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A slotted-crank mechanism with a flexibly attached slider for path generation and its dynamic synthesis
Mechanismand Machine TheoryVol. 15, pp. 233-243 Pergamon Press Ltd.. lge0. Printed in Great Britain
A Slotted-Crank Mechanism with a Flexibly Attached Slider for Path Generation and its Dynamic Synthesis A. C. Rao~r Received for publication 21 March 1980
Abstract Methods reported so far for the synthesis of 4-bar mechanisms with rigid links for path generation are very complex. In this paper a slotted link mechanism consisting of a flexibly attached slider is proposed for path generation. A pointer attached to the slider mass will trace a curve. Equation of motion for the slider mass and its solution are developed. The equation obtained for displacement of the slider mass is linear and can be used for the synthesis of a mechanism by closed form solution as well as for optimization through the method of least-squares, etc. Example problems are included. Introduction FOUR-BARlinkages are extensively treated for path-generation. Both graphical and analytical methods[I-2] have been developed for the synthesis of path generating mechanisms. Present trend is to apply mathematical programming techniques for the synthesis of path generating mechanisms. The aim of the design is to generate a curve as close to the desired curve as possible. Unlike closed form solutions optimization techniques place no limit on the number of precision points. Nechi[3] used an iterative technique to increase the number of precision points but no attention is paid to quality of the mechanism. Fox and Wilmert[4] presented a method for optimum design of curve generating linkages with inequality constraints but they have not considered the effects of clearances and tolerances. Bakthavachalam and Kimbrell[5] utilised a technique similar to that developed by Fiacco and McCormick[6] for optimum synthesis of path generating 4-bar mechanisms. They have considered both the quality of motion and the effects of clearances and tolerances. Previous work[3-5] deals with kinematic synthesis only and does not consider the effects of elastic deformation which has considerable influence on the performance of the mechanism particularly at high speeds and loads. To overcome this shortcoming, Erdman, et a/.[7] have presented a general method for kinetoelastodynamic analysis and synthesis of mechanisms. Their method is illustrated with application to a 4-bar path generator. This paper does not include the effects of clearances and tolerances, etc. Sadler and Sandor[9] have presented kinetoelastodynamic harmonic analysis of 4-bar path generating mechanisms. The work reported so far deals with conventional 4-bar linkages only and the methods developed are complex and tedious due to nonlinearity of equations that occur. Here, an inversion of a slider-crank mechanism consisting of a flexibly attached slider which lends well to the synthesis is proposed for the first time for path generation. Dynamic synthesis is reported and the application is illustrated through numerical examples. The equation presented is totally free from nonlinearity and is in a form suitable for the synthesis by both the approaches, i.e. closed form as well as iterative. It is emphasized here that the purpose of the paper is to show the feasibility of such a mechanism and to indicate its scope.
Theory Consider the linkage shown in Fig. 1. To simplify the analysis let us consider the mechanism in a horizontal plane. Link-dimensions, r and l are indicated in the Fig. 1. The angles are tProfessorof MechanicalEngineeringGovernmentEngineeringCollege,Ujjain(M.P.),India. Presentlyat Government Engineering College,Jabalpur,(M.P.),India. 233 MMT Vol. 15, No, 4--A
234
B
Figure 1. measured from a horizontal reference line through OA, the centre of rotation of the slotted lever. The fixed link OAO~ makes an angle a with the reference line. The slotted lever is driven at a constant angular speed of to rad/sec. The slider mass, m, is coupled to the point B through a spring of stiffness k. The distances of point B and the pointer attached to the mass, m, along the line of stroke are given by x~ and x respectively as shown in Fig. 1. The equation of motion for the mass, m, along the slotted lever axis can be formulated based on the following concept. Along the slot the mass, m, will be subjected to a total acceleration which is equal to the sum of the two components, (i) centripetal acceleration-toex and (ii) sliding acceleration, £. Coriolis component of acceleration which acts normal to the slot-axis will not influence the motion of slider mass along the slot axis. Therefore (a)
Net Inertia force = m[.f - to2x]. (along the slot axis)
In Fig. I, a damper is incorporated between the mass, m, and the end of the slotted-lever. Hence,
(b)
the damping force = cg where, c is the viscous damping coefficient. And,
(c)
Spring force = k(x - x¢ - b)
where, b is the free length of the spring. Combining the expressions (a), (b) and (c), the equation of motion of the mass m becomes
(1)
m[.f - to2x] + cX + k(x - xc - b) = O.
From Fig. 1 the value of xc can be expressed as follows
r2
xc = r cos (8 - 0) + I 1 - ~ sin2 (/3 - 0) = rcos(8-O)+
re
l[1-~-flsin
2
(B -
]1/2
0)],
r2 since ~ sin (B - 0) < 1.
235 Realising that, sin2 (/3 - O) = 1 - c o s 2(/3 - O) 2 xc can be written as follows xc= l - ~
+rcos(/3-0)+
cos 2(/3-0).
(2)
Using eqn (2) in eqn (1) we get
:i+pYc+qx=ton2[(lr2--~+ b)+rcos(/3-O)+~ cos 2(/3-0)]
(3)
where, to 2
k
" = m ;p
c and q = (to.2 -
=m
002).
The solution to eqn (3) consists of both the complimentary function and the particular integral. The complimentary function corresponds to the free vibration which can be expected to die out due to the damping present. Finally, noting that 0 = tot, the particular integral which represents the steady state of motion can be written as follows x = A + B cos (/3 - 0) + C sin (/3 - 0) + D cos 2(/3 - 0) + E sin 2(/3 - 0)
(4)
where
Sa,
q B =
r(q -
ooe)¢o.2
pZ¢o2+ (q _ ¢02)2
(5b)
- proJ.2 C = p2¢a2 + ( q _
oj2)2
(5c)
r2(q --4W2)C0n2 D = 4114p2w2 + (q
-
40)2)2]
(Sd)
- 2pr2 cOn2
E = 4/[4p2w2 + (q _ 4w2)2]
(5e)
The displacement eqn (4) contains five constants A, B . . . . . E. It may be noticed that these constants are related to the seven parameters r, l, to, ~o., p, b and q as indicated by the expression (a)...(e) of eqn (5). Also, there is an interrelationship between q, to and oJ. as indicated after eqn (3), i.e. q = ~o.2 - to2. Hence five precision points are sufficient to determine the five constants A , B . . . . . E and hence the five design parameters r, l, to or ~o. b and p(or c). The application of eqn (4) to the synthesis of the dynamic path generator is explained below.
Design Procedure Step B I
Let the curve in Fig. 2 be the path to be generated. Then select arbitrarily, a point OA in Fig. 2, to correspond to the point OA of Fig. 1. Select precision points 1, 2. . . . 5 on the curve. At these points the generated curve matches exactly with the desired curve. Join OA 1, OA 2, etc. and let the angles made by these radial lines with respect to a horizontal through OA be 01, 02,
236
4
Figure 2. etc. respectively. We have, now, five sets of precision points (xl, OD,..., (xs, 05). Use of the numerical values in eqn (4) gives a set of five equations as follows. A.+ B cos !/~ - 0.1)+ C sin (/3.-.01).+ D cos.2(/3.-.0|,.+ .E s!n !(/3.- 01).= x.i
A'+ B cos i¢t - 05)+ C"sin (~ -'05)'+ D cos'2(~ -'05i+ E sin 2(~ - 05).= xs.
(6) It may be observed that the above set of eqns (6) are linear in terms of the constants A, B, . . . . E, and are similar to the equations we get when synthesizing 4-bar function generators using Freudenstein's[8] equation for three precision points. Simultaneous solution of eqns (6) assuming fl, yields the numerical values of A, B . . . . . E. Once the constants A, B . . . . . E are evaluated the design parameters r, 1, b, to or ~o, and c can be determined in the following manner.
Step 2 From (h) and (c) eqn (5) we get B
(q - oJ2)
C
pro
(7)
Similarly, from (d) and (e) of eqn (5) we get D
(q - 4to 2)
E
2pro
(8)
Therefore, B E
2 ( q - ~o2)
CD=(q~" Using the relationship q = ~o.2 - to2, notation after eqn (3) and letting, (B/C)(E/D) = kl; we get ~. = k~o~,
(9)
237
where, = Lk-iTT-2]
•
(ga)
Step 3 From the relation (Sd) we can write that, re _ 4--1-
2 + ( K 2 2 - 5)2to 4]
D[4p2to
(lO)
k22(k22- 5)to4
Also, relation (7) of step 2 gives pto = ( q - t o
2 C
_(k22_2) C.to2.
(11)
Substitution of eqn (11) in (10) gives
1.2 D[ 4(k22-2)2(C)2+(k22-5) 2] 4-1 =
k22( k22 - 5)
(12)
= k3 (say) Equation (12) gives the numerical value of D are known.
re/41.Since the numerical values of k2, B, C and
Step 4 Assuming a convenient value for q and using the relations q = to2_ to2 and to. = k2to, the numerical values of to and to. can be determined. Assuming a suitable value for mass, m the spring constant k can be determined or vice versa.
Step 5 Use of relation (11) finally gives p and hence c, the damping coefficient; since c = pro.
Step 6 Use of eqn (Sb) now gives the value of r, the crank radius since values of other parameters q, to2, to, and p is known.
Step 7 Use of eqn (12) gives I.
Step 8 Use of eqn (Sa) gives b, the free length of the spring and this completes the design. It may be noted that if the design obtained is not satisfactory for any reasons the procedure may be repeated either by changing the value of/3 or by choosing a new centre of rotation for the slotted crank. The latter yields a new set of precision points and thus yields a new design.
Example 1 Design a path generator of the type shown in Fig. 1 to give the motion specified in Table 1. Table 1. Precision point
0(degrees)
x(cm)
I
0
2 3 4 5
30 60 90 120
25.75 • 27.30 28.00 27.4 25.90
238 Take, fl = 60°.
Solution Substitution of the numerical data of Table 1 in eqn (4) gives a set of five equations similar to eqns (6), e.g. A + B cos 60 ° + C sin 60 ° + D cos 120° + E sin 120° = 25.75
A + B cos ( - 6 0 °) + C sin ( - 6 0 °) + D cos (-120) + E sin (-120 °) = 25.90. Simultaneous solution of the above equations gives A = 24.47, B = 3 . l i , C = - 0 . 0 5 9 , D = 0.41 and E = - 0 . 0 1 9 . Combining eqns (7) and (8) we get,
BE
kl = ~
= 2.43.
Using eqn (9a)
[5kl - 4 ]112 k2= [ k - ~ - 2 J
=4"35
or, k22= 18.9. Using eqn (12), F k3 = ~ = 0.302.
Now, let us take q = to,2 - to 2 = 6,000 = ( 1 8 . 9 - 1)to2 = 6,000. Therefore ~ = 18.3rad/sec and to, = k2to = 4.35 x 18.3 = 78.33 rad/sec but, k
~ (S)n2
m
taking the weight of the slider mass as 2 N, we get,
239
2 k = (79.33)2 x ~ = 12.82 N/cm. Using eqn (11) C p = - ~ (k2e - 2)to p = 5.868. Then x2 c=p×m= 5.868 981 = 0.012 N/(cm/sec). Use of eqn (5b) gives, r = 2.88 cm, but k3 = ~/= 0.3. Therefore, ! = 6.92 cm. Use of eqn (Sa) finally gives b, b = 17.2 cm. Finally, the numerical values of the five design parameters are listed below, r = 2.88 cm 1 = 6.92 cm b = 17.2 cm c = 0.012 N/(cm/sec) to, = 78.33 rad/sec, hence, k = 12.82 N/cm to = 18.3 rad/sec, (corresponds to a speed of 174.75 r.p.m.). Shape of the total generated path for this mechanism is given in Fig. 3; reduced to 1/4th of the actual size. A circle with a radius equal to that at precision point--1 is drawn in broken lines just to show the desired (generated) curve is different from a circle. The displacements are calculated for 30* intervals of 0 and as such the Fig. 3 does not show the exact path between the dots. The desired path is not shown as it can be taken arbitrarily for illustration (in absence of the specific curve) but the desired curve coincides with the generated curve at the five precision points 1. . . . . 5. This indicates that the closed form solution presented above does not have control over the deviation of the desired path from the generated path in between the chosen precision points. This drawback can be overcome by formulating an objective function F in the following manner and then minimise it[10-12]. To match the generated curve more closely with the desired one, n precision points where n ~, 5 may be chosen and an objective function F may be formulated in the following manner N
R
N
F: ~ ¢2=~ (Xd,--Xsi)':~ [A +B cos ( B - 0 , ) + + D cos 2(B - 0j) + E sin 2(B - 0~)- x,~l2.
C sin(B-0,), (13)
240
IO
Figure 3. where, Xdi and xgi are the desired radius and the generated radius at ith precision point. There is no theoretical limit on n, the number of precision points. The objective function may be minimised by setting the partial derivations (~F/~A), (dF/~b), etc. equal to zero. This gives a set of five linear simultaneous equations, solution of which gives A, B . . . . . E. Once the constants A . . . . . E are evaluated, the design parameters can be determined following the stepwise procedure. The following problem illustrates this method.
Example 2 Design a mechanism of the type, Fig. 1, to coordinate the motion given in Table 2. Solution Substituting the numerical data given in Table 2 in eqn (13) and setting each of the partial derivatives of the resulting objective function with respect to A, B . . . . . E, equal to zero we get the following set of equations. 10A + 7.33B - 0.966C + 1.866D - 0.50E = 420.25 7.328A + 5.93B - 0.24C + 2.82D - 0.125E = 309.83 0.97A + 0.24B - 4.05C - 0.836D - 4.52E = 39.37
Table 2. Precision point
# (degrees)
x (cm)
1 2 3 4 5 6 7 8 9 10
0 15 30 45 60 75 90 105 120 135
40.94 41.75 42.57 43.00 43.40 42.70 42.12 41.71 41.26 40.80
241 1.259A + 2.236B + 0.679C + 3.722D + 0.129E = 57.42 0.SA + 0.125B - 4.52C- 0.433D - 5.75E = 20.09. Simultaneous solution of the above equations gives, A = 39.48, B -- 3.32, C = -0.86, d = 0.2 and E = 0.64. Using the above values the radii generated at various precision points are calculated using eqn (4). The plots of the desired and generated curves are shown in Fig. 4. Design parameters can be determined following the stepwise procedure already outlined and also example 1. Using eqns (7) and (8) we get,
BE kl = / ~ 7 = -12.32. Using eqn (9) k22 =
r(Sk,-4) L ~ J 1 = 4.58; or k2 = 2.14.
Using eqn (12)
F
ks = ~ = -0.2. Taking q = ton 2 _ ~ 2 = (k22 _ 1 ) 0 2 =
6,000.
We get, w2 = 6,000 = 1676, 3.58 rr)
c~.
-
o
15" --
10.21cm
Generoted
Figure 4.
10.24 cm Desired
242
hence = 40.93 rad/sec oJ, = k2w = 87.61 rad/sec.
Taking the weight of slider mass as 2 N k = m x ~o,~ = ~ 1 x (87.61)2 = 15.65 N/cm. Using eqn (11) p = - -~ (4.58 - 2)o~ = 27.35 D
2 c = p x m = 27.35 × ~ = 0.055 N/(cm/sec). Use of eqn 5(b) gives, r=2cm. Now using, k3 = F/41 we get 1 = -5 cm. Finally, use of eqn (5a) gives b = 35.6 cm. Negative sign of a link dimension indicates its layout in opposite sense.
Conclusions (1) Mathematical treatment is extremely simple. (2) Synthesis is very simple compared to other methods due to the absence of nonlinearity in eqn (4). (3a) The displacement eqn (4) is linear in terms of the constants A, B . . . . . E but not in terms of the actual design parameters. The nonlinearity involved in determining the design parameters from the constants A, B . . . . . E is of second degree and present absolutely no difficulty. The synthesis results in one positive and one negative value for each of the parameters ~o and oJ,. The negative values of oJ and oJ, have no particular significance and need not be considered. (3b) The mechanism is sensitive to changes in oJ, the driving speed. Change in the speed from the design speed alters the generated path; the magnitude of deviation depends upon the ratio (o&/~o), higher the ratio lesser is the deviation. (4) If the dimensions obtained are not satisfactory for any reason, say, location of pivots or quality of the motion etc., the linkage may be redesigned choosing a new position for point OA or selecting a different set of precision points on the given curve. This leads to a new set of values (0, x) and hence a new design. This does not require much effort. (5) External force, if any, on the slider mass can be included in the equation of motion. This does not pose any problem in the synthesis. (6) The angle B may also be taken as a design parameter to increase the number of precision points to six but this will introduce nonlinearity in eqn (4) and hence in the set of equations, like eqn (6) and the solution becomes slightly difficult. For greater number of precision points the optimization method suggested above is advantageous. The angle/], if not considered as a design parameter, makes the choice of pivot points O~ and O~ easy. It is desirable to select the angle B in such a way the line of centres 080A lies in the direction of largest radius. (7) The data, Tables 1 and 2, are purely arbitrary as the main purpose of this paper is to show the feasibility of the proposed mechanism. References 1. J. Hrones and G. Nelson, Analysis of the 4-Bar Linkage. The Technology press of Massachusetts Institute of Technology and Wiley, New York (1951).
2. R. S. Hartenbergand L Denavit,KinematicSynthesis of Linkages. McGraw-Hill,New York(1964).
243
3. A. J. Nechi, The path generation by a plane 4-bar chain dissertation. Department of Mechanical Engineering, Georgia Institute of Technology, Georgia (1968). 4. R. L. Fox and K. D. Wilmert, Optimum design of curve generating linkages with inequality constraints. ASME Paper No. 66-Mech.-20(1966). 5. N. Bakthavachalam and J. T. Kimbrell, Optimum synthesis of path-generating 4-bar mechanisms. ASME Paper No. 74-DET-6 (1974). 6. A. V. Fiacco and G. P. McCormick,Computational algorithm for the sequential unconstrained minimizationtechnique for nonlinear programming. Management Sci. 10(4),July (1964). 7. A. G. Erdman, G. N. Sandor and R. G. Oakberg, A general method for kinetoelastodynamicanalysis and synthesis of mechanisms. Trans. ASME ]. EngngInd., 1193-1204,Nov. (1972). 8. F. Freudenstein, Approximate synthesis of 4-bar linkages. Trans ASME 77, 853-861 (1972). 9. J. P. Sadler and G. N. Sandor, Kinetoelastodynamic harmonic analysis of 4-bar path generating mechanisms. ASME Paper No. 70-Mech.-61 (1970). 10. G. H. Sutherland and B. Roth, An improved least squares method for designing function generating mechanisms. ASME Paper No. 74-DET-4(1974). 11. R. I. Alizade, A. V. Mohan Rao and G. N. Sandor, Optimum synthesis of 4-bar and offset slider crank planar and spatial mechanisms using the penalty function approach with inequality and equality constraints. ASME Paper No. 74-DET-30 (1974). 12. C. H. Suh and Mecklenburg, Optimal design of mechanisms with the use of matrices and least squares. Mech. Mach. Theory g(4), 479-.495(1973).
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~EN~R
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A Slotted-Crank Mechanism with a Flexibly Attached Slider for Path Generation and its Dynamic Synthesis A. C. Rao~r Received for publication 21 March 1980
Abstract Methods reported so far for the synthesis of 4-bar mechanisms with rigid links for path generation are very complex. In this paper a slotted link mechanism consisting of a flexibly attached slider is proposed for path generation. A pointer attached to the slider mass will trace a curve. Equation of motion for the slider mass and its solution are developed. The equation obtained for displacement of the slider mass is linear and can be used for the synthesis of a mechanism by closed form solution as well as for optimization through the method of least-squares, etc. Example problems are included. Introduction FOUR-BARlinkages are extensively treated for path-generation. Both graphical and analytical methods[I-2] have been developed for the synthesis of path generating mechanisms. Present trend is to apply mathematical programming techniques for the synthesis of path generating mechanisms. The aim of the design is to generate a curve as close to the desired curve as possible. Unlike closed form solutions optimization techniques place no limit on the number of precision points. Nechi[3] used an iterative technique to increase the number of precision points but no attention is paid to quality of the mechanism. Fox and Wilmert[4] presented a method for optimum design of curve generating linkages with inequality constraints but they have not considered the effects of clearances and tolerances. Bakthavachalam and Kimbrell[5] utilised a technique similar to that developed by Fiacco and McCormick[6] for optimum synthesis of path generating 4-bar mechanisms. They have considered both the quality of motion and the effects of clearances and tolerances. Previous work[3-5] deals with kinematic synthesis only and does not consider the effects of elastic deformation which has considerable influence on the performance of the mechanism particularly at high speeds and loads. To overcome this shortcoming, Erdman, et a/.[7] have presented a general method for kinetoelastodynamic analysis and synthesis of mechanisms. Their method is illustrated with application to a 4-bar path generator. This paper does not include the effects of clearances and tolerances, etc. Sadler and Sandor[9] have presented kinetoelastodynamic harmonic analysis of 4-bar path generating mechanisms. The work reported so far deals with conventional 4-bar linkages only and the methods developed are complex and tedious due to nonlinearity of equations that occur. Here, an inversion of a slider-crank mechanism consisting of a flexibly attached slider which lends well to the synthesis is proposed for the first time for path generation. Dynamic synthesis is reported and the application is illustrated through numerical examples. The equation presented is totally free from nonlinearity and is in a form suitable for the synthesis by both the approaches, i.e. closed form as well as iterative. It is emphasized here that the purpose of the paper is to show the feasibility of such a mechanism and to indicate its scope.
Theory Consider the linkage shown in Fig. 1. To simplify the analysis let us consider the mechanism in a horizontal plane. Link-dimensions, r and l are indicated in the Fig. 1. The angles are tProfessorof MechanicalEngineeringGovernmentEngineeringCollege,Ujjain(M.P.),India. Presentlyat Government Engineering College,Jabalpur,(M.P.),India. 233 MMT Vol. 15, No, 4--A
234
B
Figure 1. measured from a horizontal reference line through OA, the centre of rotation of the slotted lever. The fixed link OAO~ makes an angle a with the reference line. The slotted lever is driven at a constant angular speed of to rad/sec. The slider mass, m, is coupled to the point B through a spring of stiffness k. The distances of point B and the pointer attached to the mass, m, along the line of stroke are given by x~ and x respectively as shown in Fig. 1. The equation of motion for the mass, m, along the slotted lever axis can be formulated based on the following concept. Along the slot the mass, m, will be subjected to a total acceleration which is equal to the sum of the two components, (i) centripetal acceleration-toex and (ii) sliding acceleration, £. Coriolis component of acceleration which acts normal to the slot-axis will not influence the motion of slider mass along the slot axis. Therefore (a)
Net Inertia force = m[.f - to2x]. (along the slot axis)
In Fig. I, a damper is incorporated between the mass, m, and the end of the slotted-lever. Hence,
(b)
the damping force = cg where, c is the viscous damping coefficient. And,
(c)
Spring force = k(x - x¢ - b)
where, b is the free length of the spring. Combining the expressions (a), (b) and (c), the equation of motion of the mass m becomes
(1)
m[.f - to2x] + cX + k(x - xc - b) = O.
From Fig. 1 the value of xc can be expressed as follows
r2
xc = r cos (8 - 0) + I 1 - ~ sin2 (/3 - 0) = rcos(8-O)+
re
l[1-~-flsin
2
(B -
]1/2
0)],
r2 since ~ sin (B - 0) < 1.
235 Realising that, sin2 (/3 - O) = 1 - c o s 2(/3 - O) 2 xc can be written as follows xc= l - ~
+rcos(/3-0)+
cos 2(/3-0).
(2)
Using eqn (2) in eqn (1) we get
:i+pYc+qx=ton2[(lr2--~+ b)+rcos(/3-O)+~ cos 2(/3-0)]
(3)
where, to 2
k
" = m ;p
c and q = (to.2 -
=m
002).
The solution to eqn (3) consists of both the complimentary function and the particular integral. The complimentary function corresponds to the free vibration which can be expected to die out due to the damping present. Finally, noting that 0 = tot, the particular integral which represents the steady state of motion can be written as follows x = A + B cos (/3 - 0) + C sin (/3 - 0) + D cos 2(/3 - 0) + E sin 2(/3 - 0)
(4)
where
Sa,
q B =
r(q -
ooe)¢o.2
pZ¢o2+ (q _ ¢02)2
(5b)
- proJ.2 C = p2¢a2 + ( q _
oj2)2
(5c)
r2(q --4W2)C0n2 D = 4114p2w2 + (q
-
40)2)2]
(Sd)
- 2pr2 cOn2
E = 4/[4p2w2 + (q _ 4w2)2]
(5e)
The displacement eqn (4) contains five constants A, B . . . . . E. It may be noticed that these constants are related to the seven parameters r, l, to, ~o., p, b and q as indicated by the expression (a)...(e) of eqn (5). Also, there is an interrelationship between q, to and oJ. as indicated after eqn (3), i.e. q = ~o.2 - to2. Hence five precision points are sufficient to determine the five constants A , B . . . . . E and hence the five design parameters r, l, to or ~o. b and p(or c). The application of eqn (4) to the synthesis of the dynamic path generator is explained below.
Design Procedure Step B I
Let the curve in Fig. 2 be the path to be generated. Then select arbitrarily, a point OA in Fig. 2, to correspond to the point OA of Fig. 1. Select precision points 1, 2. . . . 5 on the curve. At these points the generated curve matches exactly with the desired curve. Join OA 1, OA 2, etc. and let the angles made by these radial lines with respect to a horizontal through OA be 01, 02,
236
4
Figure 2. etc. respectively. We have, now, five sets of precision points (xl, OD,..., (xs, 05). Use of the numerical values in eqn (4) gives a set of five equations as follows. A.+ B cos !/~ - 0.1)+ C sin (/3.-.01).+ D cos.2(/3.-.0|,.+ .E s!n !(/3.- 01).= x.i
A'+ B cos i¢t - 05)+ C"sin (~ -'05)'+ D cos'2(~ -'05i+ E sin 2(~ - 05).= xs.
(6) It may be observed that the above set of eqns (6) are linear in terms of the constants A, B, . . . . E, and are similar to the equations we get when synthesizing 4-bar function generators using Freudenstein's[8] equation for three precision points. Simultaneous solution of eqns (6) assuming fl, yields the numerical values of A, B . . . . . E. Once the constants A, B . . . . . E are evaluated the design parameters r, 1, b, to or ~o, and c can be determined in the following manner.
Step 2 From (h) and (c) eqn (5) we get B
(q - oJ2)
C
pro
(7)
Similarly, from (d) and (e) of eqn (5) we get D
(q - 4to 2)
E
2pro
(8)
Therefore, B E
2 ( q - ~o2)
CD=(q~" Using the relationship q = ~o.2 - to2, notation after eqn (3) and letting, (B/C)(E/D) = kl; we get ~. = k~o~,
(9)
237
where, = Lk-iTT-2]
•
(ga)
Step 3 From the relation (Sd) we can write that, re _ 4--1-
2 + ( K 2 2 - 5)2to 4]
D[4p2to
(lO)
k22(k22- 5)to4
Also, relation (7) of step 2 gives pto = ( q - t o
2 C
_(k22_2) C.to2.
(11)
Substitution of eqn (11) in (10) gives
1.2 D[ 4(k22-2)2(C)2+(k22-5) 2] 4-1 =
k22( k22 - 5)
(12)
= k3 (say) Equation (12) gives the numerical value of D are known.
re/41.Since the numerical values of k2, B, C and
Step 4 Assuming a convenient value for q and using the relations q = to2_ to2 and to. = k2to, the numerical values of to and to. can be determined. Assuming a suitable value for mass, m the spring constant k can be determined or vice versa.
Step 5 Use of relation (11) finally gives p and hence c, the damping coefficient; since c = pro.
Step 6 Use of eqn (Sb) now gives the value of r, the crank radius since values of other parameters q, to2, to, and p is known.
Step 7 Use of eqn (12) gives I.
Step 8 Use of eqn (Sa) gives b, the free length of the spring and this completes the design. It may be noted that if the design obtained is not satisfactory for any reasons the procedure may be repeated either by changing the value of/3 or by choosing a new centre of rotation for the slotted crank. The latter yields a new set of precision points and thus yields a new design.
Example 1 Design a path generator of the type shown in Fig. 1 to give the motion specified in Table 1. Table 1. Precision point
0(degrees)
x(cm)
I
0
2 3 4 5
30 60 90 120
25.75 • 27.30 28.00 27.4 25.90
238 Take, fl = 60°.
Solution Substitution of the numerical data of Table 1 in eqn (4) gives a set of five equations similar to eqns (6), e.g. A + B cos 60 ° + C sin 60 ° + D cos 120° + E sin 120° = 25.75
A + B cos ( - 6 0 °) + C sin ( - 6 0 °) + D cos (-120) + E sin (-120 °) = 25.90. Simultaneous solution of the above equations gives A = 24.47, B = 3 . l i , C = - 0 . 0 5 9 , D = 0.41 and E = - 0 . 0 1 9 . Combining eqns (7) and (8) we get,
BE
kl = ~
= 2.43.
Using eqn (9a)
[5kl - 4 ]112 k2= [ k - ~ - 2 J
=4"35
or, k22= 18.9. Using eqn (12), F k3 = ~ = 0.302.
Now, let us take q = to,2 - to 2 = 6,000 = ( 1 8 . 9 - 1)to2 = 6,000. Therefore ~ = 18.3rad/sec and to, = k2to = 4.35 x 18.3 = 78.33 rad/sec but, k
~ (S)n2
m
taking the weight of the slider mass as 2 N, we get,
239
2 k = (79.33)2 x ~ = 12.82 N/cm. Using eqn (11) C p = - ~ (k2e - 2)to p = 5.868. Then x2 c=p×m= 5.868 981 = 0.012 N/(cm/sec). Use of eqn (5b) gives, r = 2.88 cm, but k3 = ~/= 0.3. Therefore, ! = 6.92 cm. Use of eqn (Sa) finally gives b, b = 17.2 cm. Finally, the numerical values of the five design parameters are listed below, r = 2.88 cm 1 = 6.92 cm b = 17.2 cm c = 0.012 N/(cm/sec) to, = 78.33 rad/sec, hence, k = 12.82 N/cm to = 18.3 rad/sec, (corresponds to a speed of 174.75 r.p.m.). Shape of the total generated path for this mechanism is given in Fig. 3; reduced to 1/4th of the actual size. A circle with a radius equal to that at precision point--1 is drawn in broken lines just to show the desired (generated) curve is different from a circle. The displacements are calculated for 30* intervals of 0 and as such the Fig. 3 does not show the exact path between the dots. The desired path is not shown as it can be taken arbitrarily for illustration (in absence of the specific curve) but the desired curve coincides with the generated curve at the five precision points 1. . . . . 5. This indicates that the closed form solution presented above does not have control over the deviation of the desired path from the generated path in between the chosen precision points. This drawback can be overcome by formulating an objective function F in the following manner and then minimise it[10-12]. To match the generated curve more closely with the desired one, n precision points where n ~, 5 may be chosen and an objective function F may be formulated in the following manner N
R
N
F: ~ ¢2=~ (Xd,--Xsi)':~ [A +B cos ( B - 0 , ) + + D cos 2(B - 0j) + E sin 2(B - 0~)- x,~l2.
C sin(B-0,), (13)
240
IO
Figure 3. where, Xdi and xgi are the desired radius and the generated radius at ith precision point. There is no theoretical limit on n, the number of precision points. The objective function may be minimised by setting the partial derivations (~F/~A), (dF/~b), etc. equal to zero. This gives a set of five linear simultaneous equations, solution of which gives A, B . . . . . E. Once the constants A . . . . . E are evaluated, the design parameters can be determined following the stepwise procedure. The following problem illustrates this method.
Example 2 Design a mechanism of the type, Fig. 1, to coordinate the motion given in Table 2. Solution Substituting the numerical data given in Table 2 in eqn (13) and setting each of the partial derivatives of the resulting objective function with respect to A, B . . . . . E, equal to zero we get the following set of equations. 10A + 7.33B - 0.966C + 1.866D - 0.50E = 420.25 7.328A + 5.93B - 0.24C + 2.82D - 0.125E = 309.83 0.97A + 0.24B - 4.05C - 0.836D - 4.52E = 39.37
Table 2. Precision point
# (degrees)
x (cm)
1 2 3 4 5 6 7 8 9 10
0 15 30 45 60 75 90 105 120 135
40.94 41.75 42.57 43.00 43.40 42.70 42.12 41.71 41.26 40.80
241 1.259A + 2.236B + 0.679C + 3.722D + 0.129E = 57.42 0.SA + 0.125B - 4.52C- 0.433D - 5.75E = 20.09. Simultaneous solution of the above equations gives, A = 39.48, B -- 3.32, C = -0.86, d = 0.2 and E = 0.64. Using the above values the radii generated at various precision points are calculated using eqn (4). The plots of the desired and generated curves are shown in Fig. 4. Design parameters can be determined following the stepwise procedure already outlined and also example 1. Using eqns (7) and (8) we get,
BE kl = / ~ 7 = -12.32. Using eqn (9) k22 =
r(Sk,-4) L ~ J 1 = 4.58; or k2 = 2.14.
Using eqn (12)
F
ks = ~ = -0.2. Taking q = ton 2 _ ~ 2 = (k22 _ 1 ) 0 2 =
6,000.
We get, w2 = 6,000 = 1676, 3.58 rr)
c~.
-
o
15" --
10.21cm
Generoted
Figure 4.
10.24 cm Desired
242
hence = 40.93 rad/sec oJ, = k2w = 87.61 rad/sec.
Taking the weight of slider mass as 2 N k = m x ~o,~ = ~ 1 x (87.61)2 = 15.65 N/cm. Using eqn (11) p = - -~ (4.58 - 2)o~ = 27.35 D
2 c = p x m = 27.35 × ~ = 0.055 N/(cm/sec). Use of eqn 5(b) gives, r=2cm. Now using, k3 = F/41 we get 1 = -5 cm. Finally, use of eqn (5a) gives b = 35.6 cm. Negative sign of a link dimension indicates its layout in opposite sense.
Conclusions (1) Mathematical treatment is extremely simple. (2) Synthesis is very simple compared to other methods due to the absence of nonlinearity in eqn (4). (3a) The displacement eqn (4) is linear in terms of the constants A, B . . . . . E but not in terms of the actual design parameters. The nonlinearity involved in determining the design parameters from the constants A, B . . . . . E is of second degree and present absolutely no difficulty. The synthesis results in one positive and one negative value for each of the parameters ~o and oJ,. The negative values of oJ and oJ, have no particular significance and need not be considered. (3b) The mechanism is sensitive to changes in oJ, the driving speed. Change in the speed from the design speed alters the generated path; the magnitude of deviation depends upon the ratio (o&/~o), higher the ratio lesser is the deviation. (4) If the dimensions obtained are not satisfactory for any reason, say, location of pivots or quality of the motion etc., the linkage may be redesigned choosing a new position for point OA or selecting a different set of precision points on the given curve. This leads to a new set of values (0, x) and hence a new design. This does not require much effort. (5) External force, if any, on the slider mass can be included in the equation of motion. This does not pose any problem in the synthesis. (6) The angle B may also be taken as a design parameter to increase the number of precision points to six but this will introduce nonlinearity in eqn (4) and hence in the set of equations, like eqn (6) and the solution becomes slightly difficult. For greater number of precision points the optimization method suggested above is advantageous. The angle/], if not considered as a design parameter, makes the choice of pivot points O~ and O~ easy. It is desirable to select the angle B in such a way the line of centres 080A lies in the direction of largest radius. (7) The data, Tables 1 and 2, are purely arbitrary as the main purpose of this paper is to show the feasibility of the proposed mechanism. References 1. J. Hrones and G. Nelson, Analysis of the 4-Bar Linkage. The Technology press of Massachusetts Institute of Technology and Wiley, New York (1951).
2. R. S. Hartenbergand L Denavit,KinematicSynthesis of Linkages. McGraw-Hill,New York(1964).
243
3. A. J. Nechi, The path generation by a plane 4-bar chain dissertation. Department of Mechanical Engineering, Georgia Institute of Technology, Georgia (1968). 4. R. L. Fox and K. D. Wilmert, Optimum design of curve generating linkages with inequality constraints. ASME Paper No. 66-Mech.-20(1966). 5. N. Bakthavachalam and J. T. Kimbrell, Optimum synthesis of path-generating 4-bar mechanisms. ASME Paper No. 74-DET-6 (1974). 6. A. V. Fiacco and G. P. McCormick,Computational algorithm for the sequential unconstrained minimizationtechnique for nonlinear programming. Management Sci. 10(4),July (1964). 7. A. G. Erdman, G. N. Sandor and R. G. Oakberg, A general method for kinetoelastodynamicanalysis and synthesis of mechanisms. Trans. ASME ]. EngngInd., 1193-1204,Nov. (1972). 8. F. Freudenstein, Approximate synthesis of 4-bar linkages. Trans ASME 77, 853-861 (1972). 9. J. P. Sadler and G. N. Sandor, Kinetoelastodynamic harmonic analysis of 4-bar path generating mechanisms. ASME Paper No. 70-Mech.-61 (1970). 10. G. H. Sutherland and B. Roth, An improved least squares method for designing function generating mechanisms. ASME Paper No. 74-DET-4(1974). 11. R. I. Alizade, A. V. Mohan Rao and G. N. Sandor, Optimum synthesis of 4-bar and offset slider crank planar and spatial mechanisms using the penalty function approach with inequality and equality constraints. ASME Paper No. 74-DET-30 (1974). 12. C. H. Suh and Mecklenburg, Optimal design of mechanisms with the use of matrices and least squares. Mech. Mach. Theory g(4), 479-.495(1973).
Uf~LAUF~NDE KU~{BELSCHLEIFE ~IT ELASTISCT[ G~OPPELT~A~ GLEITSTEI~ Ftt~ DIE E P ~ I G U ~ EURVEN UND DYN~IISCHE SYNTHESE
~EN~R
A.C. Rao Kurzfassum~, - F~r die aagea~he1~te Erzeugung einer gegebenen ebenen Kurve wird e~ae umlaufeade Eurbelschleife vorgeschlagea, aa derea Eoppel-Gleitste~n e3a we~terer In der Kulisse der Schleife gelagerter Gleitstein elastisch g e k o ~ e l t Jet, der den erzeugeadea Punkt trggt. Die Bewegungsglelchung fiir diesen Gle~tste~m wird aufgestellt und jhre LSsung gezeigt° Die Glelchung f/~r die erzeugte Kurve bzw. GleitsteJabewegung let linear uad kaan sowohl fgr d i e Synthese des Getriebes mittels geschleesener LSsung ale auch fGr die Optimierung mit der Methode der kle~asten Quadrate o.a° ~enutzt werdea. E~a Zahleabeispiel erl~utert die ProblemlSsu~g.