04CVT-32
Institute of Applied Mechanics, Technical University of Munich
Institute of Applied Mechanics, Technical University of Munich
Copyright © 2004 SAE International
The paper will focus on investigations which were made during a cooperation project with an important supplier of gears. In some working points of a pushing V-belt CVT an unexplainable noise occurs. To find out the reason of this phenomena a simulation model is built up which contains an elastic model of the pulley sheaves as well as a detailed description of the belt. With this model investigations are made but the results do not include the expected vibration so far. An analytical approach is used the calculate possible eigenfrequencies of the belt.
Together with the belt forces from the simulation model it is shown that the eigenfrequencies of the belt are in the same range as the measured frequencies of the noise.
In the next step the simulation model is extended by a non-constant friction law. Investigations show that if the friction coefficient in the contact between pulley and elements decreases with the relative velocity, the belt could be excited by the friction-contact.
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.
Although there exist different types of 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 of two sets of pulleys and the belt that runs inside of the two Vpulleys. In this case the power transmission is very complicated because the transmitted torque results in the compression force between the elements and the ring tension. Additionally there are sveral types of different contacts: element-element, element-ring, element-pulley and ring-ring.
At the institute of Applied Mechanics of the Technical
University of Munich a two-dimensional model was developed to analyse the complex dynamical behaviour of a pushing V-Belt CVT.
The multibody system of the gear described above consists of two movable and two fixed sheaves and a force transmitting belt like in Figure. 2.1.
Figure 1.1 model of a metal V-Belt CVT
Figure 2.1 Metal pushing V-belt
2.1 PULLEY SET – A belt CVT-system contains a driving and a driven pulley. One sheave of each pulley is axially movable by a hydraulic cylinder forcing the belt to a pretended radius. With a controller the piston pressures p dr
, p dn
are affected to adjust the speed ratio and the pulley thrust. The boundary conditions are given

by the external excitation (torque M, or angular velocity
ω
), the control pressure p and the belt contact forces f i
.
A CVT-system with its different components offers a diversified frequency spectrum which has to be taken into account in the mathematical model. To reduce the calculation time, frequencies above a defined limit are eliminated by calculating their degrees of freedom with a quasi-static formulation. This section is divided into two parts: In the first part the dynamical degrees of freedom are introduced and in the second part we will focus on the quasi-static degrees of freedom which are used to describe the deformation of the elastic bodies.
2.1.1 Rigid-body model – The global dynamics of the pulleys can be described with a rigid-body model as it is shown in Figure 2.2. Its state is qualified by four degrees of freedom. The shaft with the axial fixed sheave is supposed to be bedded rigid. The rotation is specified by the angle ϕ and the angular velocity
ω
. The axial movable sheave is bedded with a shaft-to-collar connection on the axis.
(2)
At steady state the axial components of the contact forces f i,ax
are balanced with the spring prestress F c
and the piston force 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 stiffness of tilting c
δ
.
(3)
This rigid-body model considers only the elasticity of the shaft-to-collar connection, whereby the elasticity of the sheaves are not included yet. This is done by superposing the tilting
δ x
,
δ y
and the elastic deformation of the sheaves. The deflection model is presented in the next section.
2.1.2 Deflection of pulley sheaves – The contact forces between the belt and the pulleys cause a considerable deformation of the pulleys’ sheaves as it is shown in
Figure 2.3.
Figure 2.2 Rigid body model of pulleys
Its position in axial direction is described by the distance
∆ z
0
between the fixed and the movable sheave as it is shown in Figure 2.2. Due to the elasticity and clearance of the shaft-to-collar connection the movable sheave tilts.
This is quantified by
δ x
,
δ y
with respect to an orthogonal axes system perpendicular to the rotation axis. The rotation
ω
is either forced kinematically, or results in the principle of angular momentum.
(1)
Here the mass momentum of inertia J z
contains all masses of the pulley. At steady state the external torque
M and the axial components of the torques effected by the contact forces f i
and the lever arms r i
are equilibrated. The motion of the movable sheave is strongly damped by the hydraulic oil. Using a second order differential equation results in eigenvalues being of different order of magnitude. To avoid numerical problems the order is reduced to one (PT
1
∆ z
0
). The distance
is calculated by equation (2). With a delay time T
1
.
Figure 2.3 Deflection of the sheaves
With a FEM-analysis the sheave deformation can be calculated with the Reciprocal Theorem of Elasticity
(Betty/Maxwell) [4]. This means, that one contact force
F rp,j
influences the gap g i
of all other contacts of a sheave, due to the flexibility coefficients w ij
. Together with Hooke’s law with the stiffness c rp
applied at an element of the belt this results in a Linear
Complementary Problem (LCP) in a standard form, see equations (4).
(4)

If the eigenfrequencies of the sheaves are high enough as it is in the considered case, this formulation is accurate enough to describe the radial movement of the elements in the arc of contact. A dynamical Ritzapproach with the disadvantage of 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 effects like separation of elements have to be taken into account, on the other hand the belt consists of 300 to 600 elements which are constrained by two multi-layer sets of thin endless steel rings.
Because the mass of the elements is small and the stiffness high, a discrete modeling of the elements resulted in extreme high frequencies (nearly 500kHz).
Reasonable simulation times are impossible. Thus, the behaviour of the belt is described by decoupled Ritzapproaches for the longitudinal and the transversal dynamics.
The belt force consists of the pushing forces between the links and the tensile forces of the rings. There are friction forces f rt
between the links and the steel rings. As by chains, there are also contact forces between pulley and links, see Figure 2.5. Since separation between the links can occur, the normal force N can become zero.
Therefore, the theory of unilateral contacts is applied, see [1].
Figure 2.4 Reference system of the belt
Therefore a reference system R is introduced. The position of the belt at the path coordinate s
R
is specified by the transversal displacement w(s
R
,t). Its first (w’) and second (w’’) derivatives depend on s
R
. Figure 2.4 shows the system of the belt. To calculate the (relative-) velocities of any points of the belt the derivatives w on time, the local tangential velocity of the elements v
0
and of the ring v i
are needed in addition. For the elements the pitch line B
0
is used as reference for the transversal displacement w
0
and the velocity v
0
. In case of the rings w i
and v i
belong the neutral axes B i
. Because the transversal position of elements and rings are regarded as to be coupled, the system B
0
and B i
are parallel and w’, w’’ and w equal for all layers. An infinitesimal length s
R
of the reference system R differs in the dedicated length s i in a belt system B depends on the curvature
κ i
. The transformation
R
, the transversal position w i and its first derivative w’. It is given in equation (5), where n is the number of the single rings and h is their thickness.
(5)
The path length of the system B i
is calculated by the integral of (5) along s
R
. Here the length l
0
of the reference system R is known.
(6)
Figure 2.5 link forces
At a suitable clamping force on the sheaves the difference of the belt force in the two strands is mainly induced by the change of the pushing forces. Describing the dynamics of the pushing belt, we discern between longituidinal and transversal motion. In the longitudinal direction the possibility of separation of the links has to be considered, whereas the ring dynamics is smooth. In both cases the longitudinal acceleration consists of a mean value q belt and a steady state component, due to the belt strain gradient
ε/ s, where s denotes a belt fixed longitudinal coordinate, see Figure 2.4. In the arc of contact a small modeling error of radial position leads to a devastating error in the contact forces. Using a continuous approach a smooth grid is necessary for the shape functions in the contact area of the elastic bodies.
Therefore, the method of hierarchical bases is used, which represents finite B-Spline functions with differently sized supports. So the whole belt is discretised by a rough grid (Figure 2.6 a ) and a additional refinement at the contact areas (Figure 2.6 b). The borderline of the contact areas is most critical, so more refinements are needed (Figure 2.6 c,d). But it is sufficient to calculate them quasi-statically.
Figure 2.6 hierarchical bases
2.3 MODELLING OF CONTACTS – The power transmission of a metal V-belt CVT is determined by the contacts between elements and sheaves. A tangential relative velocity coupled with a normal force generates a

friction force in the contact plane. The friction forces 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 of high contact forces the elasto- hydrodynamic oil film is squeezed out of the contact zone and mixed friction appears. Hence, in most cases applying a smooth approximation of Coulomb’s friction law (Figure 2.7) is sufficient. Sometimes, as it is shown in the next section, an approximation of a Stribeck curve is needed to get realistic results.
Figure 2.7 Approximation of Coulomb’s friction law
This section will focus on investigations which were made during a cooperation project with an important supplier of gears.
3.1 PROBLEM DEFINITION – In some working points of a certain V-belt CVT unexplainable noise occured. It was known from experiments that this phenomenon can be reproduced savely by the following boundary conditions:
•
Speed ratio iV=2.52 or with a stop in LOW
•
Rotation speed of driving pulley n=3000 rpm
•
Axial force of driven pulley > 50 kN
If the load is increased over the time by a ramp, the noise will occur between 130Nm and 200Nm. In Figure
3.1 a measurement of the acting moment of force at the driving pulley is shown. On the ordinate the time is given in [s] over the load in [Nm].
Figure 3.1 Measured moment of force at driving pulley
Definitely there is a relation between the noise and the amplitude of the moment of force. The observed frequencies are all about 500 to 700 Hz. In Figure 3.2 an analysis of the frequencies in the experiment is shown.

In the considered case the kinetic and potential energy are given in equation (8).
Figure 3.3 Measured movement of strand under load
(8)
As an approximation the length of the belt between the two pulleys is seen as constant over the time. The deflection of the belt at the position x and time t is described with w(x,t). v is the velocity of the belt in longitudinal direction and
µ
is the mass per length. To solve the equations of motion a three-dimensional RITZapproach as in equation (9) is used.
Figure 3.4 Measured movement of strand under no load
3.2 ANALYTICAL APPROACH – Before starting time intensive simulation runs it seems useful to have an analytical estimation of the possible eigenfrequencies of the belt. In this case the method of LAGRANGE II
(equation (7) ) was used to get the equations of motion.
T is the kinetic and V the potential energy of the system.
(7) q is the vector of the generalized coordinates and Q
T
NK the vector of non-conservative forces. For a speed ratio of iV=2.52 the geometry is shown in figure 3.5.
(9)
The equations of motion can be given in the form of equation (10). M is the mass matrix, C the matrix of stiffness and B the matrix of forces which depend on the velocity.
(10)
The eigenvalues of the system can be calculated with equation (11).
(11)
In the considered case the influence of the velocity v is very small and terms which depend on v can be neglected. Then the eigenvalues of the belt can be given analytically in equation (12).
Figure 3.5 Geometry for speed ratio iV=2.52
(12)

As it is shown in equation (12) the eigenfrequencies are mainly dominated by the forces F of the belt. Here the pressure forces between the elements have to be considered as well as the traction forces of the steel rings.
3.3 BELT FORCES – The forces in equation (12) can be calculated with simplified analytical approaches or with the described simulation model which was used in this case. At underdrive (iV>1.0) the rings slip in the driving pulley contrary to the rotation direction.
Figure 3.6 Tensile and compression forces 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 force P t1
is higher than the difference of the ring tension force. also rises. The second and third eigenfrequency in
Figure 3.7 are in the same range as the measured frequencies of 500-700 Hz. This matches with observation at the test rig which are shown in Figure 3.3 and 3.4 and illustrate the transversal deflection of 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 for the eigenfrequencies of the belt it is possible to indicate the vibration phenomena as a belt eigenform. But in the simulation itself the vibration phenomena can not be observed, because the excitation mechanism is not included so far.
3.4 EXCITATION MECHANISM – IT was a fact that the noise occurs in the vehicle as well as in the test rig. So the external excitation mechanism as a source of vibration has been excluded in a nearly secure way. One effect causing vibrations in friction dominated systems are non-constant friction-coefficients. Examples can be found in literature like [3]. Therefore in the simulation model for the V-belt a friction characteristic like the
Stribeck-curve with a friction coefficient that depends on the relative velocity was implemented in the contact.
Figure 3.8 Non-constant friction characteristic (Stribeck)
With a characteristic as it is shown in Figure 3.8 a simulation run was done. The external parameters for speed ratio, rotational velocity and axial forces are the same as in the experiments (section 3.1). The external load rises over the simulation time.
Figure 3.7 Calculated eigenfrequencies depending on load
The results for the strand T t1
(under load) and T t2
(under no load) are given in Figure 3.7. Because of the increasing pressure between the elements in the strand under load, the eigenfrequencies will fall if the load rises.
In contrast the eigenfrequencies of the span under no load will rise, because the tensile stress in the steel rings
Figure 3.9 Belt vibration strand under load

In Figure 3.9 the results for the tranversal movement of an element in the span under load is given over the external moment. At a certain load the belt starts to oscillate with amplitudes of about 0.01 mm. Although these amplitudes are about half of the amplitudes in the measurements in Figure 3.3 they are of the same dimension.
3.5 INFLUENCE OF THE FRICTION
CHARECTERISTIC – At this point it has to be mentioned that the exact friction characteristic for the coefficient in the contact between elements and pulley was not known till the point of time at which these investigations were made. Therefore simulations with different friction characteristics have been done.
Figure 3.10 relative velocity in the arc of contact
An analysis of the frequency of the belt in the simulation is shown in figure 3.11. The results are nearly the same as in the plot of the measurement (figure 3.2). It is supposed that the peek at the higher frequency in 3.11 belongs to the strand under no load.The dominant frequencies can be also seen in the corresponding characteristic of the relative velocity between the elements and the pulleys which is shown in figure 3.10.
Figure 3.12 Friction characteristic
With a friction characteristic as shown in Figure 3.12, the simulated movement of the belt (strand under load) is given in Figure 3.13. Obviously causes a steeper gradient of the friction curve a greater excitation of the belt. In addition to that, the friction characteristic influences the load area where vibration of the belt occurs.
Figure 3.11 frequency-analyses of simulated belt vibration
Figure 3.13 Belt vibration, strand under load
3.6 ELASTICITY OF THE SHEAVES – Furthermore the excitation mechanism is influenced by the elasticity of the sheaves. As it was mentioned in the modeling section, FE- dates are used together with the principle of
(Betti/MAXWELL) to take the elasticity of the sheaves into account. Figure 3.14 shows the simulation results with a variation of the stiffness of the sheaves. If the sheaves are modeled with the original stiffness parameters of the FE –model, the belt is oscillating in two load areas. This is also true if the stiffness of the sheaves is 20% increased. But if the stiffness is increased 50%, the oscillations between 80 and 120 Nm will disappear and the belt starts to oscillate firstly at about 200Nm. In case of non-elastic sheaves the vibration of the belt will completely disappear.

The simulation results together with the measurements show that the belt can be excited by self-induced vibrations. These vibrations are caused by a friction coefficient that decreases with increasing relative velocity of the contact partners. This can be seen as a fact, because experiments have shown that if a certain oil is used which does not allow a negative gradient of the friction law, the observed vibrations will disappear.
4.1 MECHANISM OF SELF-INDUCED VIBRATIONS –
All systems which allow self-excitation need a source of energy which supplies the oscillator with enough energy to compensate the loss of the system.
Figure 3.14 Simulated belt vibration, variation of sheave elasticity
The variation of the stiffness of the sheaves show, that the radial movement of the elements in the arc of contact have a main influence on the vibration. If they are very stiff, vibration does not appear. Experimental investigations underline this statement.
Figure 4.1 Radial movement of the elements in contact
In this case the spiral movement of the belt in the arc of contact can be seen as such a source of energy.
Because of the elasticity of 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 of the relative velocity changes and the elements move out of the sheave. This movement can be seen as a kind of trajectory with a given velocity v
0
(Figure 4-2) depending on an angle ϕ
with ϕ in
<= ϕ
<= ϕ out
which describes the azimuthal position in the arc of contact.
In Figure 4.2 simulation results are shown. The relative velocity of an element in radial direction and the resulting relative velocity which includes both, the radial and the azimuthal components of an element, are given over the time. If v
0 is high enough, the gradient of the friction characteristic will be negative. Around v
0 small movements of the elements are possible as the jagged curve of the radial relative velocity in Figure 4.2 shows.

Figure 4.2 Relative velocity between elements and sheaves in the arc of contact (secondary pulley)
4.2 SIMPLIFIED MODEL –In order to analyse the vibration, a simplified model of the movement of the elements can be established, as it is shown in Figure
4.3. Some elements can be seen as one mass with one degree of freedom x against v
0 in radial direction.
The relative velocity between elements and pulleys in the arc of contact depends on the position of the elements in the sheaves given by an angle ϕ.
If there is an oscillation in the relative movement, as far as our experience goes, the radial direction will be dominant. Hence, in the following only the movement in radial direction is taken into account.
Figure 3.4 Simplified model of the radial movement of the elements in the arc of contact
In accordance with these assumptions the acting forces are the belt forces f( ϕ
) with a component f rad in radial direction and the contact forces between the elements and the sheaves consisting of the normal forces f
P,n and the friction forces f
P,r
. The stiffness of the belt is given by c belt
, the damping coefficient is d belt
. c ele is the resulting stiffness of the elements and the sheaves and is the friction coefficient between elements and sheaves. The wedge angle of the sheaves is . Therefore the equation of motion can by given by equation (13). m x
= f rad
−
2 f
P , n sin
θ −
2 f
P , n
µ cos
θ
(13)
With the preload forces f rad,0 in radial and f ax,0
in axial direction the forces are given by equation (14). f rad f ax
=
= f rad , 0 f ax , 0
− c belt
+ c ele x
− d belt x tan
θ x f
P , n
= f ax cos
θ
(14)

4.3 CONSIDERATION OF ENERGY – In the following the energy during one period of the oscillator given by equation (13) is considered. Therefore the movement of the mass in (13) is assumed to be harmonic as in equation (15) with the angular frequency . x x
=
= cos
ω t
− x ˆ
ω sin
ω t
(15)
The movement of the mass m around the trajectory is supposed to be small and the friction coefficient in the contact between the elements and the sheaves can be linearised around the given velocity v
0
. In equation (16)
µ
0
is the friction coefficient at given velocity v
0
and
µ ′ is the gradient
∂ µ
/
∂ v .
µ = µ
0
+ µ ′ x
(16)
The energy E during one period T can be calculated by equation (17) where f includes all extern forces acting on the mass-element in equation (13).
E
=
T
0 f
⋅
( v
0
+ 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
= f rad , 0 v
0
T
− d belt x ˆ 2
−
2 f ax , 0
µ
0 v
0
T
−
2
π
2
−
2 tan
θ
T f ax , 0
µ ′ x ˆ 2
4
π
2
T f ax , 0 v
0
T
−
(18)
If there is no oscillation with balanced. But if
xˆ the terms with v
0
are
µ ′
is negative and equation (19) is valid, the energy during one period is positive and the system can be excited.
µ ′ < −
2 d belt f ax , 0
(19)
4.4 BLOCKDIAGRAM OF EXCITATION MECHANISM
– Systems which can be excited by itself typically consists of three main components: a source of energy, an oscillator and a kind of switcher that supplies the oscillator with energy depending on the state of the system. The above mentioned excitation-mechanism can be described by the block-diagram in Figure 4-4.
Figure 4.4 Block diagram of excitation mechanism
In this case the source of energy is represented by the spiral movement of the belt in the arc of contact. The oscillator is the belt (with the elements) itself. If the movement of the elements in the arc of contact is in the same direction as v
0
, a part of the energy of the spiral motion is used to supply the oscillator with energy. The reason is the friction characteristic that supports the movement in direction of v
0
because the friction forces are smaller than in the other direction. Hence the friction curve can be seen as a switcher that is activated by the state of the oscillator.
•
A dynamical multibody model for a metal pushing Vbelt CVT is presented. The deflection of the pulleys and the belt as well as all relevant contact properties are taken into consideration. The frequencies being of technical relevance are calculated dynamically.
Frequencies above the limit of 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 friction characteristic depending on the relative velocity between the elements and the pulleys can cause self-induced vibrations of the belt. The frequencies of these vibrations are nearly the same in the simulation model and in measurements. The results show furthermore that the friction characteristic and the elasticity of the sheaves mainly determine the working area where vibration occurs. These working points still differ between measurements and simulation. Unfortunately it was not possible to get measurements of the exact friction curve. If they are available for example as a two dimensional characteristic diagram with the friction coefficient over the normal contact forces 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 simplified model for the radial

movement of the elements in the arc of contact was made. The consideration of energy during one period show, that self–induced vibrations are possible if the gradient of the friction characteristic will be negative.
1. F. Pfeiffer, Ch. Glocker, “Multibody Dynamics with
Unilateral Contacts”, John Wiley & Sons New York
1996.
2. M. Bullinger, F. Pfeiffer, H. Ulbrich, “Elastic Modelling of Contacts in continuous variable transmissions”,
Proc. of ECCOMAS Thematic Conferences on advances in computational Multibody Dynamics,
Lisabon 2003.
3. K. Magnus, K. Popp, “Schwingungen”, Teubner
Studienbücher Mechanik, Stuttgart 1997.
4. K. Magnus, „Grundlagen der Technischen
Mechanik“, Teubner Studienbücher Mechanik,
Stuttgart 1990.
5. T. Fujii, T. Kitagawa, S.J. Kanehara, „A Study of
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. 930666, SAE, p. 1-11. 1993.
6. T. Ide, H. Tanaka, “Contact Force Distribution
Between Pulley Sheave and Metal Pushing V-Belt”,
Proceedings of CVT 2002 Congress, VDI-Bericht
1790 p. 343-355., VDI-Verlag, Düsseldorf 2002.