synthesis of mechanisms

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Sunder Dasika

U11ME195 SVNIT, Surat

SYNTHESIS OF MECHANISMS

1

Introduction

1 Introduction

Many machine design problems require the creation of a device with particular set of motion

characteristics. Synthesis of mechanisms means the design or creation of a mechanism to yield

a desired set of motions characteristics. Examples are, moving a tool from position A to

position B in a machine tool in a particular interval of time or tracing out a particular path in

space to insert a path into assembly. The possibilities are endless, but a common denominator

is often the need for a linkage to generate the desired motion.

1.1 Stages of Kinematic Synthesis

The synthesis of linkages consists of three primary stages. The first one is type synthesis,

followed by number synthesis and at the last dimensional synthesis.

1.1.1 Type Synthesis

In type synthesis, the kind of mechanism is selected. It may be a linkage, a geared system, a

cam-follower mechanism or a belt and pulley system. The main considerations to be taken in

this stage of design are the methods used for manufacturing, availability of materials, safety,

space and economics. The study of kinematics is only slightly involved in type synthesis.

1.1.2 Number Synthesis

Number synthesis deals with the number of links and joints required to obtain the required

mobility. The mobility of a mechanism is the number of input parameters that must be

independently controlled to bring the device into a particular position. For a mechanism having

n links joined by j1 single-degree-of-freedom pairs and j2 double-degree-of-freedom pairs, the

mobility m is given by:

= 3( 1) 21 2 (1.1)

This criteria is called Kutzbach criteria.

1.1.3 Dimensional Synthesis

In this step, the dimensions of individual links are found out. The dimensions of links be found

by either graphical or by analytical methods. Both these methods have been considered in this

report.

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Introduction

1.2 Types of Kinematic Synthesis Problems

There are three fundamental types of kinematic synthesis problems, which can be solved in a

systematic manner. However, one should remember that drawing strict borders between these

problem types is not possible and a designer should be well versed with all three types.

1.2.1 Function Generation

In these problems, the output member rotates, oscillates or reciprocates according to a specified

function of time or function of input motion. The function generated is of the form y = f(x),

where x represents the motion of input and y represents the motion of output. Conceptually,

they are a black box that deliver a predictable output motion based on the given input motion.

1.2.2 Path Generation

It is defined as control of a point in plane such that it follows some prescribed path. This is

typically accomplished with at least four bars, wherein a point on the coupler traces the desired

path. No attempt is made in path generation to control the orientation of the link that contains

the point of interest.

1.2.3 Body Guidance or Motion Generation

It is defined as control of line in space, such that it assumes some prescribed set of sequential

positions. Here the orientation of the link containing the line is important. For e.g., the bucket

in a bull dozer must assume a set of positions to dig, pick up, and dump the excavated earth.

1.3 Scope of this Report

The number of techniques available are large, some of which may be quite frustrating. Hence

only a few of the more useful approaches have been discussed in this report.

Graphical Methods

Two Position Synthesis

Three Position Synthesis

Overlay Method

Analytical Methods

Bloch's MethodFreudensteins'

Method

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Precision Positions; Structural Error; Chebychev Spacing

2 Precision Positions; Structural Error; Chebychev Spacing

Before we could actually start the synthesis of mechanisms some important considerations have

to be taken into account. The most important ones are:

Relating the function y = f(x) and the input and output motions of the linkage and;

Positioning the required output points in such a way, so as to minimize the error

2.1 Crank-Angle Relationships

The output and the input variables of a mechanism are proportionally related to the specified

function y = f(x). The input rotation of the mechanism is proportional to the independent

variable x, while the output motion g of the mechanism is proportional to the dependent variable

y.

If y = f(x), with xs x xf and ys y yf is the domain of the problem, the following relations

can be developed by simple linear interpolation:

= +

( ) (2.1a)

= +

( ) (2.1b)

Figure 2.1 - Nomenclature of 4-bar mechanism used throughout this report.

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Precision Positions; Structural Error; Chebychev Spacing

2.2 Precision Points and Structural Error

Precision points are those points on a linkage that exactly satisfy the desired function. We have

to assume that if the design fits the specifications at these few points, then it will probably

deviate only slightly from the desired function between the precision points.

Structural error is defined as the theoretical difference between the function produced by the

synthesised linkage and the function originally prescribed. For many function generation

problems the structural error in four-bar mechanism can be held to less than 4 percent.

2.3 Chebychev Spacing

The amount of structural error in the solution can be reduced by choosing appropriate positions

of precision points. Freudenstein and Sandor gave a very good trial for spacing these precision

points, called Chebychev spacing. For n precision points in range x0 x xn+1, the Chebychev

spacing is given by:

=

1

2(+1 + 0)

1

2(+1 0) cos

(2 1)

2, = 1, 2, , (2.2)

Chebychev spacing can also be conveniently found by graphical approach as described below.

Construct a circle whose diameter is equal to the range, x = xn+1 x

Inscribe a regular polygon having 2n sides in this circle

Drop perpendiculars from each jth vertex to intersect the diameter, x at precision

position value of xj

Figure 2.2 Graphical determination of

Chebychev spacing

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Graphical Methods of Synthesis

3 Graphical Methods of Synthesis

Dimensional synthesis of a linkage is the determination of the lengths of the links necessary to

accomplish the necessary desired motions. Many techniques exist to accomplish the task of

dimensional synthesis of four-bar linkages. The simplest and quickest methods are graphical.

These work well for up to four precision points, beyond which an analytical or numerical

approach is necessary.

3.1 Limiting Conditions

Linkage synthesis procedure of often only provide that the particular positions specified will

be obtained. They say nothing about the linkages behaviours between those points. Two

important considerations about this have to be kept in mind: the extreme positions of the

linkage and the transmission angle.

3.1.1 Extreme Positions

In this test, we check whether the linkage can reach all the specified design positions, without

reaching the extreme positions. The extreme positions of the linkage are determined by the

colinearity of the crank and the coupler.

Figure 3.1 Extreme positions of a four-bar linkage

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The steps to determine the extreme positions of a mechanism are:

1. Construct two arcs, one of radius r3 + r2 and another of radius r3 r2, with centre at

point O2.

2. Draw another arc of radius r4, with centre at point O4.

3. The points of intersection obtained by the two arcs drawn in step 1 and the arc drawn

in step 2 give the two extreme positions of the linkage.

3.1.2 Transmission Angle

Another test that can be quickly applied to linkage design in order to judge its quality, is to

measure its transmission angle. The transmission angle is shown in the figure below and is

defined as the angle between the output link and the coupler. It indicates the quality of force

and velocity transmission at the joint. Brodell and Soni developed an analytical method of

synthesizing the crank-rocker linkage in which the time ratio Q equals unity. The design also

satisfies:

= 180 (3.1)

To develop this method, we can apply the cosine rule to figure 3.1. This gives us two equations

cos(4 + ) =

12 + 4

2 (3 2)2

214 (3.2a)

cos 4 =

12 + 4

2 (3 + 2)2

214 (3.2b)

Then from figure 3.2 we obtain,

cos =

32 + 4

2 (1 2)2

234 (3.2c)

cos =

32 + 4

2 (1 + 2)2

234 (3.2d)

Now the above equations can be solved simultaneously to obtain the following ratios,

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31

= 1 cos

2 cos2 (3.3)

41

= 1 (

31

)2

1 (31

)2cos2

(3.4)

21

= (31

)2

+ (41

)2

1 (3.5)

Brodell and Soni plotted these results and found out that the transmission angle should be larger

for good quality motion and larger if high speeds are involved.

3.2 Two Position Synthesis

This is the most trivial case of function generation. The output function is defined as two

discrete angular positions of the rocker.

In figure 3.1, if > 180o, then = 180o, where can be obtained from the equation of time

ratio (ratio time of advance stroke and time of return stroke),

=

180 +

180 (3.6)

Figure 3.2 Minimum and maximum transmission angles

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A crank-and-rocker mechanism for specified values of and can be synthesized by following

the steps given below:

1. Locate point O4 and choose any desired rocker length r4.

2. Draw the two positions O4B1 and O4B2 of link 4 separated by the angle.

3. Through B1 construct any line X and then through B2 construct any line Y at the angle

to the line X. The intersection of these two lines defines the line the location of crank

pivot O2.

4. Next, the distance B2C is 2r2, or twice crank length. So, we bisect this distance to find

r2

5. The coupler length is r3 = O2B1 r2. This completes the synthesis of linkage.

Because the line X was chosen arbitrarily, there are infinite number of solutions possible for

this problem.

(a) (b)

Figure 3.3 Two position synthesis of four-bar mechanism

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3.3 Three Position Synthesis

In three position synthesis, inversion is used as a method of synthesis. Suppose that the rotation

of input rocker O2A through an angle 12 causes the output rocker to rotate through an angle

12. The link O4B is held stationary and the remaining links (including the frame) are permitted

to rotate and occupy the same relative positions. The link is hence moved backward through

an angle 12. The final position is therefore O2A2B2O4.

Figure 3.5 illustrates the problem and synthesized solution. The starting input angle is of the

crank is 2; and 12, 23, and 13 are the swing angles, respectively between the design positions

1 and 2, 2 and 3, and 1 and 3. Corresponding swing angles 12, 23, and 13 are desired for the

output lever. The length of link 4 and the start position 4 of the output rocker are to be

determined.

Figure 3.4 Linkage inverted in the O4B position

Figure 3.5

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The solution to the problem is based on inverting about link 4. The following systematic

procedure gives the method to synthesis the linkage:

1. Draw the input rocker O2A in the three specified positions and then locate a desired

position for point O4.

2. Because we will invert the link on link 4 in the first design position, join O4A2 and

rotate it backward through the angle 12 to locate A2.

3. Similarly obtain A3. A1 and A1 are coincident as the inversion is about this position.

4. Draw perpendicular bisectors to the lines A1A2 and A2A3.

5. These intersect at B1 and define the length of the coupler link 3 and the length and

start positions of link 4.

Figure 3.6 Three point synthesis procedure

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3.4 The Overlay Method

Synthesis of function generator using overlay method is the easiest and the quickest. It is

always not possible to obtain a solution, and sometimes the accuracy may also be less.

Theoretically, however, one can apply as many precision points as required. The procedure in

overlay method is somewhat iterative in nature and requires a little bit of intuition from the

designer. The major steps to be followed are:

1. On a sheet of tracing paper, construct all the input positions of the crank arm O2A.

2. On the same sheet choose an arbitrary length for coupler AB and draw arcs from the

end points of the crank arm positions.

3. On another piece of paper, construct the rocker arm, whose length is unknown at all

positions.

4. Through O4 draw a number or arbitrarily spaced arcs intersecting the lines. These

represent the possible lengths of the output rocker.

5. Finally, lay the tracing paper over the drawing and manipulate it in an effort to find the

fit.

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Analytical Methods

4 Analytical Methods

The synthesis techniques presented in the previous article were strictly graphical. The

analytical procedures are algebraic, rather than graphical and hence are less intuitive. However,

their algebraic nature makes them quite suitable for computerization.

4.1 Blochs Method of Synthesis

Bloch (a Russian kinematician) developed a method for synthesis of linkages for prescribed

angular velocities and accelerations of the links.

The links of the four-bar mechanism are replaced by position vectors and written in the

following

1 + 2 + 3 + 4 = 0 (4.1a)

In complex number form, the equation can be written as,

11 + 2

2 + 33 + 4

4 = 0 (4.1b)

The first and second derivative of this equation are

222 + 33

3 + 444 = 0 (4.1c)

2(2 + 22)2 + 3(3 + 3

2)3 + 4(4 + 42)4 = 0 (4.1d)

Figure 4.1 Vector representation of four-bar linkage

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Analytical Methods

If we now transform equations 4.1(a) through (c) back into vector notation, we get the

following relation

1 + 2 + 3 + 4 = 0

22 + 33 + 44 = 0

(2 + 22)2 + (3 + 3

2)3 + (4 + 42)4 = 0

This is a set of homogenous vector equations, having complex numbers as coefficients. Based

on the desired values of angular velocities and angular accelerations, the equations can be

solved for relative link dimensions.

2 =

1 1 10 3 40 3 + 3

2 4 + 42

1 1 12 3 4

2 + 22 3 + 3

2 4 + 42

(4.1e)

Similar equations can be developed for 3 and 4 . It turns out that the denominators for all three

links are same. Since we are only interested in finding the relative magnitudes of links, the

denominator terms can be neglected. The determinants when evaluated yield the following

result:

2 = 4(3 + 32) 3(4 + 4

2) (4.2a)

3 = 2(4 + 42) 4(2 + 2

2) (4.2b)

4 = 3(2 + 22) 2(3 + 3

2) (4.2c)

1 = (2 + 3 + 4 ) (4.2d)

4.2 Freudensteins Equation

If real and imaginary components of equation 4.1b are separated, we obtain two algebraic

equations

1 cos 1 + 2 cos 2 + 3 cos 3 + 4 cos 4 = 0 (4.3a)

1 sin 1 + 2 sin 2 + 3 sin 3 + 4 sin 4 = 0 (4.3b)

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Analytical Methods

From figure 4.1, sin 1 = 0 and cos 1 = 1; therefore

1 + 2 cos 2 + 3 cos 3 + 4 cos 4 = 0 (4.4a)

2 sin 2 + 3 sin 3 + 4 sin 4 = 0 (4.4b)

Eliminating 3 from the above set of equations and on simplification, we get

32 1

2 22 4

2

224+

14

cos 2 +12

cos 4 = cos(2 4) (4.5)

Freudenstein writes this equation in the form

1 cos 2 + 2 cos 4 + 3 = cos(2 4) (4.6)

Where,

1 =14

(a)

2 =12

(b)

3 =

32 1

2 22 4

2

224 (c)

Freudensteins equations enable us to find the motion of the output link based on that of input

link. Suppose that we wish the output lever of a four-bar linkage to occupy the positions 1, 2

and 3, corresponding to the angular positions 1, 2 and 3 of the input lever. Then, in equation

(4.6), we simply substitute 2 with i, 4 with i and write the three equations. This gives

1 cos1 + 2 cos1 + 3 = cos(1 1) (4.7a)

1 cos2 + 2 cos2 + 3 = cos(2 2) (4.7b)

1 cos3 + 2 cos3 + 3 = cos(3 3) (4.7c)

The three equations above are solved simultaneously for K1, K2 and K3. One of the link lengths

is chosen and the others can be found from equations 4.6 (a) through (c)

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Conclusion

5 Conclusion

In this report a few methods of synthesis of mechanisms (especially four-bar mechanism) have

been studied. The design methods were widely classified as either graphical or analytical. The

synthesis could be done for either two positions or three positions easily using graphical

methods. Higher number of precision points require analytical or numerical approach. One

graphical method, the overlay method, has been discussed which can be used for any number

of precision points. However, it suffers from a disadvantage that the designer requires a fair

amount of intuition and is iterative in nature. Analytical methods are non-intuitive but can be

easily programmed.

Most real life design problems have many more variables than the number of equations

available to describe the system. Such systems can be can be solved by iterating between

synthesis and analysis. Commercially available CAD programs like Creo 2.0, Autodesk

Inventor and AutoCAD allow rapid analysis of a proposed mechanical design.

synthesis of mechanisms

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