Modelling the Elastodynamic Behaviour of a Desmodromic Valve Train

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

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

kb ks ks kb Js J s Js J φ φ φ φ θ δ τ 7 τ 8 τ 7 τ8 k x 7 x 8 x k m m 7 m 8 k 7 k 8 x k k x J J J θ θ θ δ m δ 6 x k 5 k 6 k 5 x 6 m 6 m 5 x 5 k 6 k 6 δ 6 Figure : schematic of the lumped-parameter model model. However, this second mechanism is identical to the first one and theoretically moves in phase; the effects of the neglected mechanism are therefore approximately introduced, assuming that this mechanism applies the same forces and torques to the camshaft as the first one.

In [, ] the authors presented models of desmodromic valve trains having eight dofs; in those studies the speed of the camshafts was assumed to be constant, that is, the dynamic behaviour of the mechanical transmission driving the camshafts was not considered. Actually, experimental measurements carried out on the test bench, have shown that the oscillations of the camshaft speed around the mean value are quite important (up to ±%). Therefore, the driveline moving the camshafts is taken into account in the present work. The model is developed with the aim to include all the important effects, as well as to get a rather simple model. In particular, it takes into account the mass distribution, the elastic flexibility of all links (including the compliance of the driveline, the bending and torsional compliance of the camshaft, and the variation of rocker stiffness as a function of mechanism position), the damping effects, the variability of transmission ratios with mechanism position, and the presence of several non-linearities (e.g. the Hertzian contact stiffness, the backlashes in joints, and the lubricant squeeze effect). With reference to Fig., the known model input is the coordinate φ, representing the angular displacement of the brushless motor s pulley, reduced to the intermediate shaft axis by the transmission ratio τ. This pulley is assumed to rotate at constant speed. The compliance of the first timing belt is taken into account by introducing the coordinate φ, representing the angular position of the first pulley of the intermediate shaft (see also Fig. ). The moment of inertia Js, associated with coordinate φ, is obtained by properly lumping the inertia of the first timing belt together with the first pulley of the intermediate shaft and a portion of the shaft close to the pulley. The torsional stiffness k b represents the stiffness of the first belt transmissions. The torsional compliance of the intermediate shaft is considered by introducing the coordinates φ and φ, representing the angular displacements of the flywheel and the second pulley of the intermediate shaft, respectively. They are associated with the moments of inertia Js and Js, which are obtained by lumping the moment of inertia of the portions of the shaft together with the moments of inertia of the flywheel and the second pulley, respectively. The inertia Js also includes a portion of the second time belt. Consequently, torsional stiffness ks concerns the portion of the camshaft between the first pulley and the flywheel, while torsional stiffness ks is relative to the portion of the intermediate shaft between the flywheel and the second pulley. By means of the coordinate θ, which represents the angular position of the pulley fitted on the intake camshaft, the compliance of the second belt transmission is considered. The moment of inertia J is obtained by properly lumping the moment of inertia of the camshaft pulley together with the moment of inertia of a camshaft portion, the inertia of the exhaust camshaft, and the inertia of a portion of the timing belt. The torsional stiffness k b represents the belt transmissions between the intermediate shaft and the intake camshaft. The stiffness of both timing belts are not considered as constant during the simulation. In fact, due to the effect of the dynamic loads, the belts can release, causing the pre-load to be lost. The torsional compliance of the camshaft is taken into account by introducing the coordinates θ and θ, representing the angular displacements of the positive and negative cam discs, respectively. In fact, with reference to the cam mechanism moving the considered valve, the positive cam disc is the closest to the camshaft pulley. The moments of inertia J and J, associated with these coordinates, are obtained by lumping the moment of inertia of camshaft portions together with the inertia of the positive and negative cam discs, respectively. Consequently, torsional stiffness k concerns the portion of the camshaft between the pulley and the positive cam, while torsional stiffness k is relative to the portion between the two cam discs. In order to include the camshaft bending compliance, the coordinates x 7 and x 8 are introduced: they are the deflections of the camshaft in correspondence of the conjugate cam and in two orthogonal directions, respectively coincident and perpendicular to the direction of the valve motion, as shown in Fig.. The deflections are assumed to be the same for the two cam discs, due to their small axial distance. The masses m 7 and m 8, associated with the bending coordinates, are the equivalent masses of the whole camshaft, computed by using the assumed-mode method with a shape function coincident to the static deflection. The camshaft bending stiffness is modelled by the springs k 7 and k 8, whilst the camshaft bearing compliance is considered as negligible. All the other coordinates, and the related model parameters, are reduced to the direction of the valve motion. Thus, the model contains the kinematic relationships of valve displacement vs. cam angular displacements and camshaft deflections: the cam angular displacements, θ and θ, are reduced using the cam curve functions x =x (θ ) and x =x (θ ); the

camshaft deflections, x 7 and x 8, are reduced using the transmission ratios τ 7 =dx /dx 7, τ 7 =dx /dx 7, τ 8 =dx /dx 8 and τ 8 =dx /dx 8. The reduction of the coordinates to the direction of the valve motion is represented by a lever system in Fig.. More details on the procedure of reduction can be found, for example, in []. The linear coordinates x and x 6 correspond to the angular displacements of the positive and negative rockers (respectively indicated by θ and θ 6 in Fig. ), after the reduction to the direction of the valve motion. The masses m and m 6, which are associated to the coordinates x and x 6, are the reduced moments of inertia of the rockers. The parameters k and k are firstly computed by reducing and composing in 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 compliance between the two springs. Stiffness k 6 is obtained by reducing and composing in series the stiffness of the negative rocker arm in contact with the cam and the stiffness of the related Hertzian contact; similarly, k 6 concerns the stiffness of the negative rocker arm in contact with the adjuster and the related Hertzian contact; the bending compliance of the negative rocker pin is also included in the evaluation of k 6 and k 6. The parameter k 6 is the reduced stiffness of the helical rocker spring. It is noteworthy that the stiffnesses of the rockers are assumed to be variable as a function of the mechanism position. Moreover, also the Hertzian stiffness is variable, as it depends on the contact force; in the simulation it is evaluated instantaneously. The possibility of separation of the rockers from the cam discs and the adjuster is included, in order to appropriately model the effects of backlashes. The parameters δ, δ, δ 6, δ 6 refer to the amount of separation in the joints when the mechanism is at rest, that is during the dwell phase; they are reduced to the valve motion direction. The mass valve is lumped partly in mass m, in correspondence of the adjuster, and partly in mass m 5, in correspondence of the valve-head. The axial stiffness of the valve stem is k 5, while stiffness k 5 represents the stiffness of the valve-head in contact with the seat. In order to globally take account of structural damping, as well as other damping, a viscous damper is associated with each stiffness. The damper coefficient is taken proportional to the corresponding stiffness; consequently, the variability of some of the model stiffnesses (due, for example, to the contribution of Hertzian contacts, or the variability of rocker stiffness with mechanism position) affects the associated damper coefficients. In case there is no contact in joints with backlash and links are approaching, the damper coefficient is computed in order to represent the lubricant squeeze effect [, 5, ]. Coulomb friction forces have not been introduced into the model, due to their low value. It is worth noting that the system is highly timevarying. As a matter of fact the active cam-follower system is different in the phases of positive and negative acceleration. In addition, several model parameters change with mechanism position. As a consequence, the differential equations of motion are strongly non-linear. They are numerically integrated by using the software Simulink.. Estimation of the model parameters The model parameters were preliminarily evaluated on the basis of both design and literature data. The values of inertial parameters, stiffnesses and squeeze coefficients were computed based on the geometry and the material of mechanism links. With reference to the mechanical transmission moving the camshafts, the main uncertainty concerns the estimation of the stiffness of the two timing belts. As a matter of fact, no experimental data about belt stiffness were available. Concerning the others mechanism links, the finite element method (FEM) was employed in order to evaluate the stiffness of the rockers and the valve-head. It is worth noting that the stiffness of the rockers was computed taking into account the effect of the variation of position and direction of the contact forces cam-rocker and rocker-adjuster, as the mechanism changes its position. The FEM analysis was carried out considering only few positions of the mechanism; during the simulation the instantaneous value of rocker stiffness is computed by means of interpolation. The values of the backlashes into kinematic pairs cam-rocker and rocker-adjuster, which are important functional parameters, were measured during the experimental tests, that is, taking into account the link thermal deformation. Although the backlashes have a nominal value that is the same for the two parts of the mechanism, the positive backlash tends to increase, due to thermal deformation, whilst the negative one suffers a reduction. The damper coefficients were preliminarily established based on previous models of the desmodromic valve train developed by the authors [, ].

* Ignoring Hertzian stiffness Js = 8. 5 kg m k * =..8 Js =.5 kg m Js = 6. kg m k * 6 =. 5.6 J =.65 kg m J =. 5 kg m k * =..8 J =.9 5 kg m m =. kg k * 6 = 8.9. m =.66 kg m 5 =. kg m 6 =.6 kg m 7 = 6.6 kg m 8 = 6.6 kg k b =.8 Nm/rad ks =.765 Nm/rad ks = 5.6 Nm/rad k b =.5 Nm/rad k = 7.6 Nm/rad k =.66 5 Nm/rad k 5 k 5 k 6 k 7 k 8 = 6. MN/m = 66.8 MN/m = 6. N/m = 97. MN/m = 97. MN/m δ =.5 mm δ 6 =. mm δ =. mm δ 6 =. mm Table : Model parameters MN/m MN/m MN/m MN/m The values of some parameters were then adjusted in order to better match experimental results. In particular, due to the uncertainty in estimating the stiffness of the two timing belts, it was expected that the parameters k b and k b would be significantly adjusted. On the contrary, the other stiffnesses would be decreased to some amount, as commonly occurs in modelling [6,, ]. As a matter of fact, after the comparison between the model simulation and experimental valve motion, the stiffness of the timing belts was set to % of the preliminarily established value, while the camshaft bending stiffness, the stiffness of positive and negative rockers, and the valve-head stiffness were assumed to be the 8, 8,, and 8%, of their computed values, respectively. Concerning the viscous dampers, it was not possible to assign the same value to the proportionality constants between damper coefficients and the related stiffnesses, because of different causes of damping which take place in the various parts of the mechanism. However, the proportionality constants were set within the limited range.5 5 s.5 5 s, with the exception of the timing belts, to which was assigned the value 5 s, due to their high structural damping. The values of the model parameters are listed in Table. Experimental apparatus As previously mentioned, the experimental study is carried out on a test bench which was planned to reproduce, as well as possible, the functional conditions of the valve train. In particular, the test bench and the measurement apparatus can operate for different cylinder head type, at high camshaft speed, under high temperature of the lubrication oil, and reproducing the motorbike power belt transmission. The experimental apparatus includes a test stand, a cylinder head, an electrically powered driveline to operate the camshaft, a lubrication circuit, and measurement instrumentation []. The maximum speed available at the camshaft is rpm. In order to properly lubricate the valve train, pressurized oil is fed into the cylinder head oil galleries; oil pressure and temperature are similar to those picked up from the motorbike during the racing (i.e. 6 bar and C, respectively). It is worth noting that only the components required for the operation of the valve train are included into the system; as a consequence, no gas forces, combustion, or spurious vibrations occur. Therefore, the system response is not the actual one, that is, the response of the motorbike engine system. However, the inclusion (or exclusion) of the forces due to compressed gases does not compromise the validity of the experimental data as a modelling tool []. The measurement equipment consists of a laser vibrometer and data acquisition apparatus. The laser equipment is a Polytec s High Speed Vibrometer (HSV-), used for non-contact measurements and high velocity applications, which can measure the absolute and relative velocity and displacement up to m/s and mm respectively; the maximum frequency is 5 khz. The centre of the valve-head plane surface was chosen as measurement point, thus making it possible to minimize possible valve s flexional

Normalised Valve Acceleration Experimental - - Numerical - - 6 8 6 Figure : valve acceleration (5 camshaft rpm) Normalised Valve Acceleration Experimental - - Numerical - - 6 8 6 Figure 6: valve acceleration (65 camshaft rpm) Normalised Valve Acceleration Experimental - - Numerical - - 6 8 6 Figure 5: valve acceleration (55 camshaft rpm) vibration effects, which may negatively affect valve motion measurement. An area close to the valve seat was selected as a reference surface. The differential measurement between valve surface and reference plane permits the elimination of raw vibration effects of head cylinder support. The signals were collected by means of a National Instrument PXI data acquisition system; the sampling frequency was khz and the filter cut-off frequency was set to 5 khz in order to prevent aliasing. During the tests, both valve velocity and displacement were recorded. The digital signals were processed and analysed with MATLAB software. 5 Results and discussion This Section firstly presents, by Figs. 6, some Valve Displacement [mm]... -.... Experimental Numerical -. 6 8 Figure 7: valve displacement (65 camshaft rpm) numerical results in comparison with experimental data in terms of valve acceleration over one cam revolution. They refer to camshaft speed of 5, 55, and 65 rpm. The experimental valve acceleration is obtained by means of numerical derivative, while the numerical valve motion is represented by the acceleration of the mass m 5 of the model. The acceleration scale is made dimensionless with reference to the theoretical maximum value. Figures 6 show that a good agreement between simulated and experimental results is attained for a wide range of camshaft speed. In particular, the simulation results are very similar to the actual valve motion within both positive and negative acceleration phases. As a matter of fact, the model is able to reproduce the more important dynamic phenomena and oscillations, and the level of the acceleration peaks is globally matched, even if some discrepancies exist.

Force [kn] 6 8 6 Figure 8: wavelet transform of valve acceleration Figure : positive cam-rocker contact force Normalised Valve Acceleration - - - Experimental Numerical (constant rocker stiffness) - 6 8 6 Figure 9: valve acceleration As a further example of the simulation accuracy, the contact between the valve-head and its seat is investigated for 65 camshaft rpm. In fact, the dynamic effects at the valve closure strongly affect the engine performances and, as such, have to be taken into account in order to predict the actual gas flow dynamics. Figure 7 compares the numerical displacement of the mass m 5 to the actual valve displacement, by means of an enlarged detail at the valve closure. The comparison shows that the model is able to properly simulate the impact location of the valve-head against the seat (at camshaft angle of about 75 degrees), as well as the bounce phenomena. In particular, the amount of separation between the valve-head and the seat, due to the bounces, is correctly estimated. Actually, in the experimental result, the first bounce takes longer and the second one occurs later than in the simulation case, but the nature of the phenomena is 69 68 67 66 65 6 6 Angular speed [rpm] 6 6 8 6 Figure : angular speed of camshaft pulley clearly the same. In order to evaluate the model effectiveness from the point of view of the frequency content, a wavelet analysis of both numerical and experimental valve acceleration is presented in Fig. 8, in the case of 65 camshaft rpm. As a matter of fact, since the data have non-stationary nature, a signal processing technique able to study how the frequency content changes with time has to be used []. The wavelet analysis reported in Fig. 8 shows that the most important dynamic phenomena are the impacts in the kinematic pairs of the rockers (at about 5, 65, and 55 camshaft degrees) and between the valvehead and its seat (at about 8 cam degrees). These impacts, due to their broadband frequency content, excite the mechanism resonances leading to damped oscillations. As noted in Sect.., the dynamic system is highly time-varying; thus, the resonance frequencies are different in the various angle phases.

Force [kn] 6 8 6 Figure : positive cam-rocker contact force (increased backlash) In particular, the comparison of Fig. 8 demonstrates that the excited natural frequencies of the model generally agree with the experimental ones, with the exception of the simulated valve-head impact which has lower frequency content. With the aim to remark the importance of taking into account the stiffness variability of the rockers, their stiffness was set to a constant value, correspondent to the position of zero and maximum valve displacement, for the positive and negative rocker, respectively. Figure 9 compares the numerical valve acceleration, obtained by assuming the stiffness as constant [Fig. 9(b)], with the experimental one [Fig. 9(a)], at 65 camshaft rpm. Disagreements between actual and simulated valve motion clearly appear both for the positive and negative acceleration phases. In particular, during the negative phase the period of the oscillations is incorrectly simulated, while in the second positive phase, the oscillation amplitude is overestimated. The model can be employed as a tool for design optimisation. In fact, by means of the simulation, the magnitude of dynamic forces can be predicted, thus making it possible to determine dynamic stress levels, to verify the structural strength of mechanism links, or to compute contact pressures. As an example, Fig. reports the contact force between the positive cam (inertia J in the model) and rocker (mass m ) for 65 camshaft rpm. As expected, the contact mainly deals with the positive acceleration phases. As a further example of model application, the angular speed of the camshaft pulley (inertia J ) is computed and reported in Fig., over one cam revolution. It shows that the oscillations of the camshaft speed around its mean value are quite Valve Displacement [mm]... -.... with rocker spring without rocker spring -. 6 8 Figure : numerical valve displacement important. These are the effect of the dynamic behaviour of the driveline moving the camshaft. As expected, the angular speed of the inertia Js (i.e. the flywheel) is practically constant; thus, the camshaft oscillations are essentially due to the compliance of the timing belt reproducing the engine belt loop. From a functional point of view, the estimation of these oscillations is very important, as it makes it possible to relate the camshaft angular position and, consequently, the valve motion, to the crank shaft position. The importance of modelling the driveline is therefore justified. As a final example, the model is employed to predict the system behaviour as a consequence of changes in some parameters. Since the dynamics of the desmodromic valve train is strongly affected by the backlash value in the kinematic pairs cam-rocker and rocker-adjuster, it is useful to predict the effect of variations of this important functional parameter due, for example, to wear or improper assembling. Figure shows the prediction of the positive camrocker contact force, due to backlash values that are increased with respect to those reported in Table : δ =. mm, δ 6 =.5 mm, and δ =δ 6 = mm. The peaks of the contact force increase of about % with respect to the previously examined case (see Fig. ). Another important aspect of the desmodromic train is the presence of the helical spring acting on the negative rocker. Figure shows the effect of a breakage of the rocker spring, by comparing the simulated valve displacement in normal condition to that one in fault condition, that is, when the value of the stiffness k 6 of the model is set to zero. In the last case, the valve keeps open for a longer time, thus affecting the engine s fluid dynamics.

6 Conclusions This paper presents a model for the simulation of the kineto-elastodynamic behaviour of the desmodromic valve train of Ducati engines, as well as a test bench designed in order to closely reproduce the functional conditions of the valve train. The model - having twelve dofs describes all the experimental mechanical system, i.e. both the valve train and the driveline. The model contains some non-linear elements and it is highly time-varying. In fact, the active cam-follower system is different in the phases of positive and negative accelerations. In addition, some model parameters in particular, the rocker stiffness strongly depend on the mechanism position. By comparison with valve motion measurements picked up on the test bench, the effectiveness of the model is satisfactorily assessed: it properly reproduces all the important dynamic phenomena taking place in the system, in a wide range of camshaft speed. It is pointed out that it is essential to take into account the variability of the rocker stiffness with mechanism position for obtaining satisfactorily simulations. Finally, applications of model for both design improvement and fault diagnostics are shown with some examples. Acknowledgements The authors wish to thank DUCATI CORSE and engineers for cooperation and assistance in the construction of the experiment. References [] Erdman, A. G., Sandor, G. N., A Review of the State of the Art and Trends, Mechanism and Machine Theory, Vol. 7, (97), pp. 9. [] Koster, M. P., Vibrations of Cam Mechanism, McMillan Press, London UK, 97. [] Sandor, G. N., Erdman, A. G., Advanced Mechanism Design: Analysis and Synthesis, Prentice-Hall, New Jersey, Vol., (98), pp. 8 5. [] Dresner, T. L., Norton, R. L., Modern Kinematics: Developments in the Last Forty Years, Erdman Ed., Wiley, New York, (99), pp. 5. [5] Dalpiaz, G., Maggiore, A., Monitoring Automatic Machines, Mechanical Systems and Signal Processing, Vol. 6, (99), pp. 57 5. [6] Dalpiaz, G., Rivola, A., A Kineto-elastodynamic Model of a Mechanism for Automatic Machine, Proceedings of the Ninth World Congress on the Theory of Machines and Mechanisms, Milano, Italy, Vol., (995), 7. [7] Pisano, A. P., Freudenstein, F., An Experimental and Analytical Investigation of the Dynamic Response of a High-Speed Cam-Follower System. Part : A Combined, Lumped/ Distribuited Dynamic Model, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 5, (98), pp. 699 7. [8] Nagaya, K., Watanabe, K., Tsukahara, Y., Vibration Analysis of High Rigidity Driven Valve System of Internal Combustion Engines, Journal of Sound and Vibration, Vol. 65, No., (99), pp.. [9] Özgür, K., Pasin, F., Separation Phenomena in Force Closed Cam Mechanisms, Mechanism and Machine Theory, Vol., No., (996), pp. 87 99. [] G. Dalpiaz, and A. Rivola, A Model for the Elastodynamic Analysis of a Desmodromic Valve Train, Proceedings of the Tenth World Congress on the Theory of Machines and Mechanisms, June, Oulu, Finland, Oulu University Press, Vol., (999), pp. 5 5. [] G. Dalpiaz, and A. Rivola, A Non-Linear Elastodynamic Model of a Desmodromic Valve Train, Mechanism and Machine Theory, Vol. 5, No., (), pp. 55 56. [] A. Carlini, A. Rivola, G. Dalpiaz, and A. Maggiore, Valve Motion Measurements on Motorbike Cylinder Heads Using High Speed Laser Vibrometer, to be presented at the 5th International Conference on Vibration Measurements by Laser Techniques, 8 June,, Ancona, Italy. [] A.P. Pisano, and F. Freudenstein, An Experimental and Analytical Investigation of the Dynamic Response of a High-Speed Cam- Follower System. Part : Experimental Investigation, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 5, (98), pp.69 698. [] O. Rioul, and M. Vetterli, Wavelets and Signal Processing, IEEE Signal Processing Magazine, Vol. 8, No., (99), pp. 8.