bicycle with control will be able to pedal at the ideal force for most of the hill. The initial time required to reach the ideal pedal force on the integral control system after starting up the hill is called the "response time". -26-

The low-pass filtering effect of integral control is

important because the control system input, the component of

pedal force transmitted to the rear wheel, varies approximately sinusoidally at a frequency of twice the pedal crank rotation rate (see Chapter 4). A straight proportional controller will constantly change the shift ratio at this same rate. This ef fect is greatly reduced by the integral controller; the shift ratio will change very little over one pedal cycle (assuming the pedal force is nominally at the ideal level).

The 2% response time of the integral controller to a pedal crank speed of 60 RPM is roughly 8.5 seconds (see Chapter 6).

Although this time may appear large, it may in fact be desirable.

This is best clarified by a second physical example. Assume the bicycle rider approaches a traffic signal which appears to be

"kick" changing red. The rider will want to apply a strong to the pedals for a few cycles in order to accelerate through the intersection rapidly. The low-pass filtering effect of the con troller, determined by its relatively high response time, will permit such short-term high-power inputs; the controller will re

Ergo- spond only to longer term trends in the riding conditions. nomic studies in fact indicate that the human body is particularly

This filtering effect enabled a significant improvement over an early design of the transmission that included a mechanical compensator for the sinusoidal input. Elimination of the mechanically complex compensator had little adverse effect on the control system performance. -27-

output.8 well adapted to short-term high-power Thus, the high

response time may in fact be a desirable feature; actual verifi

cation of this assumption is probably obtainable only by road- testing the transmission.

The Robert' s-type straight line mechanism used in the chain

force sensor (Figure 2.3) is a symmetrical 4-bar linkage which

generates straight-line motion using all pin joints. The mecha

nism generates a near-perfect straight line; disregarding clear

ances in the joints, the actual path of the integrator driving

disc and floating sprocket differs from a perfect horizontal

straight line by less than 0.008 inches over the total range of

motion. Varying the position where the chain force sensing spring

attaches to the linkage varies the mechanical advantage between

"tuning" the floating sprocket and the spring. This enabled the

spring force to match both the control system and traction drive

requirements while designing the transmission.

As mentioned earlier, a standard derailleur freewheel couples the transmission to the rear wheel. This enables ordinary

back-pedalling on the bicycle. In addition, back-pedalling will downshift the transmission at the maximum possible rate. There fore, the transmission gearing can be fully reduced in the "dead start" situation without accessing the manual control knob.

o See reference 21 -28-

2. 4) Additional Features

Several additional design features are included in the

constant force bicycle transmission to enable low-cost manufac

ture, dependable operation, and simple modification. Specifical

ly, these features include bolt-on design, straightforward com

ponent design, all pinned moving joints, and modular functional

entities.

As mentioned earlier in this chapter, the transmission

assembly is designed to bolt on the rear wheel shaft of a con

ventional 10-speed touring bicycle. No modifications of the

frame are required, and the standard pedal crank and rear wheel

are used. The entire transmission is cantilevered to the chain

side of the bicycle (see Figure 2.6). The width of the basic CVT

(not including the input/output chain extensions and integrator

driven wheel assembly) is only 2-7/8 inches.

The entire transmission is designed for ease of manufacture.

fasteners Stock bearings, gears, sprockets, seals, springs, and

parts re have been used wherever possible. The only non-stock

are the CVT input/output quiring grinding in the transmission

All discs and cradle bearing rings (see Chapter 5). remaining

standard engine lathe and components are easily fabricated on a

vertical mill.

All movable joints in the entire transmission are pin

joints. No slider, cam, or screw joints are utilized (discount

ing the worm gears). This greatly reduces the chances of binding -29-

ro S-

r (J >* o

__ 4->

CZ o

cz o

to to

I B to c ro S-

QJ O s- o

cz ro

O O

__ +->

O i

-M

i (0 o D_

(D J- Z3 CD -30-

or dirt contamination and reduces manufacturing costs.

The chain force sensor linkage and integrator driven wheel

assembly are both free-standing modules that bolt to the basic

CVT. This enables the control system to be modified extensively

without affecting the basic CVT design. Chapter 3

3. Pedal Data Instrumentation

3. 1) Introduction

As described in Chapters 2 and 4, designing the constant

force bicycle transmission requires a knowledge of typical pedal

force profiles expected on a bicycle under actual riding condi

tions. Information concerning the important parameters has not

been previously published. This chapter describes an instrumenta

tion package constructed exclusively to obtain these data. The

output generated by the instrumentation described in this chapter

is presented and analyzed in Chapter 4.

The total instrumentation package consists of three stock and

three custom-built parts. The stock parts are a standard 10-speed

bicycle, a portable stereo audio cassette recorder, and a 2-channel

pedal force DC strip chart recorder. The custom-built parts are a

to voltage sensor, a pedal position sensor, and a pair of frequency

in converters. The operation of the entire system is described

Section 3.2. Each of the three custom parts are described in de

tail in Sections 3.3 - 3.5.

3. 2) System Overview

and dis The instrumentation package described here records

position of a standard 10-speed plays the pedal force and pedal

conditions. bicycle as a function of time under actual riding

-31- -32-

The recording and display systems have been broken into two

completely independent sub-systems. This enables the recording

system to be self-contained on the bicycle itself; it needs no

external connections of any kind and does not interfere with normal

in way. cycling any The second sub-system is stationary; it con

verts data collected with the portable recording package to a con

venient strip chart recorder display.

The complete recording package is shown in Figure 3.1. The

package consists of a standard Peugeot UO-8 10-speed bicycle

equipped with a custom-built pedal crank. The crank contains a

pedal force sensor, described in Section 3.3, and a pedal position

sensor, described in Section 3.4. The custom pedal crank is con

tained within the same planar area as a standard pedal crank (see

Figure 3.4); it does not interfere with normal operation of the

pedals or the stock derailleur. Both the pedal force and pedal

position sensors output a frequency-modulated signal in the audio

range. The two simultaneous signals are recorded on separate

channels of a standard portable stereo audio cassette tape record er. The tape recorder is strapped to a rear wheel carrier on the bicycle, as shown in Figure 3.1. The instrumentation does not

"feel" noticeably affect the of the bicycle.

The display system is shown in operation in Figure 3.2. The two channels of an audio cassette tape recorded on the bicycle are played back through two custom-built frequency to voltage convert ers, which re-format the pedal position and pedal force signals 33-

E OJ

4-> to >> oo

01 c

r T3 U o o

<+- O s- o.

CO "O OJ D-

cd S- Z3 O) 34-

Figure 3.2) Pedal Profile Display Package.

on the tape to a form suitable for recording on a stock 2-channel

strip chart recorder.

Note that introducing the intermediate audio recording between

the pedal crank sensor and the strip chart recorder is not abso

lutely necessary; the sensor could be designed to record directly

on a portable strip chart recorder. However, using the intermedi

ate audio recording leads to several important advantages. First,

the audio recorder is much lighter in weight than a strip chart

recorder. Second, the audio recorder is an order of magnitude

less expensive and more durable than a strip chart recorder; there fore, it is much less susceptible to costly damage when strapped to the bicycle. Third, powering the portable tape recorder is -35-

trivial; a special chart strip recorder equipped with a portable

battery pack would be required. Fourth, mechanically jarring the

strip chart recorder would introduce a great deal of noise on the

output; this effect is reduced using audio tape1. Fifth, the

audio cassette format of the output provides a clean and compact

means of storing large amounts of data; the output can then be

selectively converted to the (cumbersome) strip chart format in

the lab. as will be Finally, seen in the following section, the

audio AC output of frequency the force sensor also provides a con

venient means of transferring the signal from the rotating sprocket

to the stationary frame without slip rings.

3) Pedal Force Sensor

The pedal force sensor detects the component of the pedal force which is transmitted to the pedal sprocket and chain. The sensor outputs a sinusoidal signal in the audio frequency range

(500 - 1000 hz); the output frequency is proportional to the pedal force. The entire self-contained unit mounts to the front de railleur sprocket of the Peugeot UO-8 10-speed bicycle (see

Inspection of the strip chart recorder displays of Figure 4.1 reveal identical noise signals superimposed on both the pedal position and force traces. This was caused by the physical jarring of the audio tape recorder under actual riding condi tions, which caused fluctuations in the tape speed. Neverthe less, the noise is probably substantially less than that which would result from jarring the pen mechanism of a chart recorder under similar conditions. Note that using a better quality audio tape recorder or improving the mechanical isolation between the tape recorder and the bicycle frame would lead to improved signal quality. -36-

Figure 3.3). The unit is entirely contained in the plane of the

pedal crank so that it will not interfere with normal cycling in (see any way Figure 3.4). The signal is transferred from the ro

sprocket to tating the frame (and cassette recorder) through a

rotary transformer (see Figures 3.10 and 3.11).

Earlier attempts at determining the pedal force under

dynamic conditions (references 6 and 18) measure normal and shear

forces applied to the pedal, which are both composed of two com

ponents. The first component is transferred to the sprocket and

chain; this component actually does work in driving the rear

wheel. The second component is simply supported by the bicycle

frame; it does no useful work. The only component of interest in

designing the constant force transmission is the component actually

reaching the transmission; i.e., the component driving the pedal 2 crank sprocket and chain . Therefore, the pedal force sensor is

designed to measure this component exclusively.

The component of interest is detected by decoupling the pedal

crank and the sprocket and instrumenting a specially-designed link between them. The pedal crank sprocket of a standard 10-speed bicycle is typically bolted directly to the right pedal crank. In

sensor is contrast, the pedal crank sprocket of the pedal force mounted to a fiberglass plate fastened to the pedal crank shaft,

2 The second component is of great interest when structurally analyzing the frame itself. -37-

s- o to c (D (V)

CD O i- o

ro -o 03

4T 38-

Sensor. End View. FngurePimirp ^ 4} Pedal Force 3.4) the Sprocket.) ^aa.^o^ ^^ ^.^ ^.^ -39-

independent of the right crank (Figure 3.5). However, the plate

is not rigidly fastened to the shaft; as shown in Figure 3.5, the

plate is mounted to the outer race of a ball bearing, and the in

ner race of this bearing is in turn fastened to the shaft. The

bearing supplies planar rigidity to the sprocket while rotationally

it from the decoupling pedal crank. Therefore, if the right pedal

were replaced in Figure 3.5, the pedals would spin freely without

imparting any torque to the front sprocket (obviously, the bicycle

cannot be driven in this condition).

The pedal crank and the pedal crank sprocket are linked with

a specially-designed right pedal crank arm (Figure 3.3). The

special crank arm is equipped with a 2.25 inch extension beyond

the center of the pedal crank shaft. This is coupled to a block bolted rigidly to the fiberglass plate (carrying the sprocket)

linkage3 through a universal joint-type (Figure 3.6). Therefore, any force which is transferred from the pedal crank shaft to the sprocket must be transmitted through the right crank extension.

Figure 3.6 reveals that the right crank extension acts as a canti lever beam. A signal proportional to the component of the force

The universal joint-like link is necessary to obtain purely uni directional loading between the pedal crank extension and the block mounted to the sprocket. An earlier design utilized a simple tensile link between the pedal crank and the sprocket connecting block. The pedal force signal was obtained by mount ing strain gages directly to the tensile link. However, bend ing and twisting between the crank and sprocket made the system "tensile" unworkable; one of the gages was found to actually go into compression! -40-

+-> CD

__ O O s- o.

c o s-

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_z in

__ cz ro S- o

ro "O OJ Q-

Ol 0)

OJ 0-

cr> c

o o 0) o

CD i- Z3 cn -41-

Shaft and the Between the Pedal Crank Figure 3.6) Linking Sprocket. -42-

being transferred to the sprocket is obtained by mounting tensile and compressive strain gages to this beam (Figures 3.6 and 3.7).

The strain gage signal is converted to a proportional fre quency-modulated signal with a combination bridge/amplifier/ voltage controlled oscillator mounted directly on the sprocket.

The circuit board of the entire electronics package is visible in

Figure 3.3. Also note the sprocket-mounted 18-volt battery pack

sensor electronics used to power it. The schematic of the force

a Wheatstone is shown in Figure 3.8. The strain gages drive

the leads bridge. The bridge is coarsely balanced by re-soldering

across a bank of 10 - 1800 ohm of the fine balance potentiometer

operation is provided to facilitate this resistors; a set of pegs

voltage reference, comprised shown in Figure 3.9. A bridge base

voltage follower, is included of a voltage divider and an op amp

the strain gages (and the resultant to reduce the voltage across

voltage follower isolates the current flow through them). The

ampli impedance changes in the bridge. An reference voltage from bridge the same layout as the fier base voltage reference shares

is adjustable except the voltage divider base voltage reference,

voltage controlled voltage input to the to set up the reference

precision utilizes a low-drift amplifier itself oscillator. The DC

differential small voltage amplifier boosts the op amp. The

magnitude to increase bridge three orders of generated by the by

the voltage controlled suitable for driving it to a level -43-

O) C33 CO CD

CO s-

to cz 0>

-o CZ 2> = ro

to c CD

CZ (O s- (_)

ro "O OJ O-

Q. O

r-*

zri -44-

o!-nuJts I

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(J oo

Z O s_ +-> o OJ

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0> o s_ o

CO -a OJ C_

CD s- Z3 O) -45-

to 03 o> Q.

cD O CZ ro

ro cn

OJ CD -o

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OJ to <- ro O o

s- o to CZ

cu o i- o

CO -o OJ

0) s_ 3 cn -46-

4 oscillator . The voltage controlled oscillator itself is based on an integrated circuit designed especially for this purpose5. The bridge and frequency adjustments are set up to generate a sinusoidal signal varying from approximately 500 - 1000 hz over the normal range of the input pedal force. This signal is transmitted through a rotary transformer with a driver consisting of an isolating am plifier (voltage follower) and an impedance-matching capacitor.

The output of the tranformer is attenuated to drive one channel of the 2-channel audio tape recorder.

The rotary transformer is used to transfer the frequency modulated force sensor output signal from the rotating sprocket to the stationary frame, where it can be hard-wired to the audio tape recorder. The custom-built transformer is shown removed from the bicycle in Figure 3.10. The primary and secondary windings of the transformer are physically independent. The rotating pri mary (left) is wound on a plexiglass core fastened to the rear of the front sprocket mounting plate. The plate was fabricated from fiberglass to provide a non-conductive sidewall for the primary

on a second winding. The stationary secondary (right) is wound

4 large amplification Despite use of the precision op amp, this was reduced to caused temperature drift problems. The drift and bridge tolerable levels by shielding the precision op amp visible Figure 3.3. The resistors with plexiglass covers, in amplifier would be significantly improved stability of the DC with lower amplification by cascading two or three op amps in series.

5 Intersil 8038 -47-

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I .

- * J J

slj Ul I

s- OJ

o <+- to cz 40 s-

O OC

(D i- =3 cn -48-

plexiglass core that fastens around the pedal crank bearing mount

(see figure 3.14)- The secondary fits inside the primary (without

physically touching it) when the sprocket is mounted to the pedal

crank (Figure 3.11). Note the pedal crank bearing mount also

serves as a steel core for the transformer, which will improve

transfer of the audio frequency signal between the primary and

secondary. Therefore, the rotary transformer simultaneously pro

vides electrical coupling and mechanical decoupling between the

force sensor and the bicycle frame. The attenuator and output

jack are also visible in Figure 3.11.

The force sensor is calibrated by applying dead weight to

the pedal of the bicycle while it is held stationary in the lab

(Figure 3.12). The pedal is maintained in the horizontal posi

torque. tion, so the dead weight is completely converted to useful

The force sensor is calibrated to output a frequency of 500 hz

pedal force is when the pedal force is 0, and 1000 hz when the

100 lb6.

3. 4) Pedal Position Sensor

signal indicative The pedal position sensor outputs an audio

crank. The signal varies of the rotational position of the pedal

of when the left pedal is at the top si nusoi dally between 1000 hz

pedal is at the of the the stroke to 500 hz when the right top

6 the output of the pedal force As will be seen in Chapter 4, pedal forces exceeding 150 lb. sensor is actually useful for -49-

5 CD

i- ro OJ c_

OJ m to <

-P CD

__ O O S- Q. oo

-o OJ

-4-> CZ CD

to cz

o> s- Z3 C7> -50-

Figure 3.12) Calibrating the Pedal Force Sensor.

stroke. This is clarified with Figure 3.13: if y indicates the angular displacement of the right pedal from the top of the stroke,

is' the output frequency of the pedal position sensor

= fpp 500 hz + 500 {3_[cos(y + ir) + 1]> hz ( 3.1 )

As will be seen in Chapter 4, the output of the pedal position sensor is useful for determining if the rate of rotation of the pedal crank is constant. However, the usefulness of the sensor is limited beyond this point, since correlating the actual angu lar position of the pedal at any moment, y to the highly non linear sinusoidal output is difficult. Much more in-depth ergonomic analysis of the pedal position data would be made possible by replacing the cam-driven analog system described here with a digital shaft encoder, which could output a linear indication of the pedal position. -51-

Figure 3.13) Definition of Angular Pedal Position.

Mechanically, the force sensor consists of a simple harmonic

full rise - full return cam (eccentric) mounted to the rear of the

pedal crank sprocket, driving a linear follower, which is attached

to the frame. The simple harmonic cam is shown removed from the

bicycle in Figure 3.10; it fits over the primary of the rotary

transformer of the pedal force sensor. The follower assembly (and

the associated electronics) is shown mounted to the frame (with

the sprocket removed) in Figure 3.14. The follower is a standard

3/8 inch ball bearing constrained to a straight line path with a

linear ball bushing. Contact between the cam and follower is

bushing. pedal maintained with a spring straddling the ball The -52-

Figure 3.14) Pedal Position Sensor Follower Assembly. (Force Sensor Rotary Transformer Secondary also Visible.) -53-

position sensor is shown completely assembled in Figure 3.15.

as with Note, the pedal force sensor, the unit does not interfere

with the pedal crank or derailleur in any way.

The electronic schematic of the position sensor is shown in

Figure 3.16. The pedal position is detected from the position of

a linear (both in motion and resistance taper) potentiometer

driven by the cam follower. The physical orientation of the po

tentiometer with respect to the cam follower is clearly visible in

Figure 3.14. The potentiometer is the heart of a variable voltage

divider. The voltage divider drives a voltage controlled oscilla

tor, identical to that used on the pedal force sensor, through an

isolating amplifier (voltage follower). Adjustments are provided

in the voltage divider to set the input to the voltage controlled

oscillator to the exact range required to obtain the desired 500 -

1000 hz output. The output of the oscillator is fed directly to

the audio tape recorder through a simple adjustable attenuator

(the output jack is visible in Figure 3.14). A fourth adjustment

controlled is provided to set up the frequency of the voltage

oscillator.

3. 5) Frequency to Voltage Converter

audio signals generated As explained in Section 3.2, the by

sensors are recorded on two the pedal force and pedal position

tape recorder. Two identical fre channels of a portable stereo

audio converters are used to change the frequency quency to voltage -54-

Figure 3.15) Pedal Position Sensor. -55-

ro E O)

__ o 00

c o s-

p o OJ

s- o c/i cz OJ 00

cz o

to o

ro -o > OJ eo e g 'g | CL + WV o- 1|> AM Wr^ | IO O ^ -. -C 05._ o ill" D) OTJ CO .E a

-2^ 0) s- Z3 cn -56-

signals stored on a cassette tape to linearly proportional DC voltages. The two voltage outputs are used to drive two channels of a DC strip chart recorder. Thus, the frequency to voltage

processors" converters are used as "post to convert the frequency modulated record generated during riding the bicycle to an easy- to-understand visual record on the chart recorder.

The frequency to voltage converters are shown in operation in

Figure 3.2. A close-up of the two identical converters is pro vided in Figure 3.17; the right converter is opened to show the electronic circuitry. The electrical schematic is provided in

Figure 3.18.

Each frequency to voltage converter is based on an integrated

o circuit designed especially for this function . As mentioned earlier, the output of the pedal force and position sensors are designed to. fall in the approximate frequency range of 500 - 1000 hz. A low-pass filter is included at the input to reduce high- frequency noise superimposed on this desired signal. The signal is then amplified and passed through the frequency to voltage con verter integrated circuit. The output of the integrated circuit

- has a large ripple voltage in the 500 1000 hz input range super imposed on the desired DC output (which oscillates at a frequency

the output of about 2 hz); this is reduced substantially by

8 National Semiconductor LM2907N '- " ' (Q 1 * c -si s* i -s * n> i

* oo ,4 t*

-s (T>

-O cz ro Z3 Q

r+ O

c-t- 0J tQ ro

o o Z3 < ro -s

cM- ro -s m

-Z9- -58-

10 H" - (A 3 i_ 0.0,0) *- >- to m 2 5 =

JVW- I* -4-> cn o ro a E to OJ o __ CM > w o WO cn O oo or *" 0) o +**. 0) "AMr CO 1|> CN *cz o s- H' -t-> to 3 o O" in CD to o 0) o m o

O) +-> Hi". (0 S- > _ 0J m 3 2 > o o. -r cz o o CO o CD C33 ro

id1 i+ O

CZ 0) a Z3 CT E 0J 43) E ID s-

>* 3 a in. 00 c

QJ o. VW 1| S- o o. 3 o cn

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z -59-

low-pass filter. The signal is then ready to be input to the strip chart recorder.

No adjustments are required on the frequency to voltage converters. However, the strip chart recorder itself must be adjusted for zero reference and gain. (This can be done con veniently by recording the pedal position and pedal force sensor calibration signals on the cassette tape.) Samples of the re sulting strip chart recorder output can be examined in Figure 4.1 Chapter 4

4. Design Criteria for the Constant Force Bicycle Transmission

4. 1) Introduction

This chapter reduces the data obtained from the pedal

instrumentation described in Chapter 3 to specific design criteria

for the constant force bicycle transmission. As explained in

Chapter 3, all data was obtained from strip chart recorder records

generated under actual riding conditions by the instrumented

Peugeot UO-8 10-speed bicycle shown in Figure 3.1. Each record

consists of two traces; the first represents the pedal position

and the second represents the component of the pedal force trans

mitted to the rear wheel (see Figure 4.1).

Section 4.2 describes the tests performed to obtain the

design criteria for the constant force bicycle transmission.

Sections 4.3 - 4.6 present and verify the resulting design esti

mates for the pedal crank velocity, pedal force profile, ideal

respectively. Section pedal force, and maximum pedal force, 4.7

bear explains an estimated loading schedule used for designing

ings in the transmission.

4. 2) Overview of the Test Data

the selection of the The results of three tests used in

presented here. important transmission design parameters are

force" intended to determine The first test, the "ideal test, is

-60- -61-

the ideal pedal force to be maintained by the constant force transmission under average touring conditions. The data were generated by riding the instrumented bicycle on a smooth, flat road at a comfortable" gearing which produced the "most crank force and loading" speed. The second test, the "worst case test, is intended to indicate the worst case loading to be expected on the transmission under average touring conditions. The data were generated by pedalling the instrumented bicycle up a sharp incline. The third test, the "normal cycling" test, is intended as a verification of the first two. The data of the first two tests were generated by the author; the data for the third were generated by an experienced cyclist instructed to ride the in

"normal" strumented bicycle under conditions. (The test was run on essentially flat suburban streets.) The data were useful in

"typical" determining the data of the first and second tests to be

A sample of the strip chart records of each of the three tests is provided in Figure 4.1. Bar graph summaries of the fre quency of peak pedal forces occurring in each of the three tests are shown in Figures 4.3 - 4.5, respectively.

The data presented in this chapter are intended for use only

parameters of the prototype constant for approximating the design force bicycle transmission. The arbitrary nature of the data is

suggested to be a comprehensive obvious; it is in no way study

transmission or general of the constant force loading bicycling -62-

*">)

(a) ideal force test

liu. .In. wi.M

A A A A A A a M\ A A A / \J v V v \ v/ \J V (lb) \j V M-*. (=) (b) worst-case load test

Iru* tin* wivm

A- /" ., a A a M A A A vaAAAAaaAa/^ Cb) ol -t- +- +-? 1 () (c) normal cycling test

Records Generated the 4.1) Sample Strip Chart Recorder by Bicycle Pedal Instrumentation in Chapter 3. -63-

performance . Far more comprehensive studies will be justified

by encouraging performance of the prototype transmission.

4. 3) Estimated Pedal Crank Velocity

The pedal crank velocity is assumed constant over an entire

revolution of the pedal crank. A constant velocity of 60 RPM is

assumed for studies involving control system performance; a con

stant velocity of 75 RPM is assumed for bearing life and fatigue

calculations. The 60 RPM rate, used for the control analysis,

"optimal" is a typical published rate for touring cycling (refer

"optimal" ence 7) , a rate above the value was used to estimate

the life of the transmission components to be conservative.

The assumption of constant velocity of the pedal crank is

based on visual inspection of the pedal position traces. As

described in Chapter 3, the output of the pedal position sensor

is a sinusoidal function of the pedal angle; if the rate of

sinu- pedal crank rotation is constant, the output will vary

soidally as a function of time. Figure 4.1 shows one true sine

wave superimposed on one crank cycle of each test of the sample

As suggested in the Introduction, substantial interest exists for ergonomic studies of bicycling. Collecting and analyzing bicycling data with the instrumentation and techniques described in Chapters 3 and 4 would comprise a worthwhile project inde pendent of the constant force transmission design.

2 controller response A higher crank rate will improve the time; see Chapter 6, footnote 11. -64-

traces; the output in fact varies approximately sinusoidally

as a function of time. Note the three pedal cycles selected

correspond to various peak pedal forces (40 lb in trace [a], 104

lb in trace [b], and 50 lb in trace [c]).

The average pedal crank rate in the ideal force test is 71

RPM. The average rate in the worst case load test is 47 RPM.

The average rate in the normal cycling test is 65 RPM. The crank

rate of the first and third tests fall conservatively within the

assumed values of 60 RPM and 75 RPM. The measured values also

appear reasonable when compared with the typical published "op

timal" crank rate of 60 RPM. The crank rate of the worst-case

load test is below the assumed minimum of 60 RPM because the

average pedal force of the test (88 lb, see Section 4.6) is sub

stantially above the ideal level (35 lb, see Section 4.5).

4. 4) Pedal Force Profile

As discussed in Chapter 3, only the component of the pedal

force transmitted to the drive chain is of importance in analyz

force is as ing the constant force bicycle transmission. This

peak sumed to vary sinusoidally between 0 and a value; i.e.,

Fp(t) = F(t) [%( cos^f+1 )] (4.1)

= the peak pedal force where: F(t) a function describing as a function of time

rotational rate T = the period of the pedal crank

for complete cycle of the Note Fp(t) will have two peaks every -65-

pedals, corresponding to the contribution from each of the two

pedals.

The basis of the assumed pedal force profile is explained

in reference to Figure 4.2. The force of the rider's foot on a

180 pedal is assumed to remain approximately vertical for the

of the stroke shown. When the pedal crank is vertical (Figure

4.2 [a] and [c]), the entire force of the rider's foot on the

pedal is supported by the frame, and no torque is input to the

transmission. When the pedal is horizontal (Figure 4.2 [b]),

the full force of the rider's foot on the pedal is converted to

input torque. The component of the pedal force converted to

input torque will thus vary sinusoidally between these extremes

twice per revolution of the pedal crank (assuming constant rota

tional speed of the pedal crank).

The reasonableness of the sinusoidal input force assumption

is again substantiated by visual inspection of the pedal force

traces. Figure 4.1 shows one true sine wave superimposed on one

pedal force cycle of each test. The highlighted cycles again

correspond to various representative peak pedal forces.

4. 5) Estimated Ideal Pedal Force

The ideal peak pedal force is the peak value of the approxi

constant force mately sinusoidal input that the bicycle transmis

sion will attempt to maintain under all riding conditions. The

value selected for all following simulations of the constant -66-

torque = rpcFp torque=0

torque=0

w to

Figure 4.2) Generation of the Sinusoidal Pedal Crank Torque.

force controller is

FD = 35 lb

This result can be combined with the idealized pedal force profile, equation (4.1), to prescribe the ideal pedal force as a function of time:

FP,ldeal " FD [ + 1 ] <4-2>

The ideal force test was run with the exclusive intent of determining the ideal peak pedal force. The results of that test are summarized in the bar graph of Figure 4.3. The mean value of

295 pedal force peaks recorded in that test is 33 lb, with a standard deviation of 11 lb. -67-

,o fs.

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saouajjnooo |o -laquinN -68-

The normal cycling test was useful in verifying the ideal

pedal force level. The mean value of the 637 pedal force peaks

recorded in that test is 55 lb, with a standard deviation of 15 lb (see Figure 4.5). The mean force of this test is expected to

exceed the value determined in test 1, since the cyclist was not

instructed to adjust the gearing to maintain the "most comforta

ble" pedal force. Nevertheless, the ideal pedal force of the

constant force transmission has been made adjustable over a large

range (25 - 65 lb) to compensate for the preferences of various

riders.

4. 6) Design Maximum Pedal Force

The design maximum pedal force is the maximum peak pedal

force expected to be input to the constant force bicycle trans

mission under any conditions. This value is important for

stress calculations in the CVT; it has been selected at 136 lb.

The worst case load test was run with the exclusive intent

of determining the design maximum pedal force. The results of

this test are summarized in Figure 4.4; 136 lb is the maximum

force that was recorded in that test. The average force of the

276 peaks recorded in the test is 88 lb.

The normal cycling test is also useful for verifying the

value of the design maximum pedal force. One isolated peak was

next highest peak. recorded at 153 lb, a full 42 lb above the

Reference (18) also reports isolated peak pedal forces generated -69-

a> on CM Ol T" 1

a, cz 1 T3 ro O

4-> to S- o 3

. o CO to ^-^ OJ jo o S- "^ o

(4- o

c/) 03 o (J V CZ 0) 5- S_

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M- O

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QJ 1 I ' 5- 1 3 I I I [ I w CM o o '| en

saouejjnooo 1 JaqwniM 70-

o o

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.0 E 3

saoueJJnooo ) jaqiunN -71-

by racing cyclists exceeding the rider's weight3. Nevertheless,

the 136 lb value has been used here because the size and weight

of the constant force bicycle transmission depend directly on the

maximum design pedal force, and minimizing size and weight are

critically important factors in developing a practical transmis- 4 si on . Furthermore, 95% of the peak forces of the worst case test

(Figure 4.4) fall below 124 lb, and the overall maximum peak force

level will hopefully be lowered substantially by the automatic

shift ratio adjustment provided by the constant force transmission.

Therefore, the 136 lb value was selected as a reasonable compromise

value for the prototype design. If higher pedal force peaks are

indeed found to cause permanent damage to the actual prototype 5 transmission, either adding overload protection or sizing up the

affected components is recommended.

4. 7) Estimated Loading Schedule

In addition to estimating the overall design maximum input

3 in contrast to the This study measured the entire pedal force, component transmitted to the chain.

limits the minimum possible 4 The maximum design pedal force also 6.4. controller response time; see Section

CVT input could provide overload 5 A spring coupling in the energy. a clutch protection with no loss of Alternately, slip overload protection could could be used. Note that including design maximum force substantially in fact be used to bring the reduction in the size and weight of the below 136 lb, making a CVT possible. -72-

force to the constant force transmission, the frequency of loading

at each pedal force and shift ratio is important in properly

sizing the bearings of the transmission. The best example is pro

vided by the CVT traction ball cradle bearings (see Chapter 5).

Assuming these bearings to be loaded at the maximum level for

their entire useful life would result in substantially oversizing

them, since Figure 4.4 shows they will spend little actual time

at the maximum load, even under severe riding conditions. The

shift ratio must also be considered, since it affects the rota

tional speed of the bearings (a crank speed of 75 RPM is assumed

for the bearing calculations; see Section 4.3).

The loading schedule used in designing the constant force

bicycle transmission is shown in Table 4.1. The loading has been

prescribed as a percentage time of operation at three representa

tive pedal forces, each of which are considered at three repre

sentative shift ratios. The design maximum force is most likely

to occur under the most severe riding conditions, as the trans

mission will shift to the full downshift configuration as rapidly

as possible. Therefore, the design maximum pedal force was esti mated to occur 5% of the operating time in the maximum downshift

configuration and 1% of the time at each of the higher shift con

ditions. The second representative pedal force was chosen at 85

lb; the transmission is assumed to operate at this input a total

of 55 lb was of 59% of its life. A minimum pedal force chosen,

of the transmission life. representing a total of 34% 73-

Table 4.1

Estimated Loading Schedule for the Constant Force Bicycle Transmission

Overall Gearing Peak Pedal Force Operation Time (inches) (pounds) (per cent)

32.7 (full downshift) 136 5

1 85 23

n 55 11

70.2 (CVT @ 1:1) 136 1

n 85 18

ii 55 12

151 (full upshift) 136 1

ii 85 18

H 55 11

when compared The schedule of Table 4.1 is very conservative

Figures 4.3 - 4.5. The to the actual input forces summarized in

of lb is the middle representative pedal force 85 approximately

over 90% of the median value for the worst case loading test;

test were below 85 lb, and peaks recorded in the normal cycling

this value in the ideal force test. the pedal force never reached

"lowest" 4.1 is the median The pedal force of Table actually -74-

value of the normal cycling test. Also note the values used in the bearing calculations are the maximum values of the actual sinusoidal load profiles. Chapter 5

5. The Continuously Variable Transmission

5. 1) Introduction

The heart of the constant force bicycle transmission is a

continuously variable traction drive. The basic drive geometry

Variator1 is similar to the commercially popular Cleveland Speed ;

however, the Cleveland design has been modified extensively to

meet the restrictions of application to a bicycle. Specifically,

the weight and size have been decreased considerably, while the

efficiency and ease of manufacture have been increased. Modify

ing the Cleveland design also enabled positioning the input and

output shaft power take-offs on the same side of the transmission.

Several continuously variable transmission designs are 2 available commercially . These include infinitely-variable belt

systems, hydraulic drives, and several additional traction drive

geometries. The Cleveland Speed Variator geometry was selected

as the basis of the constant-force bicycle transmission because

it is among the best in the categories of efficiency, simplicity,

and ease of changing the shift ratio, while it also compares

favorably to the alternate designs in size, weight, and power

capacity.

1 See Chapter 2, Footnote 2.

See reference (24).

-75- -76-

This chapter begins with an explanation of the basic

operating principles of a Cleveland-type traction drive. This

is followed by a description of the differences between the

commercial Cleveland Speed Variator and the constant force

bicycle transmission. Finally, an analysis of the critical de

sign parameters of the constant force bicycle transmission CVT

is summarized.

5. 2) Principle of Operation

The basic principle of operation of the Cleveland Speed

Variator geometry is illustrated in Figure 5.1. The geometry

consists of two equal-diameter concentric discs connected by a

balls" series of "traction positioned about their periphery.

The traction balls rotate about axes independent from the discs.

The input shaft is connected to one of the discs; the output

shaft is connected to the other. Power is transmitted between

the discs through rolling contact with the traction balls.

The key to operation of the transmission consists of chang

ing the angle of the axis of rotation of the traction balls with

respect to the axis of rotation of the input and output discs.

In Figure 5.1 (a), the axes of the traction balls are positioned

parallel to the coincident input and output shaft axes. The in

put and output discs contact the traction balls at equal radii,

and a 1:1 drive ratio is realized. In Figure 5.1 (b), the axis

positioned clockwise to the coin of the upper traction ball is

contacts cident input and output shaft axes. The input disc the 77-

3 O f"> 3 o 0)

o s 10 a 3 to/ OJ >

c_> ro V S- 3 O OJ Q. n >> o +-> 0> I -a cz 10 ro ^- c O) S > O OJ o r O

Ol (O

M- O

c o

4-> a.2 C m re S- o 3 TJ OJ Q. O 3 O <4- Aib o O CB 0) I- a. 4S 44* i

s o cz 10 r i- 7157 Q- 3 U a.

traction balls at a radius larger than the output disc, and a net downshift is realized; i.e., the input shaft must rotate farther than the output shaft for one revolution of the traction balls3.

In Figure 5.1 (c), the axis of the upper traction ball is posi tioned counterclockwise to the coincident input and output shaft axes. The input disc contacts the traction balls at a radius smaller than the output disc, and a net upshift is realized.

An exact relation for the transmission shift ratio is developed in reference to Figure 5.2. The transmission is de signed such that a line drawn through the center of a traction ball and the point of contact of the traction ball with the in

45 put or output disc makes an angle of with the horizontal.

The spherical radius of the traction ball is r.. The angle of the traction ball axes with respect to the coincident input and output shaft axes is termed the shift angle, e. The radius from the traction ball axis of rotation to the point of contact with

the input disc is r. , where

(45 r. = r. sin - e) in b

Similarly, the radius from the traction ball axis of rotation to the point of contact with the output disc is rQut, where

(45 = r. sin + r + e) out b

Therefore, to conserve power, the ratio of the input torque, Tin, to the output torque, Tout> must be

3 "downshifted" Therefore, means the transmission is acting as a speed reducer and a torque multiplier. -79-

45+ 6

45- e

Tsin out

Figure 5.2) Modelling the CI eve! and-type Traction Drive. -80-

= Tin rout r. Tout in

(45 rb sin + e) (45 rb sin - 8)

45'' 45 sin cos e + cos sin e 45'' 45 sin cos e - COS sin e

OR T. = 1 + tan e in Tout 1 - tan e (5.1)

3) Unique Characteristics of the Constant Force Bicycle

Transmission CVT

Three major modifications have been made on the conventional

Cleveland Speed Variator in the constant force bicycle transmis sion. The first changes the means of retaining the traction balls, which increases the transmission efficiency. The second changes the means of developing a tractive force between the traction balls and the discs. The third simplifies the means of positioning the traction ball axes. The second and third modifi cations decrease the weight and increase the ease of manufacture of the transmission.

A large contact force is required between the traction balls and the discs to transmit force between them under conditions of pure rolling without slipping. This subjects the traction balls to a large radial force; therefore, they must be restrained

ring" radially. This is done with a "ball in the Cleveland

Speed Variator, as suggested in Figure 5.3. Note, however, that

the traction ball. the ball ring has a surface of contact with 81-

s- o +-> ro Z. ro

-a O) OJ Q. OO

-o CZ re

> OJ o> c

o OJ

re i- 4-> to OJ ca

re co

cz o T

4-3 (J re s-

C33 -82-

cz o

m m

E to cz re s-

(A?

f CO

OJ a o

+-> E re

*-> to

o o

E to

o OJ

re s- +j to 0) OL.

re co

o 10 S-

Lf) s

cn -83-

As suggested in Figure 5.3, when the shift angle is not zero, the ball ring will contact points on the traction ball having dif ferent velocities. This is likely to cause substantial frictional losses. The efficiency of the Cleveland Speed Variator in fact drops substantially as the shift angle increases (72% at

3:1, in contrast to 90% at 1:1; see Reference 9).

The traction ball restraint mechanism for the constant force bicycle transmission is shown in Figure 5.4. Each traction ball is cradled between a pair of ball bearings fixed to the frame.

Each bearing then has a line of contact with the traction ball, rather than the surface of contact of the Cleveland design. The velocity differential problem is therefore eliminated, and high efficiency should be obtained nearly independent of shift angle.

The tractive force required by the Cleveland transmission is developed by wedging the drive discs against the traction balls with the mechanism shown in Figure 5.5. A roller bearing is captured between inclined ramps on the shaft and disc. When torque is applied to the shaft or disc, the roller bearing tends

"climb" to the ramps, wedging the discs against the traction balls with a force proportional to the input torque. The thrust on the shafts is supported by angular contact bearings pressed in the massive cast iron case. Thus, the correct tractive force

and operation will be obtained regardless of tolerancing or wear,

transmission (as the at low torque will prolong the life of the tractive forces on the traction balls will be reduced). -84-

HOUSING BREATHER FILTER

SOCKET HEAD SCREW

END PLATE

ROLLER CAGE ASSEMBLY IRIS PLATE RETAINER SPLINED RING RADIAL BEARING RADIAL BEARING THRUST BEARING

CLAMPING NUT OIL SEAL OIL SEAL

DRIVE SHAFT SHAFT DRIVE SET SCREW & PLUG

BELLEVILLE SPRING CENTERING RING DRIVE DISC BALL AXLE NEEDLE ROLLERS DRIVE BALL

DOWEL PIN

DRIVE DISC ROLLER CAGE ASSEMBLY RADIAL BEARING ANGULAR CONTACT AND CENTERING RING BEARING

SPLINED RING

Figure 5.5) Tractive Force Generating Mechanism. Cleveland Speed Variator. (Figures extracted from reference 8.) -85-

c o

cz o

to co

co cz re s-

o >s o

r CQ

0) L> S_ o

+-> c re +-> CO CZ O c_>

o OJ

-p re s- O) CZ 0) CD

03 O s- o

OJ >

O re s-

O) -86-

Two problems arise in adapting the Cleveland design to use on a bicycle. First, the inclined ramps on the disc and shaft are difficult to machine. Second, the weight of the massive case cannot be tolerated. A solution has been developed in the con stant force bicycle transmission that solves both these problems, while preserving the desirable characteristics of the Cleveland geometry (see Figure 5.6). Instead of wedging the discs into the

traction balls, the balls are wedged into the discs. This alter

nate means of producing the tractive force in itself requires

three major changes from the Cleveland design.

The first change, cradling each traction ball in an inde

pendent set of bearings, has already been described. Two sets

of the traction ball cradle bearings are fixed to the frame, and

the input and output discs are in turn cradled between these two

balls. A third set of cradle bearings is then mounted on an in

dependent lever pinned to the main frame, as clarified in Figure

5.6. The input and output shafts are not rigidly fixed to the

"float" frame; they with the input and output discs. Thus, apply

torque to the end of the ing a force proportional to the input ball floating traction ball lever wedges the floating traction

into the discs and fixed traction balls, respectively.

Cleveland design is a The second change from the applying

at the end of the force proportional to the input torque floating

the torque traction ball lever, rather than measuring directly

not possible, but at the input or output disc. This is only -87-

convenient, because the force in the driving chain is being measured to run the force-sensitive control system. A tractive force is therefore developed proportional to the driving torque by simply anchoring the chain force sensor linkage spring to the lever arm (see Figure 2.1). A cable limit stop between the force sensor linkage and the lever arm maintains the proportional tractive force when the force sensor spring "bottoms out".

The third departure from the Cleveland design consists of straddling a thrust bearing between the input and output discs

(see Figure 5.7). This eliminates the need for the massive case required to support the large thrust load in the Cleveland de

1/8" sign. The cast iron case is replaced by a thick steel frame shell. Note the revised tractive force generating mechanism is simple to manufacture.

The shift angle of the traction balls is controlled by a large cam system, called an "iris plate", in the Cleveland Speed

Variator (see Figure 5.8). The large cam again presents weight

trans and manufacturing problems. The constant force bicycle mission supports the traction ball shaft in a carriage enveloping the traction ball (see Figures 5.7 and 5.9). This carriage

perpendicular to the pivots about an axis through the ball center

axis. The traction ball plane of rotation of the ball rotational

worm gear fixed to each axes are then positioned via a simple traction ball carriage. 88-

traction ball cradle bearing

thrust bearing

Figure 5.7) Cutaway View of the Input/Output Disc Assembly. -89-

Mechanism. Cleveland r.nil c o\ Angle Positioning Figure 5.8) shiftShift ^.^ (phQt0 extracted from reference 8.) -90-

axis of rotation of traction ball carriage

traction ball shaft

cradle bearings

traction ball carriage

Figure 5.9) Cutaway View of the Shift Angle Positioning Mechanism. Constant Force Bicycle Transmission. -91-

4) CVT Analysis

The tractive force on the CVT traction balls is designed to be sufficient to transmit the maximum pedal force to the rear wheel of the bicycle without slippage in the CVT. The constant force bicycle transmission CVT design is verified in three complemen tary parts. First, the tractive force required in the worst-case loading condition is derived. Second, the derivation of the normal force available for generating the tractive force is pre sented. Third, the coefficient of friction of the traction fluid is verified to be adequate to fulfill the tractive force require ments under the available normal load. Two additional analyses end the chapter. The first examines the worst state of stress occurring in the CVT. The second crudely estimates the expected transmission life.

4.1) Worst-Case Tractive Force Requirements

A simplified schematic of the input gearing to the CVT is shown in Figure 5.10. The pedal crank drives a standard 52 tooth,

Jg-inch pitch pedal crank sprocket. This sprocket in turn drives a

20 tooth, %-inch pitch sprocket on the step-up idler. This sprock et is rigidly attached to a 54 tooth, 3/8-inch pitch sprocket, which drives the 17 tooth 3/8-inch pitch input sprocket to the CVT

mechanism). Therefore: (after passi ng through the chain force sensor

Tin = c2rPCFP (5.2)

Note: c- = f| -p- = -121 -92-

C_3

to CO

ro S_

O) r O |>

OJ o s_ o

re 4-> (0

o o

O) cz

re OJ

+-> 3 CL EZ

0) s- 3 cn -93-

and: rp- = 6.693 in.

The input torque reaches its peak value when the pedal force is

at the design maximum = 136 lb): (Fp max

Tin,max = (0.121 )(6.693 in)(136 lb)

= 110 lb-in.

The largest tractive force is required when the output

torque is also at its design maximum, which occurs when the

input torque is at the design maximum and the transmission is

downshifted = fully (e -20, see Figure 5.11a). Using equation

(5.1), the output torque is then

1-tan 6 T = T out, max in,max 1+tan e

1-tan 110 lb-in. (-20) 1+tan (-20)

= 236 lb-in.

The three traction balls contact the discs at a radius of 2.25

inches, so the maximum tractive force required is (see Figure

5.11b):

_ out p TR,max 3rdis_

236 lb-in. " 3(2.25 in.)

= 35.0 lb

5. 4.2) Derivation of the Traction Ball Normal Force

The normal force of the traction balls against the input and

output discs is developed by the wedging force of the traction -94-

(0 0) u

0) >

o n

o M

3 a

3 o

to cz o ^

!->

^ -o cz o o

OJ u s- o Ll-

CU c > o + *- OJ o M re o S- a i

*. ai (O ID re A

C -i-> o to *- s- O o a o* __

OJ s- 3 O) -95-

ball cradle bearings against the traction balls (see Figure 5.14).

The expression for the total wedging force is first derived from a consideration of the forces in the input chain acting on the tractive force generating mechanism. This expression is then converted to an expression for the normal force acting on the in put and output discs by analyzing the forces acting on each

traction ball .

The input chain schematic is repeated in Figure 5.12 to clarify the operation of the tractive force generating mechanism.

Dividing input torque equation (5.2) by the pitch radius of the

CVT input sprocket yields the force in the CVT input chain:

C rSCVT

F- = or rPC F (5.3) u c9 rSCVTr Pr

(r$CVT= 1.021 in.)

Note that twice the CVT input chain force is applied to the

the mechani floating sprocket. This force is then amplified by

sensor linkage before it is cal advantage of the chain force transmitted to the force sensor linkage spring . Therefore,

4 chain force sensor linkage, The mechanical advantage of the the of a point at M A is equal to the ratio of velocity sprocket to the of the center of the floating velocity chain force sensor springs point on the coupler where the attach to it. -96-

E to

cz re

.cz o OJ

cn cz

+j re s- OJ c: OJ CD

OJ O S- o

CD >

o re s-

4->

-CZ o

-> 3 Q. CZ

-C 4->

t|- O

4-> (J re s- O) + cz

oi S-

o> -97-

= 2c2FL- FS

(M.A. = 1.21)

As described earlier in this chapter, the force sensor spring is attached directly to the floating traction ball lever. This

lever further amplifies the spring force by the factor

7.438 in. = = d'/l9, 71q ^4 2.000 in.

The effective wedging force transmitted to the floating traction

ball is therefore

= FW C4FS

or, combining the above with equation (5.4),

rpr = 2c2C4T^~ (M-A') (5'5) FW c FPY SCVT

Proving the wedging force on all three traction balls is equal is a simple matter using Figure 5.13. The traction balls

120 are spaced at intervals about the periphery of the input and output discs, so by symmetry, the force on the fixed balls

(shown as F in Figure 5.13) must be equal. Summing forces in the X-direction then yields

F cos 60 + F cos 60 = Fw

F = FW ; Q.E.D.

traction balls can be re The wedging force applied to the lated to the normal force transmitted to the input and output -98-

floating traction ball

input /output discs

Figure 5.13) Derivation of the Wedging Force on the Fixed Traction Balls.

discs by summing the forces shown in Figure 5.14. Once again,

symmetry quickly shows FN,in = FN,out

Summing forces in the y-direction yields

45 FN,insin 45 + FN,outsin = FW

= 0.707 F, N,out W or, using Equation (5.5), PC = (M.A.) (5.6) 1-414c2c4 7 Fp FN,out SCVT -99-

traction ball

input disc output disc

Figure 5.14) Derivation of the Force of the Traction Ball Normal to the Input and Output Discs.

When Fp is at the design maximum of 136 lbs,

= 1.414(0.121)(3.719) (1.21) 136 lb FN,out, max f^ff-fc = 686 lb

5. 4.3) Traction Fluid Performance

Both the normal force available at the point of contact

between the traction ball and the output disc and the tractive

force that must be obtained at the contact point have now been

derived. The transmission must operate without slipping under

the worst case load. Therefore, the minimum coefficient of fric

tion required at the contact point can be determined: -100-

ii - TR, max Mmin FM N,out, max

_ 35.0 lb " 686 lb

= 0.051

The coefficient of friction, or traction coefficient, of

the CVT lubricant, Sun Oil traction fluid Type TDF 88, is plotted

as a function of Hertz contact pressure in Figure 5.15. The

Hertz contact pressure decreases if the normal force causing

the contact pressure decreases. Equation (5.6) shows that the normal force is proportional to the pedal force. However, Figure

5.15 shows the coefficient of friction of the traction fluid in creases as the Hertz contact pressure decreases. Therefore,

if the coefficient of friction of the traction fluid is suffi cient in the worst-case loading situation examined so far, it

will also be acceptable at all lesser loads .

The Hertz contact pressure generated under the worst-case loading will now be shown to correspond to a coefficient of friction in the traction fluid meeting u-m,-n- The maximum Hertzian compressive stress between two general curved bodies is given by the formula:

= (5,7) ac,max tt'c d

Note the worst-case loading conditions of maximum design pedal force at full downshift are seldom likely to be reached in practice. 101

I I

3000 FT/MIN 200 F

350,000 400,000 500,000

Hertz Pressure (psi)

Traction Fluid as a Figure 5.15) Coefficient of Friction of the CVT extracted Function of Hertz Contact Pressure (Graph from reference 20). 102-

where P is the normal load; c and d are functions of the radii

of curvature of the bodies at the contact point, the normal load,

and the material properties of each body6. The traction ball

is modelled as a sphere of radius r., and the output disc is

modelled as a cylinder of radius r^.^. Combining this model,

the material properties of steel, and a normal force, P, of:

= F., 4. 686 lb N,out, max

leads to the following result:

c = 0.0309

d = 0.0252

= c,max 421 kPsi

Entering this maximum value of the Hertz pressure in Figure 5.15

verifies that the advertised coefficient of friction of traction

condi- fluid TDF-88 reaches .051 under these worst-case loading

4-- 7 tions .

5. 4.4) Maximum Stress in the CVT

The largest stress in the entire transmission is the Hertzian

contact stress between the traction balls and their cradle bearings,

6 Roark (reference 14), Table XIV, #8.

7 Note the graph of Figure 5.15 is for a contact velocity of 200 3000ft/min at F. The constant force transmission CVT is of about 300ft/min expected to operate at a contact velocity conditions. at room temperature under the worst-case loading of traction fluid The analysis here assumes the performance the performance under these conditions will at least equal the represented in the graph. -103-

This stress will now be examined.

The cradle bearing geometry is sketched in Figure 5.16. The

centers of the 30 cradle bearings are offset from radial lines

from extending the input and output discs through the center of

each traction ball. By symmetry, the force on each cradle bearing

will be equal. Summing forces in the y-direction yields: 2FCR cos 30 = Fw

or FCR = 0.577FW (5.8)

Using equation (5.5), the maximum wedging force at the design

maximum pedal force is:

6;^3 Fw = 2(0. 121)(3.719) ;> (1.21)(136 lb)

= 971 lb

Therefore, the maximum force between a cradle bearing and a

traction ball is:

FCR = 0.577 (971 lb)

= 560 lb

The maximum Hertzian contact stress can again be determined using equation (5.7) and the worst-case loading parameters.

Modelling each cradle bearing as a cylinder having a radius of

0.75 inch yields:

c = 0.0308

d = 0.0189

= a 459 kpsi c ,max -104-

cradle bearings

Figure 5.16) Derivation of Cradle Bearing Forces.

The theoretical Hertzian contact stress levels are clearly very high between the traction balls and the input/output discs, and between the traction balls and the cradle bearings. The trac tion balls are stock replacements for the smaller Cleveland Speed

Variators; they are made from hardened and polished AISI grade

52100 bearing quality steel. The cradle bearing rings and input/ output discs are also to be fabricated from hardened AISI 52100 steel. The ultimate strength of this bearing-quality material is only on the order of 300 Kpsi . Nevertheless, the estimated

Hertzian stress levels are probably tolerable; ball bearings of o See reference 2, p. 141. -105-

the same material are routinely tested at predicted Hertzian

stress levels of IO6 563,000 psi with average L,0 lives of 45 x

cycles . The reason such high predicted contact stresses are

tolerable is apparently not completely understood; however,

small plastic deformations in the sphere and cylinder probably

cause the actual stress to fall substantially below the theoret

ically predicted levels.

5. 4.5) Transmission Life Estimate

The cradle bearings are the most highly loaded bearings in

the transmission. Therefore, the transmission life can be approxi

mated by estimating the life expectancy of these bearings. The

basic bearings are standard 32MM (1.2598 inch O.D.) light series

radial type (maximum-capacity) ball bearings; the outer races

are stiffened with 1.5 inch O.D. sleeves to help them withstand

the point contact stress with the traction balls. The life of

the cradle bearings have been estimated with the loading schedule

summarized in Table 4.1. The bearings are expected to have an

L,0 life of 700 hours when subjected to this loading at a pedal

crank rotational speed of 75 RPM. Therefore, the transmission is

also expected to have a life comfortably exceeding 500 hours.

g See reference 10, p. 119. Chapter 6

6. The Constant Force Controller

6. 1) Introduction

The constant force bicycle transmission utilizes a totally

mechanical integral feedback controller to adjust the transmission

shift ratio. As its name suggests, the goal of the controller is

to maintain a constant force at the pedal. As explained in Chap

ter 2, integral control produces zero steady-state error to a

step input, in addition to reducing the adverse effects of the

approximate sinusoidal force input from the pedal crank.

Section 6.2 describes the operating principles of the con

stant force controller in detail. Section 6.3 develops the

basic equations for each element of the control system, which

will be used in both a numerical simulation and a linear repre

sentation of the system. Section 6.4 analyzes the shift torque

capacity of the controller, which constrains its minimum possi

ble response time. Section 6.5 defines a model for the torque

demand at the rear wheel of the bicycle, considered to be the

control system input. Section 6.6 derives a model of the control

system which accounts for the important system non-linearities;

the model is intended for numerical simulation. Section 6.7

presents an accurate simulation of the actual control system

performance, using the model developed in the previous section.

-106- -107-

Section 6.8 formulates a linear model of the control system.

The final section presents a frequency response analysis of the

system, which will verify the control system's desirable low-

pass filter characteristics.

6. 2) Principles of Operation

The control system of the constant force transmission is a

simple mechanical integrator driven by the input chain to the

CVT. A simplified schematic of the input chain is repeated in

Figure 6.1. As mentioned in Chapter 2, the pedal crank chain is

not connected to the rear wheel as in an ordinary bicycle; rather.

it drives a step-up idler which rotates freely about the rear

wheel shaft. The input chain, connecting the step-up idler to

the CVT, is then forced to traverse a "W"-shaped path before

reaching the CVT. The upper left sprocket on this path is a

floating sprocket; it is constrained to traverse a horizontal

path by the chain force sensor linkage. The coupler link of

this 4-bar mechanism is attached to the frame by a pair of

springs in parallel . Therefore, the floating sprocket will

experience a displacement proportional to the force in the chain.

A flat circular plate is mounted to the front end of the

shaft carrying the floating sprocket on the chain force sensor

linkage (see Figure 2.3). This plate serves as the driving disc

1 traction ball The springs actually attach to the floating lever; however, the lever will move only a few thousandths of an inch relative to the frame. -108-

s- OJ

o o

0) o s- o

+-> cz re +-> to

o (J

cz r- re j_

+-> 3 Q. CZ l l

4- O

+-> O re s- OJ

3 -109-

of a mechanical integrating mechanism. The operation of this

mechanism is reviewed in Figure 6.2. The integrator driving

shown disc, rotating at angular velocity u. in Figure 6.2, moves

horizontally in back of an integrator driven wheel, which is

fixed in the horizontal direction. The rate and direction of

rotation of the integrator driven wheel depends on the dis

placement, y, of the moving integrator driving disc with respect

to its center. The reference position, y = 0, is defined where

the center of the driving disc coincides with the center! ine of

the driven wheel, as shown in Figure 5.2 (b). Therefore, if

the center of the driving disc moves in the negative y-direction

from the driven wheel, the wheel rotates in the negative (CW)

direction at a rate proportional to the displacement (Figure

6.2 [a]). Conversely, if the center of the driving disc moves

in the positive direction from the driven wheel, the driven wheel

rotates in the positive (CCW) direction at a rate proportional to the displacement (Figure 6.2 [c]).

As suggested in Figure 6.3, the integrator driven wheel changes the traction ball shift angle via a large geardown con sisting of a bevel gear set, a chain reduction, and the traction ball worm gears. The geardown also reverses the direction be tween the integrator driven wheel displacement, a, and the trac tion ball shift angle, e. The control system is set up so that the relative displacement of the integrator driving disc with

= when the average respect to the driven wheel is zero (y 0) -110-

o O- n

s- o +J ro S- cn OJ

ro c_)

CZ ro

o OJ

.cz +->

cz HI o

CO S- VcsJ* O) O. a o -u

^~ O- I o cz

*r S- Q-

OJ s- 3 cn -111-

o S-

03 a o

+-> cz ro 4-3 CO CZ o o

+-> ro E OJ JCZ o oo

QJ s-

cn -112-

pedal force reaches the desired level. If the average pedal

force goes above the desired level, a chain of events occurs

to leading downshifting the CVT: y becomes positive, which

causes the integrator driven wheel to rotate in the positive

direction, which in turn causes the CVT shift angle, e, to be

come more negative. Likewise, if the average pedal force falls below the desired level, the CVT will upshift: y becomes nega tive, the integrator driven wheel rotates in the negative direc tion, and e becomes more positive.

Note that the nearly sinusoidal force profile of the input chain will cause the integrator driving disc to be moving con stantly with respect to the driven wheel; i.e., the transmission will constantly shift while the pedals are in motion. As will be shown in Section 6.9, the low-pass filtering properties of the mechanical integrator will reduce the effect of this constant shifting to a very small oscillation of the shift angle about its ideal level. Thus, the transmission will shift substantially only if the average value of y remains positive or negative over 2 a few pedal cycles .

The integrator driven wheel is a stock friction drive wheel.

The wheel consists of an 0-ring drive tire mounted on a steel core. The 0-ring provides an easily-replacable surface of contact

The performance of the control system could be further improved by introducing a damper in the force-sensing mechanism. This system. would convert the controller to a second-order -113-

with the driving disc, besides providing an attractively high

coefficient of friction between the driving disc and the driven

wheel. The contact force between the driven wheel and the driving

disc is maintained by a pair of adjustable springs (see Figure

2.5). The combination of the 0-ring and spring loading makes

the integrator insensitive to surface irregularities, small

tolerancing errors, and dirt.

The force sensing mechanism is provided with limit stops

= at y -1.25 inches and y = 1.25 inches. This prevents the

driven wheel from falling off the driving disc. However, it

also introduces saturation nonlinearities in the control system.

These nonlinearities will be ignored in the first order analysis;

however, they will be fully accounted for in the computer simula

tion. In addition, an angular limit stop is provided on the

shaft connecting the bevel gear and manual shift angle control

knob to the sprocket driving the worm shafts (Figure 2.5). This

prevents the serious damage that would occur to the transmission

if the shift angle could exceed its design limits of 20. This

introduces another small saturation nonlinearity that is ignored

in both the first order model and the numerical simulation;

however, the transmission is not likely to operate frequently

at the shift angle extremes.

Controller 6. 3) Basic Modelling of the Constant Force

elements of the The basic equations used to represent the -114-

controller in both the numerical simulation and the linear model

are developed here. The modelling begins by considering the

set-up of the chain force sensor, as shown in Figure 6.4. The

integrator driving disc should reach the reference position

= (y 0) when the pedal force is at its average ideal value. The

ideal pedal force is assumed to be described by Equation (4.2):

FP,ideal = FD [l/2(co.+l>]

The average ideal pedal force is therefore simply

= 1/2 FP, ideal, ave FD

The force in the CVT input chain is described by Equation (5.3):

C 2 rSCVT FP

As shown in Figure 6.4, the force in the chain when the pedal

force is at the average ideal level is

F =lc ^L_F C, ideal, ave 2 2 rSCVJ D

The total force applied to the floating sprocket when y = 0 should therfore be

= 2 Fset C, ideal, ave

Fset = c2^- FD (6.1) rSCVT

the chain force sensor Set-up physically consists of adjusting

until the springs will linkage spring thumbscrew (see Figure 2.1) 115-

y=0

SCVT

Figure 6.4) Set-up of the Chain Force Sensor.

Figure 6.5) Free-Body Diagram of General Loading on the Floating Sprocket Shaft. -Im

balance this force at y = 03.

The simple spring shown balancing the floating sprocket in Figure 6.1 is an idealization of two parallel springs attached

to the coupler link of the chain force sensor linkage. The spring

constant of the actual springs ii related to an equivalent spring

constant for the idealized spring as follows:

k - 2kact Keq"7JDC7 (6.2) (kact = 7.73 lb/in.)

The general expression for the displacement of the integrator driving disc is derived with the aid of Figure 6.5. In general, the force supported by the equivalent spring is simply

S,eq C

The displacement, y, is found by dividing the difference between the general force and the setup force by the equivalent spring 4 constant :

The force sensing spring adjusting screw can be linearly calibrated over the total available pedal force range at the time of manufacture.

4 The model of the controller has been simplified by ignoring the effect of the inertia of the chain force sensor linkage and the CVT input chain in this equation. The inertia! force is ex pected to be an order of magnitude smaller than the average applied chain and spring forces. (The maximum inertia! force is expected to be approximately 1 lb; F . is approximately 28 lbs.) 117-

_ S,eq set y keq

set y k (6.3) eq

Substituting the expression for the input chain force (eq. 5.3)

and the setup force (eq. 6.1) yields: rPC 2FP " FD = y 2 rSCVT keq (6-4) As mentioned in the previous section, the chain force sensor

linkage is provided with limit stops at y = 1.25 inches.

Therefore, equation (6.3) is valid over the range

- Z < y yJ y 'max 'max

where

ymax=1'25 inches

An equation representing the operation of the integrator

itself is developed in reference to Figure 6.2. No-slip rolling

contact is assumed between the integrator driving disc and the

integrator driven wheel. Therefore, the incremental displacement

of the driving disc and driven wheel must be equal at the point

of contact, or

yw-dt = r.da

(r. = 0.813 inches)

of u. is Assuming the rotational velocity the driving disc, , constant leads to a simple expression for the angular displacement -1 18-

of the driven wheel, a:

= a 7 *o- / y dt i

As mentioned earlier, the integrator driven wheel directly con

trols the shift angle, e, through a reversing geardown having a

speed reduction ratio of 1:384. Therefore,

0).

= - e p/ydt (6.5) C3 l (c3 = 384)

4) Shift Torque Requirements

The constant force controller must provide sufficient torque

to change the shift angle of the traction balls under the worst-

case loading conditions. The gearing required to produce this

torque comprises the greatest constraint on the controller re

sponse time. This section verifies that the controller torque

output is sufficient.

The loading on the traction balls is summarized in Figure 6.6.

The maximum torque required to shift the traction balls occurs when the pedal force is at the design maximum of 136 lb. The normal force of the traction ball against the input and output discs,

FN, then reaches a maximum of 686 lb (see Section 5. 4.2), while the normal force of the traction ball against each cradle bearing,

FCR, reaches a maximum of 560 lb. (see Section 5. 4.4). The maximum coefficient of friction of the CVT traction fluid under the worst-case loading conditions is assumed to be 119-

cradle bearings

axis of rotation traction ball carriage

shift output disc

Figure 6.6) Forces Causing the Traction Ball Shift Torque.

= 0.1 vMmax

(this is approximately twice the advertised value; see Section

5. 4.3). Therefore, the maximum torque required to change the

shift angle of each traction ball is -120-

= 2 shift, max ymax rb FN, max

vmax rb CR, max

= 2(0. 1)(. 8125 in.)(686 lb)

+ 2(0. 1)(. 8215 in.)(560 lb)

= 202 lb-in.

The controller will now be shown to be capable of producing this torque.

The shift torque output of the mechanical controller is developed by the frictional force of the integrator driven wheel against the integrator driving disc. As mentioned in Section 6.2, the contact force between the wheel and the plate is generated by two springs mounted between the frame and the integrator driven wheel assembly, which is hinged on the frame (see Figure 2.5).

Each of these springs is adjustable. Minimizing the contact force is desirable to reduce wear and efficiency losses as the

"drags" chain force sensor linkage the integrator driving plate

of springs in back of the driven wheel . The combined force the is adjusted to

F. = 8 lb

A coefficient of friction of

y. = 0.5

5 integrator disc and A low contact force between the driving the shift angle reaches driven wheel is especially important if then on the plate. its limits (e = 20), as the wheel must slip -121-

is assumed between the steel driving disc and the 0-ring tire of the driven wheel. The torque developed at the integrator driven wheel is therefore (see Figure 6.7):

Tn,, = y-r.F. DW ill

= (0.5)(.8125 in.)(8 lb)

= 3.25 lb-in.

The integrator driven wheel is connected to the main traction ball worm shaft through a 1:4 bevel gear set and a 15:36 chain reduction. The main worm shaft is coupled to the remaining two worm shafts through 1:1 chain gearing (see Figure 2.1). There fore, the torque supplied to each worm by the integrator is

^in^4^3-251^^

= 10.40 lb-in.

integrator driving disc

integrator driven wheel

Mechanism. Figure 6.7) Shift Torque Generation -122-

No losses efficiency are accounted for in the bevel gear or chain reductions. themselves6 The worm gears provide a 1:40 speed reduction to the traction balls. Power losses in the worms are so substantial, their effect must be considered. The worm efficiency, n, is estimated with the formula7:

cos y tan A

COS cb + y COt A where

= T pressure angle

(14-1/2)

A = lead angle

(5 43')

y = coefficient of friction

(0.10)

Therefore,

n = 0.487 and the torque available to shift each traction ball is

TW,out = 40 n TW,in

= 203 lb-in.; Q.E.D.

c Boston Gear 16 pitch single thread steel worm, Catalog #GLVH, and 40-tooth bronze gear, Catalog #G1044.

See reference (17), equation (12-26).

8 The estimated maximum coefficient of friction of the traction fluid is used here. This value is conservative according to Figure 12-18 of reference (17) (the sliding velocity at the point of contact between the worm and the gear, the independent variable of Figure 12-18, is approximately 13 FPM). -123-

The maximum shift torque requirements estimated here are likely to be very conservative. The previous analysis assumed

static contact between the traction balls and the cradle bear

ings, and between the traction balls and the input and output

discs. The actual rolling contact between these surfaces is

to likely substantially reduce the shift torque requirements.

the Reducing integrator gearing, which will reduce the controller

response time, may therefore be possible, if road testing the transmission indicates this change is desirable. However, re

the ducing response time will adversely affect the steady-state

oscillation of the shift angle (see Section 6. 9).

6. 5) Modelling the Control System Input

The constant force bicycle controller has two inputs: the

reference input of the desired pedal force, FD, and the disturb

ance input of the torque demanded at the rear wheel of the bi

cycle. The desired pedal force is easily modelled, since it is

a constant. The torque demanded at the rear wheel of the bicycle

is much more random in nature; the required torque depends

mostly on the arbitrary current riding conditions, such as head

wind and road incline.

The general nature of the torque demands at the rear wheel

can be determined with the knowledge of the pedal force profile

estimated in Chapter 4. If the component of the pedal force

transmitted to the rear wheel is assumed to vary sinusoidally -124-

between 0 and a peak value, the torque generated at the rear wheel must share this same profile; i.e.,

TRW- W*) [?(cos^+l) (6.6)

is an Tref(t) arbitrary function reflecting the current riding conditions. Note the period, T, is the same as the pedal crank period, and the rear wheel torque is related to the output torque of the CVT by the output chain geardown ratio, c-. :

= TRW cl Tout <6-7)

The performance characteristics of the control system are best estimated by analyzing the response of the system to simple inputs representative of changes likely to be encountered in ac tual operation. Two representative input types have been utilized in the simulations of Section 6. 7. The most useful choice for

T f(t) is a simple step function: Tref - Tlev ' <*> <6-8>

sud- where T, is a constant. This choice for T - represents a lev ~ct den change in torque demand, as would likely be encountered when

headwind or a hill. changing directions with respect to a reaching

' This input provides a useful estimate of both the system response

error of the con- time to realistic input and the steady-state g function : troller. The second representative input is a ramp

9 to begin at a non-zero value This equation is actually modified value after a finite time. See and level off to a constant end Section 6. 7. -125-

Tref = Tlev l (6-9)

This choice represents a steady change in torque demand. The

response to this input provides useful qualitative information

about the control system performance.

The control system simulations presented in Section 6. 7

assume the rider will supply whatever short-term pedal force is

required by the current rear wheel torque prescription and trans

mission shift ratio. This means the actual pedal force may sig

nificantly differ from the ideal pedal force during the transient

response time of the controller. The pedal crank rotational

speed is assumed to remain constant regardless of pedal force.

The simulations require a value for the magnitude of the

The reference torque, T, , for the step input definitions.

reference torque magnitude is chosen as the peak steady-state

value of the rear wheel torque at the ideal pedal force. This

of will be explained with an example. Assume the performance

upshift the controller is to be simulated in changing from full

steady- = > ). The at t 0 to at steady state (t

T-,ev. An expression state midshift condition is prescribed via

equations (5.1), for T, is obtained by first combining (6.7), lev

and (5.2):

- tan e _ 1 =_ F TRW clc2rPC 1 + tan e P

= 10 e 0 . Due to the Midshift is defined where shift angle the exact cen nature of the shift ratio (eq. 5.1), non-linear 7.55 CVT occurs at e = ter of the shift range of the actually -126-

The peak value of the pedal force, Fp, at steady state is Fp.

Therefore, the peak value of the rear wheel torque at steady state is:

1 - tan e = ss Tlev clc2rPC l + tan e *"d (6.10) ss where

= e the steady state shift angle.

Completing the example, the midshift condition corresponds to a shift angle of 0, so

= 0.176)(.121)(6.693 (35 Tlev in.) -{"FIT lb>

= 90.1 lb-in.

All values required for T, in the simulations are summarized in Table 6.1.

Table 6.1

Important Reference Torque Values (FD = 35 lb)

Steady-State Shift Angle Peak Reference Torque Magnitude

ess Tlev

(full downshift) 193.2 lb-in.

0 (midshift) 90.1 lb-in.

20 lb-in. (full upshift) 42.0 -127-

Finally, the simulations require definition of the pedal crank speed, which determines both the period, T, of the sinu soidal input in equation (6.6) and the integrator driving disc

speed, id... A crank rate of 60 RPM is used in all following

simulations (see Chapter 4); higher crank rates will decrease

the system response time without affecting the steady state

error . The period corresponding to 60 RPM is

T = 1.00 sec

uj. , the rotational speed of the integrator driving disc, is

determined by referring to Figure 6.1 and equation (5.2). The

floating sprocket has the same number of teeth as the input

sprocket of the CVT, so

= (_. iii j - (6.11) i c- pedal

The pedal crank rotational speed is assumed constant over an

entire pedal cycle, so oj. remains constant at:

-""3.

11 proportional to co, is verified A reduction in response time by constant of the linear model, eq the expression for the time section 6.9 justi (6 19) The frequency response analysis of will not change: increasing fies that the steady state error which increases corner frequency the crank rate increases to., the of the input, u>, in (_ (eq 6.28). However, frequency the is not a function creases proportionally. Since gain, K, attenuation does not change. of oo- (eq 6.18), the net -128-

6) Numerically Modelling the Constant Force Controller

The constant force controller contains several elements that

are highly nonlinear. The first of these elements is the CVT:

inspection of equation (5.1) reveals that the shift ratio is a

non-linear very function of the shift angle, e. As verified in

Section (6.. 8), a linear approximation of this equation gives

useful results near = midshift (e 0); however, it becomes very

inaccurate at the shift limits of the CVT (e = 20). The limit

stops on the force sensor linkage provide the second major non-

linearity. Since the pedal force is modelled as dropping to

zero twice every pedal cycle, the force sensor linkage normally

enters at least one of the saturation regions (y = 1.25 inches)

twice every pedal cycle. Therefore, ignoring this saturation

nonlinearity will also substantially affect the accuracy of a

prediction of the control system performance.

The results of the numerical simulation of the basic con

troller equations presented in the next section fully account

for both of these nonlinearities. As will be shown in detail,

the numerical simulation directly integrates the equation for

the shift angle, e (equation 6.5), over many small time steps;

the shift ratio, the pedal force, and the resultant force sensor linkage position are simultaneously re-evaluated at each time step.

The computer program used to implement the numerical simu lation is included as an Appendix. This standard FORTRAN program -129-

was run on a Cyber 172 timesharing system at the University of

Minnesota. The results were displayed in easy-to-understand graphical form via a Tektronix 4014 storage tube terminal.

Figures 6.9 - 6.15 are direct hard-copies of these displays.

The integration is performed with a fourth-order Runge-Kutta single-step procedure12. A library routine, "RK", available from the University of Minnesota Computing Center, was utilized13.

This routine is capable of solving a system of up to 1000 differ ential equations; the application here to only one equation is comparatively straightforward. The routine begins the integra tion at the time prescribed in the first calling argument and ends it at the time prescribed in the second; however, the total time is automatically split internally into as many smaller time steps as are required to meet user-defined accuracy constraints.

The results shown in the next section meet relative error con straints on e of 1 x 10 radians or absolute error constraints

o of 1 x 10 radians per integration step, whichever is greater.

Single-step integration procedures are susceptible to cumulative errors over many steps; however, comparison with the linear model

12 of the fourth-order See reference (3) for a derivation Runge-Kutta formulas.

13 "RK" Documentation of routine is available from: University Computer Center 227 Experimental Engineering 208 Union Street Southeast University of Minnesota Minneapolis, MN 55455 -130-

in Section 6. 9 confirms the results to be dependable.

The Runge-Kutta procedure requires an initial condition for the output parameter, e, at the integration start time and an ex pression for the derivative of the output parameter, ~, which is generally an arbitrary function of time. The equations leading de to an expression for "DIFFEQ" -^ are clearly visible in subroutine

(see the Appendix); their origin is repeated here for clarity.

As described in the previous section, the torque required at the rear wheel must be prescribed as a function of time. The rear wheel torque definition is isolated in a separate function,

"TRW", so it may be easily modified by the user. Program line

(890) directly implements equations (6.6) and (6.8). The pedal force is then determined by combining shift ratio equation (5.1),

CVT input torque equation (5.2), and rear wheel torque equation

(6.7) as follows:

1 1 + tan e ((. ,?v - c " T ^''^ hP c-c-rp- 'RW 1 - tan e

(see program line 789). The pedal force is then related to the force sensor linkage position via equation (6.4) (program line

checked if neces 799). The saturation conditions for y are and,

"IF" FORTRAN statements (lines 805 sary, corrected with simple

derivative for the shift angle, e, and 810). Finally, the time is found by differentiating equation (6.5):

de._ to (6.13) dt c3r. -131

BIKE CONTROL SIMULATION PROGRAM INPUT INTEGRATION START TINE O A A

INPUT INTEGRATION END TIME ? 7.5 INPUT INITIAL BALL ANGLE ? 29. INPUT PEAK UALUE OF REAR UHEEL TORQUE (IN-LB) ? 90.1 INPUT GRAPH IDENTIFYING LABEL ? STEP INPUT - - FULL UPSHIFT TO MIDSHIFT

Figure 6.8) Interactive Input for the Bicycle Control Simulator.

(program line 816). Note that the most basic, and accurate, equations describing each element of the control system are utilized in the numerical simulation.

An example of the interactive input required to begin the constant force controller simulation program is provided in

Figure 6.8. All system parameters are pre-set with data state

ments (lines 160 - 170). The user then must provide the simula tion starting time, the simulation end time, the initial shift angle, the reference level of the prescribed input torque, and a title for the graphical output. Several examples of the re

section. At the sulting graphs are available in the following completion of each graph, the user is provided with the option

another time interval (see of continuing the simulation for

- Figures 6.9 - 6.12 and 6.14 6.15). -132-

6. 7) Simulation of the Control System Performance

The performance of the constant force controller is accurately

simulated in this section by means of the numerical model developed

in Section 6. 6. The representations of the rear wheel torque de

veloped in Section 6. 5 are used as input. Specifically, the con

troller response to five step inputs and two ramp inputs are pre- 14

sented .

The first simulation is shown in Figure 6.9. The step re

sponse of the controller in moving from full upshift to midshift

is represented. The interactive input required to generate this

graph is shown in Figure 6.8. The full upshift condition at t = 0

is specified with the initial shift angle value of

20 6 (0) =

The step change in rear wheel torque (superimposed on a sine

wave) is pre-specified by the inclusion of equations (6.6) and

C6.8) in function TRW, as shown in the Appendix. The magnitude of

the rear wheel torque is set to the midshift condition by inter

actively inputting the appropriate value from Table 6.1:

T, = 90.1 lb-in. lev

Three separate curves are superimposed in Figure 6.9. As indi

wide dashed line cated by the key at the top of the graph, the

solid represents the prescribed rear wheel torque, the line

14 functions are As explained in Section 6. 5, the step and ramp actually superimposed on a sine wave. -133-

* THIS PAGE INTENTIONALLY LEFT BLANK * -134-

BIKE CONTROL SIMULATOR 83'0S/02. 15.69.38. STEP INPUT - - FULL UPSHIFT TO MIDSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- iee "I

0 u T s P 75 H U I T F T T O A R N se Q G U L E E

I as D N E * Q L

e -10

e.ee 1.25 2. SO 3.75 5.60 6.85 7.50 TIME (SEC) C2- CI-3. .181 176 C3-384.C FD-35.ee OMEGA-SI. 89 RI- RPC-6.693 813 RSCUT-l.eSl SCEQ-18.78 YMAX-1.8S0 -141-

* THIS PAGE INTENTIONALLY LEFT BLANK * -142-

BIKE CONTROL SIMULATOR tt'QZ/ZS. . 15.17.43. STEP INPUT FULL UPSHIFT TO FULL DOWNSHIFT ANGLE- OUTPUT TORQUE PEDAL FORCE SHIFT

S H I F T

A N G L E

D E G

e.ee 8.50 3.75 5. 6.85 TIME (SEC)

C8- CI -3. 176 .181 C3-384.0 FD-35.00 OMEGA-51.89 RI- RPC-6.693 .813 RSCUT-1.081 SCEQ-12.78 VMAX-1.850 -143-

BIKE CONTROL SIMULATOR 83/08/02. 15.19.14. STEP INPUT FULL UPSHIFT TO FULL DOWNSHIFT PEMt F0RCE SHIFT ANGLE- eH? i coo p

7.50 8.75 10.00 11.25 12.50 13.75 15.00 TIME (SEC)

CS- CI -3. 176 .121 C3-384.0 FD-35.00 OMEGA-51.89 RI- RPC-6.693 .813 RSCUT-1.081 SCEQ-12.78 VMAX-1.850

Figure 6.11) Constant Force Controller Simulation. Step Response: Full Upshift to Full Downshift. 144-

BIKE CONTROL SIMULATOR 83/08/08. 15.81.88. STEP INPUT FULL DOWNSHIFT TO FULL UPSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE-

S H I F T

A N G L E

D E G

0.00 1.85 8.50 3.75 5.00 6.85 7.50 TIME (SEC)

CI-3. C8- 176 .181 C3-384.0 FD-35.00 OMEGA-51.89 RI- RPC-6.693 .813 RSCUT-1.081 SCEQ-18.78 VMAX-1.850 -145-

BIKE CONTROL SIMULATOR 83/08/02. 15.24.53. STEP INPUT - - FULL DOWNSHIFT TO FULL UPSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE-

S H I F T

A N G L E

D E G

C2- Cl-3.176 .121 C3-384.0 FD-35 . 00 OMEGA-SI . 89 RI- RPC-6.693 .813 RSCUT-1.021 SCEQ-12.78 VMAX-1.850

Figure 6.12) Constant Force Controller Simulation. Step Response: Full Downshift to Full Upshift. -146-

typical value of 8.5 seconds stated in Chapter 2 was determined

the by averaging full upshift to midshift and full downshift to

midshift response times.

The maximum shift step conditions of full upshift to full

downshift and full downshift to full upshift are presented in

Figures 6.11 and 6.12, respectively. The 2% response times are

14.0 seconds and 8.5 seconds, respectively.

The last step input considered has a magnitude of 0 at the

midshift condition. This represents the controller steady state

performance at midshift. The results are shown in Figure 6.13.

1.2 0 The shift angle oscillates about the reference line;

i.e., the magnitude of the steady state error is 1.2. This re

sult is verified in Section 6. 9.

The last two figures for this section, Figures 6.14 and

6.15, estimate the effectiveness of the controller in responding

to a steady change in torque demand at the rear wheel. Figure

6.14 models the response of the controller as the torque demand

steadily changes from the full upshift to the full downshift

conditions over a time span of 10 seconds. This was done by modifying function TRW to supply a ramp function superimposed on a sine wave, starting at the full upshift torque demand and

levelling out to the full downshift torque demand. This was accomplished by prescribing Tref as follows: -147-

BIKE CONTROL SIMULATOR 83/08/02. 15.87.08. STEADY STATE RESPONSE OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- iee 40- H,

0 U T P 75 U T

T 0 R Q 50- U E

I 85- N * L B

0.000 0.417 0.833 1.250 1.667 TIME (SEC)

C2- FD-35.00 OMEGA-51.89 CI-3 176 .121 C3-384.0 RI- SCEQ-18.78 VMAX-1.850 RPC-6.693 -813 RSCVT-1.081

Figure 6.13) Constant Force Controller Simulation. Steady State Performance at Midshift. 135-

BIKE CONTROL SIMULATOR 83/08/02. 15.11.00. STEP INPUT FULL UPSHIFT TO MIDSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- 100 80- Hi

0 u T S F H E I D T F A T L T 0 A F R N 0 Q 50 G R L U C E E E

D L E N B G * L B

-10

15.00 7.50 8.75 10.00 11.25 18.50 13.75 TIME (SEC)

C8- C3-384.0 FD-3S.00 0MEGA-51.89 Cl-3 176 .181 RI- RSCVT-1.081 SCEQ-18.78 YHAX-1.850 RPC-6.693 .813

Simulation. Response: Figure 6.9) Constant Force Controller Step Full Upshift to Midshift. -148-

BIKE CONTROL SIMULATOR 83/08/02. 15.32.31. RAMP INPUT FULL UPSHIFT TO FULL DOWNSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE 300i 601 , 1 1 i i m r-50 H

0 U r S P P H E U I D T F 40- 200 (4 -25 T L T 0 F A R 0 N Q R G U C L E E E

100 20-

I L D N B E X G L B

0-00 1.25 8.50 3.75 5.00 6.85 7.50 TIME (SEC) C8- Cl-3.176 .181 C3-384.0 FD-35.00 OMEGA-51.89 RI- RPC-6.693 813 RSCVT- 1.021 SCEQ-12.78 VMAX- 1.250 149-

BIKE CONTROL SIMULATOR 83/08/08. 15.33.50. RAMP INPUT FULL UPSHIFT TO FULL DOWNSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- 300

S H I F T

A N G L E

D E G

7.50 8.75 10.00 11-85 18.50 13.75 15.00 TIME (SEC)

C2- CI -3. 176 .121 C3-384.0 FD-35.00 OMEGA-51.89 RI- RPC-6.693 .813 RSCVT-1.081 SCEQ-12.78 YMAX-1.850

Figure 6.14) Constant Force Controller Simulation. Ramp Response: Full Upshift to Full Downshift in 10 Seconds. -150-

BIKE CONTROL SIMULATOR 83/08/02. 15.37.35. RAMP INPUT - - FULL DOWNSHIFT TO FULL UPSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE 2001 40-

S H I F T

A N Q L E

D E G

0.00 1.25 8.50 3.75 5.00 6.85 7.50 TIME (SEC)

-3. C8- CI 176 .121 C3-384.0 FD-35.00 ONEGA-SI. 89 RI- RPC-6.693 .813 RSCVT-1.081 SCEQ-18.78 VMAX-1.850 -151-

BIKE CONTROL SIMULATOR 83/08/02. 15.38.50. RAMP INPUT FULL DOWNSHIFT TO FULL UPSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE "I 40

0 u T S Pise-, H I D T F A T L T 0 A F R N 0 Q 100 G R U L C E E E

I 50 L ii E - B x G L B

7.50 8.75 10.00 11-85 12.50 13.75 15.00 TIME (SEC)

C3- 0MEGA-51.89 Cl-3-176 .121 C3-384.0 FD-35.00 RI- VMAX-1.850 RPC-6.693 -813 RSCVT-1.021 SCEQ-12.78

Figure 6.15) Constant Force Controller Simulation. Ramp Response: Full Downshift to Full Upshift in 10 Seconds. -152-

* 193'2 42' lb"^- + 1b'inio'sec-" 1h'1ni Aef t. OSMOsec

= 193'2 Tref lb-in- t> 10 sec

1s a9ai'n obtained TRW by superimposing Tref on a sine wave with eq. (6.6). Figure 6.14 shows the corresponding pedal force rises

from the desired pedal force of 35 lb to a maximum pedal force of

60 lb at 5.5 seconds, where it again begins dropping to the ideal

pedal force.

Figure 6.15 inverts the situation, modelling the controller

response to a steady change in torque demand from the full down

shift to the full upshift levels over 10 seconds. Function T, RW

creates a 10-second sinusoidal ramp with the following prescrip tion for Tpef :

- 42.0 lb-in. 193.2 lb-in. . . t = .Q--\ 9 ih -in + n < ln 193'2 1b_ln' + t < Tref TC~sec , 0 t 10 sec

Tref = 42.0 lb-in., t > 10 sec

Note that the peak pedal force never falls below 20 lb (at 10

seconds) when responding to this input.

An infinite number of response profiles would have to be

presented to fully understand the controller performance under

all conditions. Nevertheless, the seven profiles provided in

Figures 6.9 - 6.15 give a good understanding of the ability of

the controller to respond to inputs typically expected under actual operating conditions. The response appears fully ade quate to justify construction of a prototype transmission. -153-

6. 8) Linear Model of the Constant Force Controller

The constant force controller is approximated as a linear

control system in this section. The accuracy of the linear model

is limited by the non-linear nature of the CVT shift ratio and

integrator limit stops. Nevertheless, the model is quite accu

rate when the shift angle, e, is 0. Therefore, the model is

valuable for both verifying the numerical model of Section 6. 6

and evaluating the effectiveness of the controller as a low-pass

filter in Section 6. 9.

The equations used in developing the linear model of the

constant force controller and their Laplace transforms are sum

marized in Table 6.2. All are extracted directly from previous

sections, except the equation for T. . This equation is a

linear approximation of the CVT shift ratio. The linearization

begins with equation (5.1):

+ e Tin _ 1 tan

- T . 1 tan e out

The shift angle, e, is assumed to be small.

Therefore:

tan e = e

and:

16 Note the equation for the integrator position, y( t) , caused the limit ignores the saturation nonlinearities by stops. -154-

> aj

C/> o * >l - 4- to D- C/l **_ c/> U- C c cr r- 0 1 o 0) i s- O- > C_) o c_> O- in S- d) o_ s- 4- C\J n i- s- o CJ o r-O 0 CM <\i U II o o. ii II 0 m

^- ^s 4-) CO C/l i- a> to 10 ^^ 4*^- to O- <_ >- o CD U_ u_ -CZ (

o*j to CZ ID o

< 0 > -Q s CD co cr

4-> XJ 0) -o 4J o

o o CT s- a. oo 0> o o

0 fc." oo 0) o e +> o*j S- c_ o co s- o Osl CsJ o t CD CM

-C +J

o CD U to CD

CZ o s- i- CD 44 It, +->__ ro IX) i CM co

3 Z3 IO l3 IO LCO crz: -155-

so: T.

e2 j^- - 1 + 2e + out or, ignoring the higher order term, e2:

T. in =1+26 Tout

equation for T yields Substituting (6.7) t

Tin * 1_ (TRW + 26 TRW) Cl

The final linearization makes use of the knowledge that TRW will vary sinusoidally between 0 and the peak value T, , and the frequency is relatively fast compared to the controller response time. Therefore, the effect of TRW in the second term of the previous equation will be approximated as its average value:

TRW: ITlev <6J4>

Note T, is a constant, yielding a linear expression for Tin in terms of TRW and 6:

= J_ + T. l(t. T, 6)' (6.15) 'in c, RH lev

diagram The equations of Table 6.2 are assembled into a block representation of the controller in Figure 6.16. This figure

each element in the control system. clearly shows the function of

that the system has two in Note, as explained in Section 6. 5,

desired pedal and the puts: the reference input of the force, disturbance input of the rear wheel torque. 156-

Z3 o

CD O S- o

CO T3 CD

S- CD

O S- +-> cz o C_)

CD o S- o

cz CO 4-4 to c o o

E CC s- a. CO

CJ o

CD S-

O) 136-

represents the pedal force, and the short dashed line represents

the CVT shift angle. All controller parameters are summarized

at the bottom of each graph. Note the graph is spread over two

- contiguous time segments (0 7.5 seconds and 7.5 - 15 seconds)

to prevent overcrowding.

The prescribed rear wheel torque is clearly seen to vary

from 0 to the prescribed constant magnitude of 90.1 lb-in. The

peak pedal force drops approximately exponentially from 75 lb to

the desired pedal force of 35 lb as the shift angle decreases

approximately exponentially from the full upshift angle of

20 6 = to the midshift angle of 6 = 0. Note the sinusoidal

oscillation that occurs in the shift angle due to the sinusoidal

input; this effect is examined in detail in Section 6. 9.

The response time of a first-order system to a step input

is classically defined as the time required for the output to 15 of controller reach 2% of the final value . The response the to a true step input will be examined in Section 6. 8. However, a more meaningful response time for the constant force controller can be defined here by examining the controller response to the more realistic condition of superimposing the input step on a

required sine wave. The convention of determining the time for

has been preserved. the pedal force to reach 2% of its final value

100 lb The total expected range of the pedal force is roughly (see

15 See reference (13), p. 222 -157-

Figure 6.17 shows the block diagram of Figure 6.16 rearranged to consider the shift angle, e, to be the system output. Changing the output from the pedal force, Fp, to the shift angle, 6, places both inputs after the feedback point. This enables the two inputs to be combined, as shown in Figure 6.18, where :

Figure 6.18 also clearly shows the integral action of the system in the feedforward transfer function:

O) i 1 keq c3ri S

A final simplification of the block diagram is shown in

Figure 6.19. The transfer function of the system is: clc2rPC

*{S) . ^ clc3rirSCVTKeq 2a>iTlev

first- recognizable as that of a The transfer function is readily

verifies the exponential response order control system. This step

simulations of Section 6. 7. The gain of curves obtained in the

the system is:

,, . ClC2rPC - (6.18) N o-r ^'lev

17 6.17 and 6.18 is easily verified The equivalence of Figures signal for each diagram. by computing the error 158-

4-3 , (0 CO > > -l-> fcT" a> u Z3 CO JO Q. CD U CZ

^ l i CD 1 cr u m C u CM -a CD cz

A .a E o 00 c_>

/I L- CD CD o ^y O O ^ 4^ S- s- cm +J o cz o o C_) H c_> CT CD o > CD CO u.0". CJ U CJ s- (0 S- o o \- CM u_ O > U 0. O l-> 4- cz cz at h- CO lO -l-> +J CM CO CO U cz cz o o c_) CJ

cu CD

-CZ I % -l-> CM Htt <4- o o E a. E (0 mm CO S- CM CM cn U CO o CO a 1 I

-^ '5* o o o o

CXI

(D CD s- S- :_ Z3 Ol 159-

c1c2rPC X 2Tlev t9

- c1c3ri rSCVTkeq _ | 2(UiTlev

Figure 6.19) Simplified Block Diagram of the Constant Force Controller.

and the time constant is

clc3rirSCVTkeq T_ (6.19) l lev

Substituting the numerical values of the system parameters yields:

k = 0.0143 (6.20)

T = 1.38 sec (6.21) c

lb- The above constants use a value for T-jev of 90.1 in., correspond ed 0 corre- ing to a shift angle of (. is again set to 51.89 ,

RPM). sponding to a pedal crank rate of 60

re The model just developed is valuable for the frequency

remainder of this sec sponse analysis of the next section. The

further to examine the system tion will simplify the system even

the will not be super response to a true step input; i.e., step

model is less valuable for simu imposed on a sine wave. This

is valuable for checking the lating the real system, but it

the range of accuracy of the numerical simulation and determining

linear model. -160-

The derivation for the linear model subjected to a constant magnitude step input is identical to the previous derivation, except that two compensations made for the sinusoidal inputs

used previously are eliminated. The rear wheel torque, TRW, and

the desired pedal force, FD, will be assumed constant at half

the peak values used in the sinusoidal model of Table 6.1. The

first compensation for the sinusoidal input was made in Equation

(6.1), the setup force equation for the integrator spring. The

integrator is now set up to reach the reference position (y=0)

when the pedal force reaches its full ideal level:

rPC = 2 Fset c2 r$CVT FD (6.22)

This change can be accounted for without affecting the transfer

function by changing the input to:

^7 . - 2 FD " TRW (6"23)

equation the linearized The second compensation was made in (6.15).

wheel torque is now in model of the CVT shift ratio. The rear

terms of the input cluded at its full constant value in both

torque equation:

= ( TRW + 2 TRW0) Tin c7

function to: This changes the transfer -161

ClC2rPC

G(s)- 4TRW

1 -, ClC3rirSCVTkea 4iTRW

Note that since is now half of its previous peak value of TRW T, evs

the gain and time constant are actually the same as in equations

(6.18) and (6.19).

The standard 2% response time of the exponential response

of a linear first order system to a step input is 4 time constants

= _. / ClC3rirSCVTkeg T H '2%, linear I 4^-T^

= When TRW 45.0 in-lb (corresponding to a shift angle of 0)

T2%,linear = 5'54 sec

A parallel simulation using the numerical model is shown in

Figure 6.20. The previous numerical model has been appropriately modified for a true step input by halving the reference input

torque and pedal force, and including set-up spring force equa tion (6.22) in the integrator displacement equation (program

line 799 in the Appendix). In addition, the integrator satura tion nonlinearity has been suppressed by prescribing a large

represents system value for y . The simulation the response msx to a step change in rear wheel torque from a value corresponding to a shift angle of for t < 0 to a value corresponding to a

0 The response time is shift angle of for t _ 0. 2% nearly identically 5.54 seconds. Therefore, the linear and numerical -162-

BIKE CONTROL SIMULATOR 83^08/02. IS. 51. 19. TRUE STEP INPUT INTEGRATOR SATURATION SUPPRESSED OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- 60 1

0 U S p H u I T F

40 T T 0 A R N Q G U L E E

80-

I D N E * G L 8

0J -10

0.00 3.33 5.00 6.67 8.33 10.00 TIME (SEC)

C8 FD-17.Se 0MEGA-51.89 CI -3. 176 .181 03-384. 0 RI- SCEQ-12.78 YMAX-*** RPC-6.693 .813 RSCVT-1.021

Figure 6.20) Constant Force Controller Simulation. True Step Input; Saturation of Integrator Suppressed. -163-

models each verify other at a shift angle of 0. Note the true

2% response time is only about half of the more realistic response

time obtained by superimposing the step input on a sine wave in

Section 6. 7-

An of indication the accuracy of the linear model is obtained

again by considering a true step input to the system and trans

forming the solution for the shift angle, 6, back to the time

domain. Using Equation (6.23), the step input in the Laplace

domain will be:

2 T D RW X(s) ClC2rPC where FD is again the constant desired pedal force and TRW is again the constant prescribed rear wheel torque. The solution for e(t) is "found by solving for e(s) in the Laplace domain and transforming the result back to the time domain:

e(s) = X(s) G(s)

2 -r clc2rPC - 2FnD 'RW 4T cic2rPC RW CiCr.r-

, , hc3rirSCVreq e S'TRW

STRW clc3rirSCVTkeq> 1 clc2rPC FD" e(t) 2 'RW 1 - e TrRW L clc2rPC

The steady-state value of the shift angle is therefore: 164-

ClC2rPC ' e(-) - F t 2TRW D cic2rPC RW

Table 6.3 lists various typical rear wheel torques and the cor

responding actual shift angles that will produce them when 18 = 17.5 lb Fp , along with the steady-state shift angle predicted by the previous equation for e(). This table indicates the

0 linear model is very accurate when e = and good to about 10% for the range

-5 s e s 5

The accuracy drops severely as the shift angle approaches + 20.

9) Frequency Response Analysis of the Constant Force Controller

The analysis of the constant force bicycle transmission con cludes with a frequency response analysis of the controller. A

Bode diagram will be constructed to summarize the performance of the system over a wide range of input frequencies. This diagram will verify that the pedal crank frequency is efficiently filtered by the controller, which significantly reduces the steady state oscillation of the output shift angle.

The steady state performance of a system to a sinusoidal input of arbitrary frequency, a>, is obtained by evaluating the

18 These values were obtained using Equation (6.10). -165-

Table 6.3

Actual and Linear Model Steady-State Shift Angles

Pure Step Input

F_ = 17.5 lb

TRW 6, (lb-in.) actual linear

96.5

8.58 64.3 -

5 4.59 53.6 - -

45.0 0 0.00

37.8 5 5.49

31.5 10 12.32

21.0 20 32.81

19 transfer function at :

s = jto

( J =VT)

This enables calculation of both the magnitude and phase shift of the output when the sinusoidal input is known:

6(t) = X|G(j(o)|(sin at + /G(jgj)) (6.24)

= when: x(t) X sin (wt)

19 See reference (13), Chapter 9. -166-

The Bode diagram conveniently summarizes |G(ju)| and /G(jco) as a function of the input frequency, co. The Bode diagram of the constant force controller will now be developed.

Equation (6.17), the transfer function for the linear approximation of the system subjected to a sinusoidal input, is re-written below:

S(s) = t +KT s (6.25) c where K is the gain (Equation 6.18) and T is the time constant

(Equation 6.19). Therefore:

G(ja]) = 1 +kjo>T_ <6-26>

The numerical values of K and T corresponding to a shift angle of

0 will be used here (Equations [6.20] and [6.213). As discussed in Section 6. 8, the linear model is very accurate in this range.

The Bode diagram for this transfer function is provided in

Figure 2.1. The upper curve represents the logarithmic magnitude 20 of G(jui), in decibels, as a function of to :

20 log |G(j)| = ( 20 log K - 20 log Vl + c/l^2') db (6.27)

20 rather o> Figure Note the frequency is plotted as f than in 6.21, where

2ir 137-

* THIS PAGE INTENTIONALLY LEFT BLANK * -167-

-S

S- CD 1 L.L.-. 2

o i c_>

CD o s- o ! i

cz CO 1 : 1

o ' ! ! | C_) 1 i i i CD

' ! / : : : ! : |

. y- 1 : .

< O ' ;

E i . i : . i . . . CO "Aj-H: j }---!: i- I . ! O) CO -.-__:_. -

! ' ' CD TJ 1 O

'. . . CQ i

j 'i 9 < j I e c n n **t 8 8 ft 1 ID

CD S- Z3 -168-

The curve is simply the algebraic sum of a constant horizontal line at the log magnitude Of the gain and a first-order factor 21 curve having a corner frequency of

=^4- 2fa>iTlev = u , (6.28) c 'c clc3rirSCVTkeq

Using the gain and time constant of Equations (6.20) and (6.21)

co c =0.723^sec or: f = 0.115 hz c and:

= - + 20 log |G(ji_)| ( -35.9 20 log^ 1 ) db

The lower curve represents the phase angle of G(j<.) as a function

of cd:

/G(jco) = y4 - /l - Jg>T_

tan"1 or: /G(jgj) = - uT. (6.29)

Using the time constant of Equation (6.21),

tan"1 /G(jto) = - 1.38u>

The Bode diagram identifies the controller as a low-pass filter; i.e., signals having a frequency approximately less than the corner frequency, 0.115 hz, will simply be multiplied by the gain when passing through the controller, while signals having a

attenuated. Again frequency greater than 0.115 hz will be

21 381 See reference (13), page -169-

assuming a crank speed of 60 RPM, the input signal will have a frequency of

f = 2 hz

(twice the pedal crank frequency). Inspection of Figure 6.21 reveals this frequency to be a comfortable decade into the atten uation region. Physically, this means the large oscillation of the input force will be significantly reduced to a small oscilla tion of the output shift angle. This highly desirable attribute of integral control was visible in the simulations of Section 6. 7.

This section closes with a quantitative evaluation of the filtering effect of the controller. Using the Bode diagram or

Equation (6.27), the attenuation of the input frequency of 2 hz will be:

20 log |G(j4tt)| = - 61.7 db

The magnitude of the input signal is found with Equation (6.16):

The expression for x reaches a maximum when the rear wheel torque falls to its minimum of 0:

x = 35.0 lb max

when the rear wheel torque The expression for x reaches a minimum reaches its maximum of 90.1 in-lb:

x. = 35.0 lb mm

signal is: Therefore, the amplitude of the input -170-

X = 35.0 lb

The magnitude of the steady state error of the shift angle is then determined with equation (6.24):

|e(t)|= X |G(j4tt)| / 6_L_7 20 = 35.0 \10

1.65 = 0.0288 rad =

Using the Bode diagram or Equation (6.29), the shift angle is found to differ in phase from the input, x(t), by:

/G(j4ir) = - 87

Therefore, the shift angle will lag x(t) by 87. Note from

Equation (6.16) that x(t) leads the prescribed wheel torque by

180. Thus, the shift angle oscillation is expected to lead the sinusoidal rear wheel torque prescription by 93.

The previous results have been verified with the numerical simulations developed in Section 6. 6. The results are displayed

in Figure 6.22. The effects of the saturation nonlinearity caused by the integrator limit stops have again been suppressed

ymax. Note the amplitude and by prescribing a large value for phase angle of the steady state oscillation of the shift angle

predicted in this sec agree nearly identically with the results tion.

Figure 6.13 (Section 6. 7) shows the steady state performance

conditions as Figure except of the controller under the same 6.22,

integrator have been included. the effect of the limit stops on the -171-

BIKE CONTROL SIMULATOR 83/08/02. 15.48.11. STEADV STATE INTEGRATOR SATURATION SUPPRESSED OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE

100 , t 40 H i z

O u T S P 75j H U I T F T T 0 A R N 50 Q Q U L E E

I 25 D N E X G L B

0.000 0.417 0.833 1.250 1.667 2.083 8.500 TIME (SEC)

C2- CI -3. 176 .121 C3-384.0 FD-35.00 OMEGA-51.89 RI- RSCUT- SCEQ-12.78 RPC-6.693 .813 1.021 VMAX-*XX*X

Figure 6.22) Constant Force Controller Simulation. Steady State Performance at Midshift; Saturation of Integrator Suppressed. -172-

The steady state oscillation of the shift angle is reduced to

about 1.2. Therefore, the saturation nonlinearity of the in

tegrator has a beneficial effect on the steady state performance

of the controller. Chapter 7

Conclusion

A human powered transmission designed to automatically maintain an optimal constant force demand at the pedal of a bi cycle is described. The transmission is intended to replace the derailleur mechanism on a standard lightweight bicycle frame for general touring use. The rider powers the unit via a standard front pedal crank assembly. The transmission monitors the force applied to the pedals to maintain optimal gearing regard

less of current riding conditions. Therefore, the rider is freed from manually shifting the bicycle gearing to maintain a comfortable rate of pedalling. In addition, the continuously- variable transmission utilized in the design enables stepless shifting over a range wider than that provided by an ordinary

10-speed derailleur mechanism. The transmission is a self-con tained unit that bolts on the rear wheel assembly of a standard lightweight frame; no custom modification of the stock frame is required.

The concept of combining a constant force mechanical integral controller with a continuously variable transmission is unique, as described in Chapter 1. The CVT, controller, and interaction between them have all been custom designed to meet the unique constraints of bicycling, as described in Chapter 2. A set of instrumentation designed to determine the loading on a bicycle

-173- -174-

transmission under actual riding conditions is described in

Chapter 3. The data obtained experimentally from this instrumen tation is condensed to a set of expected loading criteria for the constant force transmission in Chapter 4. The CVT design is shown to meet these criteria in Chapter 5; likewise, the per formance of the control system is validated in Chapter 6.

The actual prototype transmission has not yet been con structed at the time of this writing. Therefore, it is empha sized that the transmission is a prototype; the radical improve ments incorporated in the design will require extensive testing beyond the simulations described here before further development is practical. Specifically, road testing the actual prototype hardware will answer the following immediate questions:

1) Does a constant force at the pedal create the optimal environ

ment for a bicyclist?

"ideal" 2) Is 35 lb the pedal force? How much will this value

vary between riders?

3) Is a higher or lower response time desirable for the controller?

4) Will the traction drive demonstrate a reasonable life?

Should overload protection be provided?

Definitive answers to these questions are not available from the

estimate" literature; the prototype has been designed on a "best basis from the experimental data described in Chapter 4. The

enable extensive modification of modularity of the prototype will

other components if each system component independent of the -175-

road testing indicates such modification is necessary.

Marketing of the prototype design itself is not practical due to the size, weight, and cost of the device. However, an swering the questions outlined in the previous paragraph will also suggest the best routes for optimization of the transmis sion. The optimization process may include further experimental research of the expected loading or finite element analyses of the CVT components. Ideally, evaluation of the prototype design will lead to a second generation transmission meeting the size, weight, durability, and cost constraints of a bolt-on accessory suitable for general marketing. Acknowledgments

The author would like to acknowledge the following individuals for their helpful suggestions on improving the transmission design:

Mr. 1) Joseph Arnold (Minneapolis, MN) for suggesting elimination

of mechanical compensation for the sinusoidal input to the con

trol system and general encouragement;

2) Mr. Kenneth Hood (Rochester, NY), who was invaluable for stressing

the importance of an easy-to-manufacture design;

3) Dr. Kim Stellson (Minneapolis, MN) for assistance in analyzing

the control system;

4) Mr. Peter Stryker (Minneapolis, MN) for suggesting the addition

of damping to the control system.

The assistance of the above individuals is sincerely appreciated.

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Bicycling!, Vol. 14, No. 10, pp. 36-37, October 1973. Appendix

Constant Force Controller Simulation Program

180- -181-

00001 PROGRAM TAPE5= BCS( INPUT , OUTPUT , INPUT , 00002+ TAPE6=0UTPUT) . 00003C 00004C CONSTANT FORCE CONTROLLER SIMULATION PROGRAM 00005C THOMAS R. CHASE 00006 C 7/20/81 0O007C 00008 C PURPOSE OOO09C THIS PROGRAM SIMULATES THE RESPONSE OF A 000 10C CONSTANT FORCE BICYCLE TRANSMISSION CONTROL 0001 1C SYSTEM TO A GIVEN REAR WHEEL TORQUE INPUT 000 12C 000 1 3C THE OUTPUT CONSISTS OF A GRAPH OF THE 000 14C SPECIFIED OUTPUT (REAR WHEEL) TORQUE, 000 15C THE PEDAL FORCE REQUIRED TO PRODUCE THAT 000 16C OUTPUT TORQUE, AND THE SHIFT ANGLE AS A OOO 1 7C FUNCTION OF TIME. 00018C 00019C THE GRAPHICS ARE WRITTEN FOR 00020C TEKTRONIX 4010 SERIES HARDWARE. 0002 1C 0002 2C THE CONTROL DIFFERENTIAL EQUATION IS SOLVED 00023C USING UNIVERSITY OF MINNESOTA LIBRARY

00024C RUNGE-KUTTA INTEGRATION ROUTINE RK . 0002 5C 00026C MAINLINE INPUT 00027C C1: CHAIN REDUCTION BETWEEN THE OUTPUT OF 00028 C THE CVT AND THE REAR WHEEL OF THE 00029C BICYCLE 00030C (INPUT WITH A DATA STATEMENT) 00031C C2: TOTAL CHAIN REDUCTION BETWEEN THE PEDAL 00032C CRANK AND THE INPUT OF THE CVT 00033C (INPUT WITH A DATA STATEMENT) 00034C C3: TOTAL GEARDOWN BETWEEN THE INTEGRATOR 00035C DRIVEN WHEEL AND THE TRACTION BALL 00036C SHIFT ANGLE 00037C (INPUT WITH A DATA STATEMENT) 00038C ENDTIME: TIME AT WHICH THE FIRST SIMULATION 00039C INTERVAL WILL END. SEC 00040C FD: PEAK VALUE OF THE IDEAL PEDAL FORCE, LB 00041C (INPUT WITH A DATA STATEMENT) 00042C IANS: INPUT BUFFER FOR YES/NO INTERACTIVE 0004 3C RESPONSE FROM THE USER 00044C ( INTEGER/ALPHA-NUMERIC) 00045C IRD: READ LOGICAL UNIT NUMBER OF THE 00046C COMPUTER BEING USED 00047C (INPUT WITH A DATA STATEMENT) 0004 8C IWR: WRITE LOGICAL UNIT NUMBER OF THE 00049C COMPUTER BEING USED 00050C (INPUT WITH A DATA STATEMENT) 00051C NMPT: NUMBER OF POINTS DISPLAYED ON THE 00052 C RESPONSE CURVE 00053C (MAXIMUM OF 501) 00054C (INPUT WITH A DATA STATEMENT) 00055C OMEGA: ANGULAR VELOCITY OF THE INTEGRATOR 00056C DRIVING DISC, RAD/SEC 00057 C (INPUT WITH A DATA STATEMENT) DRIVEN 00058C RI RADIUS OF THE INTEGRATOR WHEEL, 00059C IN. 00060C (INPUT WITH A DATA STATEMENT) IN. 0006 1 C RPC: LENGTH OF THE PEDAL CRANK, 00062C (INPUT WITH A DATA STATEMENT) INPUT SPROCKET 00063C RSCVT: PITCH RADIUS OF THE 00064C OF THE CVT, IN. DATA STATEMENT) 00065C (INPUT WITH A SCEQ- CONSTANT OF THE TWO 00066C EQUIVALENT SPRING -182-

00067C PARALLEL CHAIN FORCE SENSOR LINKAGE 0006 8 C SPRINGS ACTING THROUGH THE CHAIN FORCE 00069C SENSOR LINKAGE, LB/IN. 00070C (INPUT WITH A DATA STATEMENT) 0007 1 C TIMINIT- TIME AT WHICH INTEGRATION WILL 0007 2 C BEGIN, SEC 0007 3 C TSTRTDG: INITIAL VALUE OF THE SHIFT ANGLE OF 00074C THE CVT, DEG 00075C TLEV: PEAK VALUE OF THE SINUSOIDAL TORQUE 00076C REQUIRED TO DRIVE THE REAR WHEEL OF 0007 7 C THE BICYCLE, LB-IN. 00078C YMAX: THE MAXIMUM ALLOWABLE DISPLACEMENT OF 00079C THE CHAIN FORCE SENSOR LINKAGE, IN. 00080C (INPUT WITH A DATA STATEMENT) 00081C 0008 2C RESTRICTIONS 0008 3C 1) INTENDED FOR USE WITH TEKTRONIX 4010 00084C SERIES HARDWARE ONLY 00085C (MIXED ALPHA-NUMERIC/VECTOR OUTPUT MAY 00086C CAUSE INCORRECT DISPLAY ON SOME 0O087C COMPATIBLE TERMINALS) 00088C 2) CONTAINS CALLS TO NON-STANDARD SYSTEM 00089C ROUTINES "DATE", "EOF", "RK", AND "TIME" 00090C 00091C ERROR INDICATIONS 00092C NONE 00093C 00094C KEYWORDS 00095C CVT BALL SHIFT ANGLE: THE CONSTANT FORCE 00096C BICYCLE TRANSMISSION CONTAINS 00097 C A CLEVELAND TYPE CONTINUOUSLY 00098C VARIABLE TRANSMISSION. 00099C THE BALL SHIFT ANGLE INDICATES THE 00100C SHIFT RATIO OF THE TRANSMISSION. 00101C 00102C METHOD 00103C THE NON-LINEAR CONTROL SYTEM DIFFERENTIAL 00104C EQUATION IS NUMERICALLY INTEGRATED BY THE 0O105C RUNGE-KUTTA METHOD. 00106C 00107C DESCRIPTION OF VARIABLES 00108C IYES: BUFFER FOR HOLDING "YES" INTERACTIVE 00109C RESPONSE FLAG 001 10C (INTEGER/ALPHA-NUMERIC) 00111C STARTIM: STARTING TIME FOR ONE SIMULATION 001 12C INTERVAL, SEC 00113C THETA: SHIFT ANGLE OF THE CVT, RAD 001 14C THESTRT: INITIAL VALUE OF THE SHIFT ANGLE 001 15C OF THE CVT, RAD 001 16C TSTEP: TIME STEP BETWEEN DATA POINTS FOR 00117C ONE SIMULATION INTERVAL, SEC 001 18C 00119C SUBROUTINES CALLED 00120C DATASET: ROUTINE TO CALCULATE ALL OUTPUT 00121C PARAMETERS FOR ONE SIMULATION 001 22C INTERVAL 00123C GRAPH: ROUTINE CONTAINING ALL THE GRAPHICAL 00124C COMMANDS TO GENERATE A TEKTRONIX 00125C DISPLAY OF ALL OUTPUT INFORMATION 00126C CALCULATED FOR ONE SIMULATION 00127C INTERVAL 00128C GRNAMEI: ROUTINE TO REQUEST A TITLE FOR A 00129C TEKTRONIX GRAPH AND PREPARE IT 00 1 30C FOR PRINTING BY GENERAL PLOTTING OO 1 3 1 C ROUTINE GRNAME 00132C TEKFIN: ROUTINE TO TERMINATE TEKTRONIX 183-

001 33C GRAPHICS ROUTINES 00134C TEKINIT: ROUTINE TO INITIALIZE TEKTRONIX 00135C GRAPHICS ROUTINES 00136C 00137C MAINLINE DECLARATIONS 00138 INTEGER IANS 00139 INTEGER IYES 00140 INTEGER NMPT 00141 INTEGER IRD, IWR

00142 REAL C1 , C2, C3, FD, OMEGA, 00143+ RI, RPC, RSCVT, SCEQ, YMAX 00144 REAL T, TORDATA, 00145+ FPDATA, THEDATA 00146 REAL ENDTIME, TIMINIT, TSTRTDG 00147 REAL STARTIM, THETA, THESTRT, TSTEP 00148 REAL TLEV 00149 REAL PI 00150C 00151C COMMON STATEMENTS AND DECLARATIONS

00152 COMMON/ CONST AN / C1, C2. C3, FD , OMEGA, 00153+ RI, RPC, RSCVT, SCEQ, YMAX

00154 COMMON/ DATA / T(501), TORDATA( 501 ) , 00155+ FPDATA(501), THEDATA(501) 00156 COMMON/ RWTQ / TLEV 00157 COMMON/ INOUT / IRD, IWR 00158C 00159C DATA STATEMENTS 00160 DATA C1 / 3. 176 / 00161 DATA C2 / 0. 1211 / 00162 DATA C3 / 384. / 00163 DATA FD / 35. /

00164 DATA OMEGA / 5 1 . 89 / 00165 DATA RPC / 6.693 / 00166 DATA RI / 0.8125 / 00167 DATA RSCVT / 1.021 / 00168 DATA SCEQ / 12.78 / 00169 DATA YMAX / 1.25 / 00170 DATA NMPT / 501 / 00171 DATA IRD, IWR / 5, 6 / 00172 DATA PI / 3.14159265358979 / 'Y' 00173 DATA IYES / / 00174C ***** 00175C ***** START OF EXECUTABLE CODE 00176C 00177C INITIALIZE TEKTRONIX GRAPHICS 00178C 00179 CALL TEKINIT 00180C IDENTIFIER 00181C PRINT OUT PROGRAM 00182C 00183 WRITE(IWR, 100) 001 84C START TIME FOR THE 00185C REQUEST INTEGRATION INTERVAL 00186C FIRST SIMULATION 00187C 00188 WRITE(IWR, 120) TIMINIT 00189 READURD, 140) 00190C OF FIRST SIMULATION 00191C SET STARTING TIME 00192C INTERVAL 00193C - TIMINIT 00194 STARTIM 00195C TIME FOR THE FIRST 00196C REQUEST END INTERVAL 00197C SIMULATION 00198C 184-

00199 WRITE(IWR, 160) 00200 READ(IRD, 140) ENDTIME 00201C 00202C REQUEST INITIAL CONDITION FOR SHIFT ANGLE 00203C IN DEGREES 00204C 00205 WRITE(IWR, 180) 00206 READ(IRD, 140) TSTRTDG 00207C 00208C REQUEST PEAK VALUE OF REAR WHEEL TORQUE 00209C 00210 WRITE(IWR, 190) 00211 READ(IRD, 140) TLEV 002 1 2C 002 1 3C REQUEST AND FORMAT GRAPH IDENTIFYING LABEL 00214C 00215 CALL GRNAMEI 002 16C 002 17C CONVERT INITIAL SHIFT ANGLE TO RADIANS 00218C 00219 THESTRT TSTRTDG*PI/180. 002 20C 00221C INITIALIZE SHIFT ANGLE 00222C 00223 THETA THESTRT 00224C 00225C CALCULATE TIME STEP BETWEEN DATA POINTS 00226C 00227 TSTEP (ENDTIME STARTIM)/FLOAT(NMPT- 1 ) 00228C 00229C SET GRAPH LIMITS 002 30C 00231 CALL LIMSET(NMPT, THESTRT) 00232C 00233C MAIN PROCESSING LOOP 00234C 00235 50 CONTINUE 00236C 00237C CALCULATE ALL OUTPUT PARAMETERS FOR ONE 00238C SIMULATION INTERVAL 00239C 00240 CALL DATASET(NMPT, STARTIM, TSTEP, THETA) 00241C 0O242C CREATE A TEKTRONIX DISPLAY OF ALL OUTPUT 00243C INFORMATION FOR ONE SIMULATION INTERVAL 00244C 00245 CALL GRAPH( NMPT ) 00246C 00247C RE-SET THE STARTING TIME FOR THE NEXT 00248C SIMULATION INTERVAL 00249C 00250 STARTIM = STARTIM + FLOAT(NMPT - 1)*TSTEP 0025 1C 00252C DETERMINE IF THE INTERACTIVE USER WANTS TO 00253C CONTINUE THE SIMULATION FOR ANOTHER INTERVAL 00254C 00255 WRITE(IWR,200) 00256 READ(IRD,210)IANS 00257 IF (IANS.EQ.IYES) GO TO 50 002 58C 00259C TERMINATE TEKTRONIX GRAPHICS 00260C 00261 CALL TEKFIN 00262C 00263C TERMINATE PROGRAM 00264C -138-

BIKE CONTROL SIMULATOR 83/08/08. 15.13.89. STEP INPUT FULL DOUNSHIFT TO MIDSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- 100 "1

O ft li f fi 2% response ti u i T P P

U 75-j - D T A L T "MM 0 R 4 Q U C E -I E

I 85 L N B * L B

0.00 1.25 8.50 3.75 5.00 6.85 7.50 TIME (SEC)

CI -3. C8- 176 .181 C3-384.0 FD-35.00 0MEGA-51.89 RI- RPC-6.693 .813 RSCVT-1.081 SCEQ-18.78 VMAX-1.8S0 -139-

BIKE CONTROL SIMULATOR 83/-O8/-02. 15.15.07. STEP INPUT FULL DOWNSHIFT TO MIDSHIFT OUTPUT TORQUE PEDAL FORCE SHIFT ANGLE- 100 "1

0 u T S H P 75 U I T F T T 0 A R N Q 50 G U L E E

I 25 N E G * L B

7.50 8.75 10.00 11.85 12.50 13.75 15.00 TIME (SEC)

C2- C3-384.0 FD-35.00 0MEGA-51.89 CI-3 176 .121 RI- RSCUT-1.081 SCEQ-1S.78 YMAX-1.850 RPC-6.693 .813

Response: Figure 6.10) Constant Force Controller Simulation. Step Full Downshift to Midshift. -140-

Chapter 4). Therefore, the controller response time will be defined as the time required for the peak pedal force to fall in the range

33 lb * Fp * 37 lb for a desired pedal force setting of FD = 35 lb

As highlighted in Figure 6.9, this response time is approximately

10.0 seconds for the full upshift to midshift transition.

Figure 6.10 simulates the controller step response in changing from full downshift to midshift. The approximately exponential force and shift angle profiles are again evident. The 2% response time falls to 7.0 seconds. Comparison of the shift angle profiles of Figures 6.9 and 6.10 between 0 and 2 seconds reveals less os cillation in the full downshift configuration. This is a result of the input torque model always returning to 0 in both configura tions. Physically, this means the integrator will reach both limiting values of y when the pedal force varies from 0 to 75 lb in the full upshift configuration; however, the integrator will never leave the negative range for y when the pedal force varies from 0 to 16 lb in the full downshift configuration. Therefore, the transmission upshifts continuously when the shift angle ap proaches full downshift at a light pedal load.

for The response time of the control system differs every

state conditions. The new set of prescribed initial and steady 185-

00265 CALL EXIT 00266C 00267C FORMATS FOR PROGRAM IDENTIFIER AND INPUT 00268C REQUESTS 00269C 00270 100 FORMATC BIKE CONTROL SIMULATION PROGRAM') 00271 120 FORMATC INPUT INTEGRATION START TIME') 00272 140 F0RMAT(F10.5) 00273 160 FORMATC INPUT INTEGRATION END TIME') 00274 180 FORMATC INPUT INITIAL BALL ANGLE') 00275 190 FORMATC INPUT PEAK VALUE OF REAR WHEEL ' 00276+ 'TORQUE (IN-LB)' ) 00277 200 FORMATC DO YOU WISH TO CONTINUE THE ', 00278+ 'INTEGRATION?' ) 00279 210 F0RMAT(A1 ) 00280 END 0028 1C 07-20-81 TRC; CREATED 00282C 00283C ************************* BCS.DATASET *********** 00284 SUBROUTINE DATASET ( NMPT, STARTIM. TSTEP. 00285+ THETA ) 00286C ******************** ***************************** 00287C EXPERT: TRC 00288C 00289C PURPOSE 00290C ROUTINE TO CALCULATE ALL OUTPUT PARAMETERS 0029 1C FOR ONE SIMULATION INTERVAL 00292C 00293C THIS ROUTINE FILLS THE PARAMETER ARRAYS 00294C REQUIRED FOR THE OUTPUT GRAPHS BY 00295C INTEGRATING THE MAIN CONTROL 00296C DIFFERENTIAL EQUATION 0029 7 C 00298C INPUT 00299C C1: CHAIN REDUCTION BETWEEN THE OUTPUT OF 00300C THE CVT AND THE REAR WHEEL OF THE 00301C B I CYCLE 00302C (INPUT IN COMMON BLOCK CONST AN) 00303C C2: TOTAL CHAIN REDUCTION BETWEEN THE PEDAL 00304C CRANK AND THE INPUT OF THE CVT 00305C (INPUT IN COMMON BLOCK CONSTAN) 00306C C3: TOTAL GEARDOWN BETWEEN THE INTEGRATOR 00307C DRIVEN WHEEL AND THE TRACTION BALL 00308C SHIFT ANGLE 00309C (INPUT IN COMMON BLOCK CONSTAN) 003 10C FD: PEAK VALUE OF THE IDEAL PEDAL FORCE, LB 0031 1C (INPUT IN COMMON BLOCK CONSTAN) 003 1 2C FP: PEDAL FORCE. LB 00313C NMPT- NUMBER OF POINTS DISPLAYED ON THE 003 14C RESPONSE CURVE 003 1 5C OMEGA: ANGULAR VELOCITY OF THE INTEGRATOR 003 1 6C DRIVING DISC, RAD/SEC 003 1 7C (INPUT IN COMMON BLOCK CONSTAN) 003 1 8C RI: RADIUS OF THE INTEGRATOR DRIVEN WHEEL, 003 19C IN. 00320C (INPUT IN COMMON BLOCK CONSTAN) 0032 1C RPC: LENGTH OF THE PEDAL CRANK, IN. 00322C (INPUT IN COMMON BLOCK CONSTAN) 00323C RSCVT: PITCH RADIUS OF THE INPUT SPROCKET 00324C OF THE CVT, IN. 00325C (INPUT IN COMMON BLOCK CONSTAN) 00326C SCEQ: EQUIVALENT SPRING CONSTANT OF THE TWO 00327C PARALLEL CHAIN FORCE SENSOR LINKAGE 00328C SPRINGS ACTING THROUGH THE CHAIN FORCE 00329C SENSOR LINKAGE, LB/IN. 003 30C (INPUT IN COMMON BLOCK CONSTAN) -186-

0033 1C STARTIM: STARTING TIME FOR THE SIMULATION 00332C INTERVAL, SEC 00333C THETA: SHIFT ANGLE OF THE CVT, RAD 00334C (ALSO AN OUTPUT PARAMETER) 00335C TSTEP: TIME STEP BETWEEN DATA POINTS FOR 00336C ONE SIMULATION INTERVAL, SEC 00337C YMAX: THE MAXIMUM ALLOWABLE DISPLACEMENT OF 00338C THE CHAIN FORCE SENSOR LINKAGE, IN. 00339C 00340C OUTPUT 0034 1C FPDATA: ARRAY OF PEDAL FORCE AS A FUNCTION 00342C OF TIME, LB 00343C (OUTPUT IN COMMON BLOCK DATA) 00344C T: ARRAY OF TIME, SEC 00345C (INDEPENDENT PARAMETER OF OUTPUT GRAPHS) 00346C (OUTPUT IN COMMON BLOCK DATA) 00347C THEDATA: ARRAY OF SHIFT ANGLE AS A FUNCTON 00348C OF TIME, DEG 00349C (OUTPUT IN COMMON BLOCK DATA) 00350C THETA: SHIFT ANGLE OF THE CVT, RAD 0035 1C (ALSO AN INPUT PARAMETER) 00352C TORDATA: ARRAY OF REAR WHEEL TORQUE AS A 00353C FUNCTION OF TIME. LB-IN. 003 54C (OUTPUT IN COMMON BLOCK DATA) 00355C 00356C RESTRICTIONS 00357C 1) REQUIRES UNIVERSITY OF MINNESOTA LIBRARY 00358C RUNGE-KUTTA INTEGRATION ROUTINE "RK" 00359C 2) THE SUBROUTINE CONTAINING THE MAIN 00360C CONTROL DIFFERENTIAL EQUATION. "DIFFEQ". 0036 1C MUST BE DECLARED IN AN "EXTERNAL" 00362C STATEMENT 00363C (IT MUST BE ACCESSED BY LIBRARY ROUTINE 00364C " RK " ) 00365C 00366C ERROR INDICATIONS 00367C NONE 00368C 00369C KEYWORDS 003 70C NONE 0037 1C 00372C METHOD 00373C THE NON-LINEAR CONTROL DIFFERENTIAL 00374C EQUATION IS NUMERICALLY INTEGRATED BY 00375C THE RUNGE-KUTTA METHOD 00376C 00377C DESCRIPTION OF VARIABLES 00378C ABSERR: MAXIMUM ABSOLUTE ERROR ALLOWED FOR 00379C THETA FOR ONE INTEGRATION STEP. 00380C RAD 0038 1C (THE INTEGRATION TIME WILL BE 00382C INTERNALLY ADJUSTED BY ROUTINE "RK" 00383C TO MEET THIS CONSTRAINT, 00384C OR "RELERR", WHICHEVER IS GREATER) "RK" 00385C DT- DUMMY ARGUMENT REQUIRED BY ROUTINE 00386C (RETURNS THE ACTUAL TIME INTERVAL TIME 00387C USED IN INTEGRATING OVER THE 00388C STEP SPECIFIED IN THE CALLING 00389C ARGUMENTS TO MEET THE ACCURACY WITH 00390C CONSTRAINTS SPECIFIED "ABSERR" 0039 1C AND "RELERR") 00392C I COUNTER VARIABLE ERROR ALLOWED FOR 0039 3C RELERR: MAXIMUM RELATIVE INTEGRATION 00394C THETA FOR ONE STEP, 00395C RAD TIME WILL BE 00396C (THE INTEGRATION -187-

00397C INTERNALLY ADJUSTED BY ROUTINE 00398C "RK" TO MEET THIS CONSTRAINT. 00399C OR "RELERR", WHICHEVER IS GREATER) 00400C RKSPACE: BUFFER REQUIRED BY ROUTINE "RK" 0040 1C TF: END TIME OF ONE INTEGRATION STEP OF 00402C LENGTH TSTEP. SEC 00403C THDOT: TIME DERIVATIVE OF SHIFT ANGLE 00404C THETA, RAD/SEC 00405C TINIT: START TIME OF ONE INTEGRATION STEP 00406C OF LENGTH TSTEP, SEC 00407C 00408C SUBROUTINES CALLED 00409C DIFFEQ: ROUTINE TO SUPPLY THE MAIN CONTROL 004 10C DIFFERENTIAL EQUATION 0041 1C RK: UNIVERSITY OF MINNESOTA LIBRARY 004 1 2C FOURTH-ORDER RUNGE-KUTTA SINGLE- 004 1 3C STEP INTEGRATION ROUTINE 004 1 4C TRW: FUNCTION TO SUPPLY THE PRESCRIBED 004 1 5C OUTPUT (REAR WHEEL) TORQUE OF THE 004 16C BICYCLE 004 17C 004 1 8C DECLARATIONS FOR VARIABLES REFERENCED 004 19C IN SUBROUTINE CALL 00420 INTEGER NMPT 00421 REAL STARTIM, TSTEP, THETA 00422 REAL C1. C2, C3, FD, OMEGA. 00423+ RI, RPC, RSCVT, SCEQ. YMAX 00424 REAL T, TORDATA, 00425+ FPDATA, THEDATA 00426 REAL FP 00427C 00428C LOCAL DECLARATIONS 00429 INTEGER I

00430 REAL ABSERR, DT . RELERR. RKSPACE(4), 00431+ TF, THDOT, TINIT 004 32 REAL PI 004 3 3C 00434C EXTERNAL DECLARATIONS 00435 EXTERNAL DIFFEQ 00436C 00437C COMMON STATEMENTS AND DECLARATIONS

C2 FD , OMEGA. 00438 COMMON/ CONSTAN / CI. , C3 , YMAX 00439+ RI, RPC, RSCVT, SCEQ. 501 00440 COMMON/ DATA / T(501). TORDATA( ) , 00441+ FPDATA(501). THEDATA(501) 00442 COMMON / TEMP / FP 0044 3C 00444C DATA STATEMENTS 00445 DATA RELERR / 1.0E-6 / 00446 DATA ABSERR / 1.0E-8 / 00447 DATA PI / 3.14159265358979 / 00448C EXECUTABLE CODE ***** 00449C ***** START OF 00450C ANGLE FOR THE 0045 1C SET INITIAL SHIFT (IN DEGREES) 00452C INTEGRATION INTERVAL 00453C THETA* 180. /PI 00454 THEDATA(1) 00455C OUTPUT TORQUE FOR THE 00456C INITIALIZE INTERVAL 00457C INTEGRATION 00458C TRW(STARTIM) 00459 TORDATA(I) 00460C PEDAL FORCE FOR THE 00461C CALCULATE INITIAL INTERVAL 00462C INTEGRATION -188-

00463C (FP IS CALCULATED INTERNALLY IN 00464C ROUTINE "DIFFEQ") 00465C 00466 CALL DIFFEQ( STARTIM, THETA, THDOT ) 00467 FPDATA( 1) = FP 00468C 00469C INITIALIZE TIME ARRAY 004 70C (INDEPENDENT VARIABLE FOR OUTPUT GRAPHS) 00471C 00472 T(1) STARTIM 00473C 00474C INITIALIZE END TIME FOR THE INTEGRATION STEP 00475C PRECEEDING THE FIRST CALCULATED IN THE 00476C MAIN INTEGRATION LOOP 00477C 00478 TF STARTIM 00479C 00480C START OF MAIN INTEGRATION LOOP 00481C 00482 DO 500 1=2, NMPT 00483C 00484C CALCULATE START AND END TIME OF THE CURRENT 00485C INTEGRATION STEP 004 86C 00487 TINIT - TF 00488 TF = STARTIM + TSTEP*FLOAT( I 1) 00489C 00490C UPDATE TIME ARRAY 0049 1C (INDEPENDENT VARIABLE FOR OUTPUT GRAPHS) 00492C 00493 T(I) TF 00494C 00495C CALCULATE AND STORE CURRENT OUTPUT TORQUE 00496C 00497 TORDATA(I) - TRW(TF) 00498C 004 99C INTEGRATE MAIN CONTROL DIFFERENTIAL EQUATION "TSTEP" 00500C OVER ONE TIME STEP OF LENGTH 0050 1C NOTE: TINIT IS RETURNED SET EQUAL TO TF 00502C 00503 CALL RK( TINIT. TF, THETA, THDOT. DIFFEQ, 00504+ 1, RELERR, ABSERR, 1, 00505+ RKSPACE, DT ) 00506C TF 00507C CALCULATE SHIFT ANGLE AND PEDAL FORCE AT ROUTINE 00508 C (FP IS CALCULATED INTERNALLY IN 00509C "DIFFEQ") CO510C THDOT 00511 CALL DIFFEQ( TF, THETA, ) 00512C (IN 005 1 3C STORE CURRENT SHIFT ANGLE DEGREES) 005 1 4C AND PEDAL FORCE 005 1 5C THETA* 180. /PI 00516 THEDATA(I) = FP 00517 FPDATA(I) 00518 500 CONTINUE 00519 RETURN 00520 END 0052 1C 07-01-82 TRC; CREATED

00522C( ************ *************************BCS.LIMSET 00523C THESTRT ) 00524 SUBROUTINE LIMSET( NMPT, 00525C EXPERT- 00526C TRC 00527C 005 2 8C PURPOSE -189-

00529C ROUTINE TO CALCULATE THE DATA MINIMUM 005 30C AND MAXIMUM FOR ALL PARAMETERS INCLUDED 0053 1C ON THE OUTPUT GRAPHS BASED ON A 00532C 20 SECOND SAMPLE SIMULATION INTERVAL 00533C 005 34C INPUT 00535C NMPT- NUMBER OF POINTS DISPLAYED ON THE 0O536C RESPONSE CURVE 00537C FPDATA: ARRAY OF PEDAL FORCE AS A 00538C FUNCTION OF TIME, LB 00539C (INPUT IN COMMON BLOCK DATA) 00540C THEDATA: ARRAY OF SHIFT ANGLE AS A 0054 1C FUNCTION OF TIME, DEG 00542C (INPUT IN COMMON BLOCK DATA) 00543C THESTRT: INITIAL VALUE OF THE SHIFT ANGLE 00544C OF THE CVT, RAD 00545C TORDATA: ARRAY OF OUTPUT (REAR WHEEL) 00546C TORQUE AS A FUNCTION OF 00547C TIME. LB-IN. 00548C (INPUT IN COMMON BLOCK DATA) 00549C 00550C OUTPUT 00551C FPMAX: MAXIMUM VALUE OF PEDAL FORCE, LB 00552C (OUTPUT IN COMMON BLOCK LIMITS) 0055 3C FPMIN: MINIMUM VALUE OF PEDAL FORCE, LB 00554C (OUTPUT IN COMMON BLOCK LIMITS) 00555C THEMAX: MAXIMUM VALUE OF THE SHIFT 00556C ANGLE, DEG 00557C (OUTPUT IN COMMON BLOCK LIMITS) 00558C THEMIN: MINIMUM VALUE OF THE SHIFT 00559C ANGLE. DEG 00560C (OUTPUT IN COMMON BLOCK LIMITS) 0056 1C TORMAX: MAXIMUM VALUE OF THE OUTPUT 00562C TORQUE, LB-IN. 00563C (OUTPUT IN COMMON BLOCK LIMITS) 00564C TORMIN: MINIMUM VALUE OF THE OUTPUT 00565C TORQUE, LB-IN. 00566C (OUTPUT IN COMMON BLOCK LIMITS) 0O567C 00568C RESTRICTIONS 00569C THE MINIMUM AND MAXIMUM OBTAINED IN 005 70C THE 20 SECOND SAMPLE INTEGRATION 0057 1C INTERVAL IS ASSUMED TO BE 00572C REPRESENTATIVE OF AN ARBITRARY 00573C INTEGRATION INTERVAL 00574C 00575C ERROR INDICATIONS 00576C NONE 00577C 00578C KEYWORDS 00579C NONE 00580C 00581C METHOD 00582C A SAMPLE SIMULATION IS PERFORMED OVER A 20 0058 3C 20 SECOND TIME INTERVAL. THE MINIMUM AND 00584C MAXIMUM VALUE OF EACH PARAMETER OCCURRING 00585C IN THIS INTERVAL IS OBTAINED BY DIRECT 00586C SEARCH. 0058 7C 00588C DESCRIPTION OF VARIABLES 00589C I : COUNTER VARIABLE 00590C SAMPTIM: LENGTH OF THE SAMPLE INTEGRATION 0059 1C INTERVAL USED TO DETERMINE THE 00592C MINIMUM AND MAXIMUM PARAMETER 00593C VALUES, SEC 00594C STRTLIM: INITIAL MAGNITUDE OF THE MINIMUM -190-

00595C OR MAXIMUM VALUE ASSUMED FOR EACH 00596C PARAMETER 00597C TSTEP: TIME STEP BETWEEN DATA POINTS FOR 00598C THE SAMPLE SIMULATION INTERVAL, SEC 00599C 00600C SUBROUTINES CALLED 00601C DATASET: ROUTINE TO CALCULATE ALL OUTPUT 00602C PARAMETERS FOR ONE SIMULATION 00603C INTERVAL 00604C 00605C DECLARATIONS FOR VARIABLES REFERENCED 00606C IN SUBROUTINE CALL 00607 INTEGER NMPT 00608 REAL THESTRT 00609 REAL T, TORDATA, 00610+ FPDATA, THEDATA 00611 REAL TORMIN, TORMAX , FPMIN, 00612+ FPMAX, THEMIN, THEMAX 006 1 3C 00614C LOCAL DECLARATIONS 00615 INTEGER I 00616 REAL SAMPTIM, STRTLIM, TSTEP 00617C 006 18C COMMON STATEMENTS AND DECLARATIONS 00619 DATA COMMON/ / T(501), T0RDATA(501 ) , 00620+ FPDATA(501), THEDATA(501) 00621 COMMON/ LIMITS / TORMIN. TORMAX, FPMIN, 00622+ FPMAX, THEMIN, THEMAX 006 2 3C 00624C DATA STATEMENTS 00625 DATA STRTLIM / 10000. / 00626 DATA SAMPTIM / 20.00 / 00627C 00628C ***** START OF EXECUTABLE CODE ***** 00629C 00630C CALCULATE THE TIME STEP BETWEEN THE SAMPLE 0063 1C DATA POINTS 00632C 0O633 TSTEP SAMPTIM/FLOAT(NMPT 1) 00634C 00635C SET THE INITIAL VALUE OF THE SHIFT ANGLE 00636C 00637 THETA = THESTRT 00638C 00639C INITIALIZE THE MINIMUM AND MAXIMUM VALUE 00640C OF EACH PARAMETER TO THE DEFAULT MAGNITUDE 0064 1C 00642 TORMIN - STRTLIM 00643 TORMAX tSTRTLIM 00644 FPMIN - STRTLIM 00645 FPMAX -STRTLIM 00646 THEMIN = STRTLIM 00647 THEMAX = -STRTLIM 00648C 00649C CALCULATE THE ARRAYS OF OUTPUT PARAMETERS 00650C FOR THE SAMPLE SIMULATION INTERVAL 0065 1C 00652 CALL DATASET(NMPT, 0.0, TSTEP. THETA) 00653C 00654C SEARCH LOOP TO DETERMINE THE ACTUAL MINIMUM 00655C AND MAXIMUM OF EACH PARAMETER IN THE SAMPLE 00656C TIME INTERVAI 00657C EVERY PARAMETER 00658C ...SEARCHEACH VALUE OF 00659C IN THE ARRAY 00660C -191-

00661 DO 100 I 1 , NMPT 00662C

00663C ..IF THE CURRENT VALUE IS LESS THAN THE 00664C PREVIOUSLY-DETERMINED MINIMUM. 00665C RE-SET THE MINIMUM 00666C 00667 IF (TORDATA(I).LT. TORMIN) 00668+ TORMIN TORDATA(I) 00669 IF (FPDATA(I).LT. FPMIN) 00670+ FPMIN = FPDATA(I) 00671 IF (THEDATA(I).LT. THEMIN) 0O672+ THEMIN THEDATA(I) 00673C

00674C ...IF THE CURRENT VALUE IS GREATER THAN THE 00675C PREVIOUSLY-DETERMINED MAXIMUM. 00676C RE-SET THE MAXIMUM 00677C 00678 IF (TORDATA( I ).GT. TORMAX) 00679+ TORMAX = TORDATA(I) 00680 IF (FPDATA(I).GT.FPMAX) 00681+ FPMAX = FPDATA(I) 00682 IF (THEDATA(I).GT. THEMAX) 00683+ THEMAX - THEDATA(I) 00684 100 CONTINUE 00685 RETURN 00686 END 0068 7C 07-20-81 TRC; CREATED 00688C( 00689C ************************* BCS. DIFFEQ ************ 00690 SUBROUTINE DIFFEQ( T, THETA, THDOT ) 0069 1C ************************************************* 00692C EXPERT: TRC 00693C 00694C PURPOSE 00695C THIS SUBROUTINE SUPPLIES THE MAIN CONTROL 00696C DIFFERENTIAL EQUATION REQUIRED BY SYSTEM 00697C RUNGE-KUTTA NUMERICAL INTEGRATION 00698C ROUTINE "RK". IN ADDITION, IT IS USED TO 00699C CALCULATE THE PEDAL FORCE. FP. 00700C 0070 1C INPUT 00702C CI: CHAIN REDUCTION BETWEEN THE OUTPUT OF 00703C THE CVT AND THE REAR WHEEL OF THE 00704C BICYCLE 00705C (INPUT IN COMMON BLOCK CONSTAN) 00706C C2: TOTAL CHAIN REDUCTION BETWEEN THE PEDAL 00707C CRANK AND THE INPUT OF THE CVT 00708C (INPUT IN COMMON BLOCK CONSTAN) 00709C C3: TOTAL GEARDOWN BETWEEN THE INTEGRATOR 00710C DRIVEN WHEEL AND THE TRACTION BALL 007 1 1 C SHIFT ANGLE 007 1 2C (INPUT IN COMMON BLOCK CONSTAN) 007 13C FD: PEAK VALUE OF THE IDEAL PEDAL FORCE, LB 007 1 4C (INPUT IN COMMON BLOCK CONSTAN) 007 1 5C OMEGA: ANGULAR VELOCITY OF THE INTEGRATOR 007 16C DRIVING DISC. RAD/SEC 00717C (INPUT IN COMMON BLOCK CONSTAN) 007 18C RI: RADIUS OF THE INTEGRATOR DRIVEN WHEEL, 007 1 9C IN. 0O720C (INPUT IN COMMON BLOCK CONSTAN) 0072 1C RPC: LENGTH OF THE PEDAL CRANK, IN. 00722C (INPUT IN COMMON BLOCK CONSTAN) 007 2 3C RSCVT: PITCH RADIUS OF THE INPUT SPROCKET 00724C OF THE CVT, IN. 007 25C (INPUT IN COMMON BLOCK CONSTAN) 00726C SCFQ: EQUIVALENT SPRING CONSTANT OF THE TWO 192-

00727C PARALLEL CHAIN FORCE SENSOR LINKAGE 00728C SPRINGS ACTING THROUGH THE CHAIN FORCE 00729C SENSOR LINKAGE. LB/IN. 007 30C (INPUT IN COMMON BLOCK CONSTAN) 0073 1C T: TIME (INDEPENDENT VARIABLE), SEC 00732C THETA: SHIFT ANGLE OF THE CVT, RAD 00733C YMAX: THE MAXIMUM ALLOWABLE DISPLACEMENT OF 00734C THE CHAIN FORCE SENSOR LINKAGE. IN. 00735C (INPUT IN COMMON BLOCK CONSTAN) 00736C 00737C OUTPUT 00738C THDOT: RATE OF CHANGE OF BALL SHIFT ANGLE, 007 39C D(THETA)/DT, RAD/SEC 00740C FP: PEDAL FORCE, LB 0074 1C (OUTPUT IN COMMON BLOCK TEMP) 00742C 007 4 3C RESTRICTIONS 00744C NONE 00745C 00746C ERROR INDICATIONS 00747C NONE 00748C 00749C KEYWORDS 007 50C NONE 00751C 00752C METHOD 00753C STRAIGHTFORWARD 00754C (A DERIVATION OF THE CONTROL EQUATIONS 00755C CAN BE FOUND IN SECTION 6.6 OF THE TEXT) 00756C 00757C DESCRIPTION OF VARIABLES 00758C Y- CHAIN FORCE SENSOR LINKAGE DISPLACEMENT, 00759C IN. 00760C 0076 1C SUBROUTINES CALLED 00762C TRW: FUNCTION TO SUPPLY THE PRESCRIBED 007 6 3C TORQUE AT THE REAR WHEEL OF THE 00764C BICYCLE AS A FUNCTION OF TIME 00765C 00766C DECLARATIONS FOR VARIABLES REFERENCED 00767C IN SUBROUTINE CALL 00768 REAL T, THETA, THDOT 00769 REAL FP

00770 REAL C1, C2, C3, FD , OMEGA, 00771+ RI, RPC, RSCVT, SCEQ, YMAX 00772C 00773C LOCAL DECLARATIONS 00774 REAL Y 00775C 00776C COMMON STATEMENTS AND DECLARATIONS 00777 COMMON/TEMP/FP

00778 COMMON/ CONSTAN / C1, C2 , C3, FD. OMEGA, 00779+ RI, RPC, RSCVT. SCEQ. YMAX 00780C 0078 1C DATA STATEMENTS 00782C NONE 00783C ***** 00784C ***** START OF EXECUTABLE CODE 00785C REQUIRED 00786C CALCULATION OF THE PEDAL FORCE TO 00787C PRODUCE THE SPECIFIED REAR WHEEL TORQUE 00788C * * 00789 FP l/( C1*C2*RPC ) TRW(T) 00790+ (1. + TAN(THETA))/(1. TAN(THETA)) 0079 1C CHAIN FORCE 00792C FIND DISPLACEMENT OF THE -193-

00793C SENSOR LINKAGE. . 00794C 00795C ..FIRST. ASSUME THE CHAIN FORCE SENSOR 00796C LINKAGE HAS NOT REACHED EITHER OF ITS 00797C LIMIT STOPS 00798C 00799 Y = ( C2*RPC ) / 00800+ ( RSCVT*SCEQ ) * ( 2.*FP FD ) 0080 1C 00802C ...NEXT, TEST FOR NEGATIVE BOTTOMING OUT 00803C OF THE CHAIN FORCE SENSOR LINKAGE 00804C 00805 + IF ( (Y YMAX).LE.O. ) Y -YMAX 00806C 00807C ..FINALLY, TEST FOR POSITIVE BOTTOMING OUT 00808 C OF THE CHAIN FORCE SENSOR LINKAGE 00809C

00810 IF ( Y.GE.YMAX ) Y - YMAX 0081 1C 00812C MAIN CONTROL DIFFERENTIAL EQUATION 008 1 3C (REPRESENTS THE BASIC OPERATION OF THE 00814C MECHANICAL INTEGRAL CONTROLLER) 008 15C 00816 THDOT -OMEGA / (C3*RI)*Y 00817 RETURN 00818 END 008 1 9C 08-02-81 TRC; CREATED 00820C( 00821C ************************* BCS.TRW *************** 00822 FUNCTION TRW( T ) 00823C ************************************************* 00824C EXPERT: TRC 00825C 00826C PURPOSE 00827C FUNCTION TO SUPPLY PRESCRIBED OUTPUT TORQUE 00828C 008 29C THE TORQUE OUTPUT TO THE REAR WHEEL OF THE 00830C BICYCLE IS CONSIDERED TO BE THE INPUT TO THE 0083 1C CONTROL SYSTEM. IT IS SPECIFIED TO BE ANY 0O832C DESIRED FUNCTION OF TIME IN THIS ROUTINE. 00833C 00834C INPUT 008 35C T: TIME = INDEPENDENT PARAMETER, SEC 008 36C TLEV: PEAK VALUE OF THE SINUSOIDAL TORQUE 00837C REQUIRED TO DRIVE THE REAR WHEEL 00838C OF THE BICYCLE, LB-IN. 00839C (INPUT IN COMMON BLOCK RWTQ) OO840C 0084 1C OUTPUT 0084 2C TRW: TORQUE REQUIRED TO DRIVE THE 00843C REAR WHEEL AT TIME T, LB-IN. 00844C 00845C RESTRICTIONS 00846C SOME PRESCRIBED REAR WHEEL TORQUE PROFILES 00847C MAY NOT NEED ALL THE INFORMATION PROVIDED 00848C AS INPUTS 00849C 00850C ERROR INDICATIONS 00851C NONE 00852C 00853C KEYWORDS 00854C NONE 00855C 00856C METHOD DIRECTLY 00857C THE REAR WHEEL TORQUE PROFILE IS 00858C PRESCRIBED WITH THE EQUATION SPECIFIED -194-

00859C IN THIS SUBROUTINE 00860C 0086 1C DESCRIPTION OF VARIABLES 00862C PER: PERIOD OF THE PEDAL CRANK ROTATION 00863C OF THE BICYCLE, SEC 00864C 00865C SUBROUTINES CALLED 00866C NONE 00867C 00868C DECLARATIONS FOR VARIABLES REFERENCED 00869C IN SUBROUTINE CALL 00870 REAL T 00871 REAL TLEV 00872C 008 73C LOCAL DECLARATIONS 00874 REAL PER 00875 REAL PI 00876C 0O877C COMMON STATEMENTS AND DECLARATIONS 00878 COMMON/ RWTQ / TLEV 008 7 9C 00880C DATA STATEMENTS 00881 DATA PI / 3.14159265358979 / 00882 DATA PER / 1.0 / 00883C 00884C ***** START OF EXECUTABLE CODE 00885C 00886C THE FOLLOWING IS THE EQUATION TO DESCRIBE 00887C THE TORQUE REQUIRED AT THE REAR WHEEL AS A 00888C FUNCTION OF TIME 00889C 00890 TRW TLEV * (O.5*(C0S(4. *PI*T/PER) + 1.0)) 00891 RETURN 00892 END 00893C 07-01-82 TRC; CREATED 00894C 00895C ************************* BCS. GRAPH ************* 00896 SUBROUTINE GRAPH( NMPT ) 00897C ************************************************* 00898C EXPERT: TRC 00899C 00900C PURPOSE 00901C ROUTINE CONTAINING ALL THE GRAPHICAL 00902C COMMANDS TO GENERATE A TEKTRONIX 00903C DISPLAY OF THE SPECIFIED OUTPUT 0O904C TORQUE, THE PEDAL FORCE REQUIRED 00905C TO PRODUCE THAT OUTPUT TORQUE, AND 00906C THE SHIFT ANGLE AS A FUNCTION OF 00907C OF TIME 00908C 00909C INPUT 009 10C C1: CHAIN REDUCTION BETWEEN THE OUTPUT OF 0091 1C THE CVT AND THE REAR WHEEL OF THE 00912C BI CYCLE 00913C (INPUT IN COMMON BLOCK CONSTAN) C2- 009 1 4C TOTAL CHAIN REDUCTION BETWEEN THE PEDAL CVT 009 1 5C CRANK AND THE INPUT OF THE 009 16C (INPUT IN COMMON BLOCK CONSTAN) C3- INTEGRATOR 00917C TOTAL GEARDOWN BETWEEN THE TRACTION BALL 009 18C DRIVEN WHEEL AND THE 009 1 9C SHIFT ANGLE 00920C (INPUT IN COMMON BLOCK CONSTAN) FD" IDEAL PEDAL LB 0092 1C PEAK VALUE OF THE FORCE, 00922C (INPUT IN COMMON BLOCK CONSTAN) FORCE AS A FUNCTION 00923C FPDATA: ARRAY OF PEDAL 00924C OF TIME, LB -195-

00925C (INPUT IN COMMON BLOCK DATA) 00926C FPMAX: MAXIMUM VALUE OF PEDAL FORCE, LB 00927C (INPUT IN COMMON BLOCK LIMITS) 00928C FPMIN: MINIMUM VALUE OF PEDAL FORCE, LB 00929C (INPUT IN COMMON BLOCK LIMITS) 00930C IRD: READ LOGICAL UNIT NUMBER OF THE 0093 1C COMPUTER BEING USED 00932C (INPUT IN COMMON BLOCK INOUT) 00933C IWR: WRITE LOGICAL UNIT NUMBER OF THE 00934C COMPUTER BEING USED 00935C (INPUT IN COMMON BLOCK INOUT) NMPT- 00936C NUMBER OF POINTS DISPLAYED ON THE 00937C RESPONSE CURVE 00938C OMEGA: ANGULAR VELOCITY OF THE INTEGRATOR 00939C DRIVING DISC, RAD/SEC 00940C (INPUT IN COMMON BLOCK CONSTAN) 0094 1C RI : RADIUS OF THE INTEGRATOR DRIVEN WHEEL. 0094 2C IN. 0094 3C (INPUT IN COMMON BLOCK CONSTAN) 00944C RPC: LENGTH OF THE PEDAL CRANK, IN. 00945C (INPUT IN COMMON BLOCK CONSTAN) 00946C RSCVT: PITCH RADIUS OF THE INPUT SPROCKET 00947C OF THE CVT, IN. 009 4 8C (INPUT IN COMMON BLOCK CONSTAN) 00949C SCEQ: EQUIVALENT SPRING CONSTANT OF THE TWO 00950C PARALLEL CHAIN FORCE SENSOR LINKAGE 0095 1C SPRINGS ACTING THROUGH THE CHAIN FORCE 00952C SENSOR LINKAGE. LB/IN. 00953C (INPUT IN COMMON BLOCK CONSTAN) 00954C T: ARRAY OF TIME, SEC 00955C (INDEPENDENT PARAMETER OF OUTPUT GRAPHS) 00956C (INPUT IN COMMON BLOCK DATA) 00957C THEDATA: ARRAY OF SHIFT ANGLE AS A FUNCTION 00958C OF TIME. DEG 00959C (INPUT IN COMMON BLOCK DATA) 00960C THEMAX: MAXIMUM VALUE OF THE SHIFT 0096 1C ANGLE, DEG 0096 2C (INPUT IN COMMON BLOCK LIMITS) 0096 3C THEMIN: MINIMUM VALUE OF THE SHIFT 00964C ANGLE, DEG 00965C (INPUT IN COMMON BLOCK LIMITS) 00966C TORDATA: ARRAY OF REAR WHEEL TORQUE AS A 00967C FUNCTION OF TIME, LB-IN. 00968C (INPUT IN COMMON BLOCK DATA) 00969C TORMAX: MAXIMUM VALUE OF THE OUTPUT 00970C TORQUE, LB-IN. 0097 1C (INPUT IN COMMON BLOCK LIMITS) 00972C TORMIN: MINIMUM VALUE OF THE OUTPUT 00973C TORQUE, LB-IN. 00974C (INPUT IN COMMON BLOCK LIMITS) 00975C YMAX: THE MAXIMUM ALLOWABLE DISPLACEMENT OF 00976C THE CHAIN FORCE SENSOR LINKAGE. IN. 00977C (INPUT IN COMMON BLOCK CONSTAN) 00978C 00979C OUTPUT 00980C NONE 0098 1C 00982C RESTRICTIONS 00983C 1 ) INTENDED FOR USE WITH 00984C TEKTRONIX 4010 SERIES HARDWARE ONLY ALPHA-NUMERIC/VECTOR OUTPUT 00985C (MIXED DISPLAY 00986C MAY CAUSE INCORRECT ON SOME 00987C COMPATIBLE TERMINALS) NON-STANDARD SYSTEM 00988C 2) CONTAINS CALLS TO "DATE" "TIME" 00989C ROUTINES AND GRAPH TITLE 00990C 3) THIS ROUTINE ASSUMES THE 196-

0099 1C HAS BEEN INPUT AND FORMATTED BEFORE 00992C CALLING THIS ROUTINE BY A PREVIOUS 00993C CALL TO SUBROUTINE "GRNAMEI" 00994C 00995C ERROR INDICATIONS 00996C NONE 00997C 00998C KEYWORDS 00999C NONE 01000C 01001C METHOD 01002C STRAIGHTFORWARD 01003C 01OO4C DESCRIPTION OF VARIABLES 01005C I : COUNTER VARIABLE 01006C PFLAB: BUFFER FOR ASCII REPRESENTATION OF 01007C PEDAL FORCE AXIS LABEL 01008C SALAB. BUFFER FOR ASCII REPRESENTATION OF 01009C SHIFT ANGLE AXIS TITLE 01010C TORLAB: BUFFER FOR ASCII REPRESENTATION OF 0101 1C OUTPUT TORQUE AXIS LABEL 01012C 01013C SUBROUTINES CALLED 01014C ANMODE: TEKTRONIX TCS ROUTINE TO CHANGE 01015C GRAPHICS TERMINAL FROM VECTOR TO 01016C ALPHA-NUMERIC MODE 01017C BINITT: TEKTRONIX AGII ROUTINE TO INITIALIZE 01018C THE AGII COMMON AREA 01019C CHECK: TEKTRONIX AGII ROUTINE TO ASSURE ALL 01020C NECESSARY VARIABLES HAVE BEEN SET IN 01021C THE AGII COMMON AREA BEFORE DISPLAYING 01022C A GRAPH 01023C DATE: UNIVERSITY OF MINNESOTA UTILITY 01024C ROUTINE TO REQUEST THE CURRENT DATE 01025C FROM THE OPERATING SYSTEM 01026C DINITY: TEKTRONIX AGII ROUTINE TO 01027C REINITIALIZE LABEL VALUES OF THE 01028C Y-AXIS 01029C (ALLOWS DRAWING OF AN ADDITIONAL 01030C CURVE WITH NEW LABELS) 01031C DLIMX: TEKTRONIX AGII ROUTINE TO SET 01032C X LIMITS OF THE VIEW WINDOW 01033C DLIMY: TEKTRONIX AGII ROUTINE TO SET 01034C Y LIMITS OF THE VIEW WINDOW 01035C DRWABS: TEKTRONIX TCS ROUTINE TO DRAW A 01036C VECTOR FROM THE CURRENT BEAM 01037C POSITION IN ABSOLUTE SCREEN 01038C COORDINATES 01039C DSPLAY: TEKTRONIX AGII ROUTINE FOR 01040C DISPLAYING A COMPLETE GRAPH ON 01041C THE TERMINAL SCREEN 01042C GRNAME: ROUTINE TO CENTER AND DISPLAY THE 01043C GRAPH TITLE 01044C LINE: TEKTRONIX AGII ROUTINE TO SET THE 01045C FONT OF LINES DRAWN BETWEEN 01046C THE DATA POINTS A 01047C MOVABS: TEKTRONIX TCS ROUTINE TO MAKE 01048C MOVE (INVISIBLE DRAW) IN 01049C ABSOLUTE SCREEN COORDINATES TO CLEAR 01050C NEWPAG: TEKTRONIX TCS ROUTINE 01051C THE GRAPHICS SCREEN TO SET THE 01052C NPTS: TEKTRONIX AGII ROUTINE TO BE 01O53C NUMBER OF DATA POINTS 01054C DISPLAYED ON A GRAPH ROUTINE SET 01055C SLIMX: TEKTRONIX AGII TO VIEWPORT IN 01056C X LIMITS OF THE -197-

01057C ABSOLUTE SCREEN COORDINATES 01058C TINPUT: TEKTRONIX TCS ROUTINE TO CAUSE 01059C A PAUSE IN A DISPLAY SEQUENCE 01060C UNTIL USER INPUT () 01061C IS PROVIDED 01062C TIME: UNIVERSITY OF MINNESOTA UTILITY 01063C ROUTINE TO REQUEST THE CURRENT 01064C TIME FROM THE OPERATING SYSTEM 01065C TSEND: TEKTRONIX TCS ROUTINE TO DUMP 01066C THE GRAPHICS TERMINAL BUFFER 01067C (DISPLAYS ALL BUFFERED OUTPUT 01068C ON THE SCREEN) 01069C VLABEL: TEKTRONIX AGII ROUTINE TO PRINT 01070C A VERTICAL AXIS LABEL 01071C XLAB: TEKTRONIX AGII ROUTINE TO SET THE 01072C TIC MARK LABEL TYPE FOR THE X-AXIS 01073C XNEAT: TEKTRONIX AGII ROUTINE TO 01074C REQUEST/SUPRESS "NEAT" INTERVALS 01075C BETWEEN TIC MARKS ON THE X-AXIS 01076C YFRM: TEKTRONIX AGII ROUTINE TO SET THE 01077C TIC MARK FONT FOR THE Y-AXIS 01078C YLOC: TEKTRONIX AGII ROUTINE FOR SETTING 01079C THE LOCATION OF A Y-AXIS IN RELATION 01080C TO THE LEFT AND LOWER EDGES OF 01081C THE SCREEN 01082C YLOCRT: TEKTRONIX AGII ROUTINE FOR 01083C SETTING THE LOCATION OF A Y-AXIS 01084C IN RELATION TO THE RIGHT AND UPPER 01085C EDGES OF THE SCREEN 01086C YZERO: TEKTRONIX AGII ROUTINE TO 01087C REQUEST/SUPRESS AUTOMATIC INCLUSION 01088C OF ZERO ON THE Y-AXIS 01089C 01090C DECLARATIONS FOR VARIABLES REFERENCED 01091C IN SUBROUTINE CALL 01092 INTEGER IRD, IWR

01093 REAL CI, C2, C3, FD , OMEGA. 01094+ RI. RPC, RSCVT, SCEQ, YMAX 01095 REAL T. TORDATA, 01096+ FPDATA. THEDATA 01097 REAL TORMIN, TORMAX, FPMIN, 01098+ FPMAX, THEMIN. THEMAX 01099C 01100C LOCAL DECLARATIONS 01101 INTEGER T0RLAB(21), PFLAB(16), SALAB(17) 01102 INTEGER I 01 103C DECLARATIONS 01104C COMMON STATEMENTS AND OMEGA, 01105 COMMON/ CONSTAN / C1. C2, C3, FD, RI RSCVT, SCEQ. YMAX 01 106+ , RPC, T0RDATA(501 01107 COMMON/ DATA / T(501), ) , 01108+ FPDATA(501), THEDATA(501) FPMIN, 01109 COMMON/ LIMITS / TORMIN, TORMAX, THEMAX 01110+ FPMAX, THEMIN, IWR 01111 COMMON/ INOUT / IRD, 01112C 01113C DATA STATEMENTS

011 14C IN*LB" 01 1 15C ASCII: "OUTPUT TORQUE 01116C 84, 80. 85, 84, 32, 01117 DATA TORLAB / 79, 85, 84 82, 81, 85, 69, 32. 011 18+ 79, 32, 73, 78, 42, 76, 66 / 01 1 19+ 58, 01 120C LB" 01121C ASCII: "PEDAL FORCE 01122C -198-

01 123 DATA PFLAB / 80, 69, 68. 65. 76, 32, 70, 01124+ 79, 82. 67, 69. 32, 58. 32, 01125+ 76, 66 / 01126C DEG" 01127C ASCII: "SHIFT ANGLE : 01 128C 01129 DATA SALAB / 83, 72, 73. 70. 84, 32, 65, 01130+ 78, 71, 76, 69, 32. 58, 32, 01 131 + 68, 69, 71 / 01132C 01133C ***** START OF EXECUTABLE CODE ***** 01 134C 01135C CLEAR PAGE AND RE- INITI ALI ZE AGII COMMON 01136C 01 137 CALL NEWPAG 01138 CALL BINITT 01139C 01140C RE-SET SCREEN VIEWPORT IN THE X-DIRECTION 01 141C TO ALLOW SPACE FOR MULTIPLE Y-AXES 01 142C 01143 CALL SLIMX( 200, 800 ) 01 144C 01145C INSURE INCLUSION OF 0 ON THE Y-AXIS 01146C 01147 CALL YZERO( 0 ) 01148C 01 149C SET NUMBER OF POINTS TO BE DISPLAYED ON 0 1 1 50C EACH RESPONSE CURVE 01151C 01152 CALL NPTS( NMPT ) 01 153C "NEAT" 01154C SUPRESS TEKTRONIX TIC MARK INTERVALS 01155C ON THE X-AXIS 01156C 01 157 CALL XNEAT( O ) 01158C

01159C SET DATA WINDOW. . . 0 1 1 60C X-DIRECTION: MIN AND MAX TIME 01161C Y-DIRECTION: MIN AND MAX PEDAL FORCES 01162C 01163 CALL DLIMX( T(1), T(501) ) 01 164 CALL DLIMY( FPMIN, FPMAX ) 01165C 01166C DISPLAY PEDAL FORCE VERSUS TIME GRAPH 01 167C 01168 CALL CHECK( T, FPDATA ) 01169 CALL DSPLAY( T. FPDATA ) 01170C Y-AXIS 01171C RE-INITIALIZE LABEL VALUES OF THE 01 172C 01173 CALL DINITY 01174C FAR LEFT SIDE 01175C MOVE TORQUE AXIS TO 01176C 01177 CALL YLOC( -110 ) 01 178C WIDE-DASHED LINE 01 179C PLOT TORQUE AS A 0 1 1 80C 01181 CALL LINE( 47 ) 01 182C RE-PLOTTING OF TIME AXIS 01 183C SUPRESS 01184C 01 185 CALL XLAB( O ) 01 186C ON TRANSLATED Y-AXIS 01 187C USE SHORT TICS 01 188C 199-

01189 CALL YFRM( 2 ) 01 190C 01191C RE-SET Y DATA WINDOW FOR OUTPUT TORQUE 01192C 01193 CALL DLIMY( TORMIN. TORMAX ) 01 194C 01195C DISPLAY OUTPUT TORQUE VERSUS TIME GRAPH 01196C 01197 CALL CHECK( T, TORDATA ) 01198 CALL DSPLAY( T, TORDATA ) 01199C 01200C RE-INITIALIZE LABEL VALUES OF Y-AXIS 01201C 01202 CALL DINITY 01203C 01204C DISPLAY SHIFT ANGLE AXIS AT THE 01205C RIGHT SIDE OF THE GRAPH 01206C 01207 CALL YLOCRT( 0 ) 01208C 01209C RE-PLOT TICS ON RIGHT EDGE OF 01210C SHIFT ANGLE AXIS 0121 1C 01212 CALL YFRM( 5 ) 0 1 2 1 3C 01214C PLOT SHIFT ANGLE AS A SHORT DASHED LINE 012 15C 01216 CALL LINE( 12 ) 01217C 01218C RE-SET Y DATA WINDOW FOR SHIFT ANGLE 01219C 01220 CALL DLIMY( THEMIN, THEMAX ) 01221C 01222C DISPLAY SHIFT ANGLE VERSUS TIME GRAPH 01223C 01224 CALL CHECK( T, THEDATA ) 01225 CALL DSPLAY( T, THEDATA ) 01226C 01227C MOVE BEAM TO TOP OF GRAPH AND 01228C ENTER ALPHA-NUMERIC MODE 01229C 01230 CALL MOVABS( 192, 764 ) 01231 CALL ANMODE 01232C 01233C DETERMINE CURRENT DATE AND TIME 01234C 01235 CALL DATE( IDATE ) 01236 CALL TIME( ITIME ) 01237C 01238C PRINT PROGRAM IDENTIFIER, DATE, AND TIME 01239C 01240 WRITE (IWR, 300) IDATE, ITIME 01241C 01242C PRINT GRAPH TITLE 01243C 01244 CALL GRNAME( 500, 738 ) 01245C 01246C MOVE BEAM BETWEEN TITLE AND GRAPH 01247C AND ENTER ALPHA-NUMERIC MODE 01248C 01249 CALL MOVABS( 59, 712 ) 01250 CALL ANMODE 01251C 01252C PRINT TEXT OF KEYS 01253C 01254 WRITE (IWR, 400) -200-

01255C 01256C THE NEXT 13 COMMANDS DRAW THE 01257C GRAPHICAL PORTION OF THE KEYS... 01258C 01259C ...DRAW OUTPUT TORQUE = WIDE DASHED LINE 01260C 01261 CALL MOVABS( 241 , 719 ) 01262 CALL DRWABS( 291, 719 ) 01263 CALL MOVABS( 296. 719 ) 01264 CALL DRWABS( 346, 719 ) 01265C

01266C = ..DRAW PEDAL FORCE SOLID LINE 01267C 01268 CALL MOVABS( 535, 719 ) 01269 CALL DRWABS( 640, 719 ) 01270C

01271C . . .DRAW SHIFT ANGLE 01272C = SHORT DASHED LINE 01273C 01274C MOVE TO THE STARTING POSITION 01275C OF THE SHIFT ANGLE LABEL 01276C 01277 IX = 829 01278 CALL MOVABS( IX. 719 ) 01279C 01280C COMPLETE THE DASHED SEGMENTS 01281C OF THE LABEL BY DRAWING 01282C 11 LINE SEGMENTS OF 5 RASTER UNIT 01283C LENGTH EACH 01284C 01285 DO 200 1=1,11 01286 CALL DRWABS( IX + 5, 719 ) 01287 IX = IX + 10 01288 CALL MOVABS( IX, 719 ) 01289 200 CONTINUE 01290C 01291C MOVE TO X-AXIS LABEL POSITION AND 01292C ENTER ALPHA-NUMERIC MODE 01293C 01294 CALL MOVABS( 430, 60 ) 01295 CALL ANMODE 01296C 01297C PRINT X-AXIS LABEL 01298C 01299 WRITE (IWR, 500) 01300C 01301C MOVE TO FAR LEFT Y-AXIS LABEL POSITION 01302C 01303 CALL MOVABS( O, 644 ) 01304C 0 1 305C PRINT OUTPUT TORQUE AXIS LABEL (VERTICALLY) 01306C 01307 CALL VLABEL( 21. TORLAB ) 01308C 0 1 309C MOVE TO CENTER LEFT Y-AXIS LABEL POSITION 01310C 01311 CALL MOVABS( 112, 590 ) 01312C 01313C PRINT PEDAL FORCE LABEL 01314C 01315 CALL VLABEL( 16. PFLAB ) 01316C 01317C MOVE TO RIGHT Y-AXIS LABEL POSITION 01318C 01319 CALL MOVABS( 900, 600 ) 01320C -201-

01321C PRINT SHIFT ANGLE AXIS LABEL (VERTICALLY) 01322C 01323 CALL VLABEL( 17. SALAB ) 01324C 01325C MOVE TO SECOND LAST LINE OF GRAPH 01326C AND ENTER ALPHA-NUMERIC MODE 01327C 01328 CALL MOVABS( 136, 30 ) 01329 CALL ANMODE 01330C 01331C PRINT FIRST ROW OF CONTROL SYSTEM 01332C IDENTIFYING PARAMETERS 01333C 01334 WRITE (IWR, 600) C1, C2, C3. FD, OMEGA 01335C 01336C MOVE TO LAST LINE OF GRAPH AND 01337C ENTER ALPHA-NUMERIC MODE 01338C 01339 CALL MOVABS( 108, 6 ) 01340 CALL ANMODE 01341C 01342C PRINT SECOND ROW OF CONTROL SYSTEM 01343C IDENTIFYING PARAMETERS 01344C

01345 WRITE (IWR, 650) RPC, RI , RSCVT, SCEQ, YMAX 01346C 01347C PAUSE TO DISPLAY GRAPH 01348C 01349 CALL TINPUT( ICHAR ) 01350C 01351C ERASE GRAPH IN PREPARATION FOR THE NEXT STEP 01352C 01353 CALL NEWPAG 01354C 01355C DUMP TERMINAL BUFFER 01356C 01357 CALL TSEND 01358 RETURN 01359C 01360C FORMATS FOR PROGRAM IDENTIFIER, KEY TEXT, 01361C X-AXIS LABEL, AND CONTROL SYSTEM 01362C IDENTIFYING PARAMETERS (2 LINES) O 1 363C 01364 300 FORMAT( 'BIKE CONTROL SIMULATOR', 01365+ 2( 1X. A10 ) ) 01366 400 FORMAT ( 'OUTPUT TORQUE', 10X, 01367+ 'PEDAL FORCE', 10X, 01368+ 'SHIFT ANGLE') (SEC)' 01369 500 FORMAT( 'TIME ) 01370 600 FORMAT( 'C1='. F5.3, 2X. 'C2=', F5.3, 2X , 01371+ 'C3=', F5.1. 2X, 'FD=', F5.2, 01372+ 2X, 'OMEGA=', F5.2 )

01373 650 FORMAT( 'RPC=', F5.3, 2X , 'RI=', F5.3, 01374+ 2X, 'RSCVT=', F5.3, 2X. 01375+ 'SCEQ='. F5.2. 2X. 'YMAX=', F5.3 ) 01376 END 01377C 08-13-81 TRC; CREATED 01378C( *********** 01379C ************************* BCS.GRNAMEI 01380 SUBROUTINE GRNAMEI 0138 1C ************************************************* 01382C EXPERT- TRC 01383C 01384C PURPOSE A FOR 01385C THIS ROUTINE WILL REQUEST TITLE IT FOR 01386C A TEKTRONIX GRAPH AND PREPARE -202-

01387C PRINTING BY GENERAL PLOTTING 01388C ROUTINE GRNAME 01389C 01390C INPUT 01391C IRD: READ LOGICAL UNIT NUMBER OF THE 01392C COMPUTER BEING USED 01393C (INPUT IN COMMON BLOCK INOUT) 01394C IWR: WRITE LOGICAL UNIT NUMBER OF THE 01395C COMPUTER BEING USED 01396C (INPUT IN COMMON BLOCK INOUT) 01397C 01398C OUTPUT 01399C NCHAR: THE NUMBER OF CHARACTERS IN THE 01400C TITLE, 0 <= NCHAR <= 72 01401C (OUTPUT IN COMMON BLOCK GRNAMEB) 01402C ITITLE: ARRAY OF CHARACTERS COMPRISING 01403C THE GRAPH TITLE 01404C (A MAXIMUM OF 72 CHARACTERS 01405C WILL BE READ) 01406C (ALPHA -NUMERIC/ INTEGER, 01407C PACKED IN A1 FORMAT) 01408C (OUTPUT IN COMMON BLOCK GRNAMEB) 01409C 01410C RESTRICTIONS 01411C 1) CONTAINS CALL TO NON-STANDARD SYSTEM 014 12C ROUTINE "EOF" 01413C 2) BLANKS INCLUDED AT THE START OF THE TITLE 01414C WILL BE COUNTED AS CHARACTERS 01415C 014 16C ERROR INDICATIONS 01417C NONE 01418C 01419C KEYWORDS 01420C NONE 01421C 01422C METHOD 01423C STRAIGHTFORWARD 01424C 01425C DESCRIPTION OF VARIABLES 01426C I : COUNTER VARIABLE 01427C IBLANK: BUFFER FOR HOLDING ASCII "BLANK" 01428C ( INTEGER/ALPHA-NUMERIC) 01429C 01430C SUBROUTINES CALLED 01431C EOF: CDC FORTRAN UTILITY FUNCTION TO " 01432C DETERMINE IF AN "END-OF-FILE MARK 01433C HAS BEEN ENCOUNTERED BEFORE LOCATING 01434C ANY CHARACTERS ON AN INPUT LINE 01435C (PREVENTS ABNORMAL TERMINATION 01436C OF THE PROGRAM IF A LINE CONSISTING 01437C ONLY OF A IS INPUT) 01438C 01439C DECLARATIONS FOR VARIABLES REFERENCED 01440C IN SUBROUTINE CALL 01441 INTEGER ITITLE 01442 INTEGER IRD, IWR 01443 INTEGER NCHAR 01444C 01445C LOCAL DECLARATIONS 01446 INTEGER IBLANK 01447 INTEGER I 01448C 01449C COMMON STATEMENTS AND DECLARATIONS IWR 01450 COMMON/ INOUT / I RO , 01451 COMMON/ GRNAMEB / NCHAR. ITITLE(72) 01452C 203-

01453C DATA STATEMENTS 01454 DATA IBLANK / ' ' / 01455 DATA ITITLE / 72*( 1H ) / 01456C 01457C ***** START OF EXECUTABLE CODE ***** 01458C 01459C REQUEST GRAPH TITLE 01460C 01461 WRITE (IWR, 400) 01462C 01463C READ LINE OF GRAPH TITLE CHARACTERS 01464C 01465 READ (IRD, 500) ITITLE 01466C 01467C THE FOLLOWING LINE PREVENTS ABNORMAL 01468C TERMINATION OF THE PROGRAM ON THE MERITSS 01469C TIMESHARING SYSTEM AT THE UNIVERSITY OF 01470C MINNESOTA IF A LINE CONSISTING ONLY 01471C OF A IS INPUT 01472C 01473 IF ( EOF(IRD) ) 10, 10, 10 01474 10 CONTINUE 01475C 01476C COUNT THE NUMBER OF CHARACTERS ON THE LINE. 01477C 01478 DO 100 I " 1, 72 01479C

01480C ..COUNT BACK FROM THE END OF THE 01481C INPUT LINE UNTIL A NON-BLANK 01482C CHARACTER IS FOUND 01483C 01484 IF ( ITITLE(73 I ).NE. IBLANK ) 01485+ GO TO 200 01486 100 CONTINUE 01487C

01488C ...IF AN ALL-BLANK LINE HAS BEEN INPUT, 01489C COUNT THE LINE AS ONE BLANK CHARACTER 01490C 01491 I = 72 01492 200 CONTINUE 01493C

01494C ...SET THE NUMBER OF CHARACTERS 01495C 01496 NCHAR = 73 01497 RETURN 01498C 01499C FORMATS FOR GRAPH TITLE REQUEST AND 0 1 500C TITLE INPUT CHARACTERS 01501C 01502 400 FORMATC INPUT GRAPH IDENTIFYING LABEL') 01503 500 FORMAT (72A1) 01504 END 01505C 08-13-81 TRC; CREATED 01506C ************ ************************* BCS.GRNAME 01507C IHEIGHT 01508 SUBROUTINE GRNAME( ICENTER, )

************************************************* 01509C 01510C EXPERT: TRC 01511C 015 12C PURPOSE ROUTINE 01513C GRAPH TITLE LABELLING 01514C CENTER AND PRINT AN 01515C THIS ROUTINE WILL TEKTRONIX GRAPH. 01516C ARBITRARY TITLE ON A OVERFLOW THE AVAILABLE 01517C IF THE TITLE WILL WILL AUTOMATICALLY 01518C SPACE THIS ROUTINE -204-

01519C CONCATENATE IT. 01520C 01521C INPUT 01522C ICENTER: THE X SCREEN COORDINATE OF 01523C THE CENTER OF THE TITLE SPACE 01524C OF THE GRAPH 01525C IHEIGHT: THE Y SCREEN COORDINATE WHERE THE 01526C TITLE SHOULD BE LOCATED 01527C ITITLE: ARRAY OF CHARACTERS COMPRISING 01528C THE GRAPH TITLE 01529C (ALPHA-NUMERIC/INTEGER, 01530C PACKED IN A1 FORMAT) 01531C (INPUT IN COMMON BLOCK GRNAMEB) 01532C (NOT ACTUALLY USED IN THIS ROUTINE) 01533C NCHAR: THE NUMBER OF CHARACTERS IN THE 01534C TITLE 01535C (WILL BE CHANGED TO THE NUMBER OF 01536C CHARACTERS ACTUALLY PRINTED BY 01537C GRNAME IF CONCATENATION IS REQUIRED) 01538C (ALSO AN OUTPUT PARAMETER) 01539C (INPUT IN COMMON BLOCK GRNAMEB) 01540C 0154 1C OUTPUT 01542C NCHAR: THE NUMBER OF CHARACTERS IN THE GRAPH 01543C TITLE ACTUALLY PRINTED 01544C (ALSO AN INPUT PARAMETER) 01545C (OUTPUT IN COMMON BLOCK GRNAMEB) 01546C 01547C RESTRICTIONS 01548C 1023 ADDRESSABLE RASTER UNITS ARE ASSUMED 01549C TO BE AVAILABLE ON THE DISPLAY 0155OC (COMPATIBLE WITH TEKTRONIX 4010 SERIES 01551C HARDWARE) 01552C 01553C ERROR INDICATIONS 01554C NONE 01555C 01556C KEYWORDS 01557C SCREEN COORDINATE: ABSOLUTE COORDINATE OF 01558C AN ADDRESSABLE RASTER UNIT ON 01559C THE TEKTRONIX DISPLAY 01560C 01561C METHOD 01562C STRAIGHTFORWARD 01563C 01564C DESCRIPTION OF VARIABLES 01565C ISPACE: THE X SCREEN COORDINATE OF THE 01566C CENTER OF THE SPACE AVAILABLE 01567C FOR THE GRAPH TITLE 01568C ISTART: THE X SCREEN COORDINATE OF THE 01569C TITLE STARTING POSITION 01570C 01571C SUBROUTINES CALLED 01572C GRNAME 1: ROUTINE CONTAINING THE TEKTRONIX 01573C TCS COMMANDS TO ACTUALLY DISPLAY 01574C THE GRAPH TITLE ON THE SCREEN 01575C 01576C DECLARATIONS FOR VARIABLES REFERENCED 01577C IN SUBROUTINE CALL 01578 INTEGER ITITLE 01579 INTEGER ICENTER. IHEIGHT 01580 INTEGER NCHAR 01581C 01582C LOCAL DECLARATIONS 01583 INTEGER ISPACE, ISTART 01584C 205-

01585C C0MM1STATEMENTS AND DECLARATIONS COMMON / GRNAMEB / NCHAR. 01587C ITITLE(72) 01588C DATA STATEMENTS 01589C NONE 01590C ***** START F EXECUTABLE CODE ***** 01592C ?ETERMINE THE nlllUn ABSOLUTE CENTER OF THE F RASTER UNITS avai'able o\llsc fSr The 01596C 01597 ISPACE = ICENTER IF %\\.ll-. ( (1023 ICENTER). LT. ICENTER ) ]|* ISPACE = 1023 ICENTER 01601C CONCATENATE THE TITLE, IF REQUIRED 01602C NOTE: TEKTRONIX CHARACTERS ARE 01603C 14 RASTER UNITS WIDE 01604C 01605 IF ( NCHAR. GT . ( 2*ISPACE/ 14 ) ) ]ff* NCHAR 2*ISPACE/14 01607C 01608C DETERMINE X SCREEN COORDINATE OF TITLE 01609C STARTING POSTION 01610C 01611 ISTART = ICENTER NCHAR*14/2 01612C 01613C PRINT THE TITLE 01614C 01615 CALL GRNAME 1( ISTART, IHEIGHT ) 01616 RETURN 01617 END 01618C 08-13-81 TRC; CREATED 01619C( 01620C ************************* BCS.GRNAME1 *********** 01621 SUBROUTINE GRNAME 1 ( IX, IY ) 01622C ************************************************* 01623C EXPERT: TRC 01624C 01625C PURPOSE 01626C ROUTINE CONTAINING THE TEKTRONIX PLOT 10 TCS 01627C CALLS FOR GRAPH TITLING ROUTINE GRNAME 01628C 01629C INPUT 01630C IRD: READ LOGICAL UNIT NUMBER OF THE 01631C COMPUTER BEING USED 01632C (INPUT IN COMMON BLOCK INOUT) 01633C ITITLE: ARRAY OF CHARACTERS COMPRISING 01634C THE GRAPH TITLE 01635C (ALPHA-NUMERIC/INTEGER. 01636C PACKED IN A1 FORMAT) 01637C (INPUT IN COMMON BLOCK GRNAMEB) 01638C IWR: WRITE LOGICAL UNIT NUMBER OF THE 01639C COMPUTER BEING USED 01640C (INPUT IN COMMON BLOCK INOUT) 0164 1C IX: THE X SCREEN COORDINATE WHERE THE 01642C GRAPH TITLE WILL START TO BE PRINTED 01643C IY: THE Y SCREEN COORDINATE WHERE THE 01644C GRAPH TITLE WILL BE PRINTED 01645C NCHAR: THE NUMBER OF CHARACTERS OF 01646C ARRAY ITITLE TO BE PRINTED 01647C (INPUT IN COMMON BLOCK GRNAMEB) 01648C 01649C OUTPUT 01650C NONE 206-

01651C 01652C RESTRICTIONS 01653C 1) INTENDED FOR USE WITH SUBROUTINE 01654C "GRNAME" ONLY 01655C 2) INTENDED FOR USE WITH TEKTRONIX 01656C 4010 SERIES HARDEARE ONLY 01657C (MIXED ALPHA-NUMERIC/VECTOR OUTPUT 01658C MAY CAUSE INCORRECT DISPLAY ON 01659C SOME COMPATIBLE TERMINALS) 01660C 01661C ERROR INDICATIONS 01662C NONE 01663C 01664C KEYWORDS 01665C SCREEN COORDINATE: ABSOLUTE COORDINATE OF 01666C AN ADDRESSABLE RASTER UNIT ON 01667C THE TEKTRONIX DISPLAY 01668C 01669C METHOD 01670C STRAIGHTFORWARD 01671C 01672C DESCRIPTION OF VARIABLES 01673C NONE 01674C 01675C SUBROUTINES CALLED 01676C ANMODE: TEKTRONIX TCS ROUTINE TO CHANGE 01677C THE GRAPHICS TERMINAL FROM VECTOR 01678C TO ALPHA-NUMERIC MODE 01679C MOVABS: TEKTRONIX TCS ROUTINE TO MAKE A MOVE 01680C (INVISIBLE DRAW) IN ABSOLUTE 01681C SCREEN COORDINATES 01682C TSEND: TEKTRONIX TCS ROUTINE TO DUMP THE 01683C GRAPHICS TERMINAL BUFFER 01684C (DISPLAYS ALL BUFFERED OUTPUT 01685C ON THE SCREEN) 01686C 01687C DECLARATIONS FOR VARIABLES REFERENCED 01688C IN SUBROUTINE CALL 01689 INTEGER IX, IY 01690 INTEGER ITITLE 01691 INTEGER NCHAR 01692 INTEGER IRD, IWR 01693C 01694C LOCAL DECLARATIONS 01695C NONE 01696C 01697C COMMON STATEMENTS AND DECLARATIONS 01698 COMMON/ GRNAMEB / NCHAR, ITITLE(72) 01699 COMMON/ INOUT / IRD, IWR 01700C 01701C 01702C DATA STATEMENTS 01703C NONE 01704C 01705C ***** START OF EXECUTABLE CODE ***** 01706C 01707C MOVE BEAM TO THE STARTING LOCATION 01708C OF THE TITLE 01709C 01710 CALL MOVABS( IX, IY ) 01711C 01712C ENTER ALPHA-NUMERIC MODE AND 01713C DUMP THE TERMINAL BUFFER 01714C 01715 CALL ANMODE 01716 CALL TSEND 207-

01717C 01718C PRINT TITLE 01719C 01720 WRITE (IWR, 100) (ITITLE(I), I 1. NCHAR) 01721 RETURN 01722C 01723C FORMAT FOR GRAPH TITLE 01724C 01725 100 FORMAT ( 72A1 ) 01726 END 01727C 08-13-81 TRC; CREATED 01728C

01729C ************************* BCS TEKINIT *********** 01730 SUBROUTINE TEKINIT 01731C ************************************************* 01732C EXPERT: TRC 01733C 01734C PURPOSE 01735C THIS ROUTINE INITIALIZES TEKTRONIX TCS 01736C AND AGII SOFTWARE SYSTEMS FOR 01737C 4010 SERIES HARDWARE 01738C 01739C INPUT 01740C ICPS: CPS RATE OF THE MODEM BEING USED 01741C (INPUT WITH A DATA STATEMENT) 01742C 01743C OUTPUT 01744C NONE 01745C 01746C RESTRICTIONS 01747C INTENDED FOR USE WITH TEKTRONIX 01748C 4010 SERIES HARDWARE ONLY 01749C 01750C ERROR INDICATIONS 01751C NONE 01752C 01753C KEYWORDS 01754C NONE 01755C 01756C METHOD 01757C STRAIGHTFORWARD 01758C 01759C DESCRIPTION OF VARIABLES 01760C ICPS: RECEIVE RATE OF THE COMMUNICATIONS 01761C LINE BEING USED, CPS 01762C 01763C SUBROUTINES CALLED 01764C BINITT: TEKTRONIX AGII ROUTINE TO 01765C INITIALIZE THE AGII COMMON AREA 01766C INITT: TEKTRONIX TCS ROUTINE TO 01767C INITIALIZE THE GRAPHICS TERMINAL 01768C CONTROL SYSTEM 01769C NEWPAG: TEKTRONIX TCS ROUTINE TO CLEAR 01770C THE GRAPHICS SCREEN 01771C TSEND: TEKTRONIX TCS ROUTINE TO DUMP 01772C THE GRAPHICS TERMINAL BUFFER 01773C (DISPLAYS ALL BUFFERED OUTPUT 01774C ON THE SCREEN) 01775C 01776C DECLARATIONS FOR VARIABLES REFERENCED 01777C IN SUBROUTINE CALL 01778C NONE 01779C 01780C LOCAL DECLARATIONS 01781 INTEGER ICPS 01782C -208-

01783C COMMON STATEMENTS AND DECLARATIONS 01784C NONE 01785C 01786C 01787C DATA STATEMENTS 01788 DATA ICPS / 120 / 01789C 01790C ***** START OF EXECUTABLE CODE ***** 01791C 01792C 01793C INITIALIZE TEKTRONIX TCS COMMON BLOCK 01794C 01795 CALL INITT( ICPS ) 01796C 01797C INITIALIZE TEKTRONIX AGII COMMON BLOCK 01798C 01799 CALL BINITT 01800C 01801C DUMP TERMINAL BUFFER 01802C 01803 CALL TSEND 01804 RETURN 01805 END 01806C 08-13-81 TRC; CREATED 01807C( 01808C ************************* BCS.TEKFIN ************ 01809 SUBROUTINE TEKFIN 018 10C ************************************************* 0181 1C EXPERT: TRC 018 12C 018 13C PURPOSE 01814C THIS ROUTINE TERMINATES TEKTRONIX GRAPHICS 01815C ROUTINES FOR 4010 SERIES HARDWARE 018 16C 01817C INPUT 01818C NONE 01819C 01820C OUTPUT 01821C NONE 01822C 01823C RESTRICTIONS 01824C INTENDED FOR USE WITH TEKTRONIX 01825C 4010 SERIES COMPATIBLE HARDWARE ONLY 01826C 01827C ERROR INDICATIONS 01828C NONE 01829C 01830C KEYWORDS 01831C NONE 01832C 01833C METHOD 01834C STRAIGHTFORWARD 01835C 01836C DESCRIPTION OF VARIABLES 01837C NONE 01838C 01839C SUBROUTINES CALLED 01840C FINITT: TEKTRONIX TCS ROUTINE TO 01841C TERMINATE USE OF THE SYSTEM 01842C TERMINAL CONTROL 01843C REFERENCED 01844C DECLARATIONS FOR VARIABLES 01845C IN SUBROUTINE CALL 01846C NONE 01847C 01848C LOCAL DECLARATIONS 209-

01849C NONE 01850C 01851C COMMON STATEMENTS AND DECLARATIONS 01852C NONE 01853C 01854C DATA STATEMENTS 01855C NONE 01856C 01857C ***** START OF EXECUTABLE CODE ***** 01 858C 01859C TERMINATE TEKTRONIX GRAPHICS 01860C 01861 CALL FINITT( O, 700 ) 01862 RETURN 01863 END