synthesis of mechanisms
DESCRIPTION
Seminar Report Prepared by me on synthesis of mechanismsTRANSCRIPT
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Sunder Dasika
U11ME195 SVNIT, Surat
SYNTHESIS OF MECHANISMS
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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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Graphical Methods of Synthesis
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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Graphical Methods of Synthesis
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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Graphical Methods of Synthesis
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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Graphical Methods of Synthesis
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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Graphical Methods of Synthesis
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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Graphical Methods of Synthesis
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.