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SYNTHESIS OF MECHANISMS
Sunder Dasika U11ME195 SVNIT, Surat
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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 −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.
Introduction
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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
Analytical Methods
Overlay Method
Bloch's Method
Freudensteins' Method
Introduction
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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 ϕ 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 x s ≤ x ≤ x f and y s ≤ y ≤ y f 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.
Precision Positions; Structural Error; Chebychev Spacing
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2.2
Precision Points and Structural Error
Precision points are those points on a linkage that exactly satisf y 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
= 12 + − 12+ − cos 22−1 , = 1, 2 , … ,
spacing is given by:
(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 x j
Figure 2.2 – Graphical determination of Chebychev spacing
Precision Positions; Structural Error; Chebychev Spacing
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3 Graphical Methods of Synthesis Dimensional synthesis of a linkage is the determinati on 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 linkage’s 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
Graphical Methods of Synthesis
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The steps to determine the extreme positions of a mechanism are: 1. Construct two arcs, one of radius r 3 + r 2 and another of radius r 3 – r 2, with centre at point O2. 2. Draw another arc of radius r 4, 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 γ 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° − − − 4 cos4 = 4 − 2 4 cos4 = 24 − − 4 cos = 42 −4 cos = 24
(3.1)
To develop this method, we can apply the cosine r ule to figure 3.1. This gives us two equations (3.2a) (3.2b)
Then from figure 3.2 we obtain,
(3.2c) (3.2d)
Now the above equations can be solved simultaneously to obtain the following ratios,
Graphical Methods of Synthesis
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=√ 2c 1 o−csos 1− 4 = 1− cos = √ () (4) − 1
(3.3)
(3.4)
(3.5)
Figure 3.2 – Minimum and maximum transmission angles
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 ψ > 180o, then α = ψ – 180o, where α can be obtained from the equation of time
= 118800°° −
ratio (ratio time of advance stroke and time of return stroke), (3.6)
Graphical Methods of Synthesis
8 A crank-and-rocker mechanism for specified values of ϕ and α can be synthesized by following the steps given below: 1. Locate point O4 and choose any desired rocker length r 4. 2. Draw the two positions O4 B1 and O4 B2 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 α to the line X . The intersection of these two lines defines the line the locati on of crank pivot O2. 4. Next, the distance B2C is 2r 2, or twice crank length. So, we bisect this distance to find r 2 5. The coupler length is r 3 = O2 B1 – r 2. This completes the synthesis of linkage.
(b)
(a) Figure 3.3 – Two position synthesis of four-bar mechanism
Because the line X was chosen arbitrarily, there are infinite number of solutions possible for this problem.
Graphical Methods of Synthesis
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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 O2 A through an angle ψ 12 causes the output rocker to rotate through an angle ϕ12. The link O4 B 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 ϕ12. The final position is therefore O’2A’2 B’2O4.
Figure 3.4 – Linkage inverted in the O4 B position Figure 3.5 illustrates the problem and synthesized solution. The starting input angle is of the crank is θ 2; and ψ 12, ψ 23, and ψ 13 are the swing angles, respectively between the design positions 1 and 2, 2 and 3, and 1 and 3. Corresponding swing angles ϕ12, ϕ23, and ϕ13 are desired for the output lever. The length of link 4 and the start position θ 4 of the output rocker are to be determined.
Figure 3.5 Graphical Methods of Synthesis
10 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 O2 A 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 O4 A2 and rotate it backward through the angle ϕ12 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 A1’ A2’ and A2’ A3’. 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
Graphical Methods of Synthesis
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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.
Graphical Methods of Synthesis
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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
Bloch’s 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
⃗⃗⃗⃗4= 0 4 = 0 44 = 0 44 4 = 0
(4.1a)
In complex number form, the equation can be written as,
(4.1b)
The first and second derivative of this equation are
(4.1c) (4.1d)
Figure 4.1 – Vector representation of four-bar linkage
Analytical Methods
13 If we now transform equations 4.1(a) through (c) back into vector notation, we get the
⃗ ⃗⃗ ⃗⃗ ⃗44⃗4 == 00 ⃗ ⃗ 4 4⃗4 = 0
following relation
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
−100 −1 4 144 ⃗= 1 1 14 4 4 ⃗ ⃗4 ⃗⃗== 44 4−−44 4 ⃗4= ⃗= − ⃗ −⃗⃗4
solved for relative link dimensions.
Similar equations can be developed for
and
(4.1e)
. 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:
4.2
(4.2a) (4.2b) (4.2c) (4.2d)
Freudenstein’s Equation
If real and imaginary components of equation 4.1b are separated, we obtain two algebraic equations
csoins csoins csions 44scions44 = =00
(4.3a) (4.3b)
Analytical Methods
From figure 4.1,
sin =−0 ccooss = −1 cos 4 cos4 = 0 sin sin 4 sin4 = 0 − 2 −4 − 4 4 cos cos4 = cos − 4 cos cos4 = cos − 4 = 4 − = − − 4 = 24 and
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; therefore
(4.4a) (4.4b)
Eliminating θ 3 from the above set of equations and on simplification, we get (4.5)
Freudenstein writes this equation in the form
Where,
(4.6)
(a) (b) (c)
Freudenstein’s 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 ϕ1, ϕ2 and ϕ3, corresponding to the angular positions ψ 1, ψ 2 and ψ 3 of the input lever. Then, in equation
ccooss ccooss == ccooss −− cos cos = cos −
(4.6), we simply substitute θ 2 with ψ i, θ 4 with ϕi and write the three equations. This gives (4.7a) (4.7b) (4.7c)
The three equations above are solved simultaneously for K 1, K 2 and K 3. One of the link lengths is chosen and the others can be found from equations 4.6 (a) through (c)
Analytical Methods
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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.
Conclusion
