Kinematics of Machines

Kinematics of Mechanisms

K. Analysis K. Synthesis

Given a mechanism:

the task is to analyze its motion

- displacement, velocity, acceleration

Given a desired motion:

the task is to develop a mechanism

that meets the requirements

For the study of Kinematics, a machine may be referred to as a mechanism,

….. a combination of interconnected rigid bodies capable of a predictable relative motion

Kinematic synthesis

1. Type synthesis: selection of the type

(linkages, gears, cam & follower, belt &

pulley, chain & sprocket) of mechanism to

be used, accounting for the nature of

motion transfer, velocity ratio, space

considerations, cost, reliability etc.

2. Number synthesis: the number of links and

the number of joints needed to produce the

required motion- rules to be followed.

3. Dimensional synthesis: the proportions or

lengths of the links, or angles, necessary to

satisfy the required motion characteristics.

Given a desired motion, the task is to develop a mechanism that meets the requirements

a. An odd DoF requires an even number of

links.

b. Number synthesis: For DoF=1 & given

number of total links, determine all

compatible combinations of links (the

number and order of links)

Laying out a cam to meet certain specifications

Is dimensional synthesis.

Kinematic synthesisGiven a desired motion, the task is to develop a mechanism that meets the requirements

Kinematic synthesis

Type Syn. Number Syn. Dimensional synthesis

A mechanism design frequently requires that the output link moves (rotates or oscillates) as

a specified function of the motion of the input link: Function Generation

An example:Displacement of the follower as a specified function

of the angle of rotation of the cam.

Precision points for Function Generation

•To generate a particular function, it is usually quite difficult (not possible) to accurately produce the

desired function at more than a few (input) points.

•The (input) points at which the generated and desired functions agree are known as precision

points or accuracy points.

• It is important that the precision/accuracy (input) points be such that the error generated

between these points is minimal.

Kinematic synthesis: Dimensional synthesis•The (input) points at which the generated and desired functions

agree are known as precision points or accuracy points.

• It is important that the precision/accuracy (input) points be

such that the error generated between these points is

minimal.

The number of precision points

=

The number of design parameters

at disposal

Chebyshev’s Spacing of Accuracy Points

Kinematic synthesis: Dimensional synthesisPosition of precision points: Chebyshev’s Spacing

2 polygon sides perpendicular

to the horizontal

X can be seen as the horizontal

projection of the tip of the

Input link

Synthesis: pin-jointed 4bar mechanism

For a tangible solutionAlternative representation

Freudenstein’s

equation

Length ‘d’ is given, and ‘a’,’b’,’c’ are to be found,

which is possible if K1, K2, K3 can be found

This is possible if you have 3 equations (3 unknowns)

This in turn is possible if you can get 3 sets with

{θ2,θ4} values

Synthesis: pin-jointed 4bar mechanism

Freudenstein’s

EquationYou need 3 sets: {Input, Output angle}

1.Get 3 values of ‘x’ from

Chebyshev spacing

2.Get 3 corresponding values of ‘y’ from

the desired relation with ‘x’

3.Assume linear relation b/w

‘x’ and ‘θ’; also ‘y’ and ‘φ’

4. Get 3 values of ‘θ’ for the

3 ‘x’, as the range is given

5. Get 3 values of ‘φ’ for the

3 ‘y’, as the range is given

Synthesis: pin-jointed 4bar mechanism P1

You need {θ,φ} combinations which could be used as boundary conditions for dimensional synthesis

Values of x

Values of y

Synthesis: pin-jointed 4bar mechanism P1

You need {θ,φ} combinations which could be used as boundary conditions for dimensional synthesis

Values of θ Values of φ

Synthesis: pin-jointed 4bar mechanism P1

You need {θ,φ} combinations which could be used as boundary conditions for dimensional synthesis

Remember you will need to

derive this: the governing

equation

Synthesis: offset 4bar Slider Crank LinkageP2

The figure shows a slider crank mechanism, whose synthesis calls for the displacement (s)

of the slider C to be co-ordinated with the crank angle (θ) in a specified manner.

θ3, d?

Relate ‘s’ and θ (BC)2 = (XC-XB)2 + (YC-YB)2

P2

The figure shows a slider crank mechanism, whose synthesis calls for the displacement (s)

of the slider C to be co-ordinated with the crank angle (θ) in a specified manner.

Let the displacement of the slider be proportional to the crank angle over a given interval

Relate ‘s’ and θ

Assuming a synthesis for 3 precision points: The 3 positions of the crank (θ1,θ2,θ3) can be

obtained through Chebyshev’s spacing, while the corresponding positions of the slider

(s1,s2,s3) could be obtained by using the linear proportionality, as above.

Synthesis: offset 4bar Slider Crank Linkage

(BC)2 = (XC-XB)2 + (YC-YB)2

P2

The figure shows a slider crank mechanism, whose synthesis calls for the displacement (s)

of the slider C to be co-ordinated with the crank angle (θ) in a specified manner.

Relate ‘s’ and θ

For 3 different positions of the mechanism, involving (θ1,θ2,θ3) & (s1,s2,s3), this equation

can be used.

The task reduces to solving the 3 simultaneous equations, for the unknowns k1,k2, & k3,

following which, the lengths a,b, and c can be computed.

Synthesis: offset 4bar Slider Crank Linkage

(BC)2 = (XC-XB)2 + (YC-YB)2