Modelling the Elastodynamic Behaviour of a Train

Alessandro RIVOLA (*), Andrea CARLINI (*), and Giorgio DALPIAZ (**) (*) DIEM - University of Bologna Viale Risorgimento, 2, I - 40136 Bologna, Italy e-mail: [email protected] (**) Dept. of Engineering, University of Ferrara Via Saragat, 1, I - 44100 Ferrara, Italy e-mail: [email protected]

Abstract This paper deals with a lumped-parameter model of a motorbike engine’s desmodromic valve train. The model of such an uncommon cam system is developed and validated with the aid of experimental measurements carried out on a test bench which operates the cam mechanism by means of an electrically powered driveline. The model describes the mechanical system taking into account the mass distribution, the link elastic flexibility, and the presence of several non-linearities. The model parameter estimation is discussed and the effectiveness of the model is assessed by a comparison with experimental results. In addition, the model is employed to estimate the magnitude of contact forces and to predict the system behaviour as a consequence of changes in some design parameters; therefore, it may be used as a tool both in design optimisation and diagnostics.

1 Introduction train of a Ducati motorbike engine. Only works on widely-used trains with closing spring were found in Nowadays the study of the dynamic response of literature [7–9]; in those cases the valve spring plays flexible mechanisms operating at high speed is an important role in the system dynamics. becoming more and more important and several Conversely, in the case of desmodromic valve trains studies can be found in literature [1–4]. – i.e mechanisms with positive-drive cams – the As a matter of fact, the link elastic flexibility and dynamic effects are partly different, as shown in mass distribution, as well as the effects of [10–12] by the authors. backlashes and friction in joints, may affect the In order to get insight into the dynamics of such dynamic behaviour of the system so deeply that it an uncommon cam system, and to help the may fail to properly perform its task. In addition, development and validation of the elastodynamic high accelerations and dynamic stress may occur, model, valve motion measurements were retrieved causing early fatigue failure, and high vibration and from experimental tests carried out at the DIEM noise may arise [5–6]. Laboratory of the University of Bologna on In the specific field of valve trains for engines cooperation with Ducati [12]. During the test, the operating at very high speed, the above-mentioned was operated by means of an electrically dynamic effects are particularly important, since powered driveline. they may cause serious functional troubles, such as The model presented in this paper is a lumped- wear, fatigue loads and breakage of mechanical parameter model having twelve degrees of freedom components, jump and bounce phenomena of the (dofs). It describes the desmodromic train and the valve, and alteration of the engine’s fluid dynamics. driving mechanical transmission of the test bench, in Consequently increasing attention is addressed to order to employ the experimental results for model the elastodynamic analysis, as a tool for design assessment and validation. The model takes into optimisation and fault diagnostics as well as for the account the mass distribution, the elastic flexibility estimation of the actual dynamic forces, impacts, of all links (including the compliance of the and mechanism performances. driveline), and the presence of non-linearities. In this context, this paper presents a kineto- After the description of the mechanical system elastodynamic model of the desmodromic valve and its model, the measurement apparatus is Camshaft θ 0

φ φ φ 2 3 1

Cylinder Head Timing Belt Intermediate Shaft

φ 0 τ 1 Brushless Motor

Figure 1: Schematic of the electrically powered driveline Figure 2: Schematic of the cam mechanism driving a single valve presented. Secondly, the paper compares the two (only one camshaft is shown in the numerical results to the experimental data in terms schematic of Fig. 1). Each camshaft has two of valve displacement and acceleration by using conjugate cams: one camshaft drives the two intake proper signal processing techniques. In addition, the valves and the other the two exhaust valves. importance of taking into account the stiffness Figure 2 shows the schematic of the cam variability of the rockers is pointed out. Finally, the mechanism driving a single valve: the discs of a model is employed to estimate the magnitude of conjugate cam are each in contact with a rocker; the contact forces and to predict the system behaviour two rockers are then in contact with the backlash as a consequence of changes in some design adjuster. It is therefore possible to identify two parts parameters. of the mechanism, each made up of one of the cam discs and the related rocker: one part gives valve positive acceleration, while the negative 2 The mechanical system acceleration is given to the valve by the other part of the mechanism (the positive direction is that of the This work deals with the timing system of the valve opening). twincylinder ‘L’ engines of Ducati racing motorbike A small helical spring is mounted around the which have double overhead camshafts, negative rocker pin and properly preloaded: its desmodromic valve trains and four valves per action is mainly needed during the dwell phase, [10, 11]. when separation of the cam discs from rockers takes As mentioned in the introduction, tests are place and, consequently, the contact between the carried out on an experimental test bench, where the valve-head and the seat might be lost. camshafts are moved by means an electrically powered driveline. The schematic of the driveline is shown in Fig. 1: it consists of a brushless motor 3 The elastodynamic model driving an intermediate shaft by means of a timing belt, and a second timing belt which moves the 3.1 Description of the model camshafts. The speed of the intermediate shaft is obtained by multiplying the motor velocity by the The mechanical system described above was ratio τ1 = 10/3. In order to reduce fluctuations of modelled with a twelve dofs lumped-parameter torque and velocity, the intermediate shaft is fitted model which describes the electrically powered with a flywheel. This shaft corresponds, in the driveline, the intake camshaft and the cam motorbike engine, to the shaft moved by the crank mechanism driving a single intake valve, namely the shaft with a gear transmission having ratio 1/2. The valve close to the camshaft pulley. The other cam second belt transmission, which reproduces the mechanism and the valve driven by the same engine belt loop, has ratio τ2 = 1/1 and drives the camshaft were not included in order to simplify the k ks ks k b1 12 23 b2

Js J J J 1 s2 s 3 0 φ φ φ φ θ 0 1 2 3 0

1 1 1 2 2 1

1 1 11 1 1 1 2 2 1

τ 1 1 τ δ 1 72 82 1 13 τ 1 1 τ 1 71 8 1 1

k 13 x 7 x 8 m7 m8 m3 x 3 k 7 k 8 k 34

x x 1 2 k 01 k 12 J J J 0 1 2 θ θ θ 01 2

δ 34 m 4 δ 46

x 4 k 45 k 46 k 5

x 6 m6 m5 k 6 k 26 x 5 δ 26

Figure 3: schematic of the lumped-parameter model model. However, this second mechanism is identical approximately introduced, assuming that this to the first one and theoretically moves in phase; the mechanism applies the same forces and torques to effects of the neglected mechanism are therefore the camshaft as the first one. In [10, 11] the authors presented models of the intermediate shaft between the flywheel and the desmodromic valve trains having eight dofs; in second pulley. those studies the speed of the camshafts was By means of the coordinate θ0, which represents assumed to be constant, that is, the dynamic the angular position of the pulley fitted on the intake behaviour of the mechanical transmission driving camshaft, the compliance of the second belt the camshafts was not considered. Actually, transmission is considered. The moment of inertia J0 experimental measurements carried out on the test is obtained by properly lumping the moment of bench, have shown that the oscillations of the inertia of the camshaft pulley together with the camshaft speed around the mean value are quite moment of inertia of a camshaft portion, the inertia important (up to ±10%). Therefore, the driveline of the exhaust camshaft, and the inertia of a portion moving the camshafts is taken into account in the of the timing belt. The torsional stiffness kb2 present work. represents the belt transmissions between the The model is developed with the aim to include intermediate shaft and the intake camshaft. all the important effects, as well as to get a rather The stiffness of both timing belts are not simple model. In particular, it takes into account the considered as constant during the simulation. In mass distribution, the elastic flexibility of all links fact, due to the effect of the dynamic loads, the belts (including the compliance of the driveline, the can release, causing the pre-load to be lost. bending and torsional compliance of the camshaft, The torsional compliance of the camshaft is and the variation of rocker stiffness as a function of taken into account by introducing the coordinates θ1 mechanism position), the damping effects, the and θ2, representing the angular displacements of variability of transmission ratios with mechanism the positive and negative cam discs, respectively. In position, and the presence of several non-linearities fact, with reference to the cam mechanism moving (e.g. the Hertzian contact stiffness, the backlashes in the considered valve, the positive cam disc is the joints, and the lubricant squeeze effect). closest to the camshaft pulley. The moments of With reference to Fig. 3, the known model input inertia J1 and J2, associated with these coordinates, is the coordinate φ0, representing the angular are obtained by lumping the moment of inertia of displacement of the brushless motor’s pulley, camshaft portions together with the inertia of the reduced to the intermediate shaft axis by the positive and negative cam discs, respectively. transmission ratio τ1. This pulley is assumed to Consequently, torsional stiffness k01 concerns the rotate at constant speed. The compliance of the first portion of the camshaft between the pulley and the timing belt is taken into account by introducing the positive cam, while torsional stiffness k12 is relative coordinate φ1, representing the angular position of to the portion between the two cam discs. the first pulley of the intermediate shaft (see also In order to include the camshaft bending Fig. 1). The moment of inertia Js1, associated with compliance, the coordinates x7 and x8 are coordinate φ1, is obtained by properly lumping the introduced: they are the deflections of the camshaft inertia of the first timing belt together with the first in correspondence of the conjugate cam and in two pulley of the intermediate shaft and a portion of the orthogonal directions, respectively coincident and shaft close to the pulley. The torsional stiffness kb1 perpendicular to the direction of the valve motion, represents the stiffness of the first belt as shown in Fig. 2. The deflections are assumed to transmissions. be the same for the two cam discs, due to their small The torsional compliance of the intermediate axial distance. The masses m7 and m8, associated shaft is considered by introducing the coordinates φ2 with the bending coordinates, are the equivalent and φ3, representing the angular displacements of masses of the whole camshaft, computed by using the flywheel and the second pulley of the the assumed-mode method with a shape function intermediate shaft, respectively. They are associated coincident to the static deflection. The camshaft with the moments of inertia Js2 and Js3, which are bending stiffness is modelled by the springs k7 and obtained by lumping the moment of inertia of the k8, whilst the camshaft bearing compliance is portions of the shaft together with the moments of considered as negligible. inertia of the flywheel and the second pulley, All the other coordinates, and the related model respectively. The inertia Js3 also includes a portion parameters, are reduced to the direction of the valve of the second time belt. Consequently, torsional motion. Thus, the model contains the kinematic stiffness ks12 concerns the portion of the camshaft relationships of valve displacement vs. cam angular between the first pulley and the flywheel, while displacements and camshaft deflections: the cam θ θ torsional stiffness ks23 is relative to the portion of angular displacements, 1 and 2, are reduced using the cam curve functions x1=x1(θ1) and x2=x2(θ2); the camshaft deflections, x7 and x8, are reduced using the variability of rocker stiffness with mechanism the transmission ratios τ71=dx1/dx7, τ72=dx2/dx7, position) affects the associated damper coefficients. τ81=dx1/dx8 and τ82=dx2/dx8. The reduction of the In case there is no contact in joints with backlash coordinates to the direction of the valve motion is and links are approaching, the damper coefficient is represented by a lever system in Fig. 3. More details computed in order to represent the lubricant squeeze on the procedure of reduction can be found, for effect [2, 5, 11]. Coulomb friction forces have not example, in [2]. been introduced into the model, due to their low The linear coordinates x3 and x6 correspond to value. the angular displacements of the positive and It is worth noting that the system is highly time- negative rockers (respectively indicated by θ3 and θ6 varying. As a matter of fact the active cam-follower in Fig. 2), after the reduction to the direction of the system is different in the phases of positive and valve motion. negative acceleration. In addition, several model The masses m3 and m6 , which are associated to parameters change with mechanism position. As a the coordinates x3 and x6, are the reduced moments consequence, the differential equations of motion of inertia of the rockers. The parameters k13 and k34 are strongly non-linear. They are numerically are firstly computed by reducing and composing in integrated by using the software Simulink. series the stiffness of the positive rocker and the stiffness of the Hertzian contacts cam-rocker and rocker-adjuster, and then sharing out the global 3.2 Estimation of the model parameters compliance between the two springs. Stiffness k26 is obtained by reducing and composing in series the The model parameters were preliminarily evaluated stiffness of the negative in contact with on the basis of both design and literature data. The the cam and the stiffness of the related Hertzian values of inertial parameters, stiffnesses and squeeze coefficients were computed based on the geometry contact; similarly, k46 concerns the stiffness of the negative rocker arm in contact with the adjuster and and the material of mechanism links. the related Hertzian contact; the bending compliance With reference to the mechanical transmission of the negative rocker pin is also included in the moving the camshafts, the main uncertainty concerns the estimation of the stiffness of the two evaluation of k26 and k46. The parameter k6 is the reduced stiffness of the helical rocker spring. timing belts. As a matter of fact, no experimental It is noteworthy that the stiffnesses of the rockers data about belt stiffness were available. are assumed to be variable as a function of the Concerning the others mechanism links, the mechanism position. Moreover, also the Hertzian finite element method (FEM) was employed in stiffness is variable, as it depends on the contact order to evaluate the stiffness of the rockers and the force; in the simulation it is evaluated valve-head. It is worth noting that the stiffness of instantaneously. the rockers was computed taking into account the The possibility of separation of the rockers from effect of the variation of position and direction of the cam discs and the adjuster is included, in order the contact forces cam-rocker and rocker-adjuster, to appropriately model the effects of backlashes. as the mechanism changes its position. The FEM analysis was carried out considering only few The parameters δ13, δ34, δ26, δ46 refer to the amount of separation in the joints when the mechanism is at positions of the mechanism; during the simulation rest, that is during the dwell phase; they are reduced the instantaneous value of rocker stiffness is to the valve motion direction. computed by means of interpolation. The values of the backlashes into kinematic pairs The mass valve is lumped partly in mass m4, in correspondence of the adjuster, and partly in mass cam-rocker and rocker-adjuster, which are important m , in correspondence of the valve-head. The axial functional parameters, were measured during the 5 experimental tests, that is, taking into account the stiffness of the valve stem is k45, while stiffness k5 represents the stiffness of the valve-head in contact link thermal deformation. Although the backlashes with the seat. have a nominal value that is the same for the two In order to globally take account of structural parts of the mechanism, the ‘positive’ backlash damping, as well as other damping, a viscous tends to increase, due to thermal deformation, whilst damper is associated with each stiffness. The the ‘negative’ one suffers a reduction. damper coefficient is taken proportional to the The damper coefficients were preliminarily corresponding stiffness; consequently, the established based on previous models of the variability of some of the model stiffnesses (due, for desmodromic valve train developed by the authors example, to the contribution of Hertzian contacts, or [10, 11]. * Ignoring Hertzian stiffness possible to assign the same value to the –5 2 * proportionality constants between damper Js1 = 8.00×10 kg m k = 32.2÷233.8 13 coefficients and the related stiffnesses, because of –2 2 Js1 = 2.51×10 kg m MN/m different causes of damping which take place in the various parts of the mechanism. However, the –4 2 * Js3 = 6.12×10 kg m k = 4.14÷15.6 26 proportionality constants were set within the limited range 0.5×10–5 s ÷ 1.5×10–5 s, with the exception of J = 1.65×10–4 kg m2 MN/m 0 the timing belts, to which was assigned the value –4 –5 2 * 5×10 s, due to their high structural damping. J1 = 1.21×10 kg m k = 32.2÷233.8 34 The values of the model parameters are listed in –5 2 Table 1. J2 = 4.90×10 kg m MN/m

m = 1.34×10–2 kg * ÷ 3 k = 18.9 23.3 46 4 Experimental apparatus m = 0.66×10–2 kg MN/m 4 As previously mentioned, the experimental study is m = 3.34×10–2 kg k = 36.4 MN/m 5 45 carried out on a test bench which was planned to –2 reproduce, as well as possible, the functional m = 2.62×10 kg k = 66.8 MN/m 6 5 conditions of the valve train. In particular, the test m = 6.64×10–1 kg k = 600.0 N/m bench and the measurement apparatus can operate 7 6 for different type, at high camshaft m = 6.64×10–1 kg k = 97.3 MN/m 8 7 speed, under high temperature of the lubrication oil, and reproducing the motorbike power belt k = 4.483×102 Nm/rad k = 97.3 MN/m b1 8 transmission. The experimental apparatus includes a test stand, ks = 4.765×104 Nm/rad 12 a cylinder head, an electrically powered driveline to 4 δ operate the camshaft, a lubrication circuit, and ks = 5.600×10 Nm/rad 13 = 0.25 mm 23 measurement instrumentation [12]. k = 1.053×102 Nm/rad δ The maximum speed available at the camshaft is b2 26 = 0.10 mm 10000 rpm. In order to properly lubricate the valve k = 7.622×103 Nm/rad δ 01 34 = 0.0 mm train, pressurized oil is fed into the cylinder head oil galleries; oil pressure and temperature are similar to k = 1.466×105 Nm/rad δ 12 46 = 0.0 mm those picked up from the motorbike during the Table 1: Model parameters racing (i.e. 6 bar and 130°C, respectively). It is worth noting that only the components required for the operation of the valve train are The values of some parameters were then included into the system; as a consequence, no gas adjusted in order to better match experimental forces, combustion, or spurious vibrations occur. results. Therefore, the system response is not the actual one, In particular, due to the uncertainty in estimating that is, the response of the motorbike engine system. the stiffness of the two timing belts, it was expected However, the inclusion (or exclusion) of the forces due to compressed gases does not compromise the that the parameters kb1 and kb2 would be significantly adjusted. On the contrary, the other validity of the experimental data as a modelling tool stiffnesses would be decreased to some amount, as [13]. commonly occurs in modelling [6, 10, 11]. As a The measurement equipment consists of a laser matter of fact, after the comparison between the vibrometer and data acquisition apparatus. The laser model simulation and experimental valve motion, equipment is a Polytec’s High Speed Vibrometer the stiffness of the timing belts was set to 20% of (HSV-2002), used for non-contact measurements the preliminarily established value, while the and high velocity applications, which can measure camshaft bending stiffness, the stiffness of positive the absolute and relative velocity and displacement and negative rockers, and the valve-head stiffness up to 30 m/s and 41 mm respectively; the maximum were assumed to be the 80, 80, 40, and 80%, of their frequency is 50 kHz. computed values, respectively. The centre of the valve-head plane surface was Concerning the viscous dampers, it was not chosen as measurement point, thus making it possible to minimize possible valve’s flexional 4 4 3 Experimental 3 Experimental 2 2 1 1 0 0 -1 -1 -2 -2 4 4 3 Numerical 3 Numerical 2 2 1 1 0 0

Normalised Valve Acceleration Valve Normalised -1 Acceleration Valve Normalised -1 -2 -2 0 60 120 180 240 300 360 0 60 120 180 240 300 360 Camshaft angle [deg] Camshaft angle [deg]

Figure 4: valve acceleration (4500 camshaft rpm) Figure 6: valve acceleration (6500 camshaft rpm)

4 0.3 3 Experimental Experimental 0.2 2 1 0.1 0 0 -1 -2 -0.1 4 0.3 3 Numerical Numerical 0.2 2 1 0.1

0 Displacement [mm] Valve 0 Normalised Valve Acceleration Valve Normalised -1 -2 -0.1 0 60 120 180 240 300 360 160 200 240 280 Camshaft angle [deg] Camshaft angle [deg]

Figure 5: valve acceleration (5500 camshaft rpm) Figure 7: valve displacement (6500 camshaft rpm) vibration effects, which may negatively affect valve numerical results in comparison with experimental motion measurement. An area close to the valve seat data in terms of valve acceleration over one cam was selected as a reference surface. The differential revolution. They refer to camshaft speed of 4500, measurement between valve surface and reference 5500, and 6500 rpm. The experimental valve plane permits the elimination of raw vibration acceleration is obtained by means of numerical effects of head cylinder support. derivative, while the numerical valve motion is The signals were collected by means of a represented by the acceleration of the mass m5 of the National Instrument PXI data acquisition system; model. The acceleration scale is made dimensionless the sampling frequency was 102 kHz and the filter with reference to the theoretical maximum value. cut-off frequency was set to 45 kHz in order to Figures 4–6 show that a good agreement between prevent aliasing. During the tests, both valve simulated and experimental results is attained for a velocity and displacement were recorded. The wide range of camshaft speed. In particular, the digital signals were processed and analysed with simulation results are very similar to the actual MATLAB software. valve motion within both positive and negative acceleration phases. As a matter of fact, the model is able to reproduce the more important dynamic 5 Results and discussion phenomena and oscillations, and the level of the acceleration peaks is globally matched, even if some This Section firstly presents, by Figs. 4–6, some discrepancies exist. 4

3

2 Force [kN] Force

1

0 0 60 120 180 240 300 360 Camshaft angle [deg]

Figure 8: wavelet transform of valve acceleration Figure 10: positive cam-rocker contact force

4 6900 3 Experimental Angular speed [rpm] 2 6800 1 6700 0 -1 6600 -2 4 6500 3 Numerical 2 (constant rocker stiffness) 6400 1 0 6300

Normalised Valve Acceleration Valve Normalised -1 -2 6200 0 60 120 180 240 300 360 0 60 120 180 240 300 360 Camshaft angle [deg] Camshaft angle [deg]

Figure 9: valve acceleration Figure 11: angular speed of camshaft pulley

As a further example of the simulation accuracy, clearly the same. the contact between the valve-head and its seat is In order to evaluate the model effectiveness from investigated for 6500 camshaft rpm. In fact, the the point of view of the frequency content, a wavelet dynamic effects at the valve closure strongly affect analysis of both numerical and experimental valve the engine performances and, as such, have to be acceleration is presented in Fig. 8, in the case of taken into account in order to predict the actual gas 6500 camshaft rpm. As a matter of fact, since the flow dynamics. Figure 7 compares the numerical data have non-stationary nature, a signal processing displacement of the mass m5 to the actual valve technique able to study how the frequency content displacement, by means of an enlarged detail at the changes with time has to be used [14]. The wavelet valve closure. The comparison shows that the model analysis reported in Fig. 8 shows that the most is able to properly simulate the impact location of important dynamic phenomena are the impacts in the valve-head against the seat (at camshaft angle of the kinematic pairs of the rockers (at about 15, 65, about 175 degrees), as well as the bounce and 155 camshaft degrees) and between the valve- phenomena. In particular, the amount of separation head and its seat (at about 180 cam degrees). These between the valve-head and the seat, due to the impacts, due to their broadband frequency content, bounces, is correctly estimated. Actually, in the excite the mechanism resonances leading to damped experimental result, the first bounce takes longer oscillations. As noted in Sect. 3.1, the dynamic and the second one occurs later than in the system is highly time-varying; thus, the resonance simulation case, but the nature of the phenomena is frequencies are different in the various angle phases. 4 0.3 with rocker spring 0.2

3 0.1 0 -0.1 2 0.3 without rocker spring Force [kN] Force 0.2 1 0.1

Valve Displacement [mm] Valve 0 0 -0.1 0 60 120 180 240 300 360 160 200 240 280 320 Camshaft angle [deg] Camshaft angle [deg]

Figure 12: positive cam-rocker contact force Figure 13: numerical valve displacement (increased backlash) important. These are the effect of the dynamic In particular, the comparison of Fig. 8 demonstrates behaviour of the driveline moving the camshaft. As that the excited natural frequencies of the model expected, the angular speed of the inertia Js2 (i.e. generally agree with the experimental ones, with the the flywheel) is practically constant; thus, the exception of the simulated valve-head impact which camshaft oscillations are essentially due to the has lower frequency content. compliance of the timing belt reproducing the With the aim to remark the importance of taking engine belt loop. From a functional point of view, into account the stiffness variability of the rockers, the estimation of these oscillations is very their stiffness was set to a constant value, important, as it makes it possible to relate the correspondent to the position of zero and maximum camshaft angular position and, consequently, the valve displacement, for the positive and negative valve motion, to the crank shaft position. The rocker, respectively. Figure 9 compares the importance of modelling the driveline is therefore numerical valve acceleration, obtained by assuming justified. the stiffness as constant [Fig. 9(b)], with the As a final example, the model is employed to experimental one [Fig. 9(a)], at 6500 camshaft rpm. predict the system behaviour as a consequence of Disagreements between actual and simulated valve changes in some parameters. Since the dynamics of motion clearly appear both for the positive and the desmodromic valve train is strongly affected by negative acceleration phases. In particular, during the backlash value in the kinematic pairs cam-rocker the negative phase the period of the oscillations is and rocker-adjuster, it is useful to predict the effect incorrectly simulated, while in the second positive of variations of this important functional parameter phase, the oscillation amplitude is overestimated. due, for example, to wear or improper assembling. The model can be employed as a tool for design Figure 12 shows the prediction of the positive cam- optimisation. In fact, by means of the simulation, rocker contact force, due to backlash values that are the magnitude of dynamic forces can be predicted, increased with respect to those reported in Table 1: thus making it possible to determine dynamic stress δ13=0.30 mm, δ26=0.15 mm, and δ34=δ46=0 mm. The levels, to verify the structural strength of mechanism peaks of the contact force increase of about 20% links, or to compute contact pressures. As an with respect to the previously examined case (see example, Fig. 10 reports the contact force between Fig. 10). the positive cam (inertia J1 in the model) and rocker Another important aspect of the desmodromic (mass m3) for 6500 camshaft rpm. As expected, the train is the presence of the helical spring acting on contact mainly deals with the positive acceleration the negative rocker. Figure 13 shows the effect of a phases. breakage of the rocker spring, by comparing the As a further example of model application, the simulated valve displacement in normal condition to angular speed of the camshaft pulley (inertia J0) is that one in fault condition, that is, when the value of computed and reported in Fig. 11, over one cam the stiffness k6 of the model is set to zero. In the last revolution. It shows that the oscillations of the case, the valve keeps open for a longer time, thus camshaft speed around its mean value are quite affecting the engine’s fluid dynamics. 6 Conclusions Signal Processing, Vol. 6, (1992), pp. 517–534. [6] Dalpiaz, G., Rivola, A., A Kineto-elastodynamic This paper presents a model for the simulation of the Model of a Mechanism for Automatic Machine, kineto-elastodynamic behaviour of the desmodromic Proceedings of the Ninth World Congress on the valve train of Ducati engines, as well as a test bench Theory of Machines and Mechanisms, Milano, designed in order to closely reproduce the functional Italy, Vol. 1, (1995), 327–332. conditions of the valve train. The model - having twelve dofs – describes all the experimental [7] Pisano, A. P., Freudenstein, F., An Experimental mechanical system, i.e. both the valve train and the and Analytical Investigation of the Dynamic driveline. The model contains some non-linear Response of a High-Speed Cam-Follower elements and it is highly time-varying. In fact, the System. Part 2: A Combined, Lumped/ active cam-follower system is different in the phases Distribuited Dynamic Model, Journal of of positive and negative accelerations. In addition, Mechanisms, Transmissions, and Automation in some model parameters – in particular, the rocker Design, Vol. 105, (1983), pp. 699–704. stiffness – strongly depend on the mechanism [8] Nagaya, K., Watanabe, K., Tsukahara, Y., position. Vibration Analysis of High Rigidity Driven By comparison with valve motion measurements Valve System of Internal Combustion Engines, picked up on the test bench, the effectiveness of the Journal of Sound and Vibration, Vol. 165, No. 1, model is satisfactorily assessed: it properly (1993), pp. 31–43. reproduces all the important dynamic phenomena taking place in the system, in a wide range of [9] Özgür, K., Pasin, F., Separation Phenomena in camshaft speed. It is pointed out that it is essential to Force Closed Cam Mechanisms, Mechanism take into account the variability of the rocker and Machine Theory, Vol. 31, No. 4, (1996), pp. stiffness with mechanism position for obtaining 487–499. satisfactorily simulations. Finally, applications of [10] G. Dalpiaz, and A. Rivola, A Model for the model for both design improvement and fault Elastodynamic Analysis of a Desmodromic diagnostics are shown with some examples. Valve Train, Proceedings of the Tenth World Congress on the Theory of Machines and Mechanisms, June 20–24, Oulu, Finland, Oulu Acknowledgements University Press, Vol. 4, (1999), pp. 1534–1542.

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