- Email: [email protected]
Dynamic synthesis for time Response of a flexibly coupled slider crank mechanism with pulse loading
Dynamic Synthesis for Time Response of a Flexibly Coupled Slider Crank Mechanism with Pulse Loading by .I. M.
and
GULATI
Department
of
Jabalpur (M.P.)
A.
c.
RAO
Mechanical India
Engineering,
Government
Engineering
College,
Link mechanisms designed for time-dependent output on the assumption of uniform input velocity will not perform satisfactorily due to fluctuation of input velocity
ABSTRACT:
under the actual working conditions. Synthesis, taking the input velocity fluctuations into consideration, is very difficult. Incorporation of a flexible element between the input link and the drive-motor makes the synthesis very easy. Also, a mechanism used as a timing device, for example the one used for actuating a valve at specified timings, is subjected to pulse loading. A slider-crank mechanism subjected to such a pulse loading and flexibly coupled to a drive-motor is synthesized for position coordination in the present paper. A numerical example is included.
I. Introduction Link mechanisms have been studied extensively for function generation, i.e. for coordinated motion of the cranks. A number of design procedures have been reported. Closed form synthesis is excellent as far as the geometry is concerned, but not suitable for the application of optimization techniques. Many papers using mathematical programming have been reported recently. Almost all the above work considers the links to be rigid. Winfrey (l), Erdman, Sandor and Oakberg (2) have considered the elastic deformation of the links in the analysis and synthesis of mechanisms. The procedures suggested are tedious and lengthy. Further, the input links are supposed to move with uniform velocity, which is not true in practice due to external and inherent loads. Sometimes we have to design linkages where the output is to be timedependent, say, x = f(t). Then considering the speed of the input link to be uniform, we can write x = f(e/w) = F(B), where w is the speed of the input link and 8 the input angle. In other words, the time function is reduced to displacement function and this permits kinematic synthesis of linkages for position coordination (function generators). In practice, however, since the input velocity is not uniform, the relation x = F(B) is not strictly valid. In order to design the linkage as position coordinator we must know the correct input angle corresponding to the desired output. This usually necessitates energy consideration which is very tedious. Sherwood (3) has dealt with a slider-crank mechanism for time-dependent
The Franklin Institute 00164032/81/03018747$02.00/0
187
J. M. Gulati and A. C. Rao output. If the input link is coupled rigidly to the drive-motor, the motor is put to severe test and must be designed specially to take care of the speed fluctuations of the input crank. Sherwood presumes that such motors can be designed. The performance accuracy of such motors is highly doubtful, apart from their being costly. Mahig (4-5) reported reduction of mechanism oscillation through tuning and damping. Matthew and Tesar (6-7) reported the synthesis of spring parameters to satisfy specified energy levels and to balance general forcing functions in planar mechanisms. Halter and Carson (8) and Carson and Haney (9) have considered the force system synthesis to get the desired time response. Kothari and Rao (10)have synthesized a slider-crank mechanism with a flexibly attached slider subjected to pulse loading on the basis of uniform input velocity. But the flexible coupling of the slider does not prevent the speed fluctuation of the crank. The authors, however, feel that the incorporation of a flexible element, say, a torsion spring between the input crank and the drive-motor makes the matter simple and the synthesis becomes very easy and less costly. In what follows, the above concept is discussed and a numerical example is solved to illustrate the feasibility. This technique permits the synthesis either by closed form approach or by mathematical programming. II.
Theory
As stated above a torsion spring is introduced between the drive-motor and the input link (or flywheel). This permits the drive-motor to run at uniform speed even when the speed of the input link fluctuates. The synthesis of a mechanism even for time-dependent output consists of coordinating the positions of the input and output links by, of course, taking the input speed fluctuations into consideration. In other words, the input crank angles under the dynamic, i.e. actual working conditions will be different from those on the uniform speed assumptions and as such it becomes necessary to predict the positions of the input crank for specified angles of motor shaft. Once these angles are determined, any of the established methods can be adopted for the synthesis. In this paper a slider-crank mechanism is considered. Figure 1 shows a slider-crank mechanism with a torsion spring of constant k introduced between the motor and the input crank. The angles p and 8
FIG. 1.
188
Jod of The Franklin Institute Pergmnon Press Ltd.
Time Response of a Flexibly Coupled Slider Crank Mechanism indicate respectively the position of the motor shaft and the input crank measured from the line of stroke of the slider mass. It may be noted that in the absence of the spring, the angles /3 and 8 are equal at every instant. It is desired to design the linkage so that the slider mass operates the values when the crank occupies different positions. Let, F = f(0) be the resisting force on the slider mass whose position x is the output. This force will act on the slider mass whenever it actuates a valve. In each cycle there will be as many pulses as the number of values. The loading is periodic and repeats after every revolution of the crank so that the period is 27~ for each pulse. The Fourier series for a rectangular pulse of unit height, duration h, on from c to c + h and repeated at intervals 27r, Fig. 2, can be written
(1) If there are three pulses in a cycle, on from c1 to cl+ h, c2 to c,+ h and G, to c3+ h, Fig. 3, the series for combined effect can be written F=f(0)=~+~“~l~sin
($)[cosn(B-cl-g) +cos n(B-c,--S)+cos
n(B--c3-i)].
For the mechanism, Fig. 1, the velocity and acceleration will be, approximately, sin 28 k =-WY sin O+2n I ’ [
(2)
of the slider mass m
(3)
and jl=-w2r
[
cosEJ+-
cos 28 n
1 [* --’
’
slnO+
sin 28 2n
1’
(4)
where n = l/r.
f (8)
I
I
f’ 1
+ 0
H
4
*h-3 : c
cth
-e FIG.
Vol. 311, No. 3, March 1981 Printed in Northern Ireland
2.
189
J. A.4. Gulati and A. C. Rao
ii-r Pulse 1
T”’
t-h-
Pulse 2
n-h-w
Pulse 3
+----I
r
Pulse 1
b-h+
1
0
vh
-22s
------a -e FIG. 3.
The weight of the connecting rod can be replaced by two dynamically equivalent weights, one of which can be added to the slider mass. The second mass thus obtained will not find its location at the crank pin making the analysis difficult, To overcome this situation connecting rod cap can be made heavier (11)so that the C.G. of the resultant mass is located at the crank pin. The slider mass m is thus the effective mass. Let at some instant, 8 be greater than p. Then the d’Alembert-Lagrange equation (or virtual work equation) is (12) .. . (F-mi)i+[k(p--8)--I8]8=0 (5) where I is the effective mass moment of inertia of the crank with respect to the centre of rotation. Equation (5) leads via (3) and (4) to the following equation
+m,2.$(l-~28)+(cosB;cos30)]
(6)
The motion of the crank shaft can be taken as one of oscillation about the mean position represented by /3, the position of driving motor shaft. We may assume, e=/3+Acoswt+Bcos2wt
(7)
:.i = w -A sin wt - 2Bw sin 2wt,
(8)
and ti = -Aw2 cos wt -4Bw2
cos 2wl
(9)
Substituting from (7) and (9) in (6) and replacing F by its value from (2), we 190
JcummlofTneFxankhInstitute Per&?anml Plea9 Ltd.
Time Response of a Flexibly Coupled Slider Crank Mechanism
get, [I(-Aw’
cos wt - 4Bw* cos 2wt) + k(A cos wt + B cos 2wt)] = [J-$+iilisin
($)[cosn(*-cl-i)+cosn(*-c2-z)
+cos n(O-cX-i)]]
X Y sin 8 + [
si;,28] + my2B2[Sin;28 I sit301
+_,,[(i-~2e)+(cose2C053e)l.
The solution has to be necessarily
(10)
numerical and iterative.
First iteration Let us start with the assumptions 8 = p and the velocity 6 = w (uniform velocity). Using the assumptions in (lo), a relation in terms of A and B is obtained. Second relation involving A and B is obtained from (8) by making 6 = w, which gives A sin wt + 2B sin 2wt = 0. (11) Solving the two relations obtained we get A and B, and hence 8 and 4. These values can be used as starting values for the next iteration. Iterations may be carried out till the desired convergency is accomplished. The following numerical example, with arbitrary data, will illustrate the theory. Computation is terminated after second iteration for want of a computer. Nevertheless, the method will be clear. III. Example Design a slider-crank mechanism to operate the three valves as given below: The first valve is operated when the motor shaft angle is 30”. The second valve is operated when the motor shaft has rotated to 80” corresponding to a movement of the slider by 6 cm and the third is operated when the motor shaft rotates to 120” and the slider moves further by 4cm. Each pulse may be taken as rectangular with height as 10 and duration as 6”. Moment of inertia of flywheel = 0.06 kg-cm-set* Slider weight
= 2.039 x lop3 kg-sec*/cm
Spring constant
= 20 kg-cmlrad
Motor speed
= 20 rad/sec
IV. Solution The problem can be converted into a three precision point problem by not assuming the slider position corresponding to 30” position of the motor. Let this be Vol. 311, No. 3, March 1981 Printed in Northern Ireland
191
J. M. Gulati and A. C. Rao x0 and letting (I’-- r2) = K,, we can write for a slider-crank following synthesis equations
mechanism
the
2x, cos 30 + K, = x; 2(x0 - 6)r cos 80 + K1 = (x0 - 6)2 2(x,-
10)r cos 120+K,
Solving the above three simultaneous I = 12.23 cm,
equations,
r = 6.61 cm,
(a)
= (x0- 10)2. we get
and
x0 = 17.50 cm.
First iteration For the motor shaft position p = 30”, use of (10) and (11) gives, 17.62A
+ 10.70B
OSA + 1.732B
= 39.19
(i)
=0
(ii)
Solving (i) and (ii) simultaneously, we get A = 2.69, quent use of these values in (7) and (9) gives 8 = 31.94”
and B = -0.78.
Subse-
4 = -311.24rad/sec2.
Similarly for p = 80”:
8 = 80.01”
8 = -14.87
p = 120”:
8 = 120.37”
d = -297
rad/sec2 rad/sec2
Second iteration Using the new values of 8 in the synthesis equations solving them we get, 1 = 11.44 cm,
r = 6.65 cm,
and
similiar to (a), and
x0 = 16.53 cm.
Using the values of 8 and link dimensions obtained above, the new values of 8 and 6 for various values of fi are: p = 30”:
13= 31.96
i = -359.76
rad/sec2
p = 80”:
e = 79.99”
ti = 3.35 rad/sec2
0 = 120”:
8 = 120.38”
8 = -302.85
rad/sec2.
V. Conclusion Values of 8 obtained in the second iteration are very close to the ones obtained in the first iteration and hence the computations are terminated here. Further, iterations may be carried out if more accuracy is desired. References (1) R. C. Winfrey, “Dynamics of mechanisms with elastic links”, Doctoral Dissertation, UCLA, 1968.
192
Time Response of a Flexibly Coupled Slider Crank Mechanism (2)A. G. Erdman,
G. N. Sandor, and R. G. Oakberg, “A general method for kineto-elastodynamic analysis and synthesis of mechanisms”, J. Engng. Ind., pp. 1193-1204, Nov. 1972. (3) A. A. Sherwood, “Dynamic synthesis of a mechanism with time-dependent output”, J. Mechanisms, Vol. 3, pp. 3540, 1968. (4) J. Mahig, “Minimization of mechanism oscillations through flywheel tuning”, J. Engng. Ind. Vol. 93B, No. 1, pp. 120-124, 1971. (5) J. Mahig and Y. P. Kakad, “Theoretical reduction of a tandem compressor shaft oscillation through tuning and damping”, ASME paper No. 72-Mech-56. (6) G. K. Matthew and D. Tesar, “Synthesis of spring parameters to Satisfy Specified Energy Levels in Planar Mechanisms”, ASME paper No. 76-DET-44. (7) G. K. Matthew and D. Tesar, “Synthesis of Spring Parameters to Balance General Forcing Function in Planar Mechanisms”, ASME paper No. 76-DET-45. (8) J. M. Halter and W. L. Carson, “Mechanism-force system synthesis to obtain a desired motion-time response”, Proceedings Fourth World Congress on the Theory of Machines and Mechanisms, Vol. 2, pp. 399-404, Newcastle upon Tyne, England, 1975. (9) W. L. Carson and R. S. Haney, “Force system structural synthesis by using coupler curves and interactive computer graphics”, ASME paper No. 78-DET-35. (10)M. Kothari and A. C. Rao, “A Timing Device with Pulse Loading”, accepted for publication in J. mech. Engng. Sci., IME, London. (11)F. S. Tse, I. E. Morse and R. T. Hinkle, “Mechanical Vibrations”, p. 186, Prentice Hall of India pvt. Ltd, New Delhi, 1974. (12) B. Paul, “Kinematics and Dynamics of Planar Machinery”, p. 568, Prentice Hall, 1979.
Vol. 311. No. 3, March 1981 Printed in Northern Irelaqd
193
and
GULATI
Department
of
Jabalpur (M.P.)
A.
c.
RAO
Mechanical India
Engineering,
Government
Engineering
College,
Link mechanisms designed for time-dependent output on the assumption of uniform input velocity will not perform satisfactorily due to fluctuation of input velocity
ABSTRACT:
under the actual working conditions. Synthesis, taking the input velocity fluctuations into consideration, is very difficult. Incorporation of a flexible element between the input link and the drive-motor makes the synthesis very easy. Also, a mechanism used as a timing device, for example the one used for actuating a valve at specified timings, is subjected to pulse loading. A slider-crank mechanism subjected to such a pulse loading and flexibly coupled to a drive-motor is synthesized for position coordination in the present paper. A numerical example is included.
I. Introduction Link mechanisms have been studied extensively for function generation, i.e. for coordinated motion of the cranks. A number of design procedures have been reported. Closed form synthesis is excellent as far as the geometry is concerned, but not suitable for the application of optimization techniques. Many papers using mathematical programming have been reported recently. Almost all the above work considers the links to be rigid. Winfrey (l), Erdman, Sandor and Oakberg (2) have considered the elastic deformation of the links in the analysis and synthesis of mechanisms. The procedures suggested are tedious and lengthy. Further, the input links are supposed to move with uniform velocity, which is not true in practice due to external and inherent loads. Sometimes we have to design linkages where the output is to be timedependent, say, x = f(t). Then considering the speed of the input link to be uniform, we can write x = f(e/w) = F(B), where w is the speed of the input link and 8 the input angle. In other words, the time function is reduced to displacement function and this permits kinematic synthesis of linkages for position coordination (function generators). In practice, however, since the input velocity is not uniform, the relation x = F(B) is not strictly valid. In order to design the linkage as position coordinator we must know the correct input angle corresponding to the desired output. This usually necessitates energy consideration which is very tedious. Sherwood (3) has dealt with a slider-crank mechanism for time-dependent
The Franklin Institute 00164032/81/03018747$02.00/0
187
J. M. Gulati and A. C. Rao output. If the input link is coupled rigidly to the drive-motor, the motor is put to severe test and must be designed specially to take care of the speed fluctuations of the input crank. Sherwood presumes that such motors can be designed. The performance accuracy of such motors is highly doubtful, apart from their being costly. Mahig (4-5) reported reduction of mechanism oscillation through tuning and damping. Matthew and Tesar (6-7) reported the synthesis of spring parameters to satisfy specified energy levels and to balance general forcing functions in planar mechanisms. Halter and Carson (8) and Carson and Haney (9) have considered the force system synthesis to get the desired time response. Kothari and Rao (10)have synthesized a slider-crank mechanism with a flexibly attached slider subjected to pulse loading on the basis of uniform input velocity. But the flexible coupling of the slider does not prevent the speed fluctuation of the crank. The authors, however, feel that the incorporation of a flexible element, say, a torsion spring between the input crank and the drive-motor makes the matter simple and the synthesis becomes very easy and less costly. In what follows, the above concept is discussed and a numerical example is solved to illustrate the feasibility. This technique permits the synthesis either by closed form approach or by mathematical programming. II.
Theory
As stated above a torsion spring is introduced between the drive-motor and the input link (or flywheel). This permits the drive-motor to run at uniform speed even when the speed of the input link fluctuates. The synthesis of a mechanism even for time-dependent output consists of coordinating the positions of the input and output links by, of course, taking the input speed fluctuations into consideration. In other words, the input crank angles under the dynamic, i.e. actual working conditions will be different from those on the uniform speed assumptions and as such it becomes necessary to predict the positions of the input crank for specified angles of motor shaft. Once these angles are determined, any of the established methods can be adopted for the synthesis. In this paper a slider-crank mechanism is considered. Figure 1 shows a slider-crank mechanism with a torsion spring of constant k introduced between the motor and the input crank. The angles p and 8
FIG. 1.
188
Jod of The Franklin Institute Pergmnon Press Ltd.
Time Response of a Flexibly Coupled Slider Crank Mechanism indicate respectively the position of the motor shaft and the input crank measured from the line of stroke of the slider mass. It may be noted that in the absence of the spring, the angles /3 and 8 are equal at every instant. It is desired to design the linkage so that the slider mass operates the values when the crank occupies different positions. Let, F = f(0) be the resisting force on the slider mass whose position x is the output. This force will act on the slider mass whenever it actuates a valve. In each cycle there will be as many pulses as the number of values. The loading is periodic and repeats after every revolution of the crank so that the period is 27~ for each pulse. The Fourier series for a rectangular pulse of unit height, duration h, on from c to c + h and repeated at intervals 27r, Fig. 2, can be written
(1) If there are three pulses in a cycle, on from c1 to cl+ h, c2 to c,+ h and G, to c3+ h, Fig. 3, the series for combined effect can be written F=f(0)=~+~“~l~sin
($)[cosn(B-cl-g) +cos n(B-c,--S)+cos
n(B--c3-i)].
For the mechanism, Fig. 1, the velocity and acceleration will be, approximately, sin 28 k =-WY sin O+2n I ’ [
(2)
of the slider mass m
(3)
and jl=-w2r
[
cosEJ+-
cos 28 n
1 [* --’
’
slnO+
sin 28 2n
1’
(4)
where n = l/r.
f (8)
I
I
f’ 1
+ 0
H
4
*h-3 : c
cth
-e FIG.
Vol. 311, No. 3, March 1981 Printed in Northern Ireland
2.
189
J. A.4. Gulati and A. C. Rao
ii-r Pulse 1
T”’
t-h-
Pulse 2
n-h-w
Pulse 3
+----I
r
Pulse 1
b-h+
1
0
vh
-22s
------a -e FIG. 3.
The weight of the connecting rod can be replaced by two dynamically equivalent weights, one of which can be added to the slider mass. The second mass thus obtained will not find its location at the crank pin making the analysis difficult, To overcome this situation connecting rod cap can be made heavier (11)so that the C.G. of the resultant mass is located at the crank pin. The slider mass m is thus the effective mass. Let at some instant, 8 be greater than p. Then the d’Alembert-Lagrange equation (or virtual work equation) is (12) .. . (F-mi)i+[k(p--8)--I8]8=0 (5) where I is the effective mass moment of inertia of the crank with respect to the centre of rotation. Equation (5) leads via (3) and (4) to the following equation
+m,2.$(l-~28)+(cosB;cos30)]
(6)
The motion of the crank shaft can be taken as one of oscillation about the mean position represented by /3, the position of driving motor shaft. We may assume, e=/3+Acoswt+Bcos2wt
(7)
:.i = w -A sin wt - 2Bw sin 2wt,
(8)
and ti = -Aw2 cos wt -4Bw2
cos 2wl
(9)
Substituting from (7) and (9) in (6) and replacing F by its value from (2), we 190
JcummlofTneFxankhInstitute Per&?anml Plea9 Ltd.
Time Response of a Flexibly Coupled Slider Crank Mechanism
get, [I(-Aw’
cos wt - 4Bw* cos 2wt) + k(A cos wt + B cos 2wt)] = [J-$+iilisin
($)[cosn(*-cl-i)+cosn(*-c2-z)
+cos n(O-cX-i)]]
X Y sin 8 + [
si;,28] + my2B2[Sin;28 I sit301
+_,,[(i-~2e)+(cose2C053e)l.
The solution has to be necessarily
(10)
numerical and iterative.
First iteration Let us start with the assumptions 8 = p and the velocity 6 = w (uniform velocity). Using the assumptions in (lo), a relation in terms of A and B is obtained. Second relation involving A and B is obtained from (8) by making 6 = w, which gives A sin wt + 2B sin 2wt = 0. (11) Solving the two relations obtained we get A and B, and hence 8 and 4. These values can be used as starting values for the next iteration. Iterations may be carried out till the desired convergency is accomplished. The following numerical example, with arbitrary data, will illustrate the theory. Computation is terminated after second iteration for want of a computer. Nevertheless, the method will be clear. III. Example Design a slider-crank mechanism to operate the three valves as given below: The first valve is operated when the motor shaft angle is 30”. The second valve is operated when the motor shaft has rotated to 80” corresponding to a movement of the slider by 6 cm and the third is operated when the motor shaft rotates to 120” and the slider moves further by 4cm. Each pulse may be taken as rectangular with height as 10 and duration as 6”. Moment of inertia of flywheel = 0.06 kg-cm-set* Slider weight
= 2.039 x lop3 kg-sec*/cm
Spring constant
= 20 kg-cmlrad
Motor speed
= 20 rad/sec
IV. Solution The problem can be converted into a three precision point problem by not assuming the slider position corresponding to 30” position of the motor. Let this be Vol. 311, No. 3, March 1981 Printed in Northern Ireland
191
J. M. Gulati and A. C. Rao x0 and letting (I’-- r2) = K,, we can write for a slider-crank following synthesis equations
mechanism
the
2x, cos 30 + K, = x; 2(x0 - 6)r cos 80 + K1 = (x0 - 6)2 2(x,-
10)r cos 120+K,
Solving the above three simultaneous I = 12.23 cm,
equations,
r = 6.61 cm,
(a)
= (x0- 10)2. we get
and
x0 = 17.50 cm.
First iteration For the motor shaft position p = 30”, use of (10) and (11) gives, 17.62A
+ 10.70B
OSA + 1.732B
= 39.19
(i)
=0
(ii)
Solving (i) and (ii) simultaneously, we get A = 2.69, quent use of these values in (7) and (9) gives 8 = 31.94”
and B = -0.78.
Subse-
4 = -311.24rad/sec2.
Similarly for p = 80”:
8 = 80.01”
8 = -14.87
p = 120”:
8 = 120.37”
d = -297
rad/sec2 rad/sec2
Second iteration Using the new values of 8 in the synthesis equations solving them we get, 1 = 11.44 cm,
r = 6.65 cm,
and
similiar to (a), and
x0 = 16.53 cm.
Using the values of 8 and link dimensions obtained above, the new values of 8 and 6 for various values of fi are: p = 30”:
13= 31.96
i = -359.76
rad/sec2
p = 80”:
e = 79.99”
ti = 3.35 rad/sec2
0 = 120”:
8 = 120.38”
8 = -302.85
rad/sec2.
V. Conclusion Values of 8 obtained in the second iteration are very close to the ones obtained in the first iteration and hence the computations are terminated here. Further, iterations may be carried out if more accuracy is desired. References (1) R. C. Winfrey, “Dynamics of mechanisms with elastic links”, Doctoral Dissertation, UCLA, 1968.
192
Time Response of a Flexibly Coupled Slider Crank Mechanism (2)A. G. Erdman,
G. N. Sandor, and R. G. Oakberg, “A general method for kineto-elastodynamic analysis and synthesis of mechanisms”, J. Engng. Ind., pp. 1193-1204, Nov. 1972. (3) A. A. Sherwood, “Dynamic synthesis of a mechanism with time-dependent output”, J. Mechanisms, Vol. 3, pp. 3540, 1968. (4) J. Mahig, “Minimization of mechanism oscillations through flywheel tuning”, J. Engng. Ind. Vol. 93B, No. 1, pp. 120-124, 1971. (5) J. Mahig and Y. P. Kakad, “Theoretical reduction of a tandem compressor shaft oscillation through tuning and damping”, ASME paper No. 72-Mech-56. (6) G. K. Matthew and D. Tesar, “Synthesis of spring parameters to Satisfy Specified Energy Levels in Planar Mechanisms”, ASME paper No. 76-DET-44. (7) G. K. Matthew and D. Tesar, “Synthesis of Spring Parameters to Balance General Forcing Function in Planar Mechanisms”, ASME paper No. 76-DET-45. (8) J. M. Halter and W. L. Carson, “Mechanism-force system synthesis to obtain a desired motion-time response”, Proceedings Fourth World Congress on the Theory of Machines and Mechanisms, Vol. 2, pp. 399-404, Newcastle upon Tyne, England, 1975. (9) W. L. Carson and R. S. Haney, “Force system structural synthesis by using coupler curves and interactive computer graphics”, ASME paper No. 78-DET-35. (10)M. Kothari and A. C. Rao, “A Timing Device with Pulse Loading”, accepted for publication in J. mech. Engng. Sci., IME, London. (11)F. S. Tse, I. E. Morse and R. T. Hinkle, “Mechanical Vibrations”, p. 186, Prentice Hall of India pvt. Ltd, New Delhi, 1974. (12) B. Paul, “Kinematics and Dynamics of Planar Machinery”, p. 568, Prentice Hall, 1979.
Vol. 311. No. 3, March 1981 Printed in Northern Irelaqd
193