Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 4, 39, 2001 GEARBOX DYNAMIC MODELLING Walter Bartelmus Machinery Systems Division, Wrocław University of Technology e-mail: bartel@ig.pwr.wroc.pl The paper reports on current developments in gearbox dynamic model- ling. It refers to Müller’s one-mass two-parameter (stiffness and dam- ping) gearbox model with rectilinear vibration. The paper shows that there is a need to develop a new model, which would incorporate tor- sional vibration. The paper refers to the previous papers on gearbox dynamic modelling published by the author in journals and conference proceedings. In the present paper, the influence of the clutch damping coefficient and one random parameter value from the three-parameter errormode, and that of the interaction between error parameters on the vibration generated by a gearbox system, is analyzed. Key words: gearbox model, mathematical modelling, computer simu- lation 1. Introduction Themodelling of gearbox dynamic behaviour belongs to the fundamental problems of mechanical system modelling. The problem has been the subject of many papers in Poland and abroad:Wang (1974), Mark (1978), Velex and Maatar (1996), Smith (1998). The model presented by Müller in 1979 has received a lot of attention in Poland. It was discussed, for example, by Ryś (1977), Wilk (1981), and Dąbrowski (1992). The model is a two-parameter (stiffness and damping) model in which the inertia of two wheels has been reduced to onemass equivalent to a one-stage gearbox. The author of the present paper has found that more sophisticated mo- dels are needed to describe the gearbox dynamics properly; Bartelmus (1994, 2000). Mathematical modelling and computer simulation can be applied to 990 W.Bartelmus gearbox dynamics to support diagnostic signal evaluation for diagnostic infe- rence.This is themain aimof thepresent research.The computer simulation is based on amathematicalmodel developed byBartelmus (1994, 2000). General information on gearing needed for the computer simulation of gearbox beha- viour was given by Bartelmus (2000). Some results of computer simulations supportingdiagnostic inferencewere presented byBartelmus (1996-1999). The papers show thatmathematicalmodelling and computer simulation enable the detailed investigation of the dynamic properties of a gearing system. All the basic factors such as: design, technology, operation and change of the gearing system conditions which have a bearing on the vibration generated by a ge- aring can be investigated. The causes of vibration in gearboxes aremainly the tooth errors and vibration is an indication of them. The computer simulation results are referred to the laboratory investigation results presented byRettig (1977) and to the field measurements reported by Penter (1991), Bartelmus (1992), and Tuma et al. (1994). As mentioned above, the vibration of a gearbox indicates that there are tooth errors in it. The errors appear at the production stage. The nature of the gear-wheel interaction is such that non-linear phenomena occur caused by friction, intertooth backlash, impact-like intertooth forces and periodic chan- ges in tooth stiffness. As a result, the intertooth forces may exceed the force values, which follow from the gearbox system’s rated moment. Mathematical description enabling to include these phenomena in the equation of motion is given by Bartelmus (2000). The intertooth forces increase dramatically under unstable conditions. A one-stage gear system operates under resonance conditions and is unstable when the gearbox system’s mesh frequency is equal to its natural frequency. In such conditions, the intertooth forces are two times greater than the ra- ted forces. The phenomenon of resonance has not been investigated fully for gearbox systems. Computer simulations reveal that conditions similar to the ones occurring at resonancemay result as errors (pitting, scuffingof teeth flanks and failure of bearings) increase during the service of a gearbox system. In thepresentpaper, current developments in gearboxmodelling are presentedwith reference to the previous papers by the present author. It is shown that a flexible clutch and an errormode randomparameter have an influenceon thegearbox stability (teeth separation). An error mode is described by several parameters, i.e. maximum error value, shape of error plot and random error fluctuation depth. Gearbox dynamic modelling 991 2. Modelling of gearbox system TheMüller (1979) one-stage gearbox model is shown in Fig.1. Fig. 1. One-mass, two-parametermodel of gearbox byMüller (1979), where kz – gearing’s stiffness, lz – bearing’s stiffness, εα – tooth contact index, Yb1 – input tooth contact ordinate, Yb3 – output tooth contact ordinate, pb – gear base pitch, ∆p – gear base pitch error, Ybjgp – gear pitch error, v – pitch line velocity, P – intertooth force It is a two-parameter (stiffness anddamping)model.The inertia of the two gear-wheels is reduced to onemass.Themotion of themass is equivalent to the relative motion of the two gear-wheels. The motion is caused by the relative motion of springs (having different lengths) hitting the mass. The motion of the springs with velocity v [m/s] is equivalent to the pitch-line velocity of the wheels.As one can see in themodel shown inFig.1, themotion of themasshas no influenceon the instant changeof v as inactual gearboxes.Thisweakness of theMüller (1979)model and lack of possibility of buildingmultistage gearbox 992 W.Bartelmus Fig. 2. Two-wheel, two-parametermodel of gearbox models made the author look for a new model. It is more convenient to use a model with the rotary motion of the wheels and torsional vibration, and thus to overcome the weakness of the Müller model. The simplest model of this kind is shown in Fig.2. The rotation equations of motion may be given by statements according to the second Newton’s law of motion I1pϕ̈1 =M1−r1(F +Ft)+Mzt1 (2.1) I2pϕ̈2 = r2(F +Ft)−M2−Mzt2 To get closer to reality, a more sophisticated model is considered. The model is shown in Fig.3. The mathematical model of torsional vibration of the system is given by the following equations Isϕ̈1 =Ms(ϕ̇1)−M1−M1t I1pϕ̈2 =M1+M1t−r1(F +Ft)+Mzt1 (2.2) I2pϕ̈3 = r2(F +Ft)−M2−Mzt2 Imϕ̈4 =M2−Mr Gearbox dynamic modelling 993 Fig. 3. Systemwith one-stage gearbox The values of forces andmoments are given by M1 = k1(ϕ1−ϕ2) M2 = k2(ϕ3−ϕ4) M1t =C1(ϕ̇1−ϕ̇2) (2.3) Using the functions min and max one may write a formula for the inter- tooth force F = kz(pom,g) ( max ( A− lu+E(pom,a,e,r), min ( A+ lu+E(pom,a,e,r),0 ) )) (2.4) Ft =Cz(A ′) A= r1ϕ2−r2ϕ3 where ϕ,ϕ̇,ϕ̈ – rotation angle, angular velocity, angular acceleration Ms(ϕ̇) – electric motor driven moment characteristic M1,M2 – moments of shaft stiffness 994 W.Bartelmus Is,Im – moments of inertia for electricmotor anddrivenma- chine M1t – dampingmoment of clutch/coupler C1 – damping coefficient of the coupler F,Ft – stiffness and damping intertooth forces k1,k2 – stiffness of shafts Mzt1,Mzt2 – intertooth moment of friction, Mzti =Tρi, i=1,2 T – intertooth force of friction, Fig.2 kz(pom,g) – gearing stiffness function r1,r2 – gear base radii lu – intertooth backlash E(pom,a,e,r) – error mode function pom,a,e,r – parameters of error function, pom = frac(ϕ2z1/(2π)), where ”frac” denote fractio- nal part z1 – number of teeth in the pinion. A full description of the model is given in Bartelmus (1994) and after modification – in Bartelmus (2000). For a given pair of teeth, the value of an error is random and it can be denoted by e(random)= [1−r(1− li)]e (2.5) where e – maximum error value r – coefficient of error scope, scope (0−1) li – random value, scope (0−1). A symbolic description of an error characteristic (errormode) is E(a,e,r), parameters of the errormodeweredescribedearlier above, a is given inFig.4b and Fig.4c, changes in the range (0−1), and it indicates the position of the maximum error value on the line of action. As an example, an error mode for E(0.5,10,0.3) is given in Fig.4b and for E(0.1,10,0.3) in Fig.4c. As it follows from the above discussion, gearing’s dynamic properties de- pend on several factors. The factors may be divided, according to Bartelmus (1992, 1998a), into: • design factors • technology factors • operation factors • change of condition factors. Gearbox dynamic modelling 995 Fig. 4. (a) Stiffness function for gearing, (b) error function for errormode E(0.5,10,0.3), (c) error function for error mode E(0.1,10,0.3) Numerical solutions of the differential equations are obtained bymeans of CSSP (Continuous System Simulation Program) developed by Siwicki (1992), using the England procedure of integration. The procedure is a general pro- cedure of the Runge-Kutta type and it assures the stability of integration, even in the case of discontinuity, as well as it enables error estimation and the automatic change of the integration step. 3. Results of computer simulations The results of computer simulations published by the author (Bartelmus, 1996-2000) have been compared with the results of rig investigations by Ret- tig (1977) and field investigations by Penter (1991), Bartelmus (1992), Tuma (1994). Asmentioned above, the aim of the present paperwas to study the in- fluence of random intertooth errors on the vibration generated by the gearbox system (Fig.3) for different clutch damping coefficients: C1 =0 and C1 > 0. 996 W.Bartelmus Fig. 5. (a) Plot of Kd for error modeE(0.5,12,0.3), stiffness coefficientC1 =0; (b) plot of Kd forE(0.5,15,0.3),C1 =0; (c) plot of Kd forE(0.5,15,0.3), C1 =1000; (d) plot of Kd forE(0.5,20,0.3),C1 =1000; (e) plot of Kd for E(0.5,20,0.15),C1 =1000; (f) plot of Kd forE(0.1,20,0.3),C1 =1000 Gearbox dynamic modelling 997 The random intertooth error is described by error mode E(a,e,r), where e(random) is given by (2.5). Author’s previous studies on gearing dynamics were for the error mode described by two parameters, E(a,e) and C1 = 0. According to Bartelmus (1996), an increase in intertooth error e [µm] causes a linear increase of dynamic factor Kd, but only to a certain value of e. At this value, a nonlinear effect caused by unstable running (teeth separation) is observed. Dynamic factor Kd is defined as a ratio of the current intertooth force to the rated intertooth force. The value of e at which unstable running (teeth separation effect) is observed depends also on the value of parame- ter a in error mode E(a,e). The unstable running of a gearbox occurs when Kd > 2. Also parameter r, which specifies the relative scope of the interto- oth fluctuation, is connected with the instability. In conditions when the error mode is given by E(0.5,12,0.3) and C1 =0, the system shown in Fig.3 runs unstably (Kd = 1.8, Fig.5a). If the error mode is changed to E(0.5,15,0.3) and C1 =0, the running of the gearbox systemwill be unstable, as shown in Fig.5b, for a constant rated rotation and a constant outer load. Fig.5 shows period (1) in which gear rotation increases from 0 to 107rad/s. No external load is applied to the gearbox system, the gearbox system is loaded only by the inertia forces. In period (2) the gearbox system runs without any load. In period (3) the outer load increases from 0 to the rated load. In period (4) the gearbox system runs under a stable rated load. In the present paper, pe- riod (4) of the gearbox system run is the focus of attention. Bearing in mind that the damping clutch coefficient has an influence on the gearbox system dynamics, the plot of Kd for a run at E(0.5,15,0.3) and Cs =1000Nmswill be as the one shown in Fig.5c. The plots for gearing conditions described by E(0.5,20,0.3), E(0.5,20,0.15) and C1 = 1000Nms are shown in Fig.5d and Fig.5e. The gearbox system runs then at slight instability: Kd =2.4 and 2.3. If parameter a is decreased from 0.5 to 0.1 at errormode E(0.1,20,0.15) and C1 = 1000Nms, the plot of factor Kd is like the one shown in Fig.5f, where dynamic factor Kd = 4.7. If parameter e is further increased, the value of the dynamic factor does not change. 4. Conclusions The paper has shown a detailed analysis of amutual influence of the error mode parameters and influence of a clutch damping on a dynamic factor Kd, which reveals the gearing condition. The error mode is described by three parameters: a – shape parameter, e –maximumvalue of the error, r – scope 998 W.Bartelmus of the error random change. The analysis proves that, in general conditions of a gearing change (change of e and r), the linear increase of Kd does not hold, as it is stated in Bartelmus (1992) (experiment on a real object) and 1996 (computer simulation experiment). In the experiment on a real object and in the computer simulation experiment, only the value of e was taken into consideration. References 1. BartelmusW., 1992,VibrationConditionMonitoring ofGearboxes,Machine Vibration, 1, Springer-Verlag London Limited, 178-189 2. Bartelmus W., 1994, Computer Simulation of Vibration Generated by Me- shing of ToothedWheel forAidingDiagnostic ofGearboxes,Conference Proce- edings ConditionMonitoring ’94 Swansea, UK,PINERIDGEPRESS, Swansea, UK, 184-201 3. Bartelmus W., 1996, Diagnostic Symptoms of Unstability of Gear Systems Investigated by Computer Simulation, Proceedings of 9th International Con- gress COMADEM 96 Sheffield, 51- 61 4. Bartelmus W., 1997a, Influence of Random Outer Load and Random Ge- aring Faults on Vibration Diagnostic Signals Generated by Gearbox Systems, Proceedings of 10th International Congress COMADEM 97, 58-67 5. Bartelmus W., 1997b, Visualisation of Vibration Signal Generated by Ge- aring Obtained by Computer Simulation, Proceedings of XIV IMEKO World Congress,VII, 10, 126-131 6. Bartelmus W., 1998a,Diagnostyka Maszyn Górniczych, Górnictwo Odkryw- kowe, Śląsk, Katowice 7. Bartelmus W., 1998b, New Gear Condition Measure from Diagnostic Vi- bration Signal Evaluation, Proceedings of The 2nd International Conference, Planned Maintenance Reliability and Quality, University of Oxford England 8. Bartelmus W., 1999, Transformation ofGear Inter Teeth Forces into Accele- rationandVelocity, International Journal of RotatingMachinery,5, 3, 203-218. 9. BartelmusW., 2000,MathematicalModelling ofGearboxVibration forFault Detection, Condition Monitoring and Diagnostic Engineering Management, COMADEM International, UK, 3, 4, 5-15 10. Dąbrowski Z., 1992, The Evaluation of the Vibroacoustic Activity for Needs ofConstructing andUse ofMachines,Machine Dynamics Problems, 4,Warsaw University of Technology Gearbox dynamic modelling 999 11. Mark W.D., 1978, Analysis of the Vibratory Excitation of Gear Systems: Basic Theory, J. Acoust. Soc. Am., 63, 5, 1409-1430 12. Müller L., 1979,Przekładnie zębate, WNT,Warsaw 13. Penter A. J., 1991, A Practical Diagnostic Monitoring System, Proceedings of International Conference on Condition Monitoring, Erding, Germany, Pine- ridge Press, 79-96 14. RettigH., 1977, InnereDynamischeZusatzkräftebeiZahanradgetrieben,Ant. Antriebstechnik, 16, 11, 655-663 15. Ryś J., 1977, Analiza Obciążeń Statycznych i Dynamicznych w Walcowych Przekładniach Zębatych, Zeszyty Naukowe Politechniki Krakowskiej, 6 16. Siwicki I., 1992,Manual for CSSP (in Polish),Warsaw 17. Smith J.D., 1998, Modelling the Dynamics of Misaligned Helical Gears with Loss of Contact,Proc. Instn. Mech. Engrs, 212, Part C, 217-223 18. Tuma J., Kubena K., Nykl V., 1994, Assessment of Gear Quality Conside- ring the Time Domain Analysis of Noise and Vibration Signals,Proceedings of International Gearing Conference, Newcastle 19. VelexP.,MaatarM., 1996,AMathematicalModel forAnalysing the Influ- ence of Shape Deviations and Mounting Errors on Gear Dynamic Behaviour, Journal of Sound and Vibration, 191, 5, 629-660 20. WangS.M., 1974,Analysis ofNonlinearTransientMotion of aGearedTorsio- nal, Journal of Engineering for Industry, Transactions of the ASME, February 21. Wilk A., 1981, Wpływ Parametrów Technologicznych i Konstrukcyjnych na Dynamikę Przekładni o Zębach Prostych, Zeszyty Naukowe Politechniki Ślą- skiej, 679, Gliwice Modelowanie dynamiki przekładni zębatych Streszczenie Praca przedstawia aktualny rozwój zagadnień związanych z modelowaniem dy- namiki przekładni zębatych. Nawiązuje ona do jednomasowego, dwuparametrowego (sztywność i tłumienie) modelu przekładni Müllera o ruchu prostoliniowym. Poka- zuje potrzebę opracowania nowego modelu, który uwzględni drgania skrętne. Na- wiązuje do wcześniejszych publikacji autora pracy na temat modelowania przekładni publikowanychw czasopismach imateriałach konferencyjnych.Wprezentowanej pra- cy przedstawiono analizę wpływu tłumienia w sprzęgle, wpływ losowego parametru w trójparametrowej funkcji błędu na drgania generowane przez przekładnię zębatą. Manuscript received April 13, 2000; accepted for print December 13, 2000