Analysis of self-induced vibrations in a pushing V-belt CVT

Transcription

1 4CVT-32 Analysis o sel-induced vibrations in a pushing V-belt CVT Copyright 24 SAE International Wolram Lebrecht Institute o Applied Mechanics, Technical University o Munich Friedrich Peier, Heinz Ulbrich Institute o Applied Mechanics, Technical University o Munich ABSTRACT The paper will ocus on investigations which were made during a cooperation project with an important supplier o gears. In some working points o a pushing V-belt CVT an unexplainable noise occurs. To ind out the reason o this phenomena a simulation model is built up which contains an elastic model o the pulley sheaves as well as a detailed description o the belt. With this model investigations are made but the results do not include the expected vibration so ar. An analytical approach is used the calculate possible eigenrequencies o the belt. Together with the belt orces rom the simulation model it is shown that the eigenrequencies o the belt are in the same range as the measured requencies o the noise. In the next step the simulation model is extended by a non-constant riction law. Investigations show that i the riction coeicient in the contact between pulley and elements decreases with the relative velocity, the belt could be excited by the riction-contact. Although there exist dierent types o CVT-systems, in this paper only the metal V-belt CVT is considered. As it is shown in Figure. 1.1 such a system consists o two sets o pulleys and the belt that runs inside o the two V- pulleys. In this case the power transmission is very complicated because the transmitted torque results in the compression orce between the elements and the ring tension. Additionally there are sveral types o dierent contacts: element-element, element-ring, element-pulley and ring-ring. At the institute o Applied Mechanics o the Technical University o Munich a two-dimensional model was developed to analyse the complex dynamical behaviour o a pushing V-Belt CVT. 2 MECHANICAL MODEL The multibody system o the gear described above consists o two movable and two ixed sheaves and a orce transmitting belt like in Figure INTRODUCTION Continuously variable transmissions are an interesting alternative to conventional concepts like manual or automatic transmissions. Due to a stepless variable speed ratio they have the potential to be an ideal intersection between the engine and the power train. Figure 2.1 Metal pushing V-belt Figure 1.1 model o a metal V-Belt CVT 2.1 PULLEY SET A belt CVT-system contains a driving and a driven pulley. One sheave o each pulley is axially movable by a hydraulic cylinder orcing the belt to a pretended radius. With a controller the piston pressures p dr, p dn are aected to adjust the speed ratio and the pulley thrust. The boundary conditions are given

2 by the external excitation (torque M, or angular velocity ω ), the control pressure p and the belt contact orces i. A CVT-system with its dierent components oers a diversiied requency spectrum which has to be taken into account in the mathematical model. To reduce the calculation time, requencies above a deined limit are eliminated by calculating their degrees o reedom with a quasi-static ormulation. This section is divided into two parts: In the irst part the dynamical degrees o reedom are introduced and in the second part we will ocus on the quasi-static degrees o reedom which are used to describe the deormation o the elastic bodies Rigid-body model The global dynamics o the pulleys can be described with a rigid-body model as it is shown in Figure 2.2. Its state is qualiied by our degrees o reedom. The shat with the axial ixed sheave is supposed to be bedded rigid. The rotation is speciied by the angle ϕ and the angular velocity ω. The axial movable sheave is bedded with a shat-to-collar connection on the axis. At steady state the axial components o the contact orces i,ax are balanced with the spring prestress F c and the piston orce composed by a rotation independent part Appand a rotation dependent part F ω. The tilting δ x, δ y is calculated by equation (3), using a stiness o tilting c δ. This rigid-body model considers only the elasticity o the shat-to-collar connection, whereby the elasticity o the sheaves are not included yet. This is done by superposing the tilting δ x, δ y and the elastic deormation o the sheaves. The delection model is presented in the next section. (2) (3) Delection o pulley sheaves The contact orces between the belt and the pulleys cause a considerable deormation o the pulleys sheaves as it is shown in Figure 2.3. Figure 2.2 Rigid body model o pulleys Its position in axial direction is described by the distance z between the ixed and the movable sheave as it is shown in Figure 2.2. Due to the elasticity and clearance o the shat-to-collar connection the movable sheave tilts. This is quantiied by δ x, δ y with respect to an orthogonal axes system perpendicular to the rotation axis. The rotation ω is either orced kinematically, or results in the principle o angular momentum. Here the mass momentum o inertia J z contains all masses o the pulley. At steady state the external torque M and the axial components o the torques eected by the contact orces i and the lever arms r i are equilibrated. The motion o the movable sheave is strongly damped by the hydraulic oil. Using a second order dierential equation results in eigenvalues being o dierent order o magnitude. To avoid numerical problems the order is reduced to one (PT 1 ). The distance z is calculated by equation (2). With a delay time T 1. (1) Figure 2.3 Delection o the sheaves With a FEM-analysis the sheave deormation can be calculated with the Reciprocal Theorem o Elasticity (Betty/Maxwell) [4]. This means, that one contact orce F rp,j inluences the gap g i o all other contacts o a sheave, due to the lexibility coeicients w ij. Together with Hooke s law with the stiness c rp applied at an element o the belt this results in a Linear Complementary Problem (LCP) in a standard orm, see equations (4). (4)

3 I the eigenrequencies o the sheaves are high enough as it is in the considered case, this ormulation is accurate enough to describe the radial movement o the elements in the arc o contact. A dynamical Ritzapproach with the disadvantage o high calculation time in can be neglected in this case. 2.2 MODEL OF THE BELT The metal V-belt can be described as a discrete or a continuous approach. On the one hand discrete eects like separation o elements have to be taken into account, on the other hand the belt consists o 3 to 6 elements which are constrained by two multi-layer sets o thin endless steel rings. Because the mass o the elements is small and the stiness high, a discrete modeling o the elements resulted in extreme high requencies (nearly 5kHz). Reasonable simulation times are impossible. Thus, the behaviour o the belt is described by decoupled Ritzapproaches or the longitudinal and the transversal dynamics. The belt orce consists o the pushing orces between the links and the tensile orces o the rings. There are riction orces rt between the links and the steel rings. As by chains, there are also contact orces between pulley and links, see Figure 2.5. Since separation between the links can occur, the normal orce N can become zero. Thereore, the theory o unilateral contacts is applied, see [1]. Figure 2.5 link orces Figure 2.4 Reerence system o the belt Thereore a reerence system R is introduced. The position o the belt at the path coordinate s R is speciied by the transversal displacement w(s R,t). Its irst (w ) and second (w ) derivatives depend on s R. Figure 2.4 shows the system o the belt. To calculate the (relative-) velocities o any points o the belt the derivatives w on time, the local tangential velocity o the elements v and o the ring v i are needed in addition. For the elements the pitch line B is used as reerence or the transversal displacement w and the velocity v. In case o the rings w i and v i belong the neutral axes B i. Because the transversal position o elements and rings are regarded as to be coupled, the system B and B i are parallel and w, w and w equal or all layers. An ininitesimal length s R o the reerence system R diers in the dedicated length s i in a belt system B i. The transormation depends on the curvature κ R, the transversal position w i and its irst derivative w. It is given in equation (5), where n is the number o the single rings and h is their thickness. At a suitable clamping orce on the sheaves the dierence o the belt orce in the two strands is mainly induced by the change o the pushing orces. Describing the dynamics o the pushing belt, we discern between longituidinal and transversal motion. In the longitudinal direction the possibility o separation o the links has to be considered, whereas the ring dynamics is smooth. In both cases the longitudinal acceleration consists o a mean value q belt and a steady state component, due to the belt strain gradient ε/ s, where s denotes a belt ixed longitudinal coordinate, see Figure 2.4. In the arc o contact a small modeling error o radial position leads to a devastating error in the contact orces. Using a continuous approach a smooth grid is necessary or the shape unctions in the contact area o the elastic bodies. Thereore, the method o hierarchical bases is used, which represents inite B-Spline unctions with dierently sized supports. So the whole belt is discretised by a rough grid (Figure 2.6 a ) and a additional reinement at the contact areas (Figure 2.6 b). The borderline o the contact areas is most critical, so more reinements are needed (Figure 2.6 c,d). But it is suicient to calculate them quasi-statically. (5) The path length o the system B i is calculated by the integral o (5) along s R. Here the length l o the reerence system R is known. (6) Figure 2.6 hierarchical bases 2.3 MODELLING OF CONTACTS The power transmission o a metal V-belt CVT is determined by the contacts between elements and sheaves. A tangential relative velocity coupled with a normal orce generates a

4 riction orce in the contact plane. The riction orces can be divided into two components, one in radial direction and one in azimutal direction as it is shown in Figure 2.5. Measurements show that because o high contact orces the elasto- hydrodynamic oil ilm is squeezed out o the contact zone and mixed riction appears. Hence, in most cases applying a smooth approximation o Coulomb s riction law (Figure 2.7) is suicient. Sometimes, as it is shown in the next section, an approximation o a Stribeck curve is needed to get realistic results. Figure 2.7 Approximation o Coulomb s riction law 3 VIBRATION ANALYSIS This section will ocus on investigations which were made during a cooperation project with an important supplier o gears. Figure 3.1 Measured moment o orce at driving pulley Deinitely there is a relation between the noise and the amplitude o the moment o orce. The observed requencies are all about 5 to 7 Hz. In Figure 3.2 an analysis o the requencies in the experiment is shown. 3.1 PROBLEM DEFINITION In some working points o a certain V-belt CVT unexplainable noise occured. It was known rom experiments that this phenomenon can be reproduced savely by the ollowing boundary conditions: Speed ratio iv=2.52 or with a stop in LOW Rotation speed o driving pulley n=3 rpm Axial orce o driven pulley > 5 kn I the load is increased over the time by a ramp, the noise will occur between 13Nm and 2Nm. In Figure 3.1 a measurement o the acting moment o orce at the driving pulley is shown. On the ordinate the time is given in [s] over the load in [Nm]. Figure 3.2 Analyses o measured requencies Furthermore the movement o the belt between the two pulleys could be observed. In 3.3 and 3.4 the measured movement o the belt is shown or the strand under load and no load. Because the noise and the observed belt vibrations occur in the same working points, the vibration o the belt was seen as a possible reason or the disturbing noise phenomenon.

5 In the considered case the kinetic and potential energy are given in equation (8). (8) Figure 3.3 Measured movement o strand under load As an approximation the length o the belt between the two pulleys is seen as constant over the time. The delection o the belt at the position x and time t is described with w(x,t). v is the velocity o the belt in longitudinal direction and µ is the mass per length. To solve the equations o motion a three-dimensional RITZapproach as in equation (9) is used. Figure 3.4 Measured movement o strand under no load 3.2 ANALYTICAL APPROACH Beore starting time intensive simulation runs it seems useul to have an analytical estimation o the possible eigenrequencies o the belt. In this case the method o LAGRANGE II (equation (7) ) was used to get the equations o motion. T is the kinetic and V the potential energy o the system. The equations o motion can be given in the orm o equation (1). M is the mass matrix, C the matrix o stiness and B the matrix o orces which depend on the velocity. (9) (1) (7) The eigenvalues o the system can be calculated with equation (11). q is the vector o the generalized coordinates and Q T NK the vector o non-conservative orces. For a speed ratio o iv=2.52 the geometry is shown in igure 3.5. (11) In the considered case the inluence o the velocity v is very small and terms which depend on v can be neglected. Then the eigenvalues o the belt can be given analytically in equation (12). Figure 3.5 Geometry or speed ratio iv=2.52 (12)

6 As it is shown in equation (12) the eigenrequencies are mainly dominated by the orces F o the belt. Here the pressure orces between the elements have to be considered as well as the traction orces o the steel rings. 3.3 BELT FORCES The orces in equation (12) can be calculated with simpliied analytical approaches or with the described simulation model which was used in this case. At underdrive (iv>1.) the rings slip in the driving pulley contrary to the rotation direction. Figure 3.6 Tensile and compression orces at UD By this the tension T t2 in Figure 3.6 in the strand t2 rises always above the tension T t1 in the strand t1. At normal load the compression orce P t1 is higher than the dierence o the ring tension orce. also rises. The second and third eigenrequency in Figure 3.7 are in the same range as the measured requencies o 5-7 Hz. This matches with observation at the test rig which are shown in Figure 3.3 and 3.4 and illustrate the transversal delection o the belt in both strands, the load strand und the no-load strand. Obviously in the load strand the second eigenshape is dominant whereas in the strand under no-load it is the third eigenshape. With the simulation model and the analytical approach or the eigenrequencies o the belt it is possible to indicate the vibration phenomena as a belt eigenorm. But in the simulation itsel the vibration phenomena can not be observed, because the excitation mechanism is not included so ar. 3.4 EXCITATION MECHANISM IT was a act that the noise occurs in the vehicle as well as in the test rig. So the external excitation mechanism as a source o vibration has been excluded in a nearly secure way. One eect causing vibrations in riction dominated systems are non-constant riction-coeicients. Examples can be ound in literature like [3]. Thereore in the simulation model or the V-belt a riction characteristic like the Stribeck-curve with a riction coeicient that depends on the relative velocity was implemented in the contact. Figure 3.8 Non-constant riction characteristic (Stribeck) With a characteristic as it is shown in Figure 3.8 a simulation run was done. The external parameters or speed ratio, rotational velocity and axial orces are the same as in the experiments (section 3.1). The external load rises over the simulation time. Figure 3.7 Calculated eigenrequencies depending on load The results or the strand T t1 (under load) and T t2 (under no load) are given in Figure 3.7. Because o the increasing pressure between the elements in the strand under load, the eigenrequencies will all i the load rises. In contrast the eigenrequencies o the span under no load will rise, because the tensile stress in the steel rings Figure 3.9 Belt vibration strand under load

7 In Figure 3.9 the results or the tranversal movement o an element in the span under load is given over the external moment. At a certain load the belt starts to oscillate with amplitudes o about.1 mm. Although these amplitudes are about hal o the amplitudes in the measurements in Figure 3.3 they are o the same dimension. 3.5 INFLUENCE OF THE FRICTION CHARECTERISTIC At this point it has to be mentioned that the exact riction characteristic or the coeicient in the contact between elements and pulley was not known till the point o time at which these investigations were made. Thereore simulations with dierent riction characteristics have been done. Figure 3.12 Friction characteristic Figure 3.1 relative velocity in the arc o contact With a riction characteristic as shown in Figure 3.12, the simulated movement o the belt (strand under load) is given in Figure Obviously causes a steeper gradient o the riction curve a greater excitation o the belt. In addition to that, the riction characteristic inluences the load area where vibration o the belt occurs. An analysis o the requency o the belt in the simulation is shown in igure The results are nearly the same as in the plot o the measurement (igure 3.2). It is supposed that the peek at the higher requency in 3.11 belongs to the strand under no load.the dominant requencies can be also seen in the corresponding characteristic o the relative velocity between the elements and the pulleys which is shown in igure 3.1. Figure 3.13 Belt vibration, strand under load Figure 3.11 requency-analyses o simulated belt vibration 3.6 ELASTICITY OF THE SHEAVES Furthermore the excitation mechanism is inluenced by the elasticity o the sheaves. As it was mentioned in the modeling section, FE- dates are used together with the principle o (Betti/MAXWELL) to take the elasticity o the sheaves into account. Figure 3.14 shows the simulation results with a variation o the stiness o the sheaves. I the sheaves are modeled with the original stiness parameters o the FE model, the belt is oscillating in two load areas. This is also true i the stiness o the sheaves is 2% increased. But i the stiness is increased 5%, the oscillations between 8 and 12 Nm will disappear and the belt starts to oscillate irstly at about 2Nm. In case o non-elastic sheaves the vibration o the belt will completely disappear.

8 4 EXCITATION MECHANISM The simulation results together with the measurements show that the belt can be excited by sel-induced vibrations. These vibrations are caused by a riction coeicient that decreases with increasing relative velocity o the contact partners. This can be seen as a act, because experiments have shown that i a certain oil is used which does not allow a negative gradient o the riction law, the observed vibrations will disappear. 4.1 MECHANISM OF SELF-INDUCED VIBRATIONS All systems which allow sel-excitation need a source o energy which supplies the oscillator with enough energy to compensate the loss o the system. Figure 3.14 Simulated belt vibration, variation o sheave elasticity The variation o the stiness o the sheaves show, that the radial movement o the elements in the arc o contact have a main inluence on the vibration. I they are very sti, vibration does not appear. Experimental investigations underline this statement. Figure 4.1 Radial movement o the elements in contact In this case the spiral movement o the belt in the arc o contact can be seen as such a source o energy. Because o the elasticity o the sheaves, the elements in contact move in radial direction into the sheave till the innermost point is reached as it is shown in Figure 4.1. Then the direction o the relative velocity changes and the elements move out o the sheave. This movement can be seen as a kind o trajectory with a given velocity v (Figure 4-2) depending on an angle ϕ with ϕ in <=ϕ<=ϕ out which describes the azimuthal position in the arc o contact. In Figure 4.2 simulation results are shown. The relative velocity o an element in radial direction and the resulting relative velocity which includes both, the radial and the azimuthal components o an element, are given over the time. I v is high enough, the gradient o the riction characteristic will be negative. Around v small movements o the elements are possible as the jagged curve o the radial relative velocity in Figure 4.2 shows.

9 Figure 3.4 Simpliied model o the radial movement o the elements in the arc o contact Figure 4.2 Relative velocity between elements and sheaves in the arc o contact (secondary pulley) 4.2 SIMPLIFIED MODEL In order to analyse the vibration, a simpliied model o the movement o the elements can be established, as it is shown in Figure 4.3. Some elements can be seen as one mass with one degree o reedom x against v in radial direction. The relative velocity between elements and pulleys in the arc o contact depends on the position o the elements in the sheaves given by an angle ϕ. I there is an oscillation in the relative movement, as ar as our experience goes, the radial direction will be dominant. Hence, in the ollowing only the movement in radial direction is taken into account. In accordance with these assumptions the acting orces are the belt orces (ϕ) with a component rad in radial direction and the contact orces between the elements and the sheaves consisting o the normal orces P,n and the riction orces P,r. The stiness o the belt is given by c belt, the damping coeicient is d belt. c ele is the resulting stiness o the elements and the sheaves and is the riction coeicient between elements and sheaves. The wedge angle o the sheaves is. Thereore the equation o motion can by given by equation (13). m x = rad 2 P n sinθ 2 P,, n µ cosθ (13) With the preload orces rad, in radial and ax, in axial direction the orces are given by equation (14). rad ax P, n = c x d x = rad, ax, + c ax = cos θ ele belt tan θ x belt (14)

10 4.3 CONSIDERATION OF ENERGY In the ollowing the energy during one period o the oscillator given by equation (13) is considered. Thereore the movement o the mass in (13) is assumed to be harmonic as in equation (15) with the angular requency. x = xˆ cosωt x = xˆ ω sinωt 4.4 BLOCKDIAGRAM OF EXCITATION MECHANISM Systems which can be excited by itsel typically consists o three main components: a source o energy, an oscillator and a kind o switcher that supplies the oscillator with energy depending on the state o the system. The above mentioned excitation-mechanism can be described by the block-diagram in Figure 4-4. (15) The movement o the mass m around the trajectory is supposed to be small and the riction coeicient in the contact between the elements and the sheaves can be linearised around the given velocity v. In equation (16) µ is the riction coeicient at given velocity v and µ is the gradient µ / v. µ µ + µ x = (16) The energy E during one period T can be calculated by equation (17) where includes all extern orces acting on the mass-element in equation (13). E = T ( v + x dt ) (17) Using (13), (14), (15), (16) and (17), the energy during one period can be calculated. The result is given in equation (18). E = rad, 2 v T d ax, belt xˆ µ v T 2 2 2π T ax, µ xˆ 2 tanθ 2 2 4π T ax, v T (18) I there is no oscillation with xˆ the terms with v are balanced. But i µ is negative and equation (19) is valid, the energy during one period is positive and the system can be excited. d µ < belt 2 ax, (19) Figure 4.4 Block diagram o excitation mechanism In this case the source o energy is represented by the spiral movement o the belt in the arc o contact. The oscillator is the belt (with the elements) itsel. I the movement o the elements in the arc o contact is in the same direction as v, a part o the energy o the spiral motion is used to supply the oscillator with energy. The reason is the riction characteristic that supports the movement in direction o v because the riction orces are smaller than in the other direction. Hence the riction curve can be seen as a switcher that is activated by the state o the oscillator. 5 CONCLUSION A dynamical multibody model or a metal pushing V- belt CVT is presented. The delection o the pulleys and the belt as well as all relevant contact properties are taken into consideration. The requencies being o technical relevance are calculated dynamically. Frequencies above the limit o engeneering interest are eliminated by a quasistatic approach. To analyse the acoustic phenomenon, investigations were made with the simulation model. It is shown that a riction characteristic depending on the relative velocity between the elements and the pulleys can cause sel-induced vibrations o the belt. The requencies o these vibrations are nearly the same in the simulation model and in measurements. The results show urthermore that the riction characteristic and the elasticity o the sheaves mainly determine the working area where vibration occurs. These working points still dier between measurements and simulation. Unortunately it was not possible to get measurements o the exact riction curve. I they are available or example as a two dimensional characteristic diagram with the riction coeicient over the normal contact orces and the relative velocity, it might be possible to improve the comparison between experiment and simulation. To explain the excitation phenomena, an analytical approach with a simpliied model or the radial

11 movement o the elements in the arc o contact was made. The consideration o energy during one period show, that sel induced vibrations are possible i the gradient o the riction characteristic will be negative. REFERENCES 1. F. Peier, Ch. Glocker, Multibody Dynamics with Unilateral Contacts, John Wiley & Sons New York M. Bullinger, F. Peier, H. Ulbrich, Elastic Modelling o Contacts in continuous variable transmissions, Proc. o ECCOMAS Thematic Conerences on advances in computational Multibody Dynamics, Lisabon K. Magnus, K. Popp, Schwingungen, Teubner Studienbücher Mechanik, Stuttgart K. Magnus, Grundlagen der Technischen Mechanik, Teubner Studienbücher Mechanik, Stuttgart T. Fujii, T. Kitagawa, S.J. Kanehara, A Study o Metal Pushing V-Belt Type CVT-Part 1: Relation between Transmitted Torque and Pulley Thrust, Int. Congress and Exposition Detroit, SAE Technical Paper Series, Nr , SAE, p T. Ide, H. Tanaka, Contact Force Distribution Between Pulley Sheave and Metal Pushing V-Belt, Proceedings o CVT 22 Congress, VDI-Bericht 179 p , VDI-Verlag, Düsseldor 22.