Theory of Machines & MechanismsMCT 2212

Introduction Rigid Body Mechanics

Statics

Body at rest

Body with constant velocity

Dynamics

Body with accelerated motion

under the action of forces and moments

Theory of Machines and Mechanisms

Mechanisms/Linkages

Parts/ Link

Joints

deals with the determination of the forces and motions of links in machines

Subsystems of machines to facilitate analysis

Introduction

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Links: rigid member having nodes, i.e. attachment points– Binary link: 2 nodes– Ternary link: 3 nodes– Quaternary link: 4 nodes

Links & joints

Joint: connection between two links (at their nodes) which allows motion

Classified by type of contact, number of DOF, type of physical closure, or number of links joined.

kinematic pair : Joints are also known as kinematic pair

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Joint Classification

Type of contact:line/point i.e. higher pair, area/surface i.e. lower pair

Number of DOF: full joint=1DOF, half joint=2DOF

Form closed (closed by geometry) orForce closed (needs an external force to

keep it closed)Joint order = number of links-1

Full Joint: permits one relative motion between adjacent links. All of these

kinematic pairs are referred to as one degree of freedom(DOF) pairs.

Turning pairs allow relative turning motion between two

links., e.g. bearings, pivots, or pin joints.

Rolling pairs allow relative rolling motion between two links,

e.g. pair of friction wheels For a rolling pair, it is assumed that

there is no slippage between the links.

Sliding pairs allow relative sliding motion between two links,

e.g Piston-Cylinder.

Half Joint: allows two relative motions simultaneously

between the adjacent links and referred to as two degree of

freedom pairs.

Sliding pairs

Turning pairs

Half Joint

Kinematic Pairs

Types of joints

Higher Pairs & Lower Pairs

Lower pairs: A kinematic

pair or joint with

surface/area contact.

Higher Pairs & Lower Pairs

Higher pairs: A kinematic pair or joint with point contact or line contact.

Mechanism

Mobility: The mobility of a mechanism is defined as the number of independent parameters required to specify the position of all links of the mechanism. It also specify the number of input/actuators needed to operate the mechanism.

kinematic chain : A kinematic chain is an assembly of links formed by placing kinematic pairs at each of the nodes without specifying the ground link.Kinematic chains may be either open type or close type. Mechanism: It is an assemblage of links and joints with at least one link grounded and interconnected in a way to provide controlled output motions in response to supplied input motions.

DOF of a mechanism: The number of independent ways by which a dynamical system can move without violating any constraint imposed on it. In other words, the minimum number of independent coordinates, which can specify the position of the system completely. It is the number of parameters that determine the state of a physical system.

Link classification:

Ground: fixed wrt. 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

Machine: mechanism designed to do work.

A simple machine may also be considered as a single mechanism.

Figure 1.3(b) shows a free body diagram of the system used to analyze the manual force required to generate sufficient gripping force.

Figure 1.3(a) The tongs can be considered either as a machine or as a mechanism.

Figure: Inline 4-Cylinder EngineFigure: IC Engine Demonstration

Machine & Mechanism

Figure: A Paper/ Card Punching MachineFigure: Quick return mechanism

Machine & Mechanism

Figure: Slider crank mechanism

Figure: Scotch Yoke mechanism

Mechanisms are widely used in applications where precise relative movement and transmission of force are required. Motions may be continuous or intermittent, linear and/or angular.

Mechanisms

Worm_gear Gear-gear Gear-rack

Examples of continuous motion output

Examples of intermittent motion output

Cam –follower

Sewing machine creating a lockstitch using an

oscillating a boat shuttle

Sewing machine creating a lockstitch using Allen B Wilson's rotating hookGeneva

Mechanism

Cam –follower

Ratchet Mechanism

Every mechanism has one stationary base link. All other links may move

relative to the fixed base link. From the same kinematic chain, an inversion os

a mechanism is obtained by making the originally fixed link into a moving

link and selecting an originally moving link to be the fixed link .

Mechanism Inversion

Figure 1.39 Slider crank mechanism and its three inversions

(a) slider crank mechanism (link 1 fixed),

(b) inversion #1 (link 2 fixed),

(c) inversion #2 (link 3 fixed),

(d) inversion #3 (link 4 fixed).

Planar motion is restricted to a plane. For a planar mechanism, the motions of all of its links must take place either in the same plane or in planes that are parallel to one another. The slider crank mechanism and four-bar mechanism are examples of planar mechanisms.

Planar Mechanism

Figure 1.5 Slider crank mechanismFigure 1.7 Slider crank mechanism with offset

Figure 1.8 Four-bar mechanism

The Gruebler’s equation for the mobility, m, of a planar mechanism is given as

n= number of links in the mechanismJ1 =umber of one degree of freedom pairsJ2=umber of two degree of freedom pairs

212)1(3 JJnm

If, m< 0 i.e. “–ve”, Preloaded Structure, may require force to

assemble / Indeterminate problem .

If, m= 0 , Structure.

If, m>0 i.e. “+ve”, Mechanism.

Mobility

18Figure 1.36 Examples of mobility.

If, m< 0 i.e. “–ve”, Preloaded Structure, may require force to

assemble.

If, m= 0 , Structure.

If, m>0 i.e. “+ve”, Mechanism.

Mobility

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Figure 1.36 (Continued)

n=6, j1=7, m=1

n=11, j1=14, j2=1 m=1

n=4, j1=1, j2=1, j3=2 ,m=3

n=5; J1=6; J2=0; m=0

n=5; J1=6; J2=0; m=0 but, m=1.Full Joint, Pure rolling, no sliding

In case of pure rolling,n=3; J1=3; J2=0; m=0

In case of rolling & sliding,n=3; J1=2; J2=1; m=1

Half Joint, rolling & sliding

3

1 11

24

5

1 1 1

2 4

5

3

23

1 1

Mobility Paradoxes

The Gruebler criterion pays no attention to

link sizes or shapes, it can give misleading

results in the face of unique geometric

configurations.

Paradoxical Mechanism

a sin= b sin

A spatial 4R linkage is, in general, immovable because M=-2.

However, it may have mobility one if special geometry are met.

There are two well-know paradoxical mechanisms:– Spherical four-bar mechanism (The axes of revolute joints allpass through a single point)– Bennett mechanism

Idle Degrees of Freedom

An Idle degree of freedom is one that appears (and is) present but its value has

no effect on the input – output relationships of interest

To identify Idle degrees of freedom, first identify the input and output links

–Then we must determine if a single link or combinations of links can move

without affecting the input/output link positions

–Like a connecting link rotating (about its axis) in a steering mechanism

without changing the relationship between the steering wheel and the front

tires in a vehicle

04/28/2023 ME 3230 Page 24

Note: Pin-in-slot & Cam Contact are half joints

52132)112(3

2)1(3213

12

21

2

1

JJnmJJn

Here,

The Structure has five DOF with two Idle DOF’s.

They are the roller and the cam rocker .

mActual = MTheoretical - mIdle

=5-2 = 3

Idle Degrees of Freedom

Joints: RSSRn=4, j1=2, j2=0, j3=2 ,m=2But Actual m=1

The result seems to conflict with our practical

experience since there is a unique value of for

any given value of . i.e., the orientation of link 4

can be determined when the orientation of link 2 is

specified.

Examining the mechanism carefully will reveal that

we need an extra parameter to identify the

orientation of link 3. Because this parameter

doesn't affect the input-output relationship of the

linkage, so we call it an idle degree of freedom.

Idle Degrees of Freedom (Redundant DOF)

An idle dof is one that does not affect the input-output relationship of

the linkage.

Procedures for Locating the Idle dof are

as following:

–Identify the input link and output link.

–Check to determine if a single link or a

combination of connected links can move

without altering the relative position of the

input and output links. If the answer is

positive, there are some idle dof’s Joints: n=14, j1=6, j2=0, j3=12 ,m=12But Actual m=6 and 6 idle dof, each idle dof locates in between two spherical joint.

Stewart Platform

Figure 1.29 (a) a prosthetic hand Figure 1.29(b) Fingers wrap around an object as shown in

In a spatial mechanism, links move in three dimensions. For example, in a prosthetic hand, the thumb moves in a plane that is not parallel to the planes of motion of the other four fingers.

where the subscript refers to the number of freedoms of the joint.

54321 2345)1(6 JJJJJnM

Spatial mechanism

The Kutzbach mobility equation for spatial linkages:

Example of Spatial Linkage

4 Links; 2 spherical Joints, 1cylindrical joint and 1 revolute joint.DOF of a Spherical Joint is 3DOF of a Cylindrical Joint is 2DOF of a Revolute Joint is 1

3231415)14(6

2345)1(6 54321

JJJJJnM

Four-Bar Mechanism-Grashof's Criterion

Four-bar mechanisms may be studied by distinguishing the link lengths as

follows:

s: the length of the shortest link

l: the length of the longest link

p, q: the lengths of the other two links

To assemble the kinematic chain it is necessary that,

lqps

The type of a four-bar mechanism may be determined using Grashof"s Criterion,

(i) (ii) (iii)qpls

Then, only case (i) offers all three types of a four-bar mechanisms.

qpls qpls

Class_I Class_II Class_III

(i) If s is the input link, then

the mechanism is a crank

rocker.

(ii) If s is the base link, then

the mechanism is a drag link.

(iii) If otherwise, then the

mechanism is a rocker-rocker.

Rocker_Rocker Change Point

qpls qpls qpls

Four-Bar Mechanism-Grashof's Criterion

Figure 1.43 Types of four-bar mechanisms (a) crank rocker, (b) drag link, (c) rocker-rocker.

For S+L<P+Q

Crank-rocker if either link adjacent to shortest is groundedDouble crank if shortest link is groundedDouble rocker if link opposite to shortest is grounded

For S+L>P+QAll inversions will be double rockersNo link can fully rotate

Figure: Four Bar double rockersFor S+L=P+Q (Special case Grashof)

All inversions will be double cranks or crank rockersLinkage can form parallelogram or antiparallelogramOften used to keep coupler parallel (drafting machine)

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Parallelogram form Anti parallelogram form Deltoid form

Figure 1.47 Four-bar mechanisms:crank rocker

Let the lengths of the three moving links are r2= 2.0 cm; r3=4.0 cm; r4=5.0 cm, adjusting the length of the base link we can get the following inversion of four bar mechanism.