Kinematic Analysis of a Swash-Plate Mechanism Using Transformation Matrices

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ABSTRACT

KINEMATIC ANALYSIS OF A SWASH-PLATE MECHANISM USING TRANSFORMATION MATRICES

by David Edgar Evans

The swash-plate mechanism is used to transfer rotational motion into translational motion and vice versa. Using transformation matrices to describe each joint, the displacements of the swash-plate mechanism were analyzed. The general transformation matrix model for the plane joint, not previously developed using transformation matrices, was presented. The solution was tested using a computer program. Examples of three different swash-plate angles; 10, 45, and 25° were considered. The solution was found to agree with previous solutions while offering a more complete description of the mechanism. Future work in this area could include a complete static, velocity, and/or dynamic analysis of the swash-plate mechanism based on the solution presented. KINEMATIC ANALYSIS OF A SWASH-PLATE MECHANISM USING TRANSFORMATION MATRICES

by David Edgar Evans

A Thesis Submitted to the Faculty of New Jersey Institute of Technology In Partial Fulfillment of the Requirements for the Degree of Masters of Science in Mechanical Engineering

Department of Mechanical Engineering

January 1998 BIOGRAPHICAL SKETCH

Author: David Edgar Evans

Degree: Master of Science in Mechanical Engineering

Date: January 1998

Undergraduate and Graduate Education:

• Master of Science in Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 1998

• Bachelor of Science in Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 1995

Major: Mechanical Engineering APPROVAL PAGE

KINEMATIC ANALYSIS OF A SWASH-PLATE MECHANISM USING TRANSFORMATION MATRICES

by David Edgar Evans

Dr. Ian S. Fischer, Thesis Advisor Date Associate Professor of Mechanical Engineering, New Jersey Institute of Technology

. . Sodhi, Committee Member Date Associate Professor of Mechanical Engineering, New Jersey Institute of Technology

Dr. K.A. Narh, Committee Member Date Assistant Professor of Mechanical Engineering, New Jersey Institute of Technology This thesis is dedicated to Lois Speiden, M.D. and Edgar Speiden ACKNOWLEDGMENT

The author wishes to express his deepest gratitude to Dr. Ian S. Fisher for his unselfish dedication, guidance, and support in this research effort. It was Dr. Fisher who helped the author understand the nature of a kinematic solution for generalized joints.

The author would also like to thank the owners of Tri-Power Design L.L.C.,

Michael Mastice, Robert Mastice, and Anthony LaRosa for their understanding, flexibility with the author's work schedule while finishing this thesis, knowledge and for the use of the computer facilities at Tri-Power Design L.L.C.

The author also wishes to thank his Aunt Dr. Lois Speiden, Grandmother Marie

Speiden, Mother Norma E. S. Evans, Father William Joseph Evans and the rest of his family for their tireless emotional and financial support during the duration of time in which this thesis was made.

The author would finally like to thank God for His guidance and love. Without

Him nothing would have been possible.

i TABLE OF CONTENTS

Chapter Page

1 INTRODUCTION 1

2 PREVIOUS WORK AND SCOPE 3

3 ANALYSIS 6

3.1 Description of Methodology 6

3.2 Definition of Variables 14

3.3 Formulation of Solution 15

3.4 Swash-Plate Transformations 17

3.4.1 Transformation #1 17

3.4.2 Transformation #2 18

3.4.3 Transformation #3 19

3.4.4 Transformation #4 20

3.5 Solution of the Swash-Plate Mechanism 21

3.5.1 Initial Results 21

3.5.2 Primary Elements 22

3.5.3 Dual Elements 23

3.5.4 Derivation of the Solution in its Final Form 24

4 RESULTS 26

5 DISCUSSION AND CONCLUSION 45

APPENDIX A: FORTRAN Computer Program "SWASHP" 48

BIBLIOGRAPHY 52

ii LIST OF TABLES

Table Page

1 Variables 14

2 Denavit-Hartenburg Parameter Table 15

3 Input for First Data Set (l = 1° 27

4 FORTRAN Program "SWASHP" Output for First Data Set (l = 1° 28

5 Input for Second Data Set (l = 45°) 35

6 FORTRAN Program "SWASHP" Output for Second Data Set (l = 45°) 36

7 Input for Third Data Set (l = 25°) 37

8 FORTRAN Program "SWASHP" Output for Third Data Set (l = 25°) 38

iii LIST OF FIGURES

Figure Page

1 Swash-Plate Mechanism with One Follower/Piston Shown 4

2 General Transformations for a Revolute/Prismatic Joint 8

3 General Transformations for a Plane Joint 10

4 General Transformations for a Spherical Joint 12

5 Top View of Swash-Plate Mechanism Showing Neutral Axis as a Point 16

6 Side View of Swash-Plate Mechanism Showing Neutral Axis 16

7 Swash-Plate Showing Transformation #1 17

8 Swash-Plate Showing Transformation #2 18

9 Swash-Plate Showing Transformation #3 19

10 Swash-Plate Showing Transformation #4 20

11 Output Translation as a Function of Input Rotation (aj = 10°) 29

12 Intermediate Angle and Input Rotation 91 (aj = 10°) 30

13 Intermediate Translations s2 and r (al = 10°) 31

14 Intermediate Angle 773 and Input Angle 91 (al 10°) 32

15 Intermediate Variable C3 and Input Variable Of (al = 10°) 33

16 Intermediate Variable -3 and Intermediate Variable 773 (l = 10°) 34

17 Output Translation s4 and Input Rotation 9j 39

18 Intermediate Angle and Input Rotation 1 40

19 Intermediate Translations s2 and r 41

20 Intermediate Variable 773 and Input Variable 91 42

21 Intermediate Variable -3 and Input Variable 91 43

22 Intermediate Variable 773 and Intermediate Variable -3 44

i CHAPTER 1

INTRODUCTION

The swash-plate mechanism is comprised of four links and four joints: The ground link is

connected to the shaft/swash link by a revolute joint (R). The shaft/swash link is mated

with the coupler by a plane joint (E). The coupler is connected to the follower (also

referred to as a piston) with a ball joint (S), and the follower (piston) is mated with the

ground link by a prismatic joint (P). The mechanism is defined by its joints as an RESP

mechanism.

The appeal of the swash-plate mechanism is that its pistons and shaft of rotation

are parallel, thus facilitating a compact design with low weight and high-energy transfer

efficiency. The configuration is robust because it is easily possible to completely balance the mechanism by choosing a thickness of the angled plane to offset the momentum of the pistons when there are three or more odd number of pistons as observed by Galin and

Harris. [3]

The swash-plate mechanism has been described previously by Galin and Harris [3]

and Mitchell [11] using an old style methodology. While these previous solutions were

effective in finding a specific solution for an output translation given an input rotation, they did not fully describe the displacements and/or rotations of all the links in the mechanism before proceeding with the dynamic and/or force analysis.

This work completely describes the mechanism using dual number transformation matrices. In considering the kinematics of this mechanism, a plane (E) joint is modeled for the first time using transformation matrices. Transformation matrix solutions can now be utilized to describe other mechanisms with plane and/or spherical joints, such as the

Rzeppa constant-velocity coupling, which may have no other current means of exhaustive solution. CHAPTER 2

PREVIOUS WORK AND SCOPE

James Watt invented the swash-plate mechanism in the late nineteenth century. The purpose of the mechanism was to translate rotating motion into reciprocating motion or reciprocating motion into rotating motion. It's compact and simple design made the swash-plate appealing to designers at the time it was invented. However, Maki and

DeHart report that it was not until the 1920's that the mechanism was rendered operational enough to be considered a viable alternative to the more widely used reciprocating/rotating mechanisms. [1]

In the early 1920's, A. G. M. Mitchell made the first application of the swash-plate mechanism that could compete with the crank and connecting rod mechanism. [11]

According to Galin and Harris, Mitchell's first application for the swash-plate mechanism was an air compressor. [2] However, the swash-plate was soon used in an internal combustion engine for airplanes. Other applications for the swash plate include gas compressors and air motors. Its most notable application (because of shear volume of production) is in the automotive air-conditioning compressor. Weibel and Mantey provide an elegant description of the swash-plate mechanism, as it is used in the automotive air- conditioning compressor. [6] Because of the mechanism's wide range of utilization, it is important that its kinematics be well defined for reference of machine designers.

Using transformation matrices, the equation that describes the position of the piston in a swash-plate mechanism will be derived. The equation should be of such form

3 4 that the output displacement s4 is a sole function of the input variable 9 and the mechanism's dimensions.

To properly design a balanced swash-plate type mechanism, the thickness of the swash plate can be calculated to produce a balancing moment. The balancing moment counteracts the moment produced by the cyclical force of the pistons reciprocating at a distance from the center of the plate. This configuration of swash-plate/piston balance only works for a three or more piston mechanism. Only one piston is shown in Figure 1.

OOWE/ISO A OI (S ISMAIC OI ( AE OI (E

SWAS AE (AGE AE IE O SA

1//7 SA EOUE OI ( COUE /////

GOU IK

Figure 1

Swash Plate Mechanism with One Follower/Piston Shown

Before the moment produced by the reciprocating pistons can be found, the position equation for one piston assembly must be calculated relative to the input angle of the shaft. Let us call that the first position equation. Using the first position equation, a general expression can be made to describe the position of any piston in the assembly.

Also, the velocity and change of momentum equations can be found by taking the first and second derivatives of the position equation respectively. 5

The position equation describes the displacement of a designated piston from a

defined origin based on a given shaft angle of rotation. The only variable in the first

displacement equation should be the shaft angle of rotation 01.

The kinematic solution for a swash-plate mechanism will be developed as follows:

In chapter three, the transformation matrix method of solving the mechanism will be explained (including a full solution of a generalized plane joint). In the same chapter, the variables used in the calculation and model of the mechanism will also be presented and explained. In chapter four, the solution of the basic model will be presented. Multiple iterations of data output will be listed and graphed. Finally, in chapter five, the results of the analysis will be discussed, recommendations for the use of the data in mechanism

design will be made, future work using the general solution will be highlighted and conclusions based on the presented solution will be drawn. CHAPTER 3

ANALYSIS

3.1 Description of Methodology

The motion of the swash plate mechanism is modeled using coordinate transformation matrices. Therefore, the essential elements of the Transformation Method will now be discussed.

Clifford introduced the concept of the dual number in 1873 and the name of "dual number" was coined by Study in 1903. The definition of the dual number (s) is:

6 (1 =

In analogy with complex numbers, a duplex number can be written as:

+ 6( (2)

Where:

= primary (or real) component of = dual component of

Dual numbers follow all the rules of regular algebra keeping in mind that a dual number raised to any power is equal to zero.

A dual angle can be used to specify the relative displacement and orientation between two lines in space.

The dual angle is written as:

= + ( (3)

Where is the primary rotational part (projected angle between the lines) and s is the dual

translation part (shortest distance between the lines) of the dual angle O

The trigonometric functions of dual angles are obtained by using Taylor Series expansions:

sin O = sin 0 + (

ë = ° ( sin (4)

tan ö = tan 0 + s(s)sec 0

All trigonometric identities hold for dual trigonometric functions.

Using coordinate transformation matrices, dual numbers can be used to describe a

mechanism. This is accomplished by describing the succession of joints and links found in

the mechanism with the descriptive transformation matrix of each joint. Then, the joint

parameters used are determined and shown in a Denavit-Hartenberg table. Finally, the

mechanism displacements are evaluated by performing a series of transformations through

the joints and links specific to the mechanism. In this problem, the mechanism being

considered is an RESP (Revolute, Plane, Spherical, and Prismatic) swash-plate

mechanism. There are four joints and four links in the swash-plate mechanism.

For the solution presented, all transformations are relative to moving coordinate

systems. Therefore the order of multiplication will be to the right.

The first joint encountered in the swash-plate mechanism is the revolute joint. The

revolute, cylindrical and prismatic joints can each be described in general as a

transformation through the dual angle e n = 0, + (S about the z-axis followed by a dual

angle transformation about the x-axis through the angle n n + (n In the case of the revolute joint, there is one degree of rotational freedom. Also, in the prismatic joint there is one translational degree of freedom along the shaft. 8

Figure 2

General Transformations for a Revolute/Prismatic Joint (Drawn by Dr. Ian S. Fisher)

Note that expression (5) represents the relation of frame {n} fixed on the distal end of link

n-1” to frame {n+1} fixed on the distal end of link "n". Letting a matrix M represent the general form of the revolute/prismatic transformation, this is expressed as follows:

Note: s = sine, c = cosine

n — sOn 1

Sn n n — n +n *an = O (5) 1 n n 9

Multiplying these matrices and separating the primary and dual components gives:

On —n On nOn

On nOn — n On 0 n n (6)

— n On ansans On — n On ancansOn+ s nnn

+E SnC O — nn 9 — n nn —nnOn + nnOn 0 nn —nn

The next joint encountered in the swash-plate is the plane joint denoted by E after the

German word for plane, ebene (pronounced ehb-ih-nah). This joint has three degrees of freedom; two in translation and one in rotation. The transformation that describes the general form of the plane joint consists of a translation Pn = 0 + (rn about the x-axis followed by a translation E'n = +(n along the y-axis followed by a rotation

On = On + s(0) about the z-axis, and concluding with a translation a +(n about the x-axis. 1

Figure 3 General Transformations for a Plane Joint (Drawn by Dr. Ian S. Fisher)

Start at system {i} fixed on the distal end of link n-1. Translate distance r n along

the i Translate distance s in the direction of the j -axis. Then rotate through On

about k-axis into alignment with system {n'}. Then make a screw motion through

translation a and rotation oc„ into alignment with frame {n+1 } on the distal end of link n.

Let represent the general form of the plane joint:

Note: c = cosine, s = sine

11 = Yi(e (a

0 - eS n 0 si n - n sOn 1 0 0 (7) = 0 cn — sP 1 0 O 0 can — sa

0 sn cPn sS'n an _ 1 sa can 11

Multiplying these matrices and separating the primary from the dual components gives:

„+nl 3 „ n =

con —sO 0 0 0 n9n+n ( 8 S n On 0 + —nn—rn

1 nOn —On rnn +nn +n 0

The final joint in need of description for this mechanism is a spherical or ball joint.

This joint has three degrees of rotational freedom about all three axes. The general form of transformations for this mechanism will be denoted asf. Because there are no natural axes, the motions of the ball joint are expressed with respect to arbitrarily chosen axis.

The order of motion is:

First, a screw motion about the z-axis O n = 9„ +e(s n ) followed by a screw

motion about the y-axis fin = + ( 2 then a screw motion about the x-axis

= 7 + (z 12

Figure 4

General Transformations for a Spherical Joint (Drawn by Dr. Ian S. Fisher)

The expression represents transformation from frame { } fixed on link "n-1" to frame {n+1 } fixed on the distal end of link "n".

Let L represent the general form of the plane joint:

Note: c = cosine, s = sine

=i i(17 5

Cn — sO ci 0 1 0 0 (9) = sOi c O 1 0 0 f „ — „ 0 ca n — s

1 — CrIn 0 „ - „ 0 s n 13

Multiplying these matrices and separating the primary from the dual components gives:

+1ic97„ -„a

n On rn On ‘n —On ‘n srIn cOn c‘n +sOn s‘n

= rn n n n s4 n +On ‘„ srin sOn c‘n —cOn Cn

n s‘n clin c‘n

n cri n n sCn e n crin n c‘n

+ s17,c9n (z n + an ki —sii n (z n +a,7 s n clin On — snsqnsOns‘n — n n C„ —e n srin On — sn cOn cCn +nn ‘n +sen (zn +a n )s‘n +sen (z n +a n Cn

e n cqn sOn s‘n e n crin sen cC„

+ sq n s n cOn sC n +siin sn On cCn Sn Cln COn n n n +s 7,i sOn (z n +an )c‘n +sli se (z„+a )sC„ +5 — e n srin 97 —n (zn +an )s‘n —On (z n +a n - „

n n n —s c‘ +sses-

crin (z n +an C„ n(zn+n 4-n — nn —e n sq„sC n — e n r7n c4'n

(10) 1

3.2 Definition of Variables

Table 1 Variables

VARIABLE DESCRIPTION OF VARIABLES

9, Input shaft rotation in revolute joint

swash-plate twist about i -axis

r plane joint translation in j2-axis direction

plane joint translation in i2-axis direction

9 plane-joint Rotation about k -axis

713 ball-joint rotation about j 3-axis

-3 ball joint rotation about i3 -axis

S cylindrical sliding along k-axis (piston oscillation)

C frame offset along i4-axis (perpendicular distance from geometric axis of

the swash plate to the center of the piston)

frame twist about i4-axis 15

3.3 Formulation of Solution

The first step in the transformation matrix solution process is defining and arranging the parameters that define the swash-plate mechanism. The arrangement of the joint variables and link dimensions in a table of parameters as suggested by Denavit and Hartenberg [13] is shown in Table 1.

Table 2 Denavit-Hartenburg Parameter Table

Input Shaft Swash- Plate Ball coupler 4 Follower 0 S

In the solution of this problem, a pictorial representation mixed with a

mathematical description will be used. The order of solution will be as follows: First, the position of the coordinate system on the swash plate at each transformation will be shown with the corresponding transformation matrix. Second, the transformation matrices will be arranged according to their order of multiplication. Thirdly, the matrices will be rearranged for the solution. Fourthly and finally, mathematical manipulation software

(Derive) will be used to multiply the matrices together and then set their components equal to each other for the final solution equation.

The starting point of the swash plate transformations lies on the neutral axis. The neutral axis can be thought of as a line lying on the face of the swash-plate that is 1 perpendicular to the input shaft regardless of the swash-plate angle. The piston/follower assembly is at mid-stroke when the center of the coupler contacts the swash-plate at the neutral axis. See figures 5 and 6.

r o iew o Swas-ae Mecaism Sowig eua Ais as a oi

--"-

eua Ais

/7/ 1 /

r 6 Sie iew o Swas-ae Mecaism Sowig eua Ais 17

3.4 Swash Plate Transformations

3.4.1 Transformation #1

For this transformation, the input variable 1 is shown as a rotation about the k-axis. Note the intermediate position of the coordinate axis denoted i', j' and k'. The constant swash- angle l about the i-axis is also shown. These transformation matrices will be represented as:

ic/ea3 = el

igue 7

Swas-ae Sowig asomaio 1 1

3.4.2 Transformation #2

For this transformation, the plane joint is described with the intermediate variable translations of r and s2 along the i-axis and j-axis respectively. Also described is the intermediate variable transformation acting about the k-axis. This transformation is shown as follows:

(g=i(i(si(e (12)

igue

Swas-ae Sowig asomaio 19

3.4.3 Transformation #3

For this transformation, the ball or spherical joint will be described using the intermediate

rotational variables 773 and -3 about the j-axis and i-axis respectively. The transformation

is shown as follows:

C3 173 = i7 (773 53 (13)

Figure 9

Swash-Plate Showing Transformation #3

3.4.4 Transformation #4

In this transformation, the prismatic joint is described by a translation through the output variable s along the k-axis. There is also a screw motion through intermediate variable a4 and constant c about the i-axis. The coordinate transformation is as follows:

M(a4 s4) = i(s i(a (14)

igue 1

Swas-ae Sowig asomaio

Multiplying these four transformations together forms one transformation equation:

M(1 ( S 9 (713 1 (a , C S = (15)

For purposes of the solution, the input variables from the output variables are segregated before performing the multiplication:

i"(r , S 9 13 -3 = 1C1 -1 (1 1 - (a c s (16) ▪

3.5 Solution of the Swash-Plate Mechanism

3.5.1 Initial Results

Multiplying the matrices described in section 3.4 together with some simplification gives:

02C 0 2 C 3

7 3 C° - S 4- 3 + SO SC 3

- ES 2 S 2 C4- 3 77 3 ES 2C 7 3 S C 3 + ES C

3 S 9 5 C 3 0 2 4"

C S0 - O‘ 3 ▪ C 9 C C 3

+ r 2 3 - r 2 C - £r2 7 C 3

C C c 77 3 CC 3 / ( (r20 2 4- -‘` (-202 3 - S 3 773 S S 2 CO SC 3 i - S 2 C CC 3 i ( r0- EC 3 + E + - 2 0 2 ‘ 3 + E - 2 0 2 C S 2 C + S 2 S C 3 - S S S C 3

4 0

O ( 4 0‘` 4 - - O + 4 + 4 0

( 4 0 4 4 + 4 0 0 ( ( 4 ( (4- 4 4 0 + - E + l 4 0 E - 4 0 4 4 4 0

4 0 ( 4 O l + 4 ( 4 0„

( 0 4 + 4 (17) S C S 4 4 - r" - - 0 4 " 4 0

The solution matrix primary and dual elements are set equal to each other. This gives

eighteen equations that can be used to solve for six unknown variables (r224 0 173 C3

3.5.2 Primary Elements

Equaio o e imay comoes o e eemes i Equaio (15 yie

imay mai eeme 1-1

773 = 91 (18)

imay mai eeme 1-

s173 c9 sC3 —s9 cC3 = t9 (19

imay mai eeme 1-3

s713 c9 cC3 + 3 = 9 (

imay mai eeme -1

c773 =—casq (1

imay mai eeme -

s/3 59 3 + ce c-3 = (

imay mai eeme -3

s773 s9 cC3 — 9 3 = + l (3

imay mai eeme 3-1

— r = 1 (24)

imay mai eeme 3-

c773 3 = — — 1 (5

and imay mai eeme 3 - 3

c773 c-3 = l — 91 ( 3

3.5.3 Dual Elements

Equation of the dual components of the elements of Equation (15) yield dual matrix element 1-1:

— 773 — (27) dual matrix element 1-2:

773 C = — 1 + (28) dual matrix element 1-3:

173 -3 = 91 (29) dual matrix element 2-1:

r 713 = — 1 91 (30) dual matrix element 2-2:

— r c773 4-3 =— l 1 — — l (31) dual matrix element 2-3:

— r 773 " 3 = 1 (32) dual matrix element 3-1:

r13 r r3 = 1 + (33) dual matrix element 3-2:

773 (r C 3 — 9 -3 +r 3 + 91 3 (34) =— l +t (91 + and dual matrix element 3-3:

r3 r -3 h 9 3 — r C3 — C3 = l 1 (3 5)

3.5.4 Derivation of the Solution in its Final Form

Using equation (19):

- C 3 — (36)

Using equation (22):

n3 =- l O (37)

Using equations (34) and (35), rotation angle 773 can be determined in its proper quadrant.

Substituting equation (34) into equation (16) gives:

— l O — O O

O case, (3 8) O

This reveals angle as per its proper quadrant.

Substituting equation (34) into equation (23) gives:

- 91 O

= ( + (39) 3

Substituting equation (34) into equation (24) gives:

sOI -= - O O

(l- 1 C3 (40)

Rotation -3 can now be determined in its proper quadrant using equations (37) and (38). 5

Using equation (27):

C SO1 S - (41) C713 C-3

Using equation (30):

—l 4 4 i +l 4 4 r (42) r3"

Using equation (25):

S S773 S4 (43) 51

These equations represent a full solution for the displacements and orientations of the

swash-plate mechanism's links. CHAPTER 4

RESULTS

To represent the kinematic behavior of the swash-plate mechanism according to the solution developed in chapter 3, the variation of the output variable s4 with respect to the input variable 9/ will be shown and explained first. Secondly, the variation of the intermediate variable will be considered. Thirdly, the intermediate variables r and s2 will be described and explained. Finally, the intermediate variables 4: 3 and 773 will each be described, explained and graphed with respect to the input angle 9/ .

Several standard dimensions were used in the example of the results. These include a frame offset c of 1" and a standard frame twist angle a of 180°. To show the displacement behavior of the swash-plate mechanism, the equations (34) through (41) derived in chapter 3 were incorporated in a FORTRAN computer program called

"SWASHP" (see Appendix A). The step increment of 10° was used for the input angle

The input for the first data set is shown in Table 2 and the output of the program

"SWASHP" with a swash angle al of 10° and an input angle 0, step increment of 10° can be found in Table 3.

26 7

Table 3 Input for First Data Set (l = 1°

INPUT VARIABLE VALUE DESCRIPTION UNITS

9/ initial value 0° Initial input angle value degrees

8, step increment 10° The step increment at which the program degrees SWASHP will evaluate all intermediate and output variables 1° The swash-plate angle degrees

180° The angle between the input shaft axis of degrees rotation and the output axis of translation c4 1 Perpindicular distance (or offset) between inches the input shaft axis of rotation and the output axis of translation

Table 4 FORTRAN Program "SWASHP" Output for First Data Set (ai = 10°)

EA1 EA EA3 EA3 S S 3 1 19 1 1 3519 117 19511 -1733 91 -3 33 139 191 -373 9399 -31 3 3337 199 1 -5771 3 -1 3313 17 179 -57 7 -1133 5 3135 173 1 -777 79 -1357 3313 19 153 -7939 5 -157 7 93 193913 1351 -9519 3 -159 1511 19 11753 -1 1735 -1735 9 7 19 1 -1153 -1733 1 599 19 17 -1 -1735 -1735 11 9715 193913 175 -9519 -3 -159 1 391 19 1791 -7939 -5 -157 13 957 173 173533 -777 -79 -1357 1 1957 17 1737 -57 -7 -1133 15 917 199 171317 -5771 -3 -1 1 199719 139 175919 -373 -9399 -31 17 19511 117 1719 -1733 -91 -3 1 1 1 17 -1 19 1719 177 1719 1733 -91 3 13 175951 17591 373 -9399 31 1 1537 175191 171317 5771 -3 1 1313 173591 1737 57 -7 1133 3 1335 173557 173533 777 -79 1357 1313 17135 1791 7939 -5 157 5 113 177 175 9519 -3 159 11511 17153 17 1 -1735 1735 7 9 17 1 1153 1733 7995 17153 11753 1 1735 1735 9 9717 177 1351 9519 3 159 3 59177 17135 153 7939 5 157 31 95757 173557 1 777 79 1357 3 3957 1735913 179 57 7 1133 33 919 175191 1 5771 3 1 3 197197 175951 191 373 9399 31 35 9511 177 19511 1733 91 3 3 1 19 1 29 020

4 00

Ir 000

46 —00

--00

—0

--020 0 45 90 80 22 20 315 360 EA (Inpt ttn Anl n r

Figure 11

Output Translation as a Function of Input Rotation (l = 1°

Figure 11 indicates that the output translation is periodic in nature given a constant input shaft rotation. This graph represents the physical oscillation of the piston as a result of input shaft rotation. 3

315 c a a 7

5 o

iao

135

CA - 9

E-■ 5

5 9 135 1 5 7 315 3 THETA1 (Input Rotation Angle in Degrees)

igue 1

Iemeiae Age a Iu oaio l ( = 1°

Figure 12 represents the relationship between input rotation 8, and intermediate rotation While not immediately apparent, upon close examination, it is observed that the relationship is not linear. This will be accentuated with the second data set with a larger value for l 31

1.0 (asaio i ices

igue 13

Iemeiae asaios a r (l = 1°

igue 13 represents the path of the coupler traced on the face of the swash plate.

Upon close examination, it is observed that the path is elliptical. This shape can be accentuated with a larger swash angle l This accentuated shape will be shown in the second and following data sets. 3

-• 7 1=

a o

1 a

-1 co 0

E-

I I I 4 0 135 1 5 7 315 3 THETA1 (Input Rotation Angle in Degrees)

r 4

Iemeiae Age 77 3 a Iu Age 1 ( = 1

The physical meaning of the curve graphed in Figure 14 is a representation of the

forward rocking motion or "pitch" of the spherical joint given the input rotation t9 A

single frequency fluctuation of small amplitude is observed.

tA 20 1=1

180 r

90 O

45 90 135 180 225 270 315 360 THETA.1 (Input Rotation Angle in Degrees)

Figure 15

Intermediate Variable C3 and Input Variable 01 (a1 = 10°)

The side-to-side rotation or "yaw" of the spherical joint on the end of the follower

is described by angle 47. It is noted that the behavior of angle 4- similar in nature to the behavior of angle 773 The functions that describe the single frequency fluctuation evident in these two variables are identical except for a 90° shift. This corresponds to the angle between the coordinate axis from which each is measured. 34

17 17 175 180 185 190 ETA3 (Intermediate Angle in Degrees)

igue 1

Iemeiae aiae y3 a Iemeiae aiae 773 (a = 10°)

Figure 16 is a graph describing the pattern traced by the center of the coupler pad in two-dimensional space as the swash-plate rotates. It is a perfect circle as expected. 35

For the second and third data sets, the swash angle l is varied. In the second data set it is set at 45° to represent an extreme value thus accentuating the behavior of the intermediate variables for observation. In the third data set, is set 25° to show a trend of l between 10° and 45°. The results of all three examples are combined in figures 17-

Table 5 Input for Second Data Set (l = 45°)

INPUT VARIABLE VALUE DESCRIPTION UNITS

9 initial value 0° Initial input angle value degrees

9 step increment 10° The step increment at which the program degrees SWASHP will evaluate all intermediate and output variables 45° The swash-plate angle degrees

180° The angle between the input shaft axis of degrees rotation and the output axis of translation C4 1 Perpindicular distance (or offset) between inches the input shaft axis of roation and the output axis of translation 3

ae OA ogam "SWAS" Ouu o Seco aa Se ( = 5°

EAI EA EAS EA3 S S 3 1 5 1 1 3599 1753 51 -55 91 -1735 35573 1939955 319 -39 9399 -3 3 33779 7 93 -7711 3 -5 3931 73 17537 -99 7 -79 5 319793 1797 173 -1335 79 -7 39315 1771 55 -17 5 -3 7 9733 11 1917 -1393 3 -9399 13 19511 -13973 1735 -91 9 7 5 1 -111 -1 1 55991 13 1719 -13973 -1735 -91 11 737 11 11113 -1393 -3 -9399 1 375 17713 15335 -175 -5 -3 13 1 1797 177 -1335 -79 -7 1 11 73 153 -99 -7 -79 15 77 7 1391 -7711 -3 -5 1 193 1939955 137 -39 -9399 -3 17 17171 17531 1353 -55 -91 -1735 1 1 1 135 -1 19 1799 1797 13535 55 -91 1735 15573 1 137 39 -9399 3 1 15779 15995 1391 7711 -3 5 1931 159 153 99 -7 79 3 139793 173 177 1335 -79 7 19315 13 15339 17 -5 3 5 11733 13359 11113 1393 -3 9399 1 135 1719 13973 -1735 91 7 93 135 1 111 1 75999 135 1951 13973 1735 91 9 7373 1335 1917 1393 3 9399

3 5751 - 137 55 175 5 3 31 177 17 173 1335 79 7 3 39 15959 17537 99 7 79 33 7 15995 93 7711 3 5 3 1379 15 319 39 9399 3 35 71711 1799 51 55 91 1735 3 3 1 5 1 37

Table 7 Input for Third Data Set (l = 25°)

INPUT VARIABLE VALUE DESCRIPTION UNITS

9/ initial value 00 Initial input angle value degrees

91 step increment 10° The step increment at which the program degrees SWASHP will evaluate all intermediate and output variables l 25° The swash-plate angle degrees

bl 8 FORTRAN Program "SWASHP" Output for Third Data Set (j = 25°)

EA1 EA EAS EA3 S S 3 1 5 1 1 3595 15 57 -191 91 -97 31739 1319 3 -3773 9399 -1599 3 33379 191991 199 -5519 3 -3315 3777 19577 19957 -79 7 -997 5 31799 1995 195 -5 79 -3571 39 19 19313 -95555 5 -33 7 91 3399 191 -13 3 -319 197 59 193 -1 1735 -59 9 7 5 1 -1133 -31 1 599 59 17537 -1 -1735 -59 11 119 3399 1793 -13 -3 -319 1 3751 19 1757 -95555 -5 -33 13 75 1995 1331 -5 -79 -3571 1 175 19577 13 -79 -7 -997 15 71 191991 159 -5519 -3 -3315 1 195 1319 15337 -3773 -9399 -1599 17 19795 1 155333 -191 -91 -97 1 1 1 155 -1 19 1795 1757915 155333 191 -91 97 11739 17191 15337 3773 -9399 1599 1 15379 179 159 5519 -3 3315 1777 1373 13 79 -7 997 3 13799 11115 13311 5 -79 3571 19 15531 1757 95555 -5 33 5 111 151 1793 13 -3 319 1197 1555 17537 1 -1735 59 7 9 155 1 1133 31 79937 1555 193 1 1735 59 9 119 151 191 13 3 319 3 57515 15531 19313 95555 5 33 31 7515 11115 195 5 79 3571 3 37539 1373 19957 79 7 997 33 711 179 199 5519 3 3315 3 151 17191 3 3773 9399 1599 35 97951 175791 57 191 91 97 3 1 5 1 39

1,0 = tO Oegees o t = 5 egees = 5 egees

—1.0 45 90 135 1& 225 270 315 360 THETA 1 (Iu oaio Age i egees

Figure 17

Output Translation s and Input Rotation 0/

c13 315

11 7

7Q-5 -c

KS so •

-1 135 ci 9 E

5 9 135 1 5 7 315 35 EA 1 (Iu oaio Age i egees

igue 1

Iemeiae Age a Iu oaio 1

-1 1 (asaio i Ices

Figure 19

Iemeiae asaios s2 a r2

360 -- a I 1 egees a = 5 eees ctl = 5 dr 300 z;lio

240

180 5 E t 120 cr) El 6Q

0 0 45 90 135 180 225 270 315 360 THETA 1 (Iu oaio Age i egees

Figure

Intermediate Variable 7/3 and Input Variable 0/ 3

a 3 a° a

--g 1

1 c

EI C-

5 9 135 1 5 7 315 3 EA1 (Iu oaio Age i egees

igue 1

Iemeiae aiae a Iu aiae O 44

—9

--g - 1 -

1 -

r4E-I 1 -sq

1 1 1 1 1 240 EAS (Iemeiae Age i egees

r 22

Intermediate Variable 7/3 and Intermediate Variable 4 -3 CHAPTER 5

DISCUSSION AND CONCLUSION

In the previous chapters, a study to analyze the kinematic behavior of the swash-plate mechanism was presented. Upon examination of the data provided in Chapter 4, the solution presented seemed to be exhaustive and complete. The position and/or orientation of any link in the swash-plate mechanism was found given any input angle. This solution can be expanded to describe the position and/or orientation of any number of followers simply by adding the corresponding phase angle describing the location of that follower with respect to the reference follower at which the solution originated.

Furthermore, it was proven that the general transformation matrix solutions for the ebene and spherical joints provided accurate results in the solution of the swash-plate mechanism.

In figure 17, the trend over the single frequency produced in 360 of rotation is a non-linear increase in magnitude. This can be confirmed by checking the data. Figure 18 is perhaps the most interesting graph in chapter 4. It shows how input angle 1 and intermediate angle are related with a double frequency. The variation in magnitude is observed as non-linear. Figure 19 is perhaps the most dramatic example of the nonlinear relationship between the data sets. It is obvious that the magnitude of deviation from a circle for l = 10° is not just under a fourth of the magnitude of deviation from a circle when l = 45°. The single frequency relationships of intermediate angles 773 and 4-3 and

45 46

input angle 91 in figures 20 and 21 show the direct relationship between the swash-plate angle and magnitude of those intermediate variables. The relationship is 1:1 with a 90° phase shift because of the axis used to measure the angles being different. In figure 22, the trend for intermediate variables 4'3 and 773 is to increase in magnitude proportionally to the swash-plate angle al . This trend is shown by the larger circles corresponding to the larger swash-plate angles.

The figures in chapter 4 showing the relationships of the intermediate, input and output variables indicate that there may be some new information available to machine designers. This solution could assist in optimizing the swash-plate mechanism as it is configured in the automobile air-conditioning compressor. For example, figures 12 and 18 show a double frequency relationship between 9 1 and This could explain a critical vibration that might affect the fatigue life of the mechanism. Cursory observation indicates that the critical area affected by this double frequency vibration is the junction between the input rotation shaft and the swash-plate plane.

Future work with this solution could be extensive. The equations derived in chapter 3 offer much more potential information than was presented in chapter 4. For example, the swash angle a, could be held at zero and the angle between the input axis and the piston a could be varied. This would show a different configuration for the mechanism that could be compared to the original configuration already presented. A combination of varying the swash-plate angle al , the angle between the input shaft and the output follower/piston assembly a could also be studied and used for a possible mechanism configuration optimization in the future. Another possibility for future work is 7 a dynamic analysis of the swash-plate mechanism. By taking the first and second derivatives, the velocity and change in momentum of each link in the mechanism could be studied in detail.

In conclusion, the aforementioned work presents a reasonable, accurate and complete solution for the swash-plate mechanism. The solution is consistent with previous geometric solutions in its explanation of the extreme positions of the swash plate and yet it provides a simple way to understand the motion of the mechanism more

completely. This information could prove useful to mechanism designers interested in

optimizing the behavior of the swash-plate mechanism for any of its current applications or

possibly for any new application being explored. APPENDIX A

FORTRAN Computer Program "SWASHP"

OGAM SWAS

C is ogam is esige o soe e equaios ouce C y a asomaio mai aaysis o e swas-ae mecaism C Aso a aa ie wi e ouce o oig e ucio escie y e as equaio

AAMEE (1=311595359

CAACE 1 ASWE CAACE 1 IAM

IEGE KK

EA A3C A3C EA AS AC EA3S EA3C EA3S EA3C EA A1AA1 AAA C S EA EA1 EA EA3 EA3 S

C is saeme is askig o e swas-age as eie y e use C

I Ee e swas-age i egees EA A1

C A1 is coee o aias a cage io is ue ame "AA1" C

AA1= (A1/1 I

C Swas-ae ose is ow eee is aue is i ices C

I Ee e ose o e iso om e cee o e + swas-ae ais i ices EA C

C e age ewee e iso ais a e ais o e swas-ae is eee ee C

I Age ewee iso a swas-ae ais i egees EAA

C AA is eie oug e coesio o A o aias C

AA= (A/1 I

C is saeme iiiaies e age EA1 C

EA1 = (1/1 I

C is saeme eemies e iu se ( o aa ois ewee C a 3 egees o oaio C I Ee iu age se iceme EA K

C is saeme asks e use i a ouu ie sou e ceae C I aso oms e use o e ame o e ouu ie C I Ouu ie? y/ (Suou aswe wi aeesis? EA ASWE

C is saeme is use o oe e ouu ie i wic e C aa o e ga is ace

I(ASWE EQ y E I ame o Ouu ie? (I aeesis ease EA IAM OE(UI=IE=IAMSAUS=EW WIE(EA1EAEA3 EA3 S S EI

I EA1 EA EA3 1 1 EA3 SS

OMA(7(A1 5 OMA(7(151

C is oo is use o acuay eom e cacuaios o e ouu omuas C o e swas-ae asom

O 1 K = 3K

AS=-COS (AA1 SI (EA1

AC= COS(EA1

EA= AA(ASAC

I (EA E

EA = EA + I

EA3S=-SI (AA1 SI (EA1

EA3C= COS (AA1 SI ( EA1 /SI ( EA

EA3= AA ( EA3S EA3C 50

I (EA3 E

EA3 = EA3 + I

EA3S= (SI (EA (COS (AA1 SI (AA + SI (AA1 COS (AA COS (EA / (COS (AA1 SI (EA1

A3C=-SI (EA (COS (AA COS (AA - + SI (AA SI (AA COS (EA1

A3C= COS (AA1 SI (EA1

EA3C=A3C/ A3C

EA3= AA ( EA3S EA3C

I (EA3 E

EA3 EA3 + I

S= (CCOS (AA SI (EA1 / (COS (EA3 COS (EA3

= (C (SI (AA1 SI (AA - COS (AA1 COS (AA COS (EA1 / (COS (EA3 COS ( EA3

S= SI (EA3 / (SI (AA SI (AA - + COS (AA1 COS (AA COS (EA1

I (ASWE EQ y E

C is saeme is use o wie e ogams ouu o e C aa ies o use ae y e gaig ackage

WIE (5 EA1 (1/I EA (1/I EA3(1/I EA3(1/I SS

I 5 EA1(1/I EA(1/IEA3(1/I + EA3(1/I SS

EA1=EA1+(K/1I

1 COIUE 51

C E e ogam

E IIOGAY

1) Maki, E. R., and DeHart, A. 0 "A New Look at Swash-Plate Drive Mechanisms," SAE Transactions, Vol. 80, 1971, pp. 2712-2722

2) R. Galin, and C. 0. Harris, "Balance a Swashplate Mechanism Used in an Automotive Air Conditioning System Compressor." General Motors Engineering Journal, Vol. 10, No. 4 (Fourth Quarter 1963), pp.48-49

3) R. Galin, and C. 0. Harris, "Balance a Swashplate Mechanism Used in an Automotive Air Conditioning System Compressor." General Motors Engineering Journal, Vol. 11, No. 1 (First Quarter 1964), pp.53-56

4) K.A. edge, and J. Darling, "The Pumping Dynamics of Swash Plate Piston Pumps." ASME Journal of Dynamic Systems, Measurement and Control, Vol 111, June 1989, pp. 307-312

5) Inoue, Kiyoshi Kayaba, Industry Co. Ltd., Kanagawa, Jpn., "Study on operating moment of a swash plate type axial piston pump- first report effects of dynamic characteristics of a swash plate.," Journal of Fluid Control, Delbridge Publishing Co., Vol. 22 No. 1, Stanford, CA, 1994, pp. 7-29

6) John Weibel, Jr., and Raymond N. Mantey, "The Engineering Develoment of a Compressor for Automotive Air Conditioning Systems." General Motors Engineering Journal, Vol. 10, No. 4 (Fourth Quarter 1963), pp.48-49

7) F. Freudenstein, E.R. Maki, "Kinematic Structure of Mechanisms for Fixed and Variable-Stroke Axial-Piston Reciprocating Machines", ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol 106, Sept 1984 pp.355-364

8) A.T. Yang "Displacement Analysis of Spatial Five-Link Mechanisms Using (3x3) Matrices with Dual-Number Elements." ASME Journal of Engineering for Industry. Feb 1969 pp. 152-157

9) A.T. Yang, and F. Freudenstein, "Application of Dual-Number Quaternion Algebra to the Analysis of Spatial Mechanisms" ASME Journal of Applied Mechanics, June 1964, pp. 300-308

10) A.T. Yang, Basic Questions of Design Theory: Calculus of Screws, Editor: William R. Spillers., North Holland Publishing Company, NY, 1974 pp. 264-281

5 53

11) A.G.M. Mitchell, "Mechanism for the Interconversion of Reciprocating and Rotary Motion." Patent 1,409,057, March 7, 1922 (Filed June 11, 1918 - Renewed April 28,1921).

12) Bottema, O., and B. Roth. 1979. Theoretical Kinematics, North Holland Publishing Co., NY, 1979, ISBN 0-444-85124-0

13) J. Denavit, R.S. Hartenberg, "A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices", ASME Journal of Applied Mechanics, Evanston, Ill, June 1955, pp. 215-221

14) Harris, R. M., Edge, K. A., Tilley, D. G., "Predicting the Behavior of Slipper Pads in Swashplate-Type Axial Piston Pumps", ASME (paper) Journal of Dynamic Systems Measurement and Control, New Orleans, LA, March 1996, pp. 41-47