Mechanical Systems and Signal Processing (1999) 13(1), 169–178Article No. mssp.1998.0182, available online at http://www.idealibrary.com on

SHORT PAPER

TIME DOMAIN MODELLING OF ARECIPROCATING ENGINE

H. L B. J. S

Department of Mechanical and Materials Engineering, The University of Western Australia,Australia

(Received October 1997, accepted after revisions June 1998)

This paper describes the application of a time domain systems approach to the modellingof a reciprocating engine. The engine model includes the varying inertia effects resultingfrom the motion of the piston and con-rod. The cylinder pressure measured under operatingconditions is used to force the model and the resulting motion compared with the measuredresponse. The results obtained indicate that the model is very good.

7 1999 Academic Press

1. INTRODUCTION

Crankshaft torsional vibration is inherent in reciprocating internal combustion engines.The torsional excitation of the crankshaft vibration comes from the combustion pressureforce and this is dependent on crank angle. The torsional characteristics also change withcrank angle. Thus the torsional vibration of engines is highly complex and time dependent.The design constraints on crankshafts cause most internal combustion engine systems tohave a number of natural frequencies within the operating speed range.

Excessive torsional vibration can cause crankshaft failure, or damage the front accessorybelt, the gear train or the clutch. It can also interfere with valve timing. Even if thecranskshaft torsional vibration is not large enough to damage components, it may causeexcessive translational vibration of crankshaft main journals which can, due to bearingclearances, cause impact on the main bearings to induce engine noise.

Secondary resonance resulting from the inertia variation with crank angle [1] can causecatastrophic failure for example in diesel generating sets [2] and marine propulsion engines[3]. There have been many publications in this area. Goldsbrough [4, 5] developed atheoretical model for low-speed, multi-cylinder engines to predict the critical speed rangeswithin which torsional instability occurs. In his model he assumed that the sections ofthe crankshaft between cranks were rigid. Due to this assumption the secondary effect ofthe cranks was cancelled because of the offset angle of each crank although he did includea varying inertia term for each crank assembly.

Draminsky [3, 6, 7] developed a theoretical model to explain the secondary resonanceof reciprocating machinery. In his model, Draminsky made assumptions which weresimilar to those made by Goldsbrough but he used a single-cylinder model so that thesecondary inertia variation could not be cancelled by another crank assembly inertia. Fromhis work he found that the secondary resonance effect is a natural physical phenomenonarising from the varying contribution made by the reciprocating parts of each crank

0888–3270/99/010169+10 $30.00/0 7 1999 Academic Press

Piston

Flywheel

Crankassembly

Continuous shaft Flywheel+ +

. . . 170

Figure 1. The torsional system of the engine.

assembly to the total inertia of each crank throw. The then commonly used linear modelcould not reflect this effect. The solution to Draminsky’s equation consisted of vibrationof orders n, n−2 and n+2, where n is an integer. He showed theoretically that an (n−2)-and an (n+2)-order excitation could excite an nth order vibration.

Other work done in this area can be found in a detailed review by Hesterman [8] whohas also presented an exact solution for the torsional acceleration of a reciprocatingmechanism [9] including: piston mass; conrod mass and moment of inertia; crank massand moment of inertia; gravity forces where applicable; cylinder pressure force; and torqueon crank. This model removed many of the simplifications made by other authors.However it was asssumed that all links were rigid, that motion only occurred in one plane,and the large and small end bearings were infinitely stiff. The model has been used inseveral investigations that have involved both time and frequency domain modelling[10–13]. However, experimental results have been limited to the frequency domain andmainly in the prediction of natural frequencies. There remains the need to confirmexperimentally that the model is accurate in the time domain and that is the object of thispaper. A companion paper [1] uses controlled torsional excitation to explore the accuracyof the model further.

2. BACKGROUND THEORY

Receptance modelling in the frequency domain is a well-established technique [14],where complex systems are broken into smaller sub-systems for which the receptance isknown or can be easily determined. Receptance adding techniques then allow the

Figure 2. Sub-system representation of the engine.

Sub-system

1

Sub-system

2

Sub-system

3

Sub-system

1Combined

system

y

x

θ

φ

θ

R

AO

CTCrank

Conrod

Piston

B

Line ofmotionfor piston

171

Figure 3. Addition of sub-systems.

sub-system models to be joined together again and the overall system receptance to beevaluated. The receptance is the ratio of the response over the excitation in the frequencydomain. The systems approach has been limited to the frequency domain and the systemsmodelled have to be approximated as linear systems. The time domain receptance method[15] allows transient responses and the non-linearity of systems to be modelled using asystems approach so that the modelling of complex non-linear systems is made easier.These two systems approaches will be illustrated with reference to the engine that wasmodelled and tested. A simplified view of the engine is shown in Fig. 1. For analysis inthe frequency and time domains this was divided into three subsystems, as shown in Fig. 2.

The subsystem approach then requires that addition proceeds as illustrated in Fig. 3.Two systems are joined first to make a new sub-system and then the same procedureadopted until all sub-systems are added and the whole system known. The addingapproach and relevant equations are well known for the frequency domain and may befound in [14]. The corresponding approach for the time domain may be found in [15].

For the time domain the equations of motion of the crank assembly have been derivedby Hesterman [9]. The torque acting on the crank of a single reciprocating mechanism (see

Figure 4. Schematic of a single cylinder engine. AB= l, AR= jl, OA= r, OC= hr.

850

2

–2

550Frequency (Hz)

×10–6

rad

/Nm

691

0

. . . 172

Fig. 4) which has both angular velocity and acceleration and is subjected to gravity andgas forces is given by

T= I(u)u� + 12I'(u)u� 2 + g(u)+Q(t, u) (1)

where

I(u)= IC +mC (hr)2 + IR0r cos u

l cos f12

+mP (r cos u tan f− r sin u)2 +mR (1− j)2(r cos u)2

+ mR (jr cos u tan f− r sin u)2 (2)

I'(u)=dI(u)du

=2IR$0r cos u

l cos f13

tan f−r2 cos u sin u

(l cos f)2 %−2mR (1− j)2r2 cos u sin u

+ mR (jr cos u tan f− r sin u)×$j(r cos u)2

l cos3 f− r cos u− jr sin u tan f%

+ mP (r cos u tan f− r sin u)×$(r cos u)2

l cos3 f− r cos u− r sin u tan f% (3)

g(u)= g[mP (r cos u tan f− r sin u)+mR (jr cos u tan f− r sin u)+mChr sin u] (4)

and

Q(t, u)=Q(t)(r cos u tan f− r sin u). (5)

I(u) is the system’s mass moment of inertia function, and this is dependent on the crankangle, u. I'(u) is the derivative of I(u) with respect to u. The torque applied to thecrankshaft by the gravity forces of a vertically mounted crank assembly is given by g(u),and the torque on the crankshaft due to a piston loading force is given by equation (5).

Figure 5. Frequency using the continuous model of the shaft.

173

For the time domain analysis, the continuous shaft has to be modelled by a set of inertiasand torsional springs. The number of such elements for an accurate representation of thesystem has to be determined. This decision was made based on the frequency domainmodel as described later. The flywheel is a simple inertia and its governing equations arewell known. For the frequency domain approach, Hesterman [9] used equations (1–5) withsinusoidal excitation at a particular crank angle to obtain the engine receptances. Thereceptances of undamped continuous circular shafts are given by Bishop [14] and thosefor damped shafts are given by Derry [16]. It is thus possible to compare the frequencydomain response for the system with a continuous shaft and also with a lumped inertiashaft so that the appropriate shaft model may be used in the time domain simulation.

3. PRELIMINARY MODELLING

The engine tested was a single cylinder Villiers petrol engine of displacement 119 cc. Itis a single-cylinder, four-stroke carburetted engine. It has one inlet valve and one exhaustvalve. The cam shaft is driven by the crankshaft through gears. The bearings andpiston/bore are lubricated by oil in the sump. The shear modulus of the crankshaft materialand the moment of inertia of the crank assembly were obtained by experiment. The massand moment of inertia of the conrod, crank and flywheel were measured experimentally.The large and small end bearings of the conrod are plain bearings. The large end bearingis split and assembled on to the crank pin. The small end bearing is built into the conrod.The crankshaft is supported by two rolling element bearings.

The main parameters of the engine were found to be: piston mass (including gudgeonpin and rings), mP , 0.1649 kg; conrod mass (including bearings), mR , 0.1040 kg; conrodlength (between centres of big and little ends), l, 98.47 mm; centre of gravity of conrod(from large end centre), 16.2 mm; conrod moment of inertia about centre of gravity, IR ,1.53×10−4 kgm2; crank length, r, 24.91 mm; crank moment of inertia, C, 3.21×10−4

kgm2; cylinder bore, 55 mm; and moment of inertia of flywheel, 9.35×10−3 kgm2.The system of Fig. 2 was modelled first in the frequency domain using the systems

approach and the direct receptance of the engine at co-ordinate 1 was derived and is shownin Fig. 5. The predicted first non-zero torsional natural frequency was 691 Hz. When anequivalent two-inertia and single-torsional-spring system was used in place of thecontinuous shaft, the direct receptance was changed only slightly with the naturalfrequency reduced to 690 Hz. It was concluded that it would be acceptable to model theshaft in the same way for the time domain analysis.

4. EXPERIMENTAL WORK

The experimental set-up is shown in Fig. 6. The engine was run at around 1700 rpm withno load. The engine speed could be adjusted using the throttle. The torsional velocity ofthe front end of the crankshaft was measured using a Dantec laser torsional vibrometer.A small aluminium ring was attached to the end of the crankshaft to provide a betterreflective surface and to increase the diameter of the surface of measurement since the laservibrometer required a minimum diameter of 20 mm for the rotating surface. The cylinderpressure was measured using a Kistler pressure transducer and charge amplifier.

For time domain simulation, synchronisation between the cylinder pressure and thecrank angle was necessary. A steel disc was attached to the rear end (flywheel end) of thecrankshaft. The disc had a small hole in it, which was aligned to an optical encoder whenthe piston was at top dead centre. While the engine was running, an impulse signal wasgenerated by the optical encoder every time the piston reached the top dead centre. The

Laservibrometer

Frequencytracker A/D

A/D

A/D

Engine(pressure

transducer)(optical encoder)

Chargeamplifier

Laserbeam

Crank angle

Cylinderpressure

Angularvelocity

Computer

720

5.09

0Crank angle (°)

Cyl

inde

r pr

essu

re (

MP

a)

TDCTDCTDC

Impulse at top dead centre

. . . 174

Figure 6. Experimental set-up for the engine vibration measurements.

measured cylinder pressure synchronised with the crank angle signal is shown in Fig. 7.The data were acquired using a Macintosh personal computer with a data-acquisition card.The data were then processed using LAB VIEW II, a commercially available computerpackage for data processing.

The measured and simulated torsional vibration of the engine system are shown in Figs 8and 9, respectively. All figures show a complete cycle of the single-cylinder, four-stroke

Figure 7. Cylinder pressure synchronised with top dead centre (TDC).

0.0690

1750 rpm

Time (s)

0.0690

1750 rpm

Time (s)

175

Figure 8. The measured torsional vibration of the crankshaft.

engine speed fluctuation, i.e. two revolutions. The Fast Found Transform (FFT) of themeasured torsional vibration of the crankshaft is shown in Fig. 10.

5. DISCUSSION

The measured first torsional natural frequency of the system obtained from the FFTanalysis was 689 Hz. Compared with the predicted torsional natural frequency of 690 Hz,the discrepancy between the predicted and measured values in the frequency domain is0.15%. The discrepancy between the measured result and what was predicted by thecontinuous model (691 Hz) is 0.3%. It is thus well established that the mean value of theengine inertia and the other system parameters are accurate. It should be noted that theFFT shows the n−2 and n+2 side bands associated with varying engine inertia effects.

Figure 9. Predicted torsional vibration of the crankshaft using the time domain model.

1000

0.8

0300

Frequency (Hz)

An

gula

r ve

loci

ty

800

0.4

400 500 600 700 900

689 Hz

. . . 176

Figure 10. FFT of the measured time response of the crankshaft torsional vibration.

The results obtained in the time domain also show very good agreement. It is concludedthat the engine model is thus accurate as it has already been confirmed [9] fromexperimental measurements in the frequency domain. The frequency domain work has alsoindicated the limitations that arise using simplified engine models [12], as stated in thediscussion and conclusions of that paper. ‘‘The results have shown that when an ‘exact’model is used the well known secondary inertia effects may not be present at both n+2and n−2. This is a relatively common experience with practical measurements and ispossibly the result of the interaction of various effects. However, if a simple model onlyhaving a 2u variation is used both the n+2 and n−2 components appear. The exactmodel also has significant components at one and two times rotational speed because ofthe inertia variation function of the engine. When a 2u model is used the one times enginespeed component is lost. The most interesting result is however the preservation of n+2and n−2 components when the effects of the rate of change of inertia are removed. Thisresults in no components at one and two times engine speed but the n+2 and n−2components remain. This indicates that the secondary effects are the result of the naturalfrequency varying with crank angle at 2u but not because the speed is varying at 2u.’’

6. CONCLUSIONS

The experimental work reported in this paper has shown the accuracy of the enginemodel presented previously and used in frequency domain experiment and analysis. Thetime domain results have indicated that the model is very suitable for modelling in the timedomain. The time domain receptance model used in the analysis has also been confirmed.Though not essential for the simple single-cylinder engine considered it will be ofconsiderable use for more complex engines.

REFERENCES

1. S. J. D, D. C. H and B. J. S 1998 Mechanical Systems and Signal Processingin press. The torsional excitation of variable inertia effects in a reciprocating engine.

177

2. B. J. S 1993 Experimental and Theoretical Mechanics Conference, ETM93, Bandung,Indonesia, 182–190. Torsional vibration—its importance, excitation, measurement andmodelling.

3. P. D 1965 The Marine Engineer and Naval Architect, 22–25. An introduction tosecondary resonance.

4. G. R. G 1925 Proceedings of the Royal Society 109, 99–119. Torsional vibrationsin reciprocating engine shafts.

5. G. R. G and H. B 1926 Proceedings of the Royal Society 113, 259–281. Theproperties of torsional vibrations in reciprocating engine shafts.

6. P. D 1961 Acta Polytechnica Scandinavica. Secondary resonance and subharmonics intorsional vibrations.

7. P. D 1965 Shipbuilding and Marine Engineering International 88, 180–186. Extendedtreatment of secondary resonance.

8. D. C. H and B. J. S 1995 Proceedings of the IMechE 209, 11–15. Secondary inertiaeffects in the torsional vibration of reciprocating engines—a literature review.

9. D. C. H and B. J. S 1994 Proceedings of the IMechE 208, 395–408. A systemsapproach to the torsional vibration of multi-cylinder reciprocating engines and pumps.

10. D. C. H and B. J. S 1992 Proceedings of the IMechE. Conference on Vibrationsin Rotating Machinery, 517–522. The causes of torsional resonance in reciprocating engines andpumps at other than integer multiples of the fundamental excitation frequency.

11. D. C. H and B. J. S 1994 International Mechanical Engineering Congress andExhibition, I.E.Aust. Noise and Vibration Conference 3, 1–6. A comparison of secondary inertiaeffects in a range of reciprocating engines.

12. D. C. H and B. J. S 1995 Proceedings of the 13th IMAC, 387–392. Theconsequences for torsional vibration modelling of representing reciprocating mechanisms bysimplified models.

13. D. C. H and B. J. S 1995 Vibration and Noise ’95, 151–161. Secondary resonancein a single cylinder engine.

14. R. E. D. B and D. C. J 1960 The Mechanics of Vibration. Cambridge: CambridgeUniversity Press.

15. H. L, D. C. H and B. J. S 1994 Transactions of Mechanical Engineering ME19,123–128. A systems approach for modelling in the time domain.

16. S. D and B. J. S 1994 Abstract Proceedings of the VIII International Congress onExperimental Mechanics, 72–73. The torsional vibration of a damped continuous bar.

APPENDIX A: NOMENCLATURE

Refer to Figure 4 for clarification of some of the notation.

A coupling point of crank and conrodB coupling point of conrod and pistonC position of crank’s centre of massg gravityg(u) torque about O due to the gravity forces of the crank assemblyh ratio of length OC to OA on reciprocating mechanismI mass moment of inertiaIC moment of inertia of crank about CIR moment of inertia of conrod about RI(u) variable inertia function of entire crank assembly about OI'(u) dI(u)/duj ratio of length AR to AB, 0 Q jQ 1l length of conrod ABL length of shaftmC mass of crankmP mass of pistonmR mass of conrodO centre line of shafts connecting crank assembliesQ(t) loading force on pistonQ(t, u) torque about O due to the loading force on the piston

. . . 178

r length of crank OAR position of conrod’s centre of massT torquef angular displacement of conrodu angular displacement of cranku� angular velocity of cranku� angular acceleration of crank