TOM2
Type synthesis refers to the kind of mechanism selected. The option can go in favor of gear combination, a belt Pulley combination, a linkage, or a cam mechanism. During this early stage of synthesis, one may have to consider design aspects like space consideration, safety consideration, overall economics, manufacturing process, material selection, etc. Type synthesis is thus an involved process and makes it very difficult to lay down a systematic procedure aimed at unique determination of mechanism that assures specified motion characteristics. Number synthesis is based on the most obvious external characteristics of a kinematics chain, namely, the number of links together with the number and type of joints required for a specified motion. Dimensional synthesis aims at determining significant dimensions of links and the starting position of links in a mechanism, to accomplish a specified task and prescribed motion characteristics. Hinge pin to hinge pin distances on binary, ternary, quaternary links angles between arms of bell crank levers, cam contour dimension, the diameter of a roller of follower, gear ratios, eccentricity are just a few examples of significant dimensions. In path generation, a point on the coupler link is the constraint to describe a path regarding a fixed frame of reference. Generally, the portion of the path is an Arc of the circle, an ellipse, or a straight line. Sometimes the tracing point May be required to follow a path that crosses over itself in the form of the figure of eight. A linkage mechanism whose points C on coupler is required to follow a part y = f(x) as the crank rotates, is an example of the past generation. A function generator represents a major category of synthesis problems. It is frequently required in a kinematic synthesis that an output link must rotate, oscillate or reciprocate according to a prescribed function of time or input motion. This is called function generation. For instance in a four-bar mechanism, for generating a function y=f(x), x would represent the motion of input crank and the proportions of link length of the mechanism are to be so chosen that the motion of output link represents the function y=f(x) approximately. Another example of function generation can be seen in linkage mechanism to correlate steering angles of axes of front wheels of an all-terrain vehicle with the relative speed at which each driving wheel should rotate avoiding skidding or access wear. Motion generation or rigid body guidance requires that an entire body be guided through a pre-selected sequence of motion. Such a body is guided usually as a part of the coupler. For example, in the construction industry, heavy parts such as buckets and blades of bulldozer must be moved through a series of prescribed positions. In the case of buckets of bucket loader mechanism, the tip of the bucket on coupler must have a prescribed path. Path to be followed by the tip of the bucket is very important because the tip is supposed to have a scooping trajectory followed by lifting and dumping trajectory. Function generation and path generation problems can be solved either by approximate or exact approach. The difference between the two approaches can be understood if we compare Watt approximate straight-line motion mechanism with Hart and Peaucelliar straight-line mechanism. Being an approximate straight-line motion mechanism, the tracing point of Watt's mechanism will have a wavy path, intersecting the desired path of straight-line y=mx adds a finite number of points. Thus except for the points of intersection, all other points raised by the coupler do not lie on an exact straight path. The points of intersection of the desired path with the actual path are defined as accuracy or precision points. The theme of the approximate approach of synthesis should be: There should be as many accuracy points as possible, and There should be a minimum deviation between the desired path and the actual path. When a mechanism is designed to generate specified function or trace a given curve y=F(x) over a given range 
the function is exactly generated at a finite number of points only. These are known as accuracy points and are represented in figure a by points of intersection of generated function y=f(x) with desired function y=F(x). These points are indicated as 
in the same figure. The number of these accuracy points is equal to the number of fixed parameters that may be used for synthesis & varies between three & six. Let x1 x2 x3……….. xn be the value of independent variables at precision points 
, working range 
. The difference between the two functions is called a structural error. The structural error 
is shown plotted against independent variable x in Fig (b). It must be clearly understood that structural error is inherent in an approximate synthesis. Best approximation between the function to be generated and function generated can be obtained when the absolute value of maximum structural error between any two precision points equal corresponding value at the ends 
Quite often Chebyshev spacing of precision points is used as a first guess to minimize the structural error. It must be remembered, however, that this approach is applicable in special cases when the function is symmetric. For the first guess, the best spacing of accuracy points, called Chebyshev's spacing, are given for the range 
For instance, let us assume that using Chebyshev's spacing, three precision points are to be obtained for generating a function 
in the range 
. The Chebychev accuracy points can also be obtained using a simple geometric construction. The method consists of drawing a circle on a diameter equal to the range of the function of an independent variable. Thus, The next step is to inscribe in a regular polygon of sides 2n. Where n is the number of accuracy points. The polygon must be so inscribed that at least two sides are perpendicular to the x-axis. Feet of perpendiculars dropped from the corners of this polygon on the x-axis then give accuracy points. Chebyshev accuracy points are used for solving problems on function generation. Accuracy is an important factor in designing a mechanism. It is possible to design a mechanism that will generate a given motion theoretically. The difference between the required motion & actual motion is called a structural error. The errors producing because of manufacturing problems like tolerances in the length of link & bearing clearances are called mechanical error. Structural errors are present even there are no mechanical errors. Structural errors are affected by precision point selection. Consider a four-bar mechanism ABCD as shown in figure a, in which AB=a, BC=b, CD=c, and DA=d. The link AD is fixed and lies along the x-axis . Let the links AB (input links), BC (coupler), and DC (output link) make angles 
respectively along the x-axis or fixed link AD. The relation between the angles and link lengths may be developed by considering the links as a vector. The expressions for displacement, velocity, and acceleration analysis are derived as discussed below: Displacement analysis For equilibrium of the mechanism, the sum of the components along the x-axis & along the y-axis must be equal to zero. First of all, taking the sum of the components along the x-axis as shown in figure b we have Consider a four-bar mechanism as shown in the figure. The synthesis of the four-bar mechanism, when input and output angles are specified, is discussed below: Let the three positions, angular displacement (
of the input link AB and the three positions (
of the output links as shown in figure B are known and we have to determine the dimensions a,b,c, and d of four-bar mechanism. The Freudenstein's equation For the three different positions of the mechanism, the equation (i) may be written as The equation (iii),(iv) and (v) are 3 simultaneous equation and maybe solve for 
using Cramer's rule of determinants as discussed below: Once the values of 
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.
Unit V
Synthesis of Mechanism
Q1) Explain type synthesis.
Ans.
Type synthesis
Q2) Explain number synthesis.
Ans.
Number synthesis
Q3) Explain dimensional synthesis.
Ans.
Q4) Explain the path generation.
Ans.
Other typical examples include the thread guiding eye on a sewing machine and mechanism to advance the film on camera.
Q5) Explain function generation.
Ans.
Q6) Explain motion generation.
Ans.
In past generation one is concerned with the path of tracer., In motion generation, the entire motion of the coupler link is of concern.
Q7) Explain precision points.
Ans.
Q8) Explain Chebychev's spacing with a neat sketch.
Ans.
Where 
are the accuracy points. Thus, for first, second, and third accuracy points, j is equal to 1,2, and 3 respectively.
n = total number of accuracy points.
Here. 
And n= number of accuracy points=3
First, second and third accuracy points are
And 
Corresponding values of the dependent variable by, as obtained by substituting 
values (j=1,2,3) in the function 
Q9) Write a note on a structural error & mechanical error.
Ans.
Q10) Derive Freudenstein’s eqution.
Ans.
Squaring both sides
Now taking the sum of the components along y-axis we have
Squaring both sides
Adding equations (ii) and (iv)
Let 
Equation (v) can be written as
The equation (vii) is known as the Freudenstein equation.
Q11) Explain Analytical synthesis using the kinematic coefficient in the four-bar mechanism.
Ans.
Where 
Now the values of 
are given by
0 matching results found