Synthesis of Mechanism

Theory Of Machine
Lecture 1-2-3
New chapter: Synthesis of Mechanisms
Subtopics:
Introduction of synthesis,
Types of synthesis,
Synthesis of a four bar chain,
Freudenstein’s equation for four bar mechanism,
Precision point for function generator (Chebychev
spacing method),
Bloch method.

INTRODUCTION:
Kinematic synthesis determines the size and configuration of
mechanisms
Once this is defined, then material selection, stress analysis and
dynamic analysis are used to shape the components to ensure
performance and reliability.

Types of synthesis:
Kinematic synthesis for a mechanical system is described as
having three general phases, known as
1. Type synthesis,
2. Number synthesis and
3. Dimensional synthesis

1. Type synthesis:
It is the process of determining possible mechanism structures
to perform a given task or combination of tasks without regard
to the dimensions of the components.
 Choosing the type of mechanism to perform the required
function

Example of type synthesis:
To transmit power Selection of any one drive out of a rope drive,
a belt-pulley transmission or a gear train
Given a task to be produced by a mechanism, find the type that
will best perform it, e.g., a linkage, a cam mechanism, a gear train,
or a combination of these.

2. Number synthesis:
The number of links and the number of joints needed to produce
the required motion are calculated.

3. Dimensional synthesis:
The proportions or lengths of the links, or angles, necessary to
satisfy the required motion characteristics are found out.
 The main objective here is to find the dimensions defining the
geometry of the various links and joints of the kinematic chain
 Given a task to be produced by a mechanism, find its relevant
geometric parameters.

Dimensioning involves two phases:
1. Functional dimensioning includes the determination of the
fundamental dimensions of the machine parts, prior to the
shaping of all its parts. It is the functional dimensioning
where kinematic synthesis plays a major role.
2. Mechanical dimensioning pertains to the dimensioning of
the machine elements for stress, strength, heat capacity, and
dynamic-response requirements

Synthesis is the procedure by which identification of the
specific mechanism (Type synthesis), and appropriate
dimensions of the linkages are identified
(Dimensional synthesis) to get desired motion.

Kinematic Analysis vs. Kinematic Synthesis
The fundamental problems in mechanism kinematics can be
broadly classified into:
(a) Analysis: For Given linkage find the motion of its links, for a
prescribed motion of its input joint(s).
(b) Synthesis:
For Given a task find the linkage that best performs the task.

The task at hand can be one of three, in this context:
(a) Function generation: the motion of the output joint(s) is
prescribed as a function of the motion of the input joint(s);
(b) Motion generation (rigid-body guidance): the motion of the
output link(s) is prescribed in terms of the motion of the input
link(s) or joint(s);
(c) Path generation: the path traced by a point on a floating link—a
link not anchored to the mechanism frame—is prescribed as a
curve, possibly timed with the motion of the input joint(s).

Two types of dimensional synthesis:
1. Exact synthesis: Number of linkage parameters available is
sufficient to produce exactly the prescribed motion. Problem
leads to—linear or, most frequently, nonlinear—equation
solving .
2. Approximate synthesis: Number of linkage parameters
available is not sufficient to produce exactly the prescribed
motion. Optimum dimensions are sought that approximate the
prescribed motion with the minimum error. Problem leads to
mathematical programming (optimization)

In-line slider crank mechanism
Approximate synthesis Design problem:
Determine the appropriate lengths L2 and L3 of the crank and
coupler respectively to achieve the desired stroke

From the kinematic diagram in the figure we conclude that the
center of the crank rotation is on the constrained path of the slider
and that can take on any value since it does not affect the stroke,
however a shorter connecting arm yields greater velocities and
accelerations for the slider, therefore its length should be as large
as possible

Mechanism to move a link between two positions
Some applications require a link to move between two fixed
positions, when these positions are specified the design
problem is termed two point synthesis, it can be achieved
using a single pivot or by using the coupler in a four bar
mechanism.

Two point synthesis using a pivot
Design problem:
Identify the location of the pivot and angle of rotation given the
initial and final locations of two points on the link

a. Geometrical
Synthesis:
1. Draw the lines that
connect the initial and
final locations of each
point
2. Construct the
perpendicular bisectors
s to these lines
3. The intersection is
the location of the
pivot

4. The angle between the
lines connecting the pivot
and the end locations of one
of the points is the angle of
rotation.

Two point synthesis using a four bar mechanism
Design problem:
Identify the location of the pivots and the lengths of all four
links Geometrical Synthesis:

connect the initial and
final locations of each
point
2. Construct the
s to these lines
3. The pivots can be
placed anywhere on the

4. The lengths can then be
measured graphically
Note that longer pivoting
lengths will rotate at smaller
angles, this produces larger
transmission angles and
reduces the forces required
to drive the linkage.

Mechanism to move a link between three positions
Some applications require a link to move between three fixed
positions, when these positions are specified the design problem
is termed three point synthesis and can be achieved using the
coupler in a four bar mechanism.
Design problem:
Identify the location of the pivots and the lengths of all four links

Geometrical Synthesis:
connect the initial and final
locations of each point
2. Construct the
perpendicular bisectors to
these lines
3. The intersections of two
of the perpendicular
bisectors are the pivot
locations.

4. The lengths can then be
measured graphically

The synthesis, or design, of four-bar mechanisms is important
when aiming to produce a desired output motion for a specific
input motion

By kinematic synthesis we mean the design or creation of a
mechanism to attain specific motion characteristics.
In this sense, Synthesis is the very essence of design because it
represents the creation of new hardware to meet particular
requirements of motion: displacement; velocity; acceleration;
individually or in combination.

Classifications of Synthesis Problem
The problems in synthesis can be placed in one of the following three
categories :
1. Function generation
2. Path generation
3. Body guidance
These are discussed as follows :
1. Function generation
In designing a mechanism, the frequent requirement is that the output
link should either rotate, oscillate or reciprocate according to a
specified function of time or function of the motion of input link. This
is known as function generation.

A simple example is that of designing a four bar mechanism to generate
the function y = f (x).
In this case, x represents the motion of the input link and the mechanism
is to be designed so that the motion of the output link approximates the
function y.
Note :
The common mechanism used for function generation is that of a
cam and a follower in which the angular displacement of the
follower is specified as a function of the angle of rotation of the
cam. The synthesis problem is to find the shape of the cam
surface for the given follower displacements.

2. Path generation
In a path generation, the mechanism is required to guide a point (called
a tracer point or coupler point) along a path having a prescribed shape.
The common requirements are that a portion of the path be a circular
arc, elliptical or a straight line.
Examples of such devices include James Watt's straight line linkage
3. Motion Generation
Body guidance
In body guidance, both the position of a point within a moving body and
the angular displacement of the body are specified. The problem may be
a simple translation or a combination of translation and rotation.

Output motion is a set of positions of a line defined as x, y, z successive
locations.
Guiding a rigid body through a finite set of positions and orientations.
A good example is a landing gear mechanism which must retract and
extend the wheels, having down and up locked poses with specific
intermediate poses for collision avoidance,

Function Generation using Freudenstein's Equation
A presentation of using Freudenstein's Equation in the synthesis of
four-bar linkages that will mechanically generate functions like
and almost anything else we can think of.

Question: Discuss Synthesis of Four Bar Mechanism
Freudenstein's Equation

Consider a four bar mechanism as shown in Fig.
The synthesis of a four bar mechanism, when input and output angles
are specified, is discussed below :

Once the values of k1, k2 and k3 are known, then the link lengths a, b, c
and d are determined by using equation (ii).
In actual practice, either the value of a or d is assumed to be unity to
get the proportionate values of other links.

Example: Design a four bar mechanism to co-ordinate the input and
output angles as follows :
Input angles = 15°, 30° and 45° ;
Output angles = 30°, 40° and 55°.

Assuming the length of one of the links, say a as one unit, we get
the length of the other links as follows :
We know that
K1 = d / a
or
d = K1 * a = 0.905 units

Question:
Discuss Precision Point Selection using Chebyshev Spacing
The structural error in a function generator is simply the error
between the mathematical function and the actual mechanism,
usually expressed as a percentage. A good choice of precision points
will help reduce the structural error.
One good choice for the three precision points is using Chebyshev
Spacing, which is simply a kind of equal spacing around a circle, then
projection onto the horizontal bisector of the circle With
Freudenstein's Equation we are limited to three precision points. We
also have the bounds of the interval on x as given.

Chebychev spacing Precision Points for Function Generation
In designing a mechanism to generate a particular function, it is
usually impossible to accurately produce the function at more
than a few points.
The points at which the generated and desired functions agree
are known as precision points or accuracy points
These points must be located so as to minimize the error
generated between these points.
The best spacing of the precision points, for the first trial, is
called Chebychev spacing.

According to Freudenstein and Sandor, the Chebychev spacing for
n points in the range Xs ≤ X≤ Xf (i.e. when X varies between Xs and
Xf) is given by
where Xj = Precision points
ΔX = Range in X= Xf − Xs, and
j = 1, 2, ... n
The subscripts s and f indicate start and finish positions respectively.

The precision or accuracy points may be easily obtained by using the
graphical method as discussed below--- Chebychev spacing .
1. Draw a circle of diameter equal to the range ΔX= Xf − Xs.
2. Inscribe a regular polygon having the number of sides equal to twice
the number of precision points required, i.e. for three precision points,
draw a regular hexagon inside the circle, as shown in Fig.
3. Draw perpendiculars from each corner which intersect the diagonal of
a circle at precision points X1, X2, X3.

Freudenstein's Equation
The development begins with the loop closure equation for a four-bar
linkage, as shown in Figure below:

Loop Closure Equation
The loop closure equation simply sums the position vectors around the complete four-
bar linkage, and in vector form is given by
Each link has length r and is at angle , hence the complex form may be written as
…..1
….2

We can expand (2) using Euler's identity, then separate Real and Imag
terms. Before doing that, notice that the angle of link 1 is
Finally, for conciseness, use the short form
With these substitutions, equation (2) yields the two equations
….3
….4

Since the input of our mechanism will be link 2, and the output will be
link 4,
…5

Let the minimum and maximum values of independent variable x be
called Xi and Xf .
Our three precision points X1; X2; X3 will between Xi and Xf ; the
sequence will be [ Xi X1 X2 X3 Xf]
Chebyshev solution for N points; expressed for 3 points using
notation it is

Summery
Introduction of synthesis,
Types of synthesis,
Synthesis of a four bar chain,
Freudenstein’s equation for four bar mechanism,
Precision point for function generator (Chebychev
spacing method),

Synthesis of Mechanism

More Related Content

What's hot (20)

Similar to Synthesis of Mechanism (20)

More from R A Shah (19)

Recently uploaded (20)

Synthesis of Mechanism