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Nonlinear vibrations of a slider-crank mechanism
N o n l i n e a r vibrations of a slider-crank m e c h a n i s m K. A. Ansari Gonzaga University, Spokane, Washington 99258, USA
N. U. K h a n Department of Mechanical Engineering, Universio, of Petroleum and Minerals, Dhahran, Saudi Arabia {Received April 1985)
A nonlinear vibration analysis of the slider-crank mechanism used in reciprocating machines such as internal-combustion engines and air compressors is formulated by the principles of variational mechanics. A dynamically equivalent system is employed in which the connecting rod is modelled as a pendulum oscillating about a reciprocating piston and an approximate solution is derived by an application of the Ritz averaging method. Numerical results for a typical example are presented and discussed. Key words: mathematical model, slider-crank mechanism, nonlinear vibrations
The slider-crank mechanism is a special case of the four-bar linkage finding wide application in reciprocating machines. Common examples are found in petrol and diesel engines where the gas force acts on the piston transmitting motion through the connecting rod to the crank. It is also used in air compressors where an electric motor or petrol engine drives the crank, tral~mits motion to the connecting rod and the piston which, in turn, compresses the air. Although the rotating masses in a reciprocating engine can be balanced with counter-weights, balancing of the reciprocating masses cannot, in general, be achieved totally. The resulting troublesome vibrations must, therefore, be avoided by other means, but only after a reasonable balance is provided for. Methods of minimizing these vibrations include avoiding resonance, using vibration isolation mountings, and dynamic vibration absorbers. Nonetheless, this inherent unbalance present in certain engines such as the one-, two-, three-, or fourcylinder engines can lead to undesirable and unwanted oscillations of the connecting rod. Many investigators have worked in the general area of this basic mechanism. Some 1'2 have employed various assumptions to linearize the equations of motion, thereby restricting the scope of their investigators. Neubauer a analysed the transverse vibration problem of the connecting rod by neglecting the longitudinal motion and Coriolis acceleration. Viscomi and Ayre 4 analysed the bending vibration of the connecting rod of a slider-crank mechanism and investigated the effects of such parameters as connecting rod length, mass, damping, piston force and frequency on system response. Solutions for
114 Appl. Math. Modelling, 1986, Vol. 10, April
linear as well as nonlinear forms of the equations were obtained, from which it was concluded that tile response could be closely approximated by the first mode since the second flexural mode was seen to be insignificant. Small displacements were assumed in the analysis, however, and the longitudinal vibration of the connecting rod was neglected. Jansinski et al.s studied the vibrations of an elastic connecting rod for a high speed slider-crank mechanism and developed the solution as an asymptotic expansion in terms of a small system parameter using the averaging method of Krylov and Boguliubov. Elastic stability was studied and steady-state solutions for both longitudinal and transverse vibrations were derived. The method generated small amplitude vibrations for small rotational speeds. It must be noted, however, that certain nonlinear coupling terms were assumed to be small and were dropped ill the analysis. Chu and Pan 6 derived nonlinear, coupled, partial differential equations for transverse and longitudinal vibrations of an elastic connecting rod in a slider-crank mechanism operating at high speed. Using the Kantarovich method and weighted residuals, the governing partial differential equations were transformed into ordinary differential equations which were solved by using piecewise polynomials and the fourth-order Runge-Kutta method. The dynamic response was evaluated and the effects of design parameters such as crank length/connecting rod length ratio, natural frequencies, viscous damping and crank rotational speed were investigated. The results generated can be used by designers to predict vibrations of slider-crank mechanisms under high 0307-904X/86/02114-5/$03.00 © 1986 Butterworth & Co. (Publishers) Ltd
Nonlinear vibrations of a slider-crank mechanism: K. A. Ansari and N. U. Khan
speed operating conditions. From the analysis which assumed small deformations, it was concluded that the effect of viscous damping on longitudinal vibrations is, in general, small. The problem relating to stabilty of motion of a connecting rod has been discussed elsewhere. 7,s Zhu and C b e n 9 assumed a constant cross-sectional area for the connecting rod, and using a perturbation method in which the crank radius/connecting rod length ratio was treated as a small parameter, investigated the stability of transverse motion. Using velocity ratio constraints, Myklebust et all ° investigated the dynamic response of a slider-crank mechanism during startup, and obtained useful response information such as crankshaft angular speed, connecting rod angular speed and piston speed time histories by integration of the differential equations of motion obtained. Slider-crank mechanisms have traditionally been designed based on the assumption that the individual links are rigid bodies. For high speeds of operation, however, a perturbative motion about the instantaneous position predicted by the use of the rigid-body assumption is observed. This oscillation can lead to fretting and associated wear of machine components, posing a threat to traditional slidercrank mechanism design methodology. In this paper, the nonlinear vibrations of the slider-crank mechanism are analysed and investigated by resorting to a dynamically equivalent system in which the connecting rod is modelled as a pendulum oscillating about a reciprocating piston. A solution to the problem is then obtained by an application of the Ritz averaging method. Numerical results for a typical example are generated, presented, and discussed.
Approximate values for the massesMp andM c to be lumped at the piston pin and crank pin respectively are: Mp = Dlp ffI
mcR he
1
(l) McR 2
Mc - - R
mcR hp
+- -
l
where rap, m e , and mcR are the masses of the piston plus pin, crank and connecting rod respectively, R is the crank radius, R2 is the distance of the crank centre of gravity from the crank axis, l is the connecting rod length, and h e and hp are the distances respectively of the connecting rod CG to the crank pin and piston pin. In terms of generalized coordinates x t and et, the kinetic energy T, the potential energy V and the Rayleigh dissipation function F R can be written as: T = ~Mp2~ + ~Mc(~ ~ + 12¢~ + 212,~ r coset ) V = ½(k + k e ) x 2 + Mcgl(1 - - c o s ~bt)
(2)
FR =C~ where C, k, and k e represent system damping and stiffnesses, and x t and Cr are total system displacements which are superimpositions of system oscillations, x and ¢, upon the instantaneous rigid body positions x* and ~b* as follows: X t = X "4- X*
and
t~ t =
q~+ q~*
(3)
Free vibrations Analysis Figure 1 shows the mathematical model of a slider-crank
mechanism employed for this analysis. It is a dynamically equivalent two-degree-of-freedom system consisting of two concentrated masses. Although this model has the same inertia forces as the actual connecting rod, it would have a slightly different inertia torque. This approximation has been found to be adequate and valid for ordinary connecting rods.11'12
Using Lagrange's equations, the following nonlinear coupled differential equations governing the free vibration of the system can be derived: (Mp + M e ) 2 t + M e l ( ¢ t
cos Ct --
~2 sin ~t)
+ (k + k e ) x t + 26"b~"t = 0
(4)
l ¢ t + E t cos et + g sin et = 0
where x r and q~t are not 'small'. Using a two-mode approximation, the oscillations x and ¢ can be assumed in terms of the normal modes of the linearized mechanism as follows: 2
p
x=
xt
Z X U ) @ t) /=1
(5) ,/VV~
2
_-- %u~
]=1
,3C
where {X q), c~(/)/l } denotes the ]th norlnal mode corresponding to the linearized forms of equations (4) and the q-}t) are the normal coordinates. A harmonic solution of the form:
C
qj(t) = B i sinjwt,
t
Mc Figure I
Mathematical model of slider-crank mechanism
(j = 1, 2)
(6)
is sought where B i are arbitrary parameters to be determined as functions of the system parameters and the nonlinear natural frequency 6o. Defining the average Lagrangian £ as the time integral of the Lagrangian function taken over a period, the two arbitrary parameters Bi and B2 can be evaluated by the Ritz averaging method using: a£ a£ -0 and - - = 0 (7) aB 1 aB2
Appl. Math. Modelling, 1986, Vol. 10, April
115
Nonlinear vibrations of a slider-crank mechanism: K. A. Ansari and N. U. Khan Application of equations (7) yields the following relationships for the first and second harmonics BI and B2 in terms of the nonlinear natural frequency oa and given system parameters for the fundamental nonlinear mode:
F2(Mp_+Mc)A~X) B, = l L
2
4
McA(2 l)'
-I- -2A ( 1(I)A ~--1
2k"I{1) ] F
2g
+ A (1)-~
l ''=
M~~---~A (:l ) ' j + Lloa=AT-:l)A~')j - ,,(2) ~ F2(Mp+Mc),l I "~
B==I L
+
k<:) l f
2oa2McA ~2)' j + k'2
(8) 4 +
l ''=
loa=,~2)A(=2)j
Forced vibrations There could be various sources of excitation in a reciprocating machine such as transmission of vibration through connections, initial disturbances, and random type of excitation. Since any periodic excitation can, in general, be broken up into a Fourier series of cosine and sine functfons, a simple, yet representative, forced vibration problem is formulated with a harmonic driving force Fo sin oat.t at the piston. A steady state solution is considered in the form of equations (6) in which w is now replaced by the exciting frequency oat-. Two equations can be obtained by the application of a generalized Hamilton's principle which takes into account time averages of given nonconservative forces acting on the system: a£ aBl
N u m e r i c a l results a n d discussion The numerical example attempted relates to a singlecylinder internal-combustion engine and is taken from reference 11 (ch. 20). Given data is as follows: Stroke = 8 in (20.32 cm) Speed = 1500 rpm Weight of crank and crank-pin = 8 lb (3.63 kg) Distance from crankshaft axis to CG of crank and crankpin = 2.5 in (6.35 cm) Length of connecting rod = 16 in (40.64 cm) Weight of connecting rod = 8 lb (3.63 kg) Distance from CG of connecting rod to crank-pin = 4 in (10.16 era) Weight of piston and piston-pin = 7 lb (3.18 kg) k = 10001b/ft (145.94 N cm -1) C = k e = 0 (assumed) The numerical results obtained are presented in Figures 2-4. The free vibration curves sketched in Figure 2 show that the oscillation amplitudes decrease with increase in frequency, indicating a soft-spring type behaviour. They indicate that the amplitudes B~ and B2 become quite large when the nonlinear frequency value is under the fundamental linear natural frequency of the system. From the first harmonic response curves plotted in Figure 3 for
a£ -
QI
OB2
=-Q2
(9)
where Q~ and Q2 are the average nonconservative generalized forces associated with the arbitrary parameters B1 and B2 respectively. After evaluation of these generalized forces using the virtual power concept and the average Lagrangian, equations (9) yield:
10
I
I
0
I
60
I
I
I
100
I
I
140
I
180
220
Percentoge fundomentol l ineor nsturol frequency, t0/w Ln)(xlO0)
C*lraa, + C:ra, + C~ = O, Blstr,= =
(C'r}+C~') in,
(ra, = BdBlst) (10)
(ra = e J B l s t ,
Figure 2 Variation of natural frequency with amplitude in fundamental nonlinear mode
~I = oat-/oaLm) where the following quantities are defined:
D
C ~ -- McA 20) ~BlstoaL(l)r}/-I 3 2 :2 ~ 2
104
~
~
*
10 3
D~__~~
~
-
~
C
- ------
FO -- 10 Ib Fo = lO01b
---~
F° = 5001b
C~ = --[{(Mp + M c ) A p ) + McA(i)~/A(1)2 , 1
+ 2McA O) } oa2L 0)"} + McgA (1)'/(A ~l)l) --kA~ 1)1 Blot
(1 l)
uo,,oo,o
d 10 2
I0 ~
C~' = {2(Mp +Mc)A~2)/(McA (2)~
t~
2 + 2/(A~')A (2)) + 4/.4 (2)' } l 2 oaL(,)
C * = {g/(2 ,A 7)A 7)) -- kA {2)I(2McA 7 ),) } 12
I0"
~'~
~
E
(12)
is the system static deflection obtained by putting oat-= 0 in the first of equations (9). Equations (10) give the harmonic response of the slider-crank mechanism in its fundamental nonlinear mode.
116 Appl. Math. Modelling, 1986, Vol. 10, April
Bronch BCD is
\
E
C~' = - - F o
S Xst = FA {Ol(kA {1)2 -- McgA (O=ll)
--~.~
0.1
0.2
1
2
10
Frequency ratio, rf
Figure 3 First harmonic responseof system in fundamental nonlinear mode conversion factor: Ib force = 4.44 N
Nonlinear vibrations o f a slider-crank mechanism: K. A. Ansari and N. U. Khan
lations, which for the linearized system, would normally be very large in real life and infinite in theory, are, in this case, quite small because of the energy-dissipating effect of the nonlinea,'ities.
q~(t)
o.o~o 2
,-,,.
0.005
0
r
o
,~. o ~-1
"~
-0.005 E
8
-0.010 tn
1
2
3
4
5
Time, s
Figure 4 Time history of steady state system oscillations. Conversion factor: 1 in = 2.54cm
various amplitudes of the exciting force, the following observations are made. While, in the case of the linearized system, all first harmonic response curves for varying amplitudes of the exciting force Fo can be consolidated into one single curve by plotting ra, versus rf, this is not possible in the nonlinear case. For the nonlinear problem, on the other hand, there are two possible solutions for the first harmonic in the region o f r . f < 1, but only the lower one is stable. The unstable or physically impossible branch (BCD) of the curve is indicated. For r.r> 1, there are two complex roots and one real root yielding only a single-valued first harmonic in this region. When the frequency is gradually increased, the amplitude growth will initially proceed along the curve AB and subsequently decrease along the curve BEF. The form of the first harmonic response curves in the fundamental nonlinear mode is essentially the same as that of the linearized mechanism with small damping. The curves, however, bend or sweep over to the left as in the case of a mass on a soft spring, in which the equivalent spring stiffness decreases with increasing amplitudes. The effect of the nonlinearities is thus similar to that of a soft spring which gradually becomes less stiff with increasing amplitudes. The rounding-off of the resonance peaks at B appears to be due to the velocity-dependent terms present in the differential equations of motion of the system. The effect of these terms on the forced response curve is thus the same as that of the presence of small damping on the response of the linearized system. Further, amplitudes become quite small when rf > 2, reinforces the 'small damping' effect. The amplitude at linear resonance (rf = 1) which is infinitely large in the case of the linearized system with no damping has been curtailed by the presence of the nonlinearities. The effect of the nonlinear terms on system response is thus the same as that of the presence of dissipation or damping on linear system response. This is to be expected and only reasonable in real life since some damping will always be present in any physical system. Because of the large system stiffness considered, very small second harmonic response amplitudes have been generated and, therefore, these are not presented. A typical time history of steady-state system oscillations is plotted in Figure 4 for the case where the driving frequency is equal to the fundamental natural frequency of the linearized system. The oscillations, though periodic, will not, in general, be simple harmonic. The maximum system oscil-
Conclusions The Ritz averaging method has been applied to the problem of nonlinear oscillations of a slider-crank mechanism in which the engine connecting rod has been modelled as a pendulum oscillating about a reciprocating piston. Numerical results have been generated and discussed for a typical slider-crank mechanism used in a single cylinder internalcombustion engine. The results of the nonlinear analysis indicate that system oscillations about instantaneous rigid body positions are, in general, small, even when the forcing frequency is equal to the fundamental linear natural frequency. This clearly happens as a result of the shift in the resonance peak that normally occurs in a nonlinear system. Acknowledgement The authors acknowledge the support of the University of Petroleum and Minerals, Dhalwan, Saudi Arabia. They are also grateful to Judy Jech for her accurate typing of the manuscript.
Nomenclature Aft ) Bi B1 B2 Blst C Fo FR hc hp k, k e l L Mc McR Mp qi (21 ra, , ra." r.f R R2 T V x, ¢
XO') cbq)/l
ith component ofjth normal mode for linearized mechanism undetermined parameters in Ritz method first harmonic amplitude second harmonic amplitude system static deflection system damping amplitude of exciting force Rayleigh's dissipation function distance of connecting rod CG to crank pin distance of connecting rod CG to piston pin system stiffnesses length of connecting rod Lagrangian function crank mass mass of connecting rod mass of piston plus pin normal coordinates average nonconservative generalized force associated with Bj amplitude ratios frequency ratio crank radius distance of crank CG from crank axis system kinetic energy system potential energy system perturbations about instantaneous rigid body positions
} /th normal mode for linearized system
co
nonlinear natural frequency forcing frequency OJL(1) fundamental linear natural frequency Average value of a quantity taken over a cycle is represented by a bar over that quantity. Dots denote differentiation with respect to time. coy-
A p p l . M a t h . M o d e l l i n g , 1986, V o l . 10, A p r i l
117
Nonlinear vibrations of a slider-crank mechanism: K. A. Ansari and N. U. Khan
References 1
Kozsevnyikov, S. N. 'The dynamics of machines having flexible members and divided parameters', A Nehezipari Muszaki Egyetem Magyar Nyelvu Kozlemenyei, XII Kotetebol, Miskole, Hungary, 1965, p. 379 2 Meyer zur CapeUen, W. 'Bending vibrations in the coupler of an oscillating crank mechanism', Osterreichisches hlgenieurArchly. 1962, XVI (4), 341 3 Neubauer, A. H. Jr., Cohen, R. and Hall, A. S. Jr. 'An analytical study of the dynamics of an elastic linkage', ASME J. Engng Jbr Industry 1966, 88 (3), 311-317 4 Viscomi, B. V. and Ayre, R. S. 'Nonlinear dynamics response of elastic slider-crank mechanism', ASME J. Engng Jbr bzdustt 9, 1971,93 (1), 251-264 5 Jansinski, P. W., Lee, H. C. and Sandor, G. N. 'Vibrations of elastic connecting rod of a high speed slider-crank mechanism', ASME J. Engng for bldustry 1971, 93,636-644 6 Chu, S. C. and Pan, K. C. 'Dynamic responses of a high speed
118 Appl. Math. Modelling,. 1986, Vol. 10, April
7 8 9 10 11 12
slider-crank mechanism with an elastic connecting rod', AMEJ. Engng Jbr hldustrv 1975, 97,542-550 Seevers, J. A. and Yang, A. T. 'Dynamic stability analysis of linkage with elastic members via analog simulation', Analog Comput. Tech. 1972, pp. 67-74 Badlani, M. and Kleinbenz, W. 'Dynamic stability of elastic mechanisms', ASME Paper No. 78-DET-23, 1978 Zhu, Z. G. and Chen, Y. 'The stability of the motion of the connecting rod', ASME J. Mech., TJ'ansm. Autom. Design 1983, 105,637-640 Myklebust, A., Fernandez, E. F. and Choy, T. S. 'Dynamic response of slider-crank machines during startup', J. Mech. Transm. Autom. DesigTh Trans. ASME 1984, 106, 452-457 Martin, G. H. 'Kinematics and dynamics of machines', International Student Edition, McGraw-Hill, Tokyo, 1969 Shigley, J. E. and Uicker, 1. J. Jr. 'Theory of Machines and Mechanisms', International Student Edition, McGraw-Hill, Tokyo, 1980
N. U. K h a n Department of Mechanical Engineering, Universio, of Petroleum and Minerals, Dhahran, Saudi Arabia {Received April 1985)
A nonlinear vibration analysis of the slider-crank mechanism used in reciprocating machines such as internal-combustion engines and air compressors is formulated by the principles of variational mechanics. A dynamically equivalent system is employed in which the connecting rod is modelled as a pendulum oscillating about a reciprocating piston and an approximate solution is derived by an application of the Ritz averaging method. Numerical results for a typical example are presented and discussed. Key words: mathematical model, slider-crank mechanism, nonlinear vibrations
The slider-crank mechanism is a special case of the four-bar linkage finding wide application in reciprocating machines. Common examples are found in petrol and diesel engines where the gas force acts on the piston transmitting motion through the connecting rod to the crank. It is also used in air compressors where an electric motor or petrol engine drives the crank, tral~mits motion to the connecting rod and the piston which, in turn, compresses the air. Although the rotating masses in a reciprocating engine can be balanced with counter-weights, balancing of the reciprocating masses cannot, in general, be achieved totally. The resulting troublesome vibrations must, therefore, be avoided by other means, but only after a reasonable balance is provided for. Methods of minimizing these vibrations include avoiding resonance, using vibration isolation mountings, and dynamic vibration absorbers. Nonetheless, this inherent unbalance present in certain engines such as the one-, two-, three-, or fourcylinder engines can lead to undesirable and unwanted oscillations of the connecting rod. Many investigators have worked in the general area of this basic mechanism. Some 1'2 have employed various assumptions to linearize the equations of motion, thereby restricting the scope of their investigators. Neubauer a analysed the transverse vibration problem of the connecting rod by neglecting the longitudinal motion and Coriolis acceleration. Viscomi and Ayre 4 analysed the bending vibration of the connecting rod of a slider-crank mechanism and investigated the effects of such parameters as connecting rod length, mass, damping, piston force and frequency on system response. Solutions for
114 Appl. Math. Modelling, 1986, Vol. 10, April
linear as well as nonlinear forms of the equations were obtained, from which it was concluded that tile response could be closely approximated by the first mode since the second flexural mode was seen to be insignificant. Small displacements were assumed in the analysis, however, and the longitudinal vibration of the connecting rod was neglected. Jansinski et al.s studied the vibrations of an elastic connecting rod for a high speed slider-crank mechanism and developed the solution as an asymptotic expansion in terms of a small system parameter using the averaging method of Krylov and Boguliubov. Elastic stability was studied and steady-state solutions for both longitudinal and transverse vibrations were derived. The method generated small amplitude vibrations for small rotational speeds. It must be noted, however, that certain nonlinear coupling terms were assumed to be small and were dropped ill the analysis. Chu and Pan 6 derived nonlinear, coupled, partial differential equations for transverse and longitudinal vibrations of an elastic connecting rod in a slider-crank mechanism operating at high speed. Using the Kantarovich method and weighted residuals, the governing partial differential equations were transformed into ordinary differential equations which were solved by using piecewise polynomials and the fourth-order Runge-Kutta method. The dynamic response was evaluated and the effects of design parameters such as crank length/connecting rod length ratio, natural frequencies, viscous damping and crank rotational speed were investigated. The results generated can be used by designers to predict vibrations of slider-crank mechanisms under high 0307-904X/86/02114-5/$03.00 © 1986 Butterworth & Co. (Publishers) Ltd
Nonlinear vibrations of a slider-crank mechanism: K. A. Ansari and N. U. Khan
speed operating conditions. From the analysis which assumed small deformations, it was concluded that the effect of viscous damping on longitudinal vibrations is, in general, small. The problem relating to stabilty of motion of a connecting rod has been discussed elsewhere. 7,s Zhu and C b e n 9 assumed a constant cross-sectional area for the connecting rod, and using a perturbation method in which the crank radius/connecting rod length ratio was treated as a small parameter, investigated the stability of transverse motion. Using velocity ratio constraints, Myklebust et all ° investigated the dynamic response of a slider-crank mechanism during startup, and obtained useful response information such as crankshaft angular speed, connecting rod angular speed and piston speed time histories by integration of the differential equations of motion obtained. Slider-crank mechanisms have traditionally been designed based on the assumption that the individual links are rigid bodies. For high speeds of operation, however, a perturbative motion about the instantaneous position predicted by the use of the rigid-body assumption is observed. This oscillation can lead to fretting and associated wear of machine components, posing a threat to traditional slidercrank mechanism design methodology. In this paper, the nonlinear vibrations of the slider-crank mechanism are analysed and investigated by resorting to a dynamically equivalent system in which the connecting rod is modelled as a pendulum oscillating about a reciprocating piston. A solution to the problem is then obtained by an application of the Ritz averaging method. Numerical results for a typical example are generated, presented, and discussed.
Approximate values for the massesMp andM c to be lumped at the piston pin and crank pin respectively are: Mp = Dlp ffI
mcR he
1
(l) McR 2
Mc - - R
mcR hp
+- -
l
where rap, m e , and mcR are the masses of the piston plus pin, crank and connecting rod respectively, R is the crank radius, R2 is the distance of the crank centre of gravity from the crank axis, l is the connecting rod length, and h e and hp are the distances respectively of the connecting rod CG to the crank pin and piston pin. In terms of generalized coordinates x t and et, the kinetic energy T, the potential energy V and the Rayleigh dissipation function F R can be written as: T = ~Mp2~ + ~Mc(~ ~ + 12¢~ + 212,~ r coset ) V = ½(k + k e ) x 2 + Mcgl(1 - - c o s ~bt)
(2)
FR =C~ where C, k, and k e represent system damping and stiffnesses, and x t and Cr are total system displacements which are superimpositions of system oscillations, x and ¢, upon the instantaneous rigid body positions x* and ~b* as follows: X t = X "4- X*
and
t~ t =
q~+ q~*
(3)
Free vibrations Analysis Figure 1 shows the mathematical model of a slider-crank
mechanism employed for this analysis. It is a dynamically equivalent two-degree-of-freedom system consisting of two concentrated masses. Although this model has the same inertia forces as the actual connecting rod, it would have a slightly different inertia torque. This approximation has been found to be adequate and valid for ordinary connecting rods.11'12
Using Lagrange's equations, the following nonlinear coupled differential equations governing the free vibration of the system can be derived: (Mp + M e ) 2 t + M e l ( ¢ t
cos Ct --
~2 sin ~t)
+ (k + k e ) x t + 26"b~"t = 0
(4)
l ¢ t + E t cos et + g sin et = 0
where x r and q~t are not 'small'. Using a two-mode approximation, the oscillations x and ¢ can be assumed in terms of the normal modes of the linearized mechanism as follows: 2
p
x=
xt
Z X U ) @ t) /=1
(5) ,/VV~
2
_-- %u~
]=1
,3C
where {X q), c~(/)/l } denotes the ]th norlnal mode corresponding to the linearized forms of equations (4) and the q-}t) are the normal coordinates. A harmonic solution of the form:
C
qj(t) = B i sinjwt,
t
Mc Figure I
Mathematical model of slider-crank mechanism
(j = 1, 2)
(6)
is sought where B i are arbitrary parameters to be determined as functions of the system parameters and the nonlinear natural frequency 6o. Defining the average Lagrangian £ as the time integral of the Lagrangian function taken over a period, the two arbitrary parameters Bi and B2 can be evaluated by the Ritz averaging method using: a£ a£ -0 and - - = 0 (7) aB 1 aB2
Appl. Math. Modelling, 1986, Vol. 10, April
115
Nonlinear vibrations of a slider-crank mechanism: K. A. Ansari and N. U. Khan Application of equations (7) yields the following relationships for the first and second harmonics BI and B2 in terms of the nonlinear natural frequency oa and given system parameters for the fundamental nonlinear mode:
F2(Mp_+Mc)A~X) B, = l L
2
4
McA(2 l)'
-I- -2A ( 1(I)A ~--1
2k"I{1) ] F
2g
+ A (1)-~
l ''=
M~~---~A (:l ) ' j + Lloa=AT-:l)A~')j - ,,(2) ~ F2(Mp+Mc),l I "~
B==I L
+
k<:) l f
2oa2McA ~2)' j + k'2
(8) 4 +
l ''=
loa=,~2)A(=2)j
Forced vibrations There could be various sources of excitation in a reciprocating machine such as transmission of vibration through connections, initial disturbances, and random type of excitation. Since any periodic excitation can, in general, be broken up into a Fourier series of cosine and sine functfons, a simple, yet representative, forced vibration problem is formulated with a harmonic driving force Fo sin oat.t at the piston. A steady state solution is considered in the form of equations (6) in which w is now replaced by the exciting frequency oat-. Two equations can be obtained by the application of a generalized Hamilton's principle which takes into account time averages of given nonconservative forces acting on the system: a£ aBl
N u m e r i c a l results a n d discussion The numerical example attempted relates to a singlecylinder internal-combustion engine and is taken from reference 11 (ch. 20). Given data is as follows: Stroke = 8 in (20.32 cm) Speed = 1500 rpm Weight of crank and crank-pin = 8 lb (3.63 kg) Distance from crankshaft axis to CG of crank and crankpin = 2.5 in (6.35 cm) Length of connecting rod = 16 in (40.64 cm) Weight of connecting rod = 8 lb (3.63 kg) Distance from CG of connecting rod to crank-pin = 4 in (10.16 era) Weight of piston and piston-pin = 7 lb (3.18 kg) k = 10001b/ft (145.94 N cm -1) C = k e = 0 (assumed) The numerical results obtained are presented in Figures 2-4. The free vibration curves sketched in Figure 2 show that the oscillation amplitudes decrease with increase in frequency, indicating a soft-spring type behaviour. They indicate that the amplitudes B~ and B2 become quite large when the nonlinear frequency value is under the fundamental linear natural frequency of the system. From the first harmonic response curves plotted in Figure 3 for
a£ -
QI
OB2
=-Q2
(9)
where Q~ and Q2 are the average nonconservative generalized forces associated with the arbitrary parameters B1 and B2 respectively. After evaluation of these generalized forces using the virtual power concept and the average Lagrangian, equations (9) yield:
10
I
I
0
I
60
I
I
I
100
I
I
140
I
180
220
Percentoge fundomentol l ineor nsturol frequency, t0/w Ln)(xlO0)
C*lraa, + C:ra, + C~ = O, Blstr,= =
(C'r}+C~') in,
(ra, = BdBlst) (10)
(ra = e J B l s t ,
Figure 2 Variation of natural frequency with amplitude in fundamental nonlinear mode
~I = oat-/oaLm) where the following quantities are defined:
D
C ~ -- McA 20) ~BlstoaL(l)r}/-I 3 2 :2 ~ 2
104
~
~
*
10 3
D~__~~
~
-
~
C
- ------
FO -- 10 Ib Fo = lO01b
---~
F° = 5001b
C~ = --[{(Mp + M c ) A p ) + McA(i)~/A(1)2 , 1
+ 2McA O) } oa2L 0)"} + McgA (1)'/(A ~l)l) --kA~ 1)1 Blot
(1 l)
uo,,oo,o
d 10 2
I0 ~
C~' = {2(Mp +Mc)A~2)/(McA (2)~
t~
2 + 2/(A~')A (2)) + 4/.4 (2)' } l 2 oaL(,)
C * = {g/(2 ,A 7)A 7)) -- kA {2)I(2McA 7 ),) } 12
I0"
~'~
~
E
(12)
is the system static deflection obtained by putting oat-= 0 in the first of equations (9). Equations (10) give the harmonic response of the slider-crank mechanism in its fundamental nonlinear mode.
116 Appl. Math. Modelling, 1986, Vol. 10, April
Bronch BCD is
\
E
C~' = - - F o
S Xst = FA {Ol(kA {1)2 -- McgA (O=ll)
--~.~
0.1
0.2
1
2
10
Frequency ratio, rf
Figure 3 First harmonic responseof system in fundamental nonlinear mode conversion factor: Ib force = 4.44 N
Nonlinear vibrations o f a slider-crank mechanism: K. A. Ansari and N. U. Khan
lations, which for the linearized system, would normally be very large in real life and infinite in theory, are, in this case, quite small because of the energy-dissipating effect of the nonlinea,'ities.
q~(t)
o.o~o 2
,-,,.
0.005
0
r
o
,~. o ~-1
"~
-0.005 E
8
-0.010 tn
1
2
3
4
5
Time, s
Figure 4 Time history of steady state system oscillations. Conversion factor: 1 in = 2.54cm
various amplitudes of the exciting force, the following observations are made. While, in the case of the linearized system, all first harmonic response curves for varying amplitudes of the exciting force Fo can be consolidated into one single curve by plotting ra, versus rf, this is not possible in the nonlinear case. For the nonlinear problem, on the other hand, there are two possible solutions for the first harmonic in the region o f r . f < 1, but only the lower one is stable. The unstable or physically impossible branch (BCD) of the curve is indicated. For r.r> 1, there are two complex roots and one real root yielding only a single-valued first harmonic in this region. When the frequency is gradually increased, the amplitude growth will initially proceed along the curve AB and subsequently decrease along the curve BEF. The form of the first harmonic response curves in the fundamental nonlinear mode is essentially the same as that of the linearized mechanism with small damping. The curves, however, bend or sweep over to the left as in the case of a mass on a soft spring, in which the equivalent spring stiffness decreases with increasing amplitudes. The effect of the nonlinearities is thus similar to that of a soft spring which gradually becomes less stiff with increasing amplitudes. The rounding-off of the resonance peaks at B appears to be due to the velocity-dependent terms present in the differential equations of motion of the system. The effect of these terms on the forced response curve is thus the same as that of the presence of small damping on the response of the linearized system. Further, amplitudes become quite small when rf > 2, reinforces the 'small damping' effect. The amplitude at linear resonance (rf = 1) which is infinitely large in the case of the linearized system with no damping has been curtailed by the presence of the nonlinearities. The effect of the nonlinear terms on system response is thus the same as that of the presence of dissipation or damping on linear system response. This is to be expected and only reasonable in real life since some damping will always be present in any physical system. Because of the large system stiffness considered, very small second harmonic response amplitudes have been generated and, therefore, these are not presented. A typical time history of steady-state system oscillations is plotted in Figure 4 for the case where the driving frequency is equal to the fundamental natural frequency of the linearized system. The oscillations, though periodic, will not, in general, be simple harmonic. The maximum system oscil-
Conclusions The Ritz averaging method has been applied to the problem of nonlinear oscillations of a slider-crank mechanism in which the engine connecting rod has been modelled as a pendulum oscillating about a reciprocating piston. Numerical results have been generated and discussed for a typical slider-crank mechanism used in a single cylinder internalcombustion engine. The results of the nonlinear analysis indicate that system oscillations about instantaneous rigid body positions are, in general, small, even when the forcing frequency is equal to the fundamental linear natural frequency. This clearly happens as a result of the shift in the resonance peak that normally occurs in a nonlinear system. Acknowledgement The authors acknowledge the support of the University of Petroleum and Minerals, Dhalwan, Saudi Arabia. They are also grateful to Judy Jech for her accurate typing of the manuscript.
Nomenclature Aft ) Bi B1 B2 Blst C Fo FR hc hp k, k e l L Mc McR Mp qi (21 ra, , ra." r.f R R2 T V x, ¢
XO') cbq)/l
ith component ofjth normal mode for linearized mechanism undetermined parameters in Ritz method first harmonic amplitude second harmonic amplitude system static deflection system damping amplitude of exciting force Rayleigh's dissipation function distance of connecting rod CG to crank pin distance of connecting rod CG to piston pin system stiffnesses length of connecting rod Lagrangian function crank mass mass of connecting rod mass of piston plus pin normal coordinates average nonconservative generalized force associated with Bj amplitude ratios frequency ratio crank radius distance of crank CG from crank axis system kinetic energy system potential energy system perturbations about instantaneous rigid body positions
} /th normal mode for linearized system
co
nonlinear natural frequency forcing frequency OJL(1) fundamental linear natural frequency Average value of a quantity taken over a cycle is represented by a bar over that quantity. Dots denote differentiation with respect to time. coy-
A p p l . M a t h . M o d e l l i n g , 1986, V o l . 10, A p r i l
117
Nonlinear vibrations of a slider-crank mechanism: K. A. Ansari and N. U. Khan
References 1
Kozsevnyikov, S. N. 'The dynamics of machines having flexible members and divided parameters', A Nehezipari Muszaki Egyetem Magyar Nyelvu Kozlemenyei, XII Kotetebol, Miskole, Hungary, 1965, p. 379 2 Meyer zur CapeUen, W. 'Bending vibrations in the coupler of an oscillating crank mechanism', Osterreichisches hlgenieurArchly. 1962, XVI (4), 341 3 Neubauer, A. H. Jr., Cohen, R. and Hall, A. S. Jr. 'An analytical study of the dynamics of an elastic linkage', ASME J. Engng Jbr Industry 1966, 88 (3), 311-317 4 Viscomi, B. V. and Ayre, R. S. 'Nonlinear dynamics response of elastic slider-crank mechanism', ASME J. Engng Jbr bzdustt 9, 1971,93 (1), 251-264 5 Jansinski, P. W., Lee, H. C. and Sandor, G. N. 'Vibrations of elastic connecting rod of a high speed slider-crank mechanism', ASME J. Engng for bldustry 1971, 93,636-644 6 Chu, S. C. and Pan, K. C. 'Dynamic responses of a high speed
118 Appl. Math. Modelling,. 1986, Vol. 10, April
7 8 9 10 11 12
slider-crank mechanism with an elastic connecting rod', AMEJ. Engng Jbr hldustrv 1975, 97,542-550 Seevers, J. A. and Yang, A. T. 'Dynamic stability analysis of linkage with elastic members via analog simulation', Analog Comput. Tech. 1972, pp. 67-74 Badlani, M. and Kleinbenz, W. 'Dynamic stability of elastic mechanisms', ASME Paper No. 78-DET-23, 1978 Zhu, Z. G. and Chen, Y. 'The stability of the motion of the connecting rod', ASME J. Mech., TJ'ansm. Autom. Design 1983, 105,637-640 Myklebust, A., Fernandez, E. F. and Choy, T. S. 'Dynamic response of slider-crank machines during startup', J. Mech. Transm. Autom. DesigTh Trans. ASME 1984, 106, 452-457 Martin, G. H. 'Kinematics and dynamics of machines', International Student Edition, McGraw-Hill, Tokyo, 1969 Shigley, J. E. and Uicker, 1. J. Jr. 'Theory of Machines and Mechanisms', International Student Edition, McGraw-Hill, Tokyo, 1980