A NON LINEAR MODEL OF THE GEARTRAIN OF THE TIMING SYSTEM OF THE DUCATI RACING MOTORBIKE

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1 A NON LNEAR MOEL OF THE GEARTRAN OF THE TMNG SYSTEM OF THE UCAT RACNG MOTORBKE Gabriele BENN (*), Alessandro RVOLA (*), Giorgio ALPAZ (**), Emiliano MUCCH (**) (*) EM University of Bologna, Viale Risorgimento, Bologna, taly (**) ip. di ngegneria University of Ferrara, Via Saragat, Ferrara, taly ABSTRACT This work deals with the elastodynamic model of the geartrain operating the desmodromic valve trains of a fourcylinder L engine of the ucati racing motorbikes. The geartrain dynamic behaviour is simulated by means of a lumped parameter model which takes into account the mass moment of inertia of gears and shafts, the torsional shaft compliance, the damping effect, the parametric excitation due to the time-varying meshing stiffness, and the presence of backlash between meshing teeth. n particular, the possibility of tooth contact on both contact lines is included in the model. The comparison between numerical results and experimental measurements shows that the effectiveness of the model is satisfactorily assessed. The model allows the prediction and interpretation of the actual dynamic behaviour of the system, so to be used in design optimisation of the geartrain. Keywords: elastodynamic modelling, geartrain, timing system, desmodromic mechanism. NTROUCTON n the field of racing motorbike engines, high speed and reliability are continuously growing requirements. Results obtained for the competitive motorbike engine can also be utilized for common production engines, where conformity to vibration and noise standards must be fulfilled. As known, the engine performance is strongly affected by the valve motion. t is, therefore, essential to assure an accurate valve timing and to operate for valve vibration lowering. n the case of the high speed engine, a geartrain is often adopted for transmitting power and motion from the crankshaft to the camshafts, that is, for driving the valve train system. As a matter of fact, it is understood that one advantage of the geartrain is the more precise valve timing at high engine speeds compared with a chain or a belt transmission. However, the presence of the backlash in the geartrain strongly affects its dynamics and may lead to performance deterioration when valve timing is lost in case the rotational behaviour becomes unstable. ncorrect behaviour of the geartrain results in additional dynamic forces, reduced durability and stability, and becomes a source of noise as well as other undesirable phenomena (e.g. gear rattle). The development of an elastodynamic model of the geartrain allows the estimation of the actual dynamic forces, impacts, and mechanism performances, as well as the design optimisation and fault diagnostics.

2 This work deals with the geartrain of the timing system of the ucati racing motorbike. The motorbike is equipped with a fourcylinder L engine which has double overhead camshafts, desmodromic valve trains and four valves per cylinder. This research activity is carried out by the EM of the University of Bologna in cooperation with ucati Corse. Previous researches by the authors [ 4], have demonstrated that the desmodromic train, which is a mechanism with positive-drive cams, exhibits different dynamic behaviour in comparison with the widely-used trains having a closing spring [5, 6]. n addition, the authors inspected the influence of the driving transmission to the torsional behaviour of the camshafts, but in those cases the engine adopted a belt transmission to drive the valve train [ 4]. Conversely, since the present engine adopts a geartrain for driving the valve system, in this work a non-linear lumped elastodynamic model of the geartrain is developed with the aim of studying and analysing the influence of the geartrain dynamics on the valve timing. n literature several papers deal with the dynamic modelling of gears: a review of mathematical models used in gear dynamics is given in [7] and a discussion about lumped parameter models, continuous system model and finite element model in gear dynamics is reported in [8]. A rotatory model for spur gear dynamics was developed in [9], where the possibility of tooth separation due to the backlash was inspected. The calculation of the gear meshing stiffness is discussed in [0, ]. n this paper the geartrain dynamic behaviour is simulated by means of a lumped parameter model which takes into account the mass moment of inertia of gears and shafts, the torsional shaft compliance, the damping, the parametric excitation due to the timevarying meshing stiffness and the presence of backlash between meshing teeth. n particular, the possibility of tooth contact on both contact lines is included in the model. The gears are considered as mounted on stiff bearings and the bending compliance of the geartrain is neglected, that is, only the gear torsional vibrations are examined. The presence of the desmodromic mechanisms is approximately taken into account by the corresponding inertial torques acting on the camshafts operated by the geartrain. The comparison between numerical results and experimental measurements shows that the effectiveness of the model is satisfactorily assessed. The results show that the model can be an useful tool in order to predict and understand the actual dynamic behaviour of the system, so to be used in both design optimisation and diagnostics of the valve timing system. THE MECHANCAL SYSTEM (THE GEARTRAN SYSTEM) The ucati racing motorbike is equipped with a fourcylinder L engine which has double overhead camshafts, desmodromic valve trains and four valves per cylinder. The geartrain adopted to transfer power and motion from the crankshaft to the camshafts is schematically shown in Fig.. t consists of twelve spur gears of involute tooth profile: the crankshaft pinion R 0, the gear-wheel R, the two pinions R H and R V which respectively drive the horizontal and the vertical bank, the idler gears R H(V), R 3H(V), and the four gears fitted to the camshafts, namely the gears R 4H(V)E and R 4H(V), where symbols E and denote the exhaust and the intake camshaft, respectively. n particular, the three gears R, R H and R V are fitted to the same gear-shaft. The geartrain dynamic behaviour is affected by the excitation due to both the crankshaft dynamics and the motion of the desmodromic mechanisms. n addition, a parametric excitation due to the time-varying meshing stiffness is present. n order to get insight into the dynamics of the geartrain system and help the development of the elastodynamic model, several experimental tests were carried out at the ucati Laboratory. n particular, the angular velocity of the crankshaft pinion R 0, the gear-wheel

3 R, and the gear mounted on the exhaust camshaft of the vertical bank (i.e. gear R 4VE in Fig. []) were picked up during the tests and have been used for model validation. Fig. Schematic of the geartrain. 3 MOEL ESCRPTON 3. General description of the model The model of the geartrain is developed with the aim of including all the important effects, as well as to get a rather simple model. Since the geartrain system consists of spur gears, one has to take into account only transversal plane dynamics. n particular, in the developed model, the gears are considered as mounted on stiff bearings and the bending compliance of the geartrain is neglected, that is, only the gear torsional vibrations are examined. The geartrain system described in Section is modelled by means of a lumped parameter model with degrees of freedom which takes into account the mass moment of inertia of gears and shafts, the torsional shaft compliance, the damping effect, the parametric excitation due to the time-varying meshing stiffness and the possibility of tooth separation due to the presence of backlash between meshing teeth. The schematic of the lumped parameter model of the geartrain is shown in Fig.. The known model input is the coordinate θ 0, representing the angular displacement of the crankshaft pinion R 0. The pinion R 0 can be assumed as rotating at constant speed or, alternatively, it can be driven by the experimentally measured angular displacement, if available. The coordinate of the generic gear i, is the angular displacement θ i. A mass moment of inertia is associated to each coordinate. The torsional stiffness k T concerns the

4 portion of the shaft between the gear-wheel R and the pinion R H, while torsional stiffness k T is relative to the portion of the shaft between the pinions R H and R V. Since the disks reported in Fig. correspond to the base cylinders of the gears, the springs acting along the disk tangent line represent the meshing stiffness evaluated along the gear contact line. As an example, the stiffness k 3H represent the meshing stiffness between the idler gears R H and R 3H. The possibility of tooth contact on both contact lines is included in the model, as will be discussed in Section 3.. The model takes into account the possibility of tooth separation due to the presence of backlash between meshing teeth. The parameters δ 0, δ H, δ V, δ 3H, δ 3V, δ 34V(E,), and δ 34H(E,) refers to the amount of tooth backlash measured on the common normal between the profile of adjacent teeth of the related gear pair. Fig. Schematic of the lumped parameter model of the geartrain. n order to globally consider the structural damping, as well as other damping effects, a viscous damper is associated with each stiffness. The dampers are not reported on Fig. for the sake of simplicity. The damper coefficient is taken proportional to the corresponding stiffness; consequently, the variability of the meshing stiffness affects the associated damper coefficient. n case there is no contact in a gear pair, the damper coefficient does not include the structural damping, but is still considered different from zero in order to consider other dissipative effects. The presence of the desmodromic mechanisms is approximately included by the corresponding inertial torques T HE(), and T VE() acting on the exhaust (intake) camshafts operated by the geartrain: their computation will be reported in Section 3.4. t is worth noting that the system is highly time-varying. As a matter several model parameters change during the geartrain motion, and backlashes are present. As a consequence, the differential equations of motion are strongly non-linear. They are numerically integrated by using the software Simulink.

5 3. Model of meshing gears The model of a generic meshing gear pair of the geartrain is shown in Fig. 3(a). The symbols θ j and M j denote the angular displacement of the gear j and the torque acting on it, respectively. The base circle of the gear j has radius R bj. There exist two tangent lines of the two basis circles: let irect Contact Line (CL) be the line of action when the torque M acting on the gear has the same direction as the angular displacement θ, and nverse Contact Line (CL) be the line of action when the torque M reverses direction. At the nominal position given in Fig. 3(b), a designated tooth of gear coincides with the center line O O, and the backlashes δ are equal at both side of this tooth with respect to corresponding teeth of gear. The backlash δ is measured on the common normal between the profiles of adjacent teeth. By denoting the mass moment of inertia of the gear j as J j, and let k and c be the meshing stiffness and the damping coefficient along CL, k and c be the meshing stiffness and the damping coefficient along CL, the dynamic equations of gear pair during meshing appear as follows: J && θ = R J && θ = R b ( F b ( F + F ) + M F ) + M () where the meshing force acting on CL is indicated as F, while F refers to the CL. J O O R b θ M θ δ R b δ irect Contact Line K C nverse Contact Line θ M R b CL CL R b O J O θ Fig. 3(a) Model of a generic meshing gear pair. Fig. 3(b) Meshing gears at the nominal position. Those meshing forces are due to both elastic and viscous contribution. ue to the backlash, when the driving gear slows down, stops, or reverses direction, gear teeth may lose contact and impacts between teeth will occur. From the mathematical point of view, the backlash introduces non-linear terms into the Equation (); in particular:

6 if ( R θ R θ ) > δ F b b ( R θ R θ δ ) + c ( R θ& R θ& ) = k ; F = 0 () b b b b if R b F θ Rb θ δ ( R θ & R θ& ) = F = c (3) b b if ( R θ R θ ) < δ F b b ( R θ R θ + δ ) c ( R θ& R θ& ) = k ; F = 0 (4) b b where the damper coefficients c and c are taken proportional to the corresponding meshing stiffness. n case there is no contact, the elastic term becomes null, while the viscous term is still considered different from zero in order to take into account other dissipative effects. n this case, the damper coefficient, namely c, is estimated during the model validation. 3.3 Meshing stiffness t is well known that for involute gears under normal operating conditions, the main excitation to the system originates from the periodic change in tooth stiffness due to the alternating engagement of single and double pairs of teeth. Several variable gear mesh stiffness models have been developed by a number of investigators (e.g. [0]). Let K s be the stiffness of a single tooth pair, p be the pitch of the gear pair, and ε be the gear contact ratio, the stiffness K s can be calculated by []: b b K s ( x) K m.8.8 = x + x ε ( ε p) ( ε p) (5) where x is the coordinate travelling along the contact arc εp, and K m is evaluated according to SO The gears of the considered geartrain have contact ratio, i.e. ε, below.0. Therefore, the number of meshing teeth pairs will alternate between one and two. The meshing stiffness of two meshed gears appears as shown in Fig. 4: when two gear pairs engage, the mesh stiffness is evaluated by parallel superposition of the corresponding tooth stiffness of the pairs in contact. The resulting meshing stiffness is periodic with respect to the pitch of the gear pair, that is, the fundamental frequency of the parametric excitation due to the timevarying meshing stiffness is the inverse of the time meshing period T=π/(z j Ω j ), where z j is the number of teeth for the gear j and Ω j is its angular velocity. n order to take into account the influence of both rim section design and gear webs on the tooth meshing stiffness, a finite element analysis was performed with the aim of evaluating the torsional compliance of the gear body. As an example, Fig. 5 shows the results of the finite element analysis carried out on gear R. Let K and K* be the tooth meshing stiffness of the gear pair, calculated by neglecting and taking into account the torsional compliance of gear discs, respectively. By denoting as K Tj the torsional stiffness of the gear disc j due to the rim section design and/or gear webs, in this work the tooth meshing stiffness K* is calculated by combining the stiffness K, K T, and K T as springs connected in series, that is: = + + (6) K * K K K T T

7 Fig. 4 Tooth meshing stiffness. Fig. 5 Analysis of the torsional compliance of gear R. 3.4 The inertial contribution of the desmodromic valve mechanisms The model of the geartrain is developed with the aim of obtaining a rather simple model. For this reason, at the present stage of model development, the presence of the desmodromic mechanisms operated by the geartrain is approximately taken into account by the corresponding inertial torques acting on the camshafts. On the other hand, since the authors have already developed models of the desmodromic mechanism [, 3 4], such a models could be combined with the geartrain model in further research work. The desmodromic valve train is a mechanism with positive-drive cams. Figure 6(a) shows the schematic of the cam mechanism driving a single valve, while Fig. 6(b) shows the assembly of two camshafts and the valve trains which drive the eight valves of two cylinders of an engine bank. With reference to Fig. 6(a), the discs of a conjugate cam are each in contact with a rocker; the two rockers are then in contact with the backlash adjuster. t is therefore possible to identify two parts of the mechanism: one part, consisting of one cam disc and the related rocker, gives valve positive acceleration, while the negative acceleration is given to the valve by the other cam disc with the associated rocker (the positive direction is that of the valve opening). The coordinate θ indicates the camshaft angular position, and s=s(θ) is the valve displacement function. When the mechanism is in motion, inertial torques and forces will arise. The inertial force F reduced to the direction of the valve motion can be calculated as follows: F = ( m + m + m + m a (7) P N A V ) where m P and m N, are the mass moments of inertia of the rockers reduced to the direction of the valve motion, m A is the mass of the backlash adjuster, m V is the valve mass, and a is the valve acceleration. The corresponding inertial torque T, i.e. the inertial force F reduced to the camshaft axis, can be written as: T ds( θ) = F (8) dθ By denoting the camshaft angular velocity as Ω, the following expression holds:

8 a = d s( ϑ ) ds( ϑ) Ω + dϑ d ϑ dω dt (9) the substitution of the Equations (8, 9) into the Equation (7) leads to: T = ( m P + m N + m A + m V ds( θ ) d s( θ ) ds( θ ) dω ) Ω + d θ dθ d θ dt (0) which is the inertial contribution of a single desmodromic mechanism. Fig. 6(a) Schematic of the cam mechanism driving a single valve. Fig. 6(b) The double overhead camshafts driving eight desmodromic mechanisms. Fig. 7 Normalised inertial torque acting on the exhaust camshaft of the vertical engine bank.

9 Each camshaft drives four desmodromic trains; therefore, four inertial torques will be applied on it. Figure 7 shows the normalised total inertial torque acting on the exhaust camshaft of the vertical engine bank, i.e. T VE (see Fig. ), calculated by neglecting the camshaft angular acceleration. 4 RESULTS AN SCUSSON The model has been validate based on experimental tests carried out at the ucati Laboratory. n particular, the angular velocity of the gear-wheel R, and the gear mounted on the exhaust camshaft of the vertical bank (i.e. gear R 4VE ) were picked up during the tests. Those measurements were referred to the angular position of the crankshaft pinion R 0. Figure 8 shows the comparison between experimental and numerical ratio of the angular velocity of gear-wheel R and gear R 4VE with respect to the angular velocity of the crankshaft pinion R 0, at 0000 crankshaft rpm and over two crankshaft revolutions (i.e. one camshaft revolution). The comparison shows that the model is able to predict the dynamic behaviour of the geartrain system quite well, even if some discrepancies subsist. n particular, for gear R disagreement occurs in the range crankshaft degrees, whilst the model generally underestimates the oscillation amplitude of the gear R 4VE. From a functional point of view, the estimation of those ratios is very important, because provides information on the dynamic behaviour of the geartrain with respect of the valve timing. n particular the analysis of the angular velocity ratio between gear R 4VE and the pinion R 0, makes it possible to relate the camshaft angular position and, consequently, the valve motion, to the crankshaft position. Fig. 8 Experimental ( ) and simulated ( ) angular velocity ratio: (a) gear-wheel R to pinion R 0 ; (b) gear of exhaust camshaft of vertical engine bank (gear R 4VE ) to pinion R 0.

10 5 CONCLUSONS An elastodynamic model of the geartrain operating the timing system of a fourcylinder L engine of the ucati racing motorbikes is presented. The geartrain dynamic behaviour is simulated by means of a lumped parameter model which takes into account the mass moment of inertia of gears and shafts, the torsional shaft compliance, the damping effect, the parametric excitation due to the time-varying meshing stiffness, and the presence of backlash between meshing teeth. The possibility of tooth contact on both contact lines is included in the model. The comparison between numerical results and experimental measurements shows that the effectiveness of the model is satisfactorily assessed. The model provides information on the dynamic behaviour of the geartrain with respect of the valve timing. As a further research work, models of the desmodromic mechanisms, which have been previously developed by the authors, as well as a crankshaft model, will be combined with the geartrain model. Acknowledgements The authors wish to thank ucati Corse S.r.l. for supplying experimental data and the engineers of this Company for advice and discussions during the course of the research. REFERENCES. alpiaz G., and Rivola A., A Non-Linear Elastodynamic Model of a esmodromic Valve Train, Mechanism and Machine Theory, Vol. 35, No., pp , (000).. Carlini A., Rivola A., alpiaz G., and Maggiore A., Valve Motion Measurements on Motorbike Cylinder Heads Using High Speed Laser Vibrometer, 5th nternational Conference on Vibration Measurements by Laser Techniques: Advances and Applications, Ancona, taly, pp , (00). 3. Rivola A., Carlini A., and alpiaz G., Modelling the elastodynamic behaviour of a desmodromic valve train, SMA 00 nternational Conference on Noise & Vibration Engineering, Leuven, Belgium, pp , (00). 4. Carlini A., and Rivola A., A non linear elastodynamic model of a camshaft supported by journal bearings, Proceedings of AMETA 03 6th AMETA Congress of Theoretical and Applied Mechanics, 9- September 003, Ferrara, taly, (003). 5. Nagaya, K., Watanabe, K., Tsukahara, Y., Vibration Analysis of High Rigidity riven Valve System of nternal Combustion Engines, Journal of Sound and Vibration, Vol. 65, No., pp. 3 43, (993). 6. Özgür, K., Pasin, F., Separation Phenomena in Force Closed Cam Mechanisms, Mechanism and Machine Theory, Vol. 3, No. 4, pp , (996). 7. H.N. Ozguven,.R. Houser, 988, Mathematical models Used in Gear ynamics A Review, Journal of Sound and Vibration, Vol., No.3, pp H.N. Ozguven, 99, Assessment of some recently developed mathematical models in gear dynamics, Proceedings of the Eighth World Congress on the Theory of Machines and Mechanisms, 6-3 August 99, Prague, Czechoslovakia. 9..C.H. Yang, Z.S. Sun, 985, A Rotatory Model for Spur Gear ynamics, Journal of Mechanisms, Transmissions, and Automation in esign, Vol. 07, pp J.H. Kuang and Y.T. Yang, 99, An estimate of mesh stiffness and load sharing ratio of a spur gear pair, Proceedings of the ASME nternational Power Transmission and Gearing Conference, Scottsdale, Arizona, Vol., pp. -9, (E Vol. 43-).. Y. Cai and T. Hayashi, 994, The linear approximated equation of vibration of a pair of spur gears (theory and experiment), Transactions of the ASME, Journal of Mechanical esign, Vol. 6, pp