FR3119000A1 - Pendulum damping device and associated design method - Google Patents

Abstract

Pendulum damping device (100) for mobility device comprising: - a support (1), - a plurality of N pendulums (5) characterized in that at least one pendulum (5) verifies the following equation: Figure for the abstract: Figure 2

Description

Pendulum damping device and associated design method

The field of the present invention relates to coupling systems intended to couple an energy-generating member of a mobility device to a distributing member of this energy, such as clutch mechanisms or torque converters.

Rotating machinery is often subject to torque fluctuations, causing rotor vibration and ultimately causing premature fatigue and noise pollution. In the automotive industry, vibrations of the transmission system generated by the torque fluctuations of a heat engine can be reduced by using pendulum damping devices (in English “centrifugal pendulum vibration absorbers” CPVA). These passive devices consist of several masses oscillating along a given trajectory with respect to the rotor at their support. Previous studies have revealed that the dynamics of these devices are subject to instabilities. There is a need to improve the dynamic stability of pendular damping devices.

To meet this need, the invention according to one of its aspects proposes a pendular damping device for a mobility device, said device comprising:

- a support,,

- a plurality of pendulums

and characterized in that at least one pendulum verifies the following equation:

According to the variants, the pendular damping device according to the invention has all or part of the following characteristics:

• at least one pendulum has a profile for rolling without slipping on a roller of the device.
• the roller is fixed relative to the support
• the order response of at least one pendulum verifies the following equation:

• the pendulum damping device comprises two single-wire pendulums
• the pendular damping device comprises two bifilar pendulums
• the pendular damping device has parameters verifying at least one of the relationships below:

η is positive, even 0< η <2, η representing

0.05 < µ < 0.2 , µ representing

α _[1] is positive, even 0< α _[1] <1, µ representing

n _p <1, µ representing

-5<x _[4] <5), µ representing

Another subject of the invention, according to another of its aspects, is a transmission system for a mobility device comprising a pendular damping device as described.

Advantageously, the pendulums according to the invention admit a significant rotational movement with respect to the support, in addition to the traditional translational movement. The efficiency of such devices is optimal for a perfect synchronous movement of oscillating masses. Nevertheless, due to non-linearities, the unison of masses can be broken in favor of an energy localization on a given damper, which generates a loss of attenuation efficiency. To assess the stability of such devices, a dynamic model based on an analytical perturbation method is established. The purpose of this model is to analytically predict a localization and response jumps. The validity of the model was confirmed by comparison with both a numerical resolution of the system dynamics and an experimental study.

The invention can be better understood on reading the following description of examples of its implementation and on examining the appended drawing in which:

- the represents a system according to an implementation of the invention,

- the schematically represents a pendular damping device according to the invention,

- the illustrates the order response of unison pendulums and their stability,

- the represents the response of the pendulums and the response of the support obtained through the analytical and numerical models

- the illustrates the time signals of the pendulums and the support, and

- Figure 6 illustrates an example of a pendular damping device shown schematically in

Within the framework of the reduction of polluting emissions and of the fuel consumption of vehicles with internal combustion engines, automobile manufacturers are trying to reduce the cylinder capacity and the speed of the engine. These developments lead to a significant increase in rotational irregularities called “acyclisms”, mainly due to higher combustion pressure. One of the main characteristics of these reciprocating engines is the linear dependence of the acyclic frequency on the average engine speed. The proportionality coefficient is called the engine (or ignition) order and depends only on the architecture of the engine. For four-stroke engines, the engine order is half the number of cylinders. During the acceleration phase, the motor sweeps through a wide range of frequencies containing certain driveline torsion modes. This situation can lead to significant levels of noise and vibration in the passenger compartment and premature wear of driveline components. Pendulum damping devices have been used for many years to minimize automotive powertrain acyclisms in engine order. These passive devices consist of oscillating masses (pendulums) moving along particular paths relative to a primary inertia (support) as shown in Fig. . The operating principle of the pendulum damping device is that of a tuned mass damper whose rigidity is proportional to the average engine speed. That allows it to stay tuned to the motor sequence to reduce torsional vibrations from the mounting.

The system studied is represented in FIG. 1. It consists of a flywheel J _r rotating around its center O and serving as a support 1 for a plurality of N pendulums 5, with here N=2 pendulums. Its total angular position is where t is time, Ω is the average rotational speed of the flywheel, and θ is the fluctuation part of the rotation. A couple is applied to this steering wheel, where represents the total angular position of the steering wheel, i.e. . can be decomposed into a constant torque and a periodic couple . The constant torque balances with the torque resulting from the damping of the flywheel, thus giving the average rotational speed Ω.

In the model, an equivalent linear viscous damping is used for the steering wheel. N mass pendulums and inertia (around their center of gravity) oscillate on their trajectory . Their position on these trajectories is given by the curvilinear abscissa and their distance from the center of rotation of the steering wheel is . The characteristic dimension represents the position of the pendulums at rest (when so that they are perfectly centrifuged). In addition to the traditional translational movement, the present study considers that the pendulums rotate relative to the flywheel according to the rotation function . As with the flywheel, an equivalent linear viscous damping is used to model the damping between the flywheel and the i ^th pendulum. The pendulums and their associated functions of trajectory and rotation are considered hereafter to be identical and the index "i" is therefore dropped with regard to the parameters of the pendulums. Using the relationship between average torque, support damping and average rotational speed, , by introducing the non-dimensional parameters and variables

and changing the independent variable of t by , the equations of motion can be written as follows:

where represents a derivation from . Thus the functions of trajectory and rotation can be expressed in the form of polynomials such as:

where is the tuning order of the pendulums and are trajectory and rotation coefficients. Note that, in the case of , the trajectories of the pendulums are epicycloids, which correspond to the tautochrone trajectory for .

It can be assumed that the torque fluctuation applied to the support contains only one harmonic whose non-dimensional form is , where n is the motor order. For a car engine, n is the number of strokes per revolution of the crankshaft. At this stage, is not a periodic forcing term but a nonlinear term because it depends on θ. Two methods are proposed below to transform it into a periodic forcing term.

Method 1 . It is known to use the change of independent variable proposed by S. Shaw et al [10] to directly transform into a periodic forcing term.

Method 2 . Another method is to express the torque fluctuation as a function of for writing where is the angular frequency of the acyclism. For a car engine, the excitation frequency must correspond to the number of strokes per second, which gives where refers to a derivation with respect to time. It can be assumed that the crankshaft rotational speed fluctuation is much lower than the average rotational speed, i.e. . It follows that , which generates a banal transformation of the torque into a periodic forcing term.

A linear analysis of the system is performed. After transforming the external torque into a periodic forcing term according to method 2 (above, one can linearize equations (1) and (2) and then use a known property of arrowhead matrices to find that the eigenvalues and the eigenvectors of the system are:

is the efficient chord order, is the linear rotation coefficient and the exponent T indicates the transposition. is a rigid body mode for which only the support is excited. are N - 1 degenerate modes for which the pendulums i and i + 1 are out of phase while the other modules are stationary (i = 1, …, N-1). They are associated with the eigenvalue n ₁₀ , which has a multiplicity N – 1 and the support is a node of these modes. Note that, in the case N = 2, is not degenerate and simply corresponds to an out-of-phase motion of the pendulums. is a mode for which the pendulums move in unison but in phase opposition with respect to the support. corresponds to the proper order of the pendulums in the case , and it is related to the chord order so that :

When a torque fluctuation is applied to the support, the pendulums generate an antiresonance on order (in the conservative case). Thus, for a torque fluctuation of order n, one must choose to minimize support vibrations (which can be extended to the non-conservative case since the damping is low).

This application proposes a linear analysis of the pendular damping device. Nevertheless, it can be seen from equations (1) and (2) that the dynamics of the system is subject to several sources of nonlinearity. An analytical model allowing the visualization and prediction of nonlinear phenomena will be derived hereafter.

It is known that the construction of the model can start with a scaling of the parameters. This allows a simplification of the equations of motion so that the dynamics of the pendulums become decoupled from that of the support. Since it is considered that the movement of the pendulums is small, the trajectory and rotation functions (3) are truncated so that:

be used in the following calculations. These functions are assumed to be symmetric (respectively antisymmetric) with respect to s _i = 0 as is the case in practice due to design constraints. The fluctuation of the rotation of the support is a priori made up of several harmonics so that it can be extended as follows:

Where and are the first and second harmonics respectively, and HOT stands for Higher Order Terms.

Scaling

In this paragraph, the goal is to scale the weight of certain parameters and variables in order to capture the desired physical phenomena. The following assumptions and remarks govern scaling:

• the optimal configuration of the system corresponds to low damping (both of the support and of the pendulums); in this way, the vibration amplitude of the support at its antiresonance is very low;

• torque fluctuation is small compared to the kinetic energy of the support (which is equilibrium) ; that implies that is small ;

• the total geometric inertia of the pendulums around the point O, , is considered to be small compared to the inertia of the total rotational system, , so that µ is small;

• the inertia of the support being significant, the fluctuation in rotation speed is low compared to the average rotational speed; note that this assumption has already been adopted in chapter 3 to transform the external torque into a periodic excitation;

• the amplitude of motion of the pendulums is small compared to their distance from the center of rotation O, so that be small;

• the trajectory function chosen (cf. equation (6)) is an epicycloid perturbed by ; the disturbance being considered to be weak, is a priori small;

• the chosen rotation function (cf. equation (6)) differs from a linear rotation by the term ; the rotation being considered to be mainly linear, is a priori small.

From the above, by introducing the small parameter , the following scaled parameters are introduced:

where and are scaling coefficients to be determined. To give physical meaning to , we can choose to set .

Support dynamics

The goal here is to obtain an equation governing the dynamics of the support as a function of the pendulum torque and the external torque. By introducing the trajectory and rotation functions (6), the extended form of θ (7) and the scaled parameters (8) into the support equation (1) and setting , we can write :

Note that the external torque has been transformed into a periodic forcing term using method 2 (see chapter 2). Equation (9) uses the first-order pendulum equation (supposing ).

Equation (9) is similar to that obtained if the change in independent variable has been used. The two differences are that (•)' represents a derivation from and that the torque term is . Nevertheless, using the chain rule, we can observe that:

so that, at the order retained in equation (9), derivatives with respect to and are equivalent. In addition, by extending the term torque, we can notice that:

This means that, at the order retained in equation (9) and assuming that , it is equivalent to transform the torque into a periodic forcing term using the change of independent variables (cf. method 1 in chapter 2) or assuming that (see method 2 in chapter 2).

Pendulum dynamics

The goal here is to decouple the dynamics of the pendulum from that of the support. For this, the reduced equation (9) of the support is introduced into the equation (2) of the pendulum. Then, using the trajectory and rotation functions (6) and choosing the following set of scaling coefficients:

the pendulum equation reduces to:

The set of scaling parameters was chosen to decouple the pendulum equation from the support dynamics while retaining the effect of nonlinearities arising from trajectory and rotation functions. Equation (13) includes the influence of external torque, coupling between pendulums through the sum over N, and nonlinearities arising from both path and rotation. It is also known that since the pendulums are identical and coupled, the system exhibits 1:1 internal resonances. Equations (13) are weakly coupled because the pendulums are indirectly coupled through the support and their effect on the support is small since their relative inertia is small. Also, these equations are weakly nonlinear because it is assumed that is small and the trajectory and rotation functions chosen are close to an epicycloid and a linear rotation, which gives a linear behavior for small fluctuations in the rotational speed.

In the present study, it was chosen to solve the reduced equations of the pendulums by using the known method with multiple scales. Two rotation scales are introduced, and . As previously done for the support, the pendulum displacement is extended so that:

where and represent the first and third harmonics respectively. Plugging equation (14) into equation (13), we get a solution for in the form of :

where and are governed by the system with 2N equations:

and are vectors containing respectively and , and D ₁ represents a derivation with respect to . Functions and are given by:

where σ is a disagreement term such that . Within the framework of the study presented in this document, only the constant state regime is taken into account. This implies that the amplitude and the phase of the pendulums with respect to the excitation are invariable over time, so that:

where represents the phase shift of pendulum i with respect to excitation and takes the form of . Using this phase shift, the first-order response can be written in the more classical form:

The introduction of equation (19) and in equation (16) gives:

is a vector containing . and are respectively and , expressed in terms of coordinates and . The system (21) must be solved to determine the pendulum response.

It was observed in chapter 3 that, in a linear regime, a single mode participates in the response of the pendulums. This linear mode, , is called "unison mode" because it describes a movement in which all pendulums have the same amplitude and phase. Since the behavior considered in this chapter is weakly nonlinear, we can expect the pendulums to move in unison. It is therefore assumed that:

By introducing equation (22) into equation (21), one can find the following implicit order response of unison pendulums:

The backbone of the nonlinear unison mode can be derived from equation (23) as: . It is interesting to notice that the hardening or softening behavior of pendulums is governed by the sign of . This property will later be used to specify design rules (see chapter 7). By inserting equation (22) into equation (21), one can also find the following implicit amplitude response:

The calculation of the amplitude response provides access to the phase response, given by:

Then, using the results for and and the reduced equation (9) of the support, we can calculate the amplitude of the first two harmonics of the support at the retained orders, as follows:

Equation (27) indicates that the nonlinear effects induced by the pendulums generate higher order harmonics of the support (at the retained order, only the ^2nd harmonic is present).

The critical amplitude of the pendulums response for which the unison stability changes is calculated using and the unison condition (22), which gives:

where the index d refers to a “desynchronization”. Similarly, critical amplitudes for which the stability of the periodic response changes are given by:

where the index j refers to a "jump". It is interesting to notice that the critical amplitudes and tend to infinity because tends to zero. Furthermore, these amplitudes are independent of the forcing amplitude .

The preceding developments will now be applied to a system of two pendulums. An example of a pendular damping device 100 according to the invention is represented in FIG. 2 and its parameters are given in table 1. Its low tuning order makes it well suited for filtering vibrations arising from cylinder deactivation. The pendular damping device comprises two single-wire pendulums, only one of which is shown in FIG. 2, the architecture being symmetrical. The profile of a pendulum rolls without slipping on a roller of radius r ₁ (fixed on the support). The shape of the profile makes it possible to control the trajectory followed by the center of gravity of the pendulum. For such pendulums, the rotation function is governed by functional geometry. The first two rotation coefficients are:

Table 1: Parameters of the pendulum damping device shown in the .

In practice, the order of excitation is fixed so that the order n is constant. Nevertheless, pendulums may not be tuned exactly to excitement. This disagreement may be intentional or it may arise from material imperfections. In all cases, the variation of the order of excitation is similar to the introduction of a detuning (provided that the detunings of the pendulums are identical) and it is therefore relevant to study the effect of the detuning on the system response [18]. The order response of the unison pendulums and their stability are shown on the for several torque amplitudes. The bending of the response indicates that pendulums exhibit hardening behavior. The bifurcation zones: unstable unison in zones I and II and unstable periodic response in zones II and III, delimited by the bifurcation curves (29) and (30), are represented. The zone corresponding simultaneously to the unstable unison and to the unstable periodic responses is represented in zone II, whereas the stable responses are represented in zone IV. For small torque amplitudes the response is nearly linear so that no change in stability occurs. As the torque amplitude increases, the unison response intersects the bifurcation curves, resulting in a different state of stability.

Figure 3 allows a visualization of the stability of the pendulums in the design space will now be presented. This space allows to evaluate the stability of pendulums of the same type but with different sets of parameters . In the general case, it is a 3D space. Nevertheless, for the pendular damping device shown in Figure 2, the rotation is imposed by the functional geometry, which brings to be linked to . Also, due to space constraints, it is convenient to evaluate designs for a fixed ratio value. . Thus, using equations (31) and (32), one can get in terms of , which reduces the design space to 2D space.

The curves in continuous lines and the framework curve Ba make it possible to define different zones of stability: zone I corresponds to pendulums in unison in an unstable manner, zone III to pendulums with an unstable periodic response, zone II corresponds to pendulums in unison in an unstable way and with an unstable periodic response and zone IV corresponds to a zone of stability with the response curves in continuous lines.

The analytical model is here compared to an exact resolution of equations (1) and (2). The response of the pendulums obtained through the analytical and numerical models is presented in figure (4a). The unison mode loses its stability at a forked bifurcation in favor of a localized response. The BP bifurcation point, theoretically located at the intersection of the black curve and the unison response, is accurately predicted by the model. After this bifurcation, the response splits into two branches so that the energy is localized on one of the two pendulums. The pendulum located on the upper branch has a greater amplitude of movement than what is predicted by the unison mode, which can pose a problem since the maximum amplitude is limited by the nature of the chosen trajectory.

Figure 4 illustrates a comparison of the pendulum response, Figure 4a, and the support response, Figure 4b, obtained analytically and numerically. The color coding of the analytical results is the same as that used in Figure 3. Circles represent the numerical results. In Figure 4a, one of the pendulums is represented by light circles, the other by darker circles.

Again, the analytical model allows a precise prediction of the response of the support on its stable portions, corresponding to zone IV previously illustrated. Surprisingly, portions for which pendulums are not in unison are also well predicted by the model, even if the model only uses the unison mode. This point is interesting because it tends to show that a location does not necessarily influence the dynamics of the support.

Design process

According to another of its aspects, the subject of the invention is a transmission system for a mobility device comprising a pendular damping device as described.

According to yet another of its aspects, the subject of the invention is a method for designing a pendular damping device comprising a support and a plurality of pendulums, said method comprising at least one step of studying the sign of .

. As discussed earlier, a predicted unison motion of pendulums can be broken down into more complex regimes. When this is the case, a desynchronization leads to undesirable behavior and possible impact noise problems due to energy localization on a pendulum. According to the embodiments, the invention proposes several design rules making it possible to avoid system instabilities.

Said method comprises at least one step of studying the sign of . As discussed earlier, the tightening or softening behavior of pendulums is governed by the sign of . Setting this term to zero makes the response of the pendulums linear (cf. equations (23) and (24)) so that the limits of the zones of instability tend to infinity (cf. equations (29) and (30 )). In particular, the process follows the following design rule:

Physically, this rule means that nonlinearities arising from trajectory and rotation cancel each other out to make the system linear. Note that the case is studied above: it corresponds to an epicyclic trajectory and a linear rotation function. The relation (33) can be represented by a line in maps, a line which delimits an inner region corresponding to softening behaviors while the outer region corresponds to hardening behaviors. Equation (33) is also shown in Figure 5, along with other values of giving softening and hardening behaviors.

Alternatively, the method uses in particular the vertical and horizontal tangents of the bifurcation curves (29) and (30). The orders corresponding to this variant are given by:

The indices d and j are related respectively to the desynchronization and to the jumps of the response.

Figure 5 illustrates response behavior as a function of the value of (a) relaxation, , (b) linear, , (c) hardening, . The limits of the bifurcation zones in the undamped case are represented by dotted lines. and system parameters are given in Table 1. The black and gray dots indicate the dots corresponding respectively to and . The black and gray diamonds indicate the points corresponding respectively to and .

The exponents υ and h indicate a vertical tangent and a horizontal tangent respectively. Solutions + and – correspond respectively to behaviors of hardening and softening. is the backbone of the unison mode evaluated for zero displacement of the pendulums (cf. equation (23)). Points corresponding to equation (34) are represented by black and gray dots and diamonds in Figure 5. Vertical tangents indicate the minimum level of detuning for the response to become unstable, while horizontal tangents indicate the minimum amplitude at which the response enters an area of instability. Note that the torque level corresponding to these tangents can be obtained using equations (34), (29), (30) and (24). By focusing on the vertical tangents, we can see that, in the case of hardening, , while in the case of relaxation, Where . This means that hardening pendulums do not exhibit instabilities provided that (i.e. they are perfectly tuned or over-tuned). Similarly, easing pendulums do not exhibit instabilities in unison provided that (i.e. they are perfectly tuned or under-tuned). These easing pendulums are nevertheless very susceptible to exhibiting unstable periodic responses even if they are perfectly tuned or under-tuned. This happens if , which is generally the case when the damping is low. These remarks are illustrated on figure 5 on which one can note that the limit of the zones not in unison (black curves) does not cross the vertical line , while the limit of the unstable periodic response (gray curves) intersects it in the easing case. Furthermore, it can be seen from equation (34) that and tend to , when the damping is reduced. In other words, the low damping configuration is the most critical because a very low level of detuning can lead to a loss of unison. Likewise, and tend to when damping is reduced. This is visible on the on which the limit of the instability zones corresponding to the undamped case is represented by dotted lines.

Three different tuning levels (over-tuning, under-tuning, and pendulum perfect tuning) can also be represented by regions of instability in design space on maps, not shown in this application,

An experimental study on the pendular damping device 100 illustrated in FIG. 6 was carried out to evaluate the accuracy of the analytical and numerical results. This pendular damping device 100 is of the type shown in FIG. 2. It comprises a flywheel serving as a support 1 for two identical single-wire pendulums 5 specially designed to present energy localization. The extended shape of the pendulums 5 aims to increase their inertia, which increases the impact of rotation. Support 1 is fixed on an inertia bench using an inertial adaptation system . Thereby, must be replaced by the equivalent inertia in previous developments. The damping coefficients are found experimentally. The parameters of the system are given in Table 2. The amplitude of movement of the pendulums 5 was obtained experimentally as follows: the pendulum damping device 100 is centrifuged and an oscillating torque is applied while a high-speed camera records a film of the rotating system. Then, a point tracking algorithm is used to extract the pendulum signals from the film. Finally, the signal is decomposed into Fourier series to extract the amplitudes of the pendulums from the time signals. This procedure was repeated for four torque amplitudes, , and for and .

Table 2: CPVA parameters studied experimentally.

In the scaling step described earlier, µ was proposed as the weak perturbation parameter of the perturbation analysis. Thus, we can choose , which gives an order of magnitude not to be exceeded for each weak parameter.

The invention is not limited to two pendulums, the device according to the invention may comprise a greater number of pendulums. These can be monofilar or bifilar.

Claims (10)

Pendulum damping device (100) for a mobility device comprising:
- a support (1),
- a plurality of N pendulums (5)
characterized in that
at least one pendulum (5) verifies the following equation:
Pendulum damping device according to claim 1, at least one pendulum (5) having a profile (6) for rolling without slipping on a roller (7) of the device. Pendulum damping device according to the preceding claim, the roller (7) being fixed relative to the support (1) Pendulum damping device according to any one of the preceding claims, the order response of at least one pendulum (5) verifies the following equation:
Pendulum damping device according to any one of the preceding claims, comprising two monofilament pendulums. Pendulum damping device according to
η is positive, even 0< η <2,
0.05 < µ < 0.2 ,
α _[1] is positive, even 0< α _[1] <1,
n _p <1,
-5<x _[4] <5). Transmission system for a mobility device comprising a pendular damping device according to any one of the preceding claims. Method for designing a pendular damping device comprising a support (1) and a plurality of N pendulums (5)
said method comprising at least one step of studying the sign of . Method according to claim 8, the nonlinear tuning coefficient x _[4] satisfying the equation: (33) Method according to claim 8, the corresponding commands being given by:
The indices d and j being related respectively to the desynchronization and to the jumps of the response.