Gruebler's Equation

Faculty of Engineering Department of Mechanical Engineering EMEC 3302, Theory of Machines

Instructor: Dr. Anwar Abu-Zarifa Office: IT Building, Room: I413 Tel: 2821 eMail: [email protected] Website: http://site.iugaza.edu.ps/abuzarifa Office Hrs: see my website

SAT 09:30 – 11:00 Q412

MON 09:30 – 11:00 Q412

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 2 Text Book: R. L. Norton, Design of Machinery “An Introduction to the Synthesis and Analysis of Mechanisms and Machines”, McGraw Hill Higher Education; 3rd edition

Reference Books:

. John J. Uicker, Gordon R. Pennock, Joseph E. Shigley, Theory of Machines and Mechanisms . R.S. Khurmi, J.K. Gupta,Theory of Machines . Thomas Bevan, The Theory of Machines . The Theory of Machines by Robert Ferrier McKay . Engineering Drawing And Design, Jensen ect., McGraw-Hill Science, 7th Edition, 2007 . Mechanical Design of Machine Elements and Machines, Collins ect., Wiley, 2 Edition, 2009

Course Description:

The course provides students with instruction in the fundamentals of theory of machines. The Theory of Machines and Mechanisms provides the foundation for the study of displacements, velocities, accelerations, and static and dynamic forces required for the proper design of mechanical linkages, cams, and geared systems.

Students combine theory, graphical and analytical skills to understand the Engineering Design. Upon successful completion of the course, the student will be able:

. To develop the ability to analyze and understand the dynamic (position, velocity, acceleration, force and torque) characteristics of mechanisms such as linkages and cams. . To develop the ability to systematically design and optimize mechanisms to perform a specified task. . To increase the ability of students to effectively present written, oral, and graphical solutions to design problems. . To increase the ability of students to work cooperatively on teams in the development of mechanism designs.

The subject Theory of Machines may be defined as that branch of Engineering-science, which deals with the study of relative motion between the various parts of a machine, and forces which act on them. The knowledge of this subject is very essential for an engineer in designing the various parts of a machine.

Kinematics: The study of motion without regard to forces

More particularly, kinematics is the study of position, displacement, rotation, speed, velocity, and acceleration.

A mechanism: is a device that transforms motion to some desirable pattern and typically develops very low forces and transmits little power.

A machine: typically contains mechanisms that are designed to provide significant forces and transmit significant power.

Any machine or device that moves contains one or more kinematic elements such As linkages, … gears…. belts and chains.

Bicycle is a simple example of a kinematic system that contains a chain drive to provide Torque.

the transmission is full of gears….

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 10 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 11 Chapter 2 DEGREES OF FREEDOM (MOBILITY)

• DOF: Number of independent parameters (measurements) needed to uniquely define position of a system in space at any instant of time. • A mechanical system’s mobility (M) can be classified according to the number of degrees of freedom (DOF). • DOF is defined with respect to a selected frame of reference (ground).

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 13  Rigid body in a plane has 3 DOF: x,y,z  Rigid body in 3D-space has 6 DOF, 3 translations & 3 rotations three lengths (x, y, z), plus three angles (θ, φ, ρ).  The pencil in these examples represents a rigid body, or link.

• Pure rotation: the body possesses one point (center of rotation) that has no motion with respect to the “stationary” frame of reference. All other points move in circular arcs. • Pure translation: all points on the body describe parallel (curvilinear or rectilinear) paths. • Complex motion: a simultaneous combination of rotation and translation.

Linkage design: . Linkages are the basic building blocks of all mechanisms . All common forms of mechanisms (cams, gears, belts, chains) are in fact variations on a common theme of linkages. • Linkages are made up of links and joints.

• Links: rigid member having nodes • Node: attachment points

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 18 1. Binary link: 2 nodes 2. Ternary link: 3 nodes 3. Quaternary link: 4 nodes Joint: connection between two or more links (at their nodes) which allows motion; (Joints also called kinematic pairs)

Joints can be classified in several ways: 1.By the type of contact between the elements, line, point, or surface. 2.By the number of degrees of freedom allowed at the joint. 3.By the type of physical closure of the joint: either force or form closed. 4.By the number of links joined (order of the joint).

A more useful means to classify joints (pairs) is by the number of degrees of freedom that they allow between the two elements joined.

• Type of contact: line, point, surface • Number of DOF: full joint=1DOF, half joint=2DOF • Form closed (closed by geometry) or Force closed (needs an external force to keep it closed) • Joint order

Joint order = number of links-1

The six lower pairs

Form closed (closed by geometry) or Force closed

 A joint (also called kinematic pair) is a connection between two or more links at their nodes, which may allow motion between the links.

 A lower pair is a joint with surface contact;ahigher pair is a joint with point or line contact.

 A full joint has one degree of freedom; a half joint has two degrees of freedom. Full joints are lower pairs; half-joints are higher pairs and allow both rotation and translation (roll-slide).

 A form-closed joint is one in which the links are kept together form by its geometry;aforce-closed joint requires some external force to keep the links together.

 Joint order is the number of links joined minus one (e.g. 1st order means two links).

• Kinematic chain: links joined together for motion • Mechanism: grounded kinematic chain • Machine: mechanism designed to do work • Link classification:

. Ground: any link or links that are fixed, nonmoving with respect to the reference frame . Crank: pivoted to ground, makes complete revolutions . Rocker: pivoted to ground, has oscillatory motion . Coupler: link has complex motion, not attached to ground

Elements: 0: Ground (Casing, Frame) 1: Rocker 2: Coupler 3: Crank

. When studying mechanisms it is very helpful to establish a fixed reference frame by assigning one of the links as “ground”.

. The motion of all other links are described with respect to the ground link.

. For example, a fourbar mechanism often looks like a 3-bar mechanism, where the first “bar” is simply the ground link.

Two unconnected links: 6 DOF (each link has 3 DOF)

When connected by a full joint: 4 DOF (each full joint eliminates 2 DOF)

Gruebler’s equation for planar mechanisms: DOF = 3L-2J-3G Where: L: number of links J: number of full joints G: number of grounded links

• Gruebler’s equation for planar mechanisms M= 3L-2J-3G • Where M = degree of freedom or mobility L = number of links J = number of full joints (half joints count as 0.5) G = number of grounded links =1

M  312LJ

Kutzbach’s modification of Gruebler’s equation

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 31 The Cylindrical (cylindric) joint - two degrees of freedom It permits both angular rotation and an independent sliding motion (C joint)

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 32 The Spherical (spheric) - Three degree of freedom It permits rotational motion about all three axes, a ball-and-socket joint (S joint)

Gruebler’s equation can be used to L = 2 determine the mobility of planar J = 1 mechanisms. G = 1 Link 1 DOF = 1 3 DOF Gruebler’s Equation

DOF = mobility L = number of links 1 DOF J = number of revolute joints or prismatic joints G = number of grounded links

DOF (M) = 3*L – 2* J – 3 *G Link 2 = 3 (L-1) – 2 * J 3 DOF

This example applies Gruebler’s equation to the determine the mobility of a vise grip plier. Each revolute joint removes two DOF. The screw joint 4 1 5 removes two DOF. 1 3 L = 5 4 2 J = 4 (revolute) 3 2 J = 1 (screw) G = 1 (your hand)

DOF = 3*5 - 2*5 -1*3 = 2

The mobility of the plier is two. Link 3 can be moved relative link1 when you squeeze your hand and the jaw opening is controlled by rotating link 5.

Slider-Crank Mechanism

Joint Formed between links Joint type As designated in the figure, there are four Number Revolute links link 1, link 2, link 3 and link 4. Link 1 1 Link 4 and Link 1 acts as a crank. Link 2 acts as (or Pin) Revolute connecting link, link 3 is the slider and 2 Link 1 and Link 2 link 4 is ground. (or Pin) Revolute 3 Link 2 and Link 3 (or Pin) Translatio 4 Link 3 and Link 4 nal or (Slider)

. If DOF > 0, the assembly of links is a mechanism and will exhibit relative motion

. If DOF = 0, the assembly of links is a structure and no motion is possible.

. If DOF < 0,then the assembly is a preloaded structure, no motion is possible, and in general stresses are present.

• Greubler criterion does not include geometry, so it can give wrong prediction • We must use inspection

L=5 J=6 E-quintet G=1 M=3*5-2*6-3*1=0

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 40 Rolling cylinders even without slip (The joint between the two wheels can be postulated to allow no slip, provided that sufficient friction is available) is an example in which the ground link is exactly the same length as the sum of two other links. If no slip occurs, then this is a one-freedom, or full, joint that allows only relative angular motion (Δθ) between the wheels. With that assumption, there are 3 links and 3 full joints, The equation predicts DOF =0(L=3,

J1=3), but the mechanism has DOF =1.

Others paradoxes exist, so the designer must not apply the equation blindly.

• Leonardo da Vinci (1452, 1519), Codex Madrid I. • Industrial Revolution was the boom age of linkages: cloth making, power conversion, speed regulation, mechanical computation, typewriting and machining

. In many applications linkages have been replaced by electronics. . Still linkages can have a cost advantage over electronic solutions: Couple different outputs by a mechanism rather than using one motor per output and electronics to achieve the coupling. . Current applications: Sports Equipment, Automotive (HVAC modules), Precision Machinery (Compliant Mechanisms), Medical Devices

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 44 Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement.

Transmission System

Gear Linkage consistent translation Inconsistent translation linear transfer function non-linear transfer function

consistent translation linear transfer function

The pushing movement of the piston (crank mechanism) is transferred into a swinging movement of the shovel.

 Twobar has -1 degrees of freedom (preloads structure)  Threebar has 0 degrees of freedom (structure)  Fourbar has 1 degree of freedom . The fourbar linkage is the simplest possible pin-jointed mechanism for single degree of freedom controlled motion

One link is grounded in each case

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 49 . The fourbar linkage is the simplest possible pin-jointed mechanism for controlled motion with one degree of freedom.

. Changing the relative lengths of the links can create a wide variety of motions.

• Ground Link • Links pivoted to ground:

– Crank B – Rocker Coupler A Rocker • Coupler Crank Link 1, length d Ground Link

Pivot 02 Pivot 04

• Provide more complex motion • See Watt’s sixbar and Stephenson’s sixbar mechanisms in the textbook

. Grashof condition predicts behavior of linkage based only on length of links . S=length of shortest link . L=length of longest link . P,Q=length of two remaining links  If S+L ≤ P+Q the linkage is Grashof :at least one link is capable of making a complete revolution  Otherwise the linkage is non-Grashof : no link is capable of making a complete revolution

Rotatability is defined as the ability of at least one link in a kinematic chain to make a full revolution with respect to the other links and defines the chain as Class I, II or III.

Revolvability refers to a specific link in a chain and indicates that it is one of the links that can rotate.

I. If S + L < P + Q (Class I), the linkage is Grashof and at least one link will be capable of making a full revolution with respect to ground.

II. If S + L > P + Q (Class II), the linkage is non-Grashof and no link will be capable of making a full revolution with respect to any other link.

III. If S + L = P + Q (Class III), the linkage is special-case Grashof and although at least one link will be capable of making a full revolution.

The crank-slider (right) is a transformation of the fourbar crank rocker, by replacing the revolute joint at the rocker pivot by a joint, maintaining the same one degree of freedom.

. A cam follower is a mechanism that appears to have only two moving links (apart from ground), but it has 1 DOF. . It has a fourbar equivalent if the coupler (Link 3) is viewed as a link of variable length.

. There are many factors that need to be considered to create good- quality designs. . The choice of joint type can have a significant effect on the ability to provide good, clean lubrication over the lifetime of the machine.

Pin Joints versus Sliders and Half Joints

A. Pin Joint  Easy to lubricate ( with hydrodynamic lubrication)  Can use relatively inexpensive bearings B. Slider  Requires carefully machined straight slot or rod  Custom made bearings  Lubrication is difficult to maintain

pin joint is the clear winner

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 57 Sleeve or journal bearing, the geometry of pin-in-hole traps a lubricant film within its annular interface by capillary action and promotes a condition called hydrodynamic lubrication in which the parts are separated by a thin film of lubricant .

Seals can easily be provided at the ends of the hole, wrapped around the pin, to prevent loss of the lubricant.

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 58 Relatively inexpensive ball and roller bearings are commercially available in a large variety of sizes for revolute joints.

Their rolling elements provide low-friction operation and good dimensional control.

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 59 For revolute joints pivoted to ground, several commercially available bearing types, Pillow blocks and flange-mount bearings.

. Unless manually operated, a mechanism will require some type of driver device to provide the input motion and energy.

. A motor is the logical choice to create the input.

. Motors come in a wide variety of types. The most common energy source for a motor is electricity, but compressed air and pressurized hydraulic fluid are also used to power air and hydraulic motors.

Electrical Motors  AC  DC  Servo  Stepping

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 61 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 62 Chapter 4 Design of Linkage Systems

1. Synthesis

2. Analysis

Design a mechanism to obtain a specified motion or force.

•Type Synthesis given the required performance, what type of mechanism is suitable? Linkages, gears, cam and follower, belt and pulley and chain and sprocket.

•Number Synthesis How many links should the mechanism have? How many degrees of freedom are desired?

•Dimensional Synthesis deals with determining the length of all links, gear diameter, cam profile.

• The creation of potential solutions in the absence of a well-defined algorithm which configures or predicts the solution and also judge its quality. • Several tools and techniques exist to assist you in this process. The traditional tool is the drafting board, on which you layout, to scale, multiple orthographic views of the design, and investigate its motions by drawing arcs, showing multiple positions, and using transparent, movable overlays. • Commercially available programs such as SolidWork and Working Model allow rapid analysis of a proposed mechanical design. The process then becomes one of qualitative design by successive analysis which is really an iteration between synthesis and analysis. Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 66 TYPE SYNTHESIS

•Thedefinition of the proper type of mechanism best suited to the problem and is a form of qualitative synthesis. • This is perhaps the most difficult task for the student as it requires some experience and knowledge of the various types of mechanisms which exist and which also may be feasible from a performance and manufacturing standpoint. • An engineer can do, with one dollar, what any fool can do for ten dollars. Cost is always an important constraint in engineering design.

•Thedetermination of the proportions (lengths) of the links necessary to accomplish the desired motions and can be a form of quantitative synthesis if an algorithm is defined for the particular problem, but can also be a form of qualitative synthesis if there are more variables than equations.

CAD program  SolidWorks

• Once a potential solution is found, it must be evaluated for its quality. There are many criteria which may be applied. However, one does not want to expend a great deal of time analyzing, in great detail, adesignwhichcanbeshowntobeinadequate by some simple and quick evaluations.

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 70 TOGGLE: One important test consist in to check that the linkage can in fact reach all of the specified design positions without encountering a limit or toggle position, also called a stationary configuration.

The toggle positions are determined by the colinearity of two of the moving links.

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 71  TRANSMISSION ANGLE: The transmission angle μ is defined as the angle between the output link and the coupler.  It is usually taken as the absolute value of the acute angle of the pair of angles at the intersection of the two links.

. Radial component only increases friction at pivot O4.

. Tangential (normal to Link 4) produces torque. – μ = 90o is optimal. – In design, keep μ > 40o

To promote smooth running and good force transmission.

Ideally, as close to 90° as possible

Excel or other program

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 Graphical Synthesis – Motion Generation Mechanism Two positions, coupler as the output B1 1. Draw the link AB in its two desired positions, A1B1 and A2B2 A A1 2 2. Connect A1 to A2 and B1 to B2.

3. Draw two lines perpendicular to B2

A1 A2 and B1B2 at the midpoint O2 (midnormals). O 4. Select two fixed pivot points, O2 4 and O4, anywhere on the two midnormals. 5. Measure the length of all links,

O2A = link 2, AB = link 3,

O4B = link 4 and O2 O4 = link 1

Same procedure as for two positions. A2 1. Draw the link AB in three desired A B1 positions. 1

2. Draw the midnormals to A1A2 and A O 3 O A2A3, the intersection locates the 2 4 fixed pivot point O2. Same for point B to obtain second pivot point O4. B2 3. Check the accuracy of the mechanism, Grashof condition and the transmission angle.

B3 4. Change the second position of link AB to vary the locations of the fixed points

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 Graphical Synthesis – Motion Generation Mechanism Two positions Grashof 4-Bar mechanism with rocker as the output

1. Draw the link CD in its two desired D1 positions, C1D1 and C2D2

2. Connect C1 to C2 and D1 to D2 and C C1 2 draw two midnormals to C1C2 and D1D2 B B D2 3. The intersection of the two A2 1 2 midnormals is the fixed pivot point O2

O4.

4. Select point B1 anywhere on link O2A = B1B2 / 2 O4 O4C1 and locate B2 so O4B1= O4B2

5. Connect B1 to B2 and extend. Select any location on this line for fixed

pivot point O2.

6. Draw a circle with radius B1 B2 / 2, 7. Measure the length of all links, O2A = link point A is the intersection of the 2, AB = link 3, O4CD = link 4 and O2 O4 = circle with the B1 B2 extension. link 1

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 84 Coupler Curves

 A coupler in a linkage in general has complex motion and provides the greatest variety of paths that can be traced.  The Hrones and Nelson Atlas of Fourbar Coupler Curves is a good reference

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 85 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 86 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 87 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 88 Chapter 5 Velocity Analysis

Definitions

Position of Point P

 i RPA  pe

Velocity of Point P    V  V R PA P PA j d j  pje  pje Link in pure rotation dt

RPA as a complex number in polar form P is the scalar length J is the complex operator (constant)

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 91 Imaginary Vector r can be written as: rei  rcos i sin  r cos Euler's formula Multiplying by i gives: i ire r sin  i cos  r r sin  Real Multiplying by i rotates a vector 90° r sin r cos

Given 2. Find 3 and 4

Numerical Example

Position Analysis

• Acceleration is the rate of change of velocity with respect to time.

   Linear acceleration A  R  V

d   Angular acceleration  dt     

A link PA in pure rotation, pivoted at point A in the xy plane

 i RpeP 

 ii VpeipeiPA      ii2  APAVpeipei PA ()    2 peii  i pe  An At Acceleration has 2 components: normal & tangential PA PA

If point A is moving

 AAAPAPA  2 ii  AA pe  i pe

Given 2. Find 3 and 4

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 103 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 104 Numerical Example

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 105 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 106 Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 107 Human Tolerance for Acceleration

Humans are limited in the level of acceleration they can tolerate.

Machines are limited by the stresses in the parts, e.g. automobile piston 40g’s at idle, 700g’s at highway 2000g’s peak.

g defined as the acceleration due to gravity g= 9.8 m/sec2

. The time derivative of acceleration is called jerk, pulse or shock.

   dA linear jerk: J  R  V  dt   d angular jerk:      dt Controlling and minimizing jerk in machine design is often of interest, especially if low vibration is desired.

roller coaster !!!

High jerk High acceleration

Chains Belts Gears

. Gear analysis is based on rolling cylinders . External gears rotate in opposite directions . Internal gears rotate in same direction

Internal Set: External Set: Opposite Movement in the same Movement direction

. Spur Gears . Bevel Gears . Helical Gears . Worm Gears . Rack and Pinion

Rack and Pinion

. Straight teeth . Noisy since all of the tooth contacts at one time . Low Cost . High efficiency (98-99%)

• Slanted teeth to smooth contact • Axis can be parallel or crossed • Has a thrust force • Efficiency of 96-98% for parallel and 50-90% for crossed

. Based on rolling cones . Bevel gears are most often mounted on shafts that are 90 degrees apart, but can be designed to work at other angles as well.

. High gear ratio . Impossible to back drive . 40-85% efficient

. Generates linear motion . Teeth are straight

Source: ZF Friedrichshafen AG

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 120 Fundamental Law of Gearing . The angular velocity ratio between 2 meshing gears remains constant throughout the mesh

. Angular velocity ratio (mV)

. Torque ratio (mT) is mechanical advantage (mA)

Input Pinion ωout rin din mV      v  ωr ωin rout dout ωinr in ω outr out ωin rout dout mT      ωout rin din

Output The positive or negative sign accounts for internal or Gear external cylinder sets

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 121  r N m  out   in  in Input Pinion V in rout N out  = angular velocity of output gear v  ωr out in = angular velocity of input gear ωinr in ω outr out rin = pitch radius of input gear

rout= pitch radius of output gear N = number of teeth on input gear Output in N = number of teeth on output gear Gear out 1  r N m   in   out  out . This means that torque is exchanged for A m  r N velocity V out in in

. Gear Ratio, mG, is what is commonly referred to when specifying gear trains . It is the magnitude of either the velocity ratio or torque ratio, whichever is > 1. mG  mV

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 123 The common normal of the tooth profiles, at all contact points within the mesh, must always pass through a fixed point on the line of centers, called the pitch point.

. Diametral Pitch (in 1/inch): pd=N/d=/pc . Module (in mm): m=d/N

ωin 2 NN24 ωωin out NN35 2 N ωω2  If N2=N4 and N3=N5 in out 3 4 N3 2 ω N in  3 Reduction ratio ωout ωout N2 5 Will be used to determine the no. of stages given a reduction ratio 2 stages

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 Reverted Compound Train • Input and output shafts are aligned • For reverted gear trains: R +R =R +R Commercial three stage reverted 2 3 4 5 compound train D2+D3=D4+D5

N2+N3=N4+N5 • Gear ratio is ω N N out  2 4 ωin N3 N5

• Conventional gearset has one DOF • If you remove the ground at gear 3, it has two DOF

• It is difficult to access 3

Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 Planetary Gearset with Ring Gear Output • Two inputs (sun and arm) and one output (ring) all on concentric shafts