Formulation of the equations of dynamic motion including the effects of variable inertia on the torsional vibrations in reciprocating engines, part I

Journal

ofSoundand

FORMULATION INCLUDING TORSIONAL

Vibration ( 1979) 66( 2) I 8 lp 186

OF THE EQUATIONS THE EFFECTS

VIBRATIONS

OF DYNAMIC MOTION

OF VARIABLE

INERTIA ON THE

IN RECIPROCATING

ENGINES, PART I

M. S. PASRICHA Department qf Mechanical Engineering, The Papua New Guinea University of Technology, Papua New Guinea

AND W. D. CARNEGIE Department of Mechanical Engineering, University of Surrey, Guildford GU2 SXH, England (Received 18 November 1978, and in revisedform

30 March 1979)

The torsional vibration problem in reciprocating engines is described by linear differential equations with constant coefficients: that is to say that all the masses are constant, and all the damping forces are proportional to velocity. However, such a representation is only approximate since a crankshaft-connecting rod system is a vibrating system with varying inertia. The total effective inertia of the crank assembly varies twice during each revolution of the crankshaft. Large variations in inertia torques can give rise to the phenomenon of secondary resonance in the torsional vibration of modern marine diesel engines. In recent years several cases of fractures in the crankshafts of large multi-cylinder marine engine systems have been attributed to the phenomenon of secondary resonance. Simplified theories predicted these designs of diesel engines as safe in practice. In view of the importance of the subject of torsional vibration in engineering practice, the formulation of the equations of dynamic motion for a multi-cylinder engine, allowing for variable inertia, is given in the present paper.

1. INTRODUCTION In recent years several cases of secondary resonance have been found to occur in the torsional vibrations in crankshaft systems of multi-cylinder diesel engines. Fractures have occurred in the crankshafts of some nine, ten and twelve cylinder marine propulsion engines which have been attributed to this phenomenon of secondary resonance: that is to say the possibility of an nth order critical of small equilibrium amplitude occurring at or near resonance with the service speed being excited by large resultant engine excitations of order (n-2) and (n+2). Measurements taken in such cases indicated torsional vibration stresses at the service speed much greater than those calculated by the normal linear vibration methods with the effect of variable inertia neglected. In view of the importance of the torsional vibration phenomena, Goldsbrough [l, 21 carried out a theoretical and experimental study to examine the effect of the reciprocating parts in producing or modifying the vibrations. After a gap of several years Gregory [3] 181 0022460X:79/1 30181f06 $02.00/O @ I979 Academic Press Inc. (London) Limited

182

M. S. PASRICHA

AND W. D. CARNEGIE

carried out further investigations. He constructed approximate solutions of the non-linear equation for an idealized single cylinder engine system and confirmed his theory by experimental tests. An interesting feature of his analysis was that under certain circumstances the amplitude of oscillation was predicted to be a multiple-valued function of the speed of rotation. Expressions for characteristic numbers in free motion defining the unstable regions were also derived. In unstable regions characteristically non-linear phenomena such as oscillation hysteresis and jumps in amplitude and phase were found to occur if the damping was light. Brook [4] demonstrated experimentally in a single degree of freedom variable inertia system the existence of regimes of shock-excited motions whose amplitudes were greater than those which occurred spontaneously under normal conditions of excitation. Porter [S] used a variant of Kryloff and Bogoliuboff’s method to analyze the periodic vibrations of a non-linear two-degree-of-freedom system which was an idealization of the dynamic system of a two-cylinder-in-line reciprocating engine. It was shown that there were two principal critical speed ranges associated with each normal mode of the system within which periodic harmonic or subharmonic vibrations of large amplitude could occur as a result of variable inertia excitation. Draminsky [6] explained the phenomenon of secondary resonance by the use of a nonlinear theory and stated that it was large second order variations in inertia which were responsible for some failures in crankshafts of multi-cylinder marine engines. He [7, S] suggested that in practice secondary resonance in torsional vibration occurred only for resonance with the lower order secondary component, (n-2) and not the higher order component, (n+2), and a method was described to calculate the nth order stress in the shaft including this effect. Meanwhile Archer [9] added further information by citing examples of crankshaft failures in large ten-cylinder and twelve-cylinder marine engines and the Draminsky calculations in such cases clearly demonstrated the existence of a phenomenon whereby an otherwise innocuous critical in the region of the service speed could be considerably magnified by interaction with a powerful, but non-resonance excitation of the (n-2) harmonic type. Failures have not occurred in all cases of engines in service which were considered to be susceptible to secondary resonance and it was found that Draminsky’s work served only to indicate, in very broad terms, the circumstances in which adverse secondary resonance effects could be anticipated. Carnegie and Pasricha [lo] examined a ten-cylinder, two stroke cycle engine with suspected secondary resonance which turned out to be a case in which the predicted Draminsky resonance magnification failed to appear when measurements were taken. It is against this background that Carnegie and Pasricha studied work done by the researchers on the non-linear equations predicting the width of the regions of instability and the existence of the non-linear nature of motion in the unstable regions. Pasricha and Carnegie [ 1l-l 51 provided a systematic appraisal of the problem by use of numerical analysis techniques and in their investigations predicted amplitudes, frequencies, critical speed ranges and the shapes of the complex waveforms at different speeds of the idealized-single-cylinder engine. Investigations were carried out to study the effects of forcing terms which had direct relevance to the study of secondary resonance phenomenon. They also offered a possible explanation as to why secondary resonance contributed to the failures only in some multi-cylinder engines and not in others. But the aim of obtaining a full understanding of the phenomenon of secondary resonance in multi-cylinder engines, without simplifying the system into an equivalent single-cylinder system as suggested by Draminsky, remains to be achieved. Further, it will

TORSIONAL

VIBRATIONS

IN ENGINES,

183

I

be extremely useful to present the results in a graphical form in terms of harmonic order numbers or differences. It is felt that a “graphical plot” showing the variables, inertia ratio and speed, together with the corresponding responses in harmonic order form would clearly indicate the regions of practical interest relating to secondary resonance. As a first step in this direction the equations of motion for a multi-cylinder engine are presented and expressed in a form convenient for solution by computer. The purpose of this paper is thus to present such a formulation of the equations of motion for multicylinder engines. The present paper is thus an interim report by the authors of progress of the work to date.

2. KINETIC ENERGY AND POTENTIAL ENERGY OF A MULTI-CYLINDER RECIPROCATING ENGINE Although the failures of crankshafts have mainly been reported in eight, ten and twelve cylinder engines in service, a multi-cylinder reciprocating engine having II cylinders, driving a heavy flywheel, is considered here for the derivation of equations of motion. The radius of each crank is a and the connecting rod to crank ratio is assumed to be large such that the reciprocating masses move with simple harmonic motion. With the mass of the connecting rod divided into two lumped masses, one at the crank pin and the other at the piston cross-head assembly, the total mass of the reciprocating parts per cylinder is denoted by M and the total moment of inertia of the rotating parts by I. Since the inertia of the flywheel, IA, is very large, its angular velocity, o, can be assumed to be constant. It is convenient to measure the rotations of the cranks relative to the unstrained equilibrium configuration of the system which exists when the crank in the cylinder line nearest to the flywheel is in its top dead centre position. At any time t when the system is rotating and vibrating, the rotations of the cranks from their datum positions will be denoted by &. Ci represents torsional stiffnesses of the crankshaft as shown in Figure 1 and 4 is the phase angle between the cranks. (A list of notation is given in the Appendix.) The kinetic energy Tof the system is thus given by T = +ZAco2+

Z + $Ma2 - iMa

cos 2 (?+

th)]f?.

(1)

The potential energy U of the system is U = ~ 4Ci(8i-8j_l)2. i=l

Figure

1. Diagram

of the crankshaft.

(2)

184

M. S. PASRICHA

AND W.

D. CARNEGIE

The equations of Lagrange for the co-ordinates Bi are (3) Therefore the generalized inertia forces associated with Bi are given by -gg+g

=

[-{I+~Ma’-~Ma’cos2(~+ei)}8i -{Ma2sin2(++tIi)j~~]+[+Ma2Bisin2(++Bi)]. (4)

Similarly the generalized elastic forces are

--au = -ci(ei-ei-,) aei

+

(5)

ci+l(ei+~-ei),

the term Ci + 1(di + 1 - Oi) being omitted in the nth equation. Substituting the relations (4) and (5) in equation (3) gives -

I+$Mrr”-tMa2cos2(2++B,)}Di-{Ma2sin2(++Bi)}~~]

[i

+[~A4~2&sin2(~+ei)]-ci(ei-ei_l~+ci+l~ei+l-ei~ i =

Equation (6) reduces to

= 0, (6)

1,2 ,..., n.

r+*Mui-fMa2c0s2(++f3,)}8,++{Muisin2(~+Ri)~8~ i +Ci(8,-Oi_,)

Since

8, = ot,

- Ci+,(ei+,-0,)

= 0.

(7)

ei-l = Wt -yi_i,

8, = Ot + yi,

ei+l = Wt +yi+l,

(8)

equation (7) becomes 27ci-1 I + $Ma2 - $Ma2 cos 2 ~ + COt+ yi n

2ni-1 ____ n

-Ci+1(0t+yi+,-_ot-_yi)

+ CUt+ yi

i;i

(O+ji)’ + Ci(Ut+yi-wt-yi-,)

= 0.

(9)

When the second and higher order terms are neglected equation (9) can be linearized into the form I+iMa2-fMa2cos2(~+cX)~Yi

+ ci(Yi-Yi-l)-

ci+I(Yi+l -Yi) = -_to’wisin2(2q+ot).

(10)

TORSIONALVIBRATIONSIN ENGINES, I

18.5

Upon making use of the relationships & = fMa2/(Z +$Va2),

l/v? = Ci/02(Z +@fa2),

l/v?+ I = ci+ l/h12(I +$kfa2),

(11)

changing the independent variable to r = at and letting dashes represent differentiation with respect to r, equation (10) becomes {l -,,os,(~+~)},l+i,,sin2(~+~)ili +jZ,~os2(~+i)}7i+(~i-v~i-l)_(~~:~+~~i)

= -.ssinZ?(++r),

(12)

where y0 = 0. Also it may be noted that the term -(yi+ 1 -y,)/$+ when writing the nth equation as (i + 1) becomes greater than n.

1 should be omitted

3. CONCLUSIONS The general equations of dynamic motion are derived for multi-cylinder engines in linearized form. These equations take into account the second-order variation in inertia and govern the vibratory motion in multi-cylinder engines when the gas pressure in the cylinders is omitted. The inertia and the elastic forces associated with generalized coordinates have also been derived. Solutions of the equations of motion for the multi-cylinder engine case will be used in a series of later papers to verify the validity of the procedure of using an equivalent value of inertia ratio and thus applying to a multi-cylinder engine the methods used for the investigations on a single cylinder engine. Solution of the equations will also provide a basic understanding of the secondary resonance effect in multi-cylinder reciprocating engines.

REFERENCES R. GOLDSBROUCH 1925 Proceedings of the Royal Society 190,99-106. Torsional vibrations in reciprocating engine shafts. G. R. GOLDSBROUGH1926 Proceedings qf the Royal Society 113, 259-264. The properties of torsional vibration in reciprocating engine shafts. R. W. GREGORY1954 Ph.D. Thesis, University ofDurham. Non-linear oscillation of a system having variable inertia. D. L. BROOK1958 Ph.D. Thesis, University of Cambridge. Shock-excited non-linear oscillation of a system having variable inertia. B. PORTER 1965 Journal Mechanical Engineering Science 7, 101-113. Non-linear torsional vibration of a two-degree-of-freedom system having variable inertia. P. DRAMINSKY1961 Acta Polytechnica, Scandinavica, Me 10, Copenhagen. Secondary resonance and subharmonics in torsional vibration. P. DRAMXNSKY 1965 Shipbuilding and Marine Engineering International 88, 22-26. An introduction to secondary resonance. P. DRAMINSKY1965 Shipbuilding and Marine Engineering International 88, 180-l 86. Extended treatment of secondary resonance.

1. G.

2. 3. 4. 5. 6. I. 8.

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M. S. PASRICHA AND W. D. CARNEGIE

9. S. ARCHER 1964 Transactions of the Institute of Marine Engineers 76, 73-134. Some factors influencing the life of marine crankshafts. 10. W. CARNEGIEand M. S. PASR~CHA1971 Transactions of the Institute of Marine Engineers 84, 16&167. An examination of the effects of variable inertia on the torsional vibrations of marine engine systems. 11. W. CARNEGIEand M. S. PASRICHA1973 Shipbuilding and Marine Engineering International 96, 583-584. Secondary resonance in marine engine systems. 12. M. S. PASRICHAand W. D. CARNEGIE1976 Journal of Ship Research 20, 32-39. Torsional vibrations in reciprocating engines. 13. W. CARNEGIEand M. S. PASRICHA1974 Journal of Ship Research 18, 131-138. Effect of forcing terms and general characteristics of torsional vibrations of marine engine systems with variable inertia. 14. M. S. PASRICHAand W. D. CARNEGIE1974 Conference of the Institute of Physics, University of Aston in Birmingham. Application of the WKBJ approximation processes for the analysis of the torsional vibrations of diesel engine systems. 15. M. S. PASRICHAand W. D. CARNEGIE1976 Journal of Soundand Vibration 46,339-345. Effects of variable inertia on the damped torsional vibrations of diesel engine systems.

APPENDIX: a crank radius I, moment of inertia of flywheel A Z moment of inertia of the rotating parts per cylinder M mass of reciprocating parts per cylinder E &Ma’/(Z +*Ma’) Bi angular displacement of ith crank from its datum position

NOTATION B,, angular displacement of flywheel A Ci torsional stiffness between the ith and (i - 1)th crank (see Figure 1) o steady angular velocity of crankshaft yi displacement of torsional motion of the ith crank n number of cylinders