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Effects of variable inertia on the damped torsional vibrations of diesel engine systems
Journal of Sound and Vibration (1976) 46(3), 339-345
EFFECTS OF VARIABLE TORSIONAL VIBRATIONS
INERTIA ON THE DAMPED OF DIESEL ENGINE SYSTEMS
M. S. PASRICHA Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221005, India AND W. D. CARNEGIE Department of Mechanical Engineering, University of Surrey, Guildford G U2 5 XH, England (Received 25 September 1975)
In the past the effects of ignoring the variable inertia characteristics of reciprocating engines on the accuracy of torsionalvibration calculations were considered to be negligible. The associated secondary resonances and regions of instability tended to be dismissed by most engineers as interesting but of no importance. The situation changed in recent years since there was evidence of the existence of the secondary resonance effects which could have contributed to a number of otherwise inexplicable crankshaft failures in large slow speed marine engines. The cyclic variation of the polar moment of inertia of the reciprocating parts during each revolution causes a periodic variation of frequency and corresponding amplitude of vibration of reciprocating engine systems. It also causes an increase in the speed range over which resonance effectsareexperienced. In the present paper a study of the effects of damping on the motion of the variable inertia system has been carried out. When the variable inertia effect is allowed for, the equation of motion including the effects of damping is non-linear. For small displacements, the equation can be linearized to predict important characteristics of the motion. Computer methods making use of a numerical analysis process, namely the modified Euler’s equations, have been applied in the investigations. The waveform responses are studied at different speeds of engine rotation. The effects of damping and of variations in the ratio of the average reciprocating inertia to average total inertia on the regions of instability have been investigated. 1. INTRODUCTION
The influence of the reciprocating parts on small torsional vibrations is a maximum when the crank is at the position corresponding to mid-stroke, and is zero when the crank is at top or bottom dead centre. It is common practice to allow for this variation by including one-half of the mass of the reciprocating parts with the mass of the rotating parts to give an equivalent rotating mass concentrated at the crankpin. In reality, a slider crank mechanism is a vibrating system with varying inertia because the effective inertia of the total oscillating mass of each crank assembly varies twice per revolution of the crankshaft. This variation in inertia of the reciprocating parts causes vibratory phenomena which cannot be explained by conventional theory incorporating only the mean values of the varying inertias. Draminsky [l] explained the phenomenon of secondary resonance? in multi-cylinder engines by use of a non-linear theory and showed that this phenomenon occurred due to the t The term “secondary resonance” is used by investigators in this field as a general description for the phenomenon of torsional vibration with variable inertia. 339
340
M. S. PASRICHA AND W. D. CARNEGIE
variation in inertia torques of the system arising from the motion of the reciprocating parts. Carnegie and Pasricha [2] carried out further investigations to explain the secondary resonance phenomenon and an effort was made to understand and present the effects of variable inertia on the vibratory motion of the crank-connecting rod system more clearly than in the past. In recent years several cases of secondary resonance have been found in torsional vibrations in crankshaft systems of multi-cylinder marine diesel engines. Archer [3] and Draminsky [4] have cited typical examples of crankshaft failures of this kind from service. Failures have not occurred in all cases of engines in service which were considered to be susceptible to secondary resonance. In view of this, Carnegie and Pasricha [5] examined the torsional vibration behaviour of a ten-cylinder two-stroke-cycle engine with suspected secondary resonance. It was found that stress magnification due to secondary resonance failed to appear. These authors further investigated the effect on the motion due to variation of inertia ratio and the effects of external excitations acting on a single cylinder engine system and offered an explanation as to why secondary resonance contributed to the failures only in some multi-cylinder engines and not in others [6, 71. The results of the present paper predict the characteristics of motion at different speeds of engine rotation and also the critical speed ranges, the engine being considered as a variable inertia system including the effect of damping. 2. THEORETICAL
CONSIDERATIONS
Figure 1 shows diagrammatically a single cylinder reciprocating engine driving a heavy flywheel A of moment of inertia IA (a list of symbols is given in the Appendix). The crank radius is a and the connecting-rod to crank ratio is assumed to be large such that the reciprocating mass moves with simple harmonic motion. With the mass of the connecting-rod divided into two lumped masses, one at the crank pin Cand the other at the piston crosshead assembly B, the total mass of the reciprocating parts is denoted by A4 and the total moment of inertia of the rotating parts by I. Since the inertia of the flywheel A is very large, its angular velocity can be assumed to be constant. The equation of Lagrange for the co-ordinate B is d(aTji%)/dt
- aTj?W = Qe,
(1)
where T is the kinetic energy of the system and QB includes both conservative and nonconservative forces. The conservative force Qec can be derived from the potential energy
Figure 1. Diagrammatic
arrangement
of engine running gear.
VARIABLE INERTIA TORSIONAL VIBRATIONS
expression V. The non-conservative external forces. Hence where eBC=
341
force Qenc may include internal dissipative forces and
QO= Qee+ Qene,
(2)
-a vlae,
(3)
Qene= -PO.
The kinetic energy, T, is given by T = +&(Z + +Ma’ - $Wz2 COS28)+ +Z,,02.
(4)
The potential energy V of the system is v = +j.i(e - ely.
(5)
Substituting the relations (3), (4) and (5) in equation (1) reduces it to &z + gt4a2 - +a2
c0s 28) + *MU” 8’ sin 20
+ pfi +
p(e - e,) = 0.
(6)
If one makes use of the equations E = $Vu”/( 1 + @M),
I/r2 = j+D’( 1 + &l!fu”),
2CW”= p/(Z + +%fu2),
(7)
where c is the damping ratio, equation (6) becomes 8(1 - Ecos 28) + ad2 sin 28 + 2~0~8 + of(e - e,) = 0.
(8)
Since
e,=ot
and
e=ot+y,
(9)
equation (8) simplifies to y”{l- 8 cos 2(ot + y)} i- E(O+ Jo)’sin 2(ot + y) + 2co,(o + Jo)+ wf y = 0.
(10)
Equation (10) is further simplified by changing the independent variable to r = cot and letting dashes represent differentiation with respect to r : y”{ 1 - Ecos (27 + 2r) + s( 1 + Y’)~sin (27 + 2r) + (2c/r) (1 + y’) + y/r 2 = 0.
(11)
When the second and higher order terms are neglected equation (11) can be linearized into the form ~“(1 -ecos2~)+(2~sin22+2c/r)y’+((l/r~)+2scos22}y=-(2c/r)-~sin22.
(12)
With the effect of damping neglected, c = 0, equation (12) reduces to (1 - scos2r)~” + (2ssin2r)y’ + {(l/r’) + 2ccos22)y = -8sin22.
(13)
3. APPLICATION TO A TYPICAL MARINE ENGINE The following data for a typical single cylinder engine was used to determine the response of the system at different speeds of rotation : equivalent inertia of rotating and reciprocating parts, Z + 3Mu2, (lb in s”) 93,930 900 bore diameter (mm) 1,550 stroke (mm) 169.5 mip (lb/in2) 2,900 bhp 0.544. E With use being made of the modified Euler’s relations of reference [6], equations (12) and (13) were programmed in Algol for solution by a digital computer with the initial conditions y=l atz=Oandy’=Oatz=O.
34’
hl. S. PASRICHA AND M,. 1). CARNEGIE
E
0.6
-
0.5
-
0.4
-
@30.2
-
0.1
/ 2.0
0. I.0
I 3.0
----e-Jo
r
Figure 2. Stability diagram (E - r) for the region Y2 I. with the effect of damping neglected.
I
0.50
I
I
0.55
/
0.60
0.65
0.m
Figure 3. Stability diagram (E - r) for the region r 2 l/2, with the effect of damping neglected
0.6
I.0
1.4
I.8
2.2
26
3.0
3.4
3.8
4.2
4.6
Figure 4. Stability diagram (E - r) for the region Yz 1, with the effect of damping included.
VARIABLE INERTIA TORSIONAL VIBRATIONS
-c=
343
O-006 0.004
1fl-P
Figure 5. Stability diagram (E N r) for the region r N l/2, with the effect of damping included.
Figure 6. Waveform relationship
of y N t for I-= 0.06, s = 0.544 and c = 0.008.
Figure 7. Waveform relationship
of y - t for
r =
0.08, e = 0344 and c = 0.008.
In Figures 2 and 3 are presented two regions of instability in the neighbourhood of r = 1 and r = O-5 as determined from equation (13), the effects of damping being neglected. It was shown [5] that the frequency of vibration in the unstable region in the vicinity of r = 1 is 1 cycle/rev of the crankshaft and in the region of instability for r N 3 the frequency of vibration is 2 cycles/rev of the crankshaft. Outside these regions of instability the responses showed a modulation of amplitude and instantaneous frequency. The authors have further determined the boundaries between stable and unstable regions from equation (12) which includes the effect of damping. The regions of instability are investigated for different values of the coefficient c and for Eranging from 0 to 0% The results of the investigations are shown in Figures 4 and 5. The time responses at two specific speeds of crankshaft rotation corresponding to r = O-06 and O-08 for s = O-544 when c = O-008 are presented in Figures 6 and 7, respectively.
4. DISCUSSION
OF RESULTS
In a system with constant inertia there are a series of well defined critical speeds corresponding to each mode and each order of vibration, but a perusal of Figures 2 to 5 shows that in a system with variable inertia these critical speeds are replaced by a number of regions of instability within which the amplitude of vibration may be limited by the amount of positive damping in the system. The regions of instability occur at approximately r ~1 and r 1: O-5. Engines having substantially heavy reciprocating parts are particularly unsatisfactory from
21. S. PASRICHA
344
FigtIre 8.
Waveform
relakionship
AND
\\‘. I>. C.ARNt_GIE
of ;’ L t tnr I :- 2. c = 0~544 and c = 0.008.
the point of view of minimizing the effects of the reciprocating parts in exciting regions of instability. Figures 4 and 5 show that the stabilizing effect of damping turns out to be much more pronounced when I’: 0.5 than when Y 2 1. These investigations also show that if the value of the damping factor (’ is increased the ranges of instability become narrower. The results closely agree with Ciregory’s [8] observations that when I’2 1 the motion is always stable for the values of E between 0 and 4c and when r z 0.5 the range of E for stable motion is defined byr:=OtoE=42./Z. Although there are no externally applied excitations. Figures 6 and 7 show. after the starting transient, a motion with sustained amplitude at two specific crankshaft speeds corresponding to I’y=0.06 and Y = 0.08 for c = 0.544 and c = 0.008. Further analysis of such theoretical records at different speeds of the engine for c = 0.544 shows that there are two vibration cycles in one revolution of the crankshaft for all values of Y below the unstable region r E 1. The time responses at higher values of V. namely, r = 2, 4, 6. etc., are repetitive complex waveforms. One such complex waveform is shown in Figure 8 for r = 2, c = 0.544 and c = 0.008. The complex waveform motions at higher values of Y, beyond the upper bound of the region of instability for r z 1, are not of great practical interest. The secondary resonance evoked by variable inertia is considered to be detrimental only in cases of diesel engines which are a step towards having have their service speeds in the range of r < 0.2. These investigations a clearer understanding of the behaviour of a variable inertia system. Thus it can be seen that, when r = 1, for a damping ratio c = 0.02 and when Eis greater than 0.1, the motion is unstable, which means that the vibration amplitude can become large. Further investigations are required to study the effect of forcing terms on the system within the unstable region when r = 1 and where the order number is unity, and also with r < 0.2 where the higher order components of the forcing impulse become predominant. 5.
CONCLUSIONS
The results presented in this paper give the analysis of the response of the variable inertia system representing a single-cylinder engine with the effect of damping included. Due to the effect of the cyclic variation of engine inertia of the reciprocating parts, there are two principal regions in which unstable conditions can occur over an appreciable range of engine r.p.m. without any externally applied excitation such as harmonic torque components arising from the cylinder gas pressure. These effects are particularly likely to occur in cases where the moment of inertia of the reciprocating parts is a large proportion of the total moment of inertia. After the starting transient a motion with sustained amplitude is obtained at specific speeds of the engine outside the instability regions. At higher speeds beyond the upper bound of the instability region for r 2 1 the time responses become complex in form. For E = 5.044, the frequency of vibration is two cycles per revolution of the crankshaft for all speeds below the lower bound ofthe unstable region of r 2: 1. in the region of instability for r 2 1, the frequency of vibration is one cycle per revolution of the crankshaft.
VARIABLE INERTIA TORSIONAL
VIBRATIONS
345
The effect of limiting the amplitudes by the positive damping torques which might be present in the system is more pronounced when r 2: O-5than when r 2: 1 and also if the value of the damping factor c is increased the ranges of instability become narrower and the range of E increases for which the motion is always stable. REFERENCES 1. P. DRAMINSKY1961 Acta Polytechnica, Scandinavia (Me lo), Copenhagen. Secondary resonance
and subharmonics in torsional vibration. 2. W. CARNEGIEand M. S. PASRICHA1973 Shipbuilding and Marine Engineering International %, 583-584. Secondary resonance in marine engine systems. 3. S. ARCHER1964 Transactions of the Institute of Marine Engineers 76,73-134. Some factors influencing the life of marine crankshafts. 4. P. DRAMINSKY1965 Marine Engineer and Naval Architect 88, 180-186. Extended treatment of secondary resonance. 5. W. CARNEGIEand M. S. PASRICHA1971 Transactions of the Institute of Marine Engineers 84, 160-167. An examination of the effects of variable inertia on the torsional vibrations of marine engine systems. 6. W. D. CARNEGIEand M. S. PASRICHA1974Journal of Ship Research 18,131-l 38. Effect of forcing terms and general characteristics of torsional vibrations of marine engine systems with variable inertia. 7. M. S. PASRICHAand W. D. CARNEGIE1974 Institute of Physics Stress Analysis Group Annual Conference, 17-19 September 1974. Application of the WKBJ approximation processes for the analysis of the torsional vibrations of diesel engines systems. 8. R. W. GREC~~RY1954 Ph.D. Thesis, University of Durham Non-linear oscillations of a variable inertia system. 9. W. KER WILSONPractical Solution of Torsional Vibration Problems, Vol2. London: Chapman and Hall Limited. APPENDIX
: NOTATION
crank radius moment of inertia of flywheel A moment of inertia of the rotating parts mass of reciprocating parts ratio of the steady angular velocity o to the natural frequency w. of the system neglecting variable inertia effects +Ma’/(Z + *Ma’) angular displacement of crank from top dead centre angular displacement of flywheel A (see Figure 1) torsional stiffness of the crankshaft displacement of torsional motion steady angular velocity natural frequency of the system with variable inertia effects neglected, =@/(I+ $Ma2)]‘tz viscous damping coefficient
EFFECTS OF VARIABLE TORSIONAL VIBRATIONS
INERTIA ON THE DAMPED OF DIESEL ENGINE SYSTEMS
M. S. PASRICHA Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221005, India AND W. D. CARNEGIE Department of Mechanical Engineering, University of Surrey, Guildford G U2 5 XH, England (Received 25 September 1975)
In the past the effects of ignoring the variable inertia characteristics of reciprocating engines on the accuracy of torsionalvibration calculations were considered to be negligible. The associated secondary resonances and regions of instability tended to be dismissed by most engineers as interesting but of no importance. The situation changed in recent years since there was evidence of the existence of the secondary resonance effects which could have contributed to a number of otherwise inexplicable crankshaft failures in large slow speed marine engines. The cyclic variation of the polar moment of inertia of the reciprocating parts during each revolution causes a periodic variation of frequency and corresponding amplitude of vibration of reciprocating engine systems. It also causes an increase in the speed range over which resonance effectsareexperienced. In the present paper a study of the effects of damping on the motion of the variable inertia system has been carried out. When the variable inertia effect is allowed for, the equation of motion including the effects of damping is non-linear. For small displacements, the equation can be linearized to predict important characteristics of the motion. Computer methods making use of a numerical analysis process, namely the modified Euler’s equations, have been applied in the investigations. The waveform responses are studied at different speeds of engine rotation. The effects of damping and of variations in the ratio of the average reciprocating inertia to average total inertia on the regions of instability have been investigated. 1. INTRODUCTION
The influence of the reciprocating parts on small torsional vibrations is a maximum when the crank is at the position corresponding to mid-stroke, and is zero when the crank is at top or bottom dead centre. It is common practice to allow for this variation by including one-half of the mass of the reciprocating parts with the mass of the rotating parts to give an equivalent rotating mass concentrated at the crankpin. In reality, a slider crank mechanism is a vibrating system with varying inertia because the effective inertia of the total oscillating mass of each crank assembly varies twice per revolution of the crankshaft. This variation in inertia of the reciprocating parts causes vibratory phenomena which cannot be explained by conventional theory incorporating only the mean values of the varying inertias. Draminsky [l] explained the phenomenon of secondary resonance? in multi-cylinder engines by use of a non-linear theory and showed that this phenomenon occurred due to the t The term “secondary resonance” is used by investigators in this field as a general description for the phenomenon of torsional vibration with variable inertia. 339
340
M. S. PASRICHA AND W. D. CARNEGIE
variation in inertia torques of the system arising from the motion of the reciprocating parts. Carnegie and Pasricha [2] carried out further investigations to explain the secondary resonance phenomenon and an effort was made to understand and present the effects of variable inertia on the vibratory motion of the crank-connecting rod system more clearly than in the past. In recent years several cases of secondary resonance have been found in torsional vibrations in crankshaft systems of multi-cylinder marine diesel engines. Archer [3] and Draminsky [4] have cited typical examples of crankshaft failures of this kind from service. Failures have not occurred in all cases of engines in service which were considered to be susceptible to secondary resonance. In view of this, Carnegie and Pasricha [5] examined the torsional vibration behaviour of a ten-cylinder two-stroke-cycle engine with suspected secondary resonance. It was found that stress magnification due to secondary resonance failed to appear. These authors further investigated the effect on the motion due to variation of inertia ratio and the effects of external excitations acting on a single cylinder engine system and offered an explanation as to why secondary resonance contributed to the failures only in some multi-cylinder engines and not in others [6, 71. The results of the present paper predict the characteristics of motion at different speeds of engine rotation and also the critical speed ranges, the engine being considered as a variable inertia system including the effect of damping. 2. THEORETICAL
CONSIDERATIONS
Figure 1 shows diagrammatically a single cylinder reciprocating engine driving a heavy flywheel A of moment of inertia IA (a list of symbols is given in the Appendix). The crank radius is a and the connecting-rod to crank ratio is assumed to be large such that the reciprocating mass moves with simple harmonic motion. With the mass of the connecting-rod divided into two lumped masses, one at the crank pin Cand the other at the piston crosshead assembly B, the total mass of the reciprocating parts is denoted by A4 and the total moment of inertia of the rotating parts by I. Since the inertia of the flywheel A is very large, its angular velocity can be assumed to be constant. The equation of Lagrange for the co-ordinate B is d(aTji%)/dt
- aTj?W = Qe,
(1)
where T is the kinetic energy of the system and QB includes both conservative and nonconservative forces. The conservative force Qec can be derived from the potential energy
Figure 1. Diagrammatic
arrangement
of engine running gear.
VARIABLE INERTIA TORSIONAL VIBRATIONS
expression V. The non-conservative external forces. Hence where eBC=
341
force Qenc may include internal dissipative forces and
QO= Qee+ Qene,
(2)
-a vlae,
(3)
Qene= -PO.
The kinetic energy, T, is given by T = +&(Z + +Ma’ - $Wz2 COS28)+ +Z,,02.
(4)
The potential energy V of the system is v = +j.i(e - ely.
(5)
Substituting the relations (3), (4) and (5) in equation (1) reduces it to &z + gt4a2 - +a2
c0s 28) + *MU” 8’ sin 20
+ pfi +
p(e - e,) = 0.
(6)
If one makes use of the equations E = $Vu”/( 1 + @M),
I/r2 = j+D’( 1 + &l!fu”),
2CW”= p/(Z + +%fu2),
(7)
where c is the damping ratio, equation (6) becomes 8(1 - Ecos 28) + ad2 sin 28 + 2~0~8 + of(e - e,) = 0.
(8)
Since
e,=ot
and
e=ot+y,
(9)
equation (8) simplifies to y”{l- 8 cos 2(ot + y)} i- E(O+ Jo)’sin 2(ot + y) + 2co,(o + Jo)+ wf y = 0.
(10)
Equation (10) is further simplified by changing the independent variable to r = cot and letting dashes represent differentiation with respect to r : y”{ 1 - Ecos (27 + 2r) + s( 1 + Y’)~sin (27 + 2r) + (2c/r) (1 + y’) + y/r 2 = 0.
(11)
When the second and higher order terms are neglected equation (11) can be linearized into the form ~“(1 -ecos2~)+(2~sin22+2c/r)y’+((l/r~)+2scos22}y=-(2c/r)-~sin22.
(12)
With the effect of damping neglected, c = 0, equation (12) reduces to (1 - scos2r)~” + (2ssin2r)y’ + {(l/r’) + 2ccos22)y = -8sin22.
(13)
3. APPLICATION TO A TYPICAL MARINE ENGINE The following data for a typical single cylinder engine was used to determine the response of the system at different speeds of rotation : equivalent inertia of rotating and reciprocating parts, Z + 3Mu2, (lb in s”) 93,930 900 bore diameter (mm) 1,550 stroke (mm) 169.5 mip (lb/in2) 2,900 bhp 0.544. E With use being made of the modified Euler’s relations of reference [6], equations (12) and (13) were programmed in Algol for solution by a digital computer with the initial conditions y=l atz=Oandy’=Oatz=O.
34’
hl. S. PASRICHA AND M,. 1). CARNEGIE
E
0.6
-
0.5
-
0.4
-
@30.2
-
0.1
/ 2.0
0. I.0
I 3.0
----e-Jo
r
Figure 2. Stability diagram (E - r) for the region Y2 I. with the effect of damping neglected.
I
0.50
I
I
0.55
/
0.60
0.65
0.m
Figure 3. Stability diagram (E - r) for the region r 2 l/2, with the effect of damping neglected
0.6
I.0
1.4
I.8
2.2
26
3.0
3.4
3.8
4.2
4.6
Figure 4. Stability diagram (E - r) for the region Yz 1, with the effect of damping included.
VARIABLE INERTIA TORSIONAL VIBRATIONS
-c=
343
O-006 0.004
1fl-P
Figure 5. Stability diagram (E N r) for the region r N l/2, with the effect of damping included.
Figure 6. Waveform relationship
of y N t for I-= 0.06, s = 0.544 and c = 0.008.
Figure 7. Waveform relationship
of y - t for
r =
0.08, e = 0344 and c = 0.008.
In Figures 2 and 3 are presented two regions of instability in the neighbourhood of r = 1 and r = O-5 as determined from equation (13), the effects of damping being neglected. It was shown [5] that the frequency of vibration in the unstable region in the vicinity of r = 1 is 1 cycle/rev of the crankshaft and in the region of instability for r N 3 the frequency of vibration is 2 cycles/rev of the crankshaft. Outside these regions of instability the responses showed a modulation of amplitude and instantaneous frequency. The authors have further determined the boundaries between stable and unstable regions from equation (12) which includes the effect of damping. The regions of instability are investigated for different values of the coefficient c and for Eranging from 0 to 0% The results of the investigations are shown in Figures 4 and 5. The time responses at two specific speeds of crankshaft rotation corresponding to r = O-06 and O-08 for s = O-544 when c = O-008 are presented in Figures 6 and 7, respectively.
4. DISCUSSION
OF RESULTS
In a system with constant inertia there are a series of well defined critical speeds corresponding to each mode and each order of vibration, but a perusal of Figures 2 to 5 shows that in a system with variable inertia these critical speeds are replaced by a number of regions of instability within which the amplitude of vibration may be limited by the amount of positive damping in the system. The regions of instability occur at approximately r ~1 and r 1: O-5. Engines having substantially heavy reciprocating parts are particularly unsatisfactory from
21. S. PASRICHA
344
FigtIre 8.
Waveform
relakionship
AND
\\‘. I>. C.ARNt_GIE
of ;’ L t tnr I :- 2. c = 0~544 and c = 0.008.
the point of view of minimizing the effects of the reciprocating parts in exciting regions of instability. Figures 4 and 5 show that the stabilizing effect of damping turns out to be much more pronounced when I’: 0.5 than when Y 2 1. These investigations also show that if the value of the damping factor (’ is increased the ranges of instability become narrower. The results closely agree with Ciregory’s [8] observations that when I’2 1 the motion is always stable for the values of E between 0 and 4c and when r z 0.5 the range of E for stable motion is defined byr:=OtoE=42./Z. Although there are no externally applied excitations. Figures 6 and 7 show. after the starting transient, a motion with sustained amplitude at two specific crankshaft speeds corresponding to I’y=0.06 and Y = 0.08 for c = 0.544 and c = 0.008. Further analysis of such theoretical records at different speeds of the engine for c = 0.544 shows that there are two vibration cycles in one revolution of the crankshaft for all values of Y below the unstable region r E 1. The time responses at higher values of V. namely, r = 2, 4, 6. etc., are repetitive complex waveforms. One such complex waveform is shown in Figure 8 for r = 2, c = 0.544 and c = 0.008. The complex waveform motions at higher values of Y, beyond the upper bound of the region of instability for r z 1, are not of great practical interest. The secondary resonance evoked by variable inertia is considered to be detrimental only in cases of diesel engines which are a step towards having have their service speeds in the range of r < 0.2. These investigations a clearer understanding of the behaviour of a variable inertia system. Thus it can be seen that, when r = 1, for a damping ratio c = 0.02 and when Eis greater than 0.1, the motion is unstable, which means that the vibration amplitude can become large. Further investigations are required to study the effect of forcing terms on the system within the unstable region when r = 1 and where the order number is unity, and also with r < 0.2 where the higher order components of the forcing impulse become predominant. 5.
CONCLUSIONS
The results presented in this paper give the analysis of the response of the variable inertia system representing a single-cylinder engine with the effect of damping included. Due to the effect of the cyclic variation of engine inertia of the reciprocating parts, there are two principal regions in which unstable conditions can occur over an appreciable range of engine r.p.m. without any externally applied excitation such as harmonic torque components arising from the cylinder gas pressure. These effects are particularly likely to occur in cases where the moment of inertia of the reciprocating parts is a large proportion of the total moment of inertia. After the starting transient a motion with sustained amplitude is obtained at specific speeds of the engine outside the instability regions. At higher speeds beyond the upper bound of the instability region for r 2 1 the time responses become complex in form. For E = 5.044, the frequency of vibration is two cycles per revolution of the crankshaft for all speeds below the lower bound ofthe unstable region of r 2: 1. in the region of instability for r 2 1, the frequency of vibration is one cycle per revolution of the crankshaft.
VARIABLE INERTIA TORSIONAL
VIBRATIONS
345
The effect of limiting the amplitudes by the positive damping torques which might be present in the system is more pronounced when r 2: O-5than when r 2: 1 and also if the value of the damping factor c is increased the ranges of instability become narrower and the range of E increases for which the motion is always stable. REFERENCES 1. P. DRAMINSKY1961 Acta Polytechnica, Scandinavia (Me lo), Copenhagen. Secondary resonance
and subharmonics in torsional vibration. 2. W. CARNEGIEand M. S. PASRICHA1973 Shipbuilding and Marine Engineering International %, 583-584. Secondary resonance in marine engine systems. 3. S. ARCHER1964 Transactions of the Institute of Marine Engineers 76,73-134. Some factors influencing the life of marine crankshafts. 4. P. DRAMINSKY1965 Marine Engineer and Naval Architect 88, 180-186. Extended treatment of secondary resonance. 5. W. CARNEGIEand M. S. PASRICHA1971 Transactions of the Institute of Marine Engineers 84, 160-167. An examination of the effects of variable inertia on the torsional vibrations of marine engine systems. 6. W. D. CARNEGIEand M. S. PASRICHA1974Journal of Ship Research 18,131-l 38. Effect of forcing terms and general characteristics of torsional vibrations of marine engine systems with variable inertia. 7. M. S. PASRICHAand W. D. CARNEGIE1974 Institute of Physics Stress Analysis Group Annual Conference, 17-19 September 1974. Application of the WKBJ approximation processes for the analysis of the torsional vibrations of diesel engines systems. 8. R. W. GREC~~RY1954 Ph.D. Thesis, University of Durham Non-linear oscillations of a variable inertia system. 9. W. KER WILSONPractical Solution of Torsional Vibration Problems, Vol2. London: Chapman and Hall Limited. APPENDIX
: NOTATION
crank radius moment of inertia of flywheel A moment of inertia of the rotating parts mass of reciprocating parts ratio of the steady angular velocity o to the natural frequency w. of the system neglecting variable inertia effects +Ma’/(Z + *Ma’) angular displacement of crank from top dead centre angular displacement of flywheel A (see Figure 1) torsional stiffness of the crankshaft displacement of torsional motion steady angular velocity natural frequency of the system with variable inertia effects neglected, =@/(I+ $Ma2)]‘tz viscous damping coefficient