Kinematics of machines can involve either analyzing an existing mechanism's motion or synthesizing a new mechanism to achieve a desired motion. Kinematic synthesis involves selecting the type of mechanism, determining the number of links needed, and defining the link dimensions. Dimensional synthesis aims to develop link dimensions such that the mechanism's output motion matches the desired motion at select precision points, often spaced using Chebyshev's method to minimize error between points. Slider-crank mechanisms can be synthesized by relating the slider displacement to crank angle at precision points defined using Chebyshev spacing.

1. 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

2. 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.

3. Kinematic synthesis
Given 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.

4. 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

5. Kinematic synthesis: Dimensional synthesis
Position 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

6. Synthesis: pin-jointed 4bar mechanism
For a tangible solution
Alternative 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

7. Synthesis: pin-jointed 4bar mechanism
Freudenstein’s
Equation
You 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

8. Synthesis: pin-jointed 4bar mechanismP1
You need {θ,φ} combinations which could be used as boundary conditions for dimensional synthesis
Values of x
Values of y

9. Synthesis: pin-jointed 4bar mechanismP1
You need {θ,φ} combinations which could be used as boundary conditions for dimensional synthesis
Values of θ Values of φ

10. Synthesis: pin-jointed 4bar mechanismP1
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

11. 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

12. 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

13. 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