A finite element model reduction based on super-elementsfor brake squeal study

O. Fazio 1,2, S. Nacivet 1, J-J. Sinou 2

1 PSA Peugeot Citroen, Centre technique de La Garenne Colombes,18 rue des Fauvelles 92250 La Garenne Colombes, Francee-mail: olivier.fazio@mpsa.com

2 Laboratoire de Tribologie et Dynamique des Systemes, UMR CNRS 5513,Ecole Centrale de Lyon, 36 avenue Guy de Collongue 69134 Ecully Cedex, France

AbstractThe problem of brake squeal in automotive was a subject of great interest for many researchers. The problemof squeal modeling has been extensively studied in recent years and a lot of theories have been formulated anddeveloped in order to explain squeal phenomena. Nowadays, it is still difficult to numerically predict brakesqueal early enough to reduce development costs. In the present study, we propose to define an efficientreduction method in order to reduce the number of contact nodes at the frictional interface. The reducedmodel is defined using super-elements with reduced node to node contact at the disc/pad interface. Thenexperimental data allows the reduced model to take in account a cubic stiffness for the contact simulation.Application for an industrial representative finite element model is proposed. First of all, the proposedreduction was validated both on non-linear static and modal quantities. Secondly, the efficiency of theformulation is undertaken by performing a stability analysis around a non-linear equilibrium point. In thepresent study, the effect of the number of contact nodes at the frictional interface will be investigated forvarious operating conditions. It will be illustrated that a reduced model based on a modal reduction andsuper-elements at the frictional interface can be developed in order to well represent squeal instabilities(versus the frequencies and the associated unstable modes). Encouraging results will be given on stabilityanalysis and highlights the huge importance of contact condition on instability appearance.

1 The general problem

1.1 Numerical Model

For this study a finite element model of an industrial brake system that has been developed in Abaqus isconsidered. It can be decomposed in several parts as detailed in Figure 1(a). This system uses a floatingcaliper technology. The caliper that holds the two brake pads can move with respect to the disc, along aline parallel to the axis of rotation of the disc. During a brake operation, hydraulic pressure is applied onthe piston, which pushes the inner brake pad until it makes contact with the disc. Then, reaction force pullsthe caliper body with the outer brake pad against the other side of the disc. Considering the finite elementmodel, a volume mesh is realized using 10-node quadratic tetrahedron (see Figure 1(b)) . It represents a totalof 176956 nodes for 100710 elements. For further estimation of our reduction performance it is interestingto notice that it is equivalent to 1 143 132 degrees of freedom (DOF) including Lagrange multiplier.

Contact and friction modeling are key parameters for squeal simulation. Here, ∗CONTACT PAIR card isused for node to surface contact modelling. The principal components of the brake system in contact with

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(a) Exploded view (b) Finite element model

Figure 1: Brake system details

the pads are the disc, the piston, the caliper and the bracket. For all these contacts, pads are always defined asslave parts. A linear penalty method is used for the contact constraint enforcement and the classic Coulomb’slaw is used for all the frictional interfaces. Also depending on the contact considered, separation can be takeninto account or not. All these parameters are summed up in Table 1.

Contact Friction coefficient SeparationDisc/Pad 0.5 Yes

Piston/Pad 0.15 NoBracket/Pad 0.15 YesCaliper/Pad 0.15 No

Table 1: Contact modelling

1.2 The non-linear problem

The brake system in frictional contact state is governed by Eq.1

Mx + Cx + Kx + Fnl (x) = Fext (1)

Where M, C and K are respectively the mass, damping and stiffness matrices. x corresponds to the vectorof displacements and dot stands for time derivative. Fext is the external force applied to the piston and thecaliper representing a brake operation. Fnl represents the non-linear forces applied to the system. Thesenon-linear forces can be given as follows:

Fnl (x) = Fcontact (x) + Ffriction (x) (2)

Where Fcontact represents the normal contact force and is governed by the following law:

Fcontact =

{Fc(δ) ∀δ ∈ R+

0 otherwise(3)

Where δ is the contact penetration.The friction force, Ffriction, is computed using the Coulomb’s law (4), with a constant friction coefficient µ,and considering a permanent sliding state at the disc/pad interface.

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Ffriction = µFcontactsign(vr) (4)

Where vr corresponds to the relative velocity between the two surfaces in contact.

1.3 Stability analysis

Stability analysis is a classic method to study brake squeal [5, 6, 3]. Complex eigenvalue analysis (CEA) isperformed for the linearized system at the vicinity of its non-linear static equilibrium. Due to the frictionforces, the problem is unsymmetric which can lead to the appearance of instabilities. This section explainsthe different steps of a stability analysis for the brake system under study.First, the non-linear static equilibrium U0 is obtained by solving the following relation.

KU0 + Fnl (U0) = Fext (5)

The brake system is loaded as follows: a nodal force is uniformly distributed on the piston surface and itsopposite is distributed on the caliper as well. A velocity field corresponding to the disc rotation is imposedon the disc surface nodes to compute friction.

Then, the system is linearized around the non-linear static equilibrium and the associated eigenvalue problemis solved as detailed in Eq.6.

(λ2M + λC + (K + Jnl))φ = 0 (6)

Where Jnl corresponds to non-linear forces linearization, around the non-linear static equilibrium point. Thecomplex eigenvalues can then be written as follows:

λi = ai + jωi (7)

Where ωi represents the pulsation of the associated eigenmode φi.

If the real part ai of an eigenvalue is positive, the corresponding eigenmode is considered as unstable andthus can generate brake squeal. Figure 2(a) represents the evolution of both real parts and frequencies, byincreasing the friction coefficient, µ, of the disc/pad interface, for the undamped industrial representativemodel used for our study, on the 0-6kHz range. The circle surface increases with the positive real part value.

Figure 2(b) illustrates the evolution of the frequencies and their associated real parts by increasing the frictioncoefficient of the disc/pad interface for a specific instability (i.e. 1.9kHz). While the disc/pad interfacefriction coefficient increases the two frequencies get closer until a µ value where they become equal, thispoint is called the Hopf bifurcation. Their associated real parts, which are nearly zero before the bifurcation,separate after this point: one becomes positive as the other is negative. The mode coupling, responsible forbrake squeal, can then be observed as shown in Figure 2(b).

Figure 3 gives an illustration of the five main instabilities observed on the frequency range of interest.

2 Model reduction method

Previous research works were carried out to define a model reduction method to apply stability analysis onindustrial representative models [2, 4]. On the other hand, Villard et al. [7] used an assembly of super-elements (SE) with reduced contact interface. However, some questions remained after his study particularlyconcerning the simplified contact modeling. In this section a new development brought to improve thereduction method proposed by Villard et al. will be presented.

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(a) (b)

Figure 2: Stability analysis and mode coupling phenomenon

2.1 Global strategy and simplified contact modelling

The idea put forward by Villard et al. [7] is to create an assembly of super-elements (SE) at the contact zonesdefining a node to node contact. Due to the fact that industrial meshing process drives to non-coincidentnodes at the frictional interfaces, Villard suggested to generate new contact interfaces for SE assembly usingnode to node contact with no need to remesh the model. Moreover this process allows to reduce the numberof degrees of freedom (DOF) at the frictional interfaces of the initial brake system.

The objective of simplified contact modelling is to reduce the number of nodes involved in the severalcontacts to define a node to node contact between the SE (whereas a node to surface contact is used inthe reference model). In so far as this contact reduction must not affect the static solution Villard et al.[7]suggest to distribute strain around these interface nodes. The *DISTRIBUTING and *COUPLING optionsare therefore used in Abaqus. Figure 4 illustrates this new contact modelling. A reference node is created inthe plane of the initial contact surface. It allows to distribute contact force on the nodes of the initial meshedsurface involved in the contact. This reference node can not directly be used due to an incompatibility. Dueto the fact that SE reduction can not be directly applied to the reference node, an additional spring is createdwith arbitrary stiffness (i.e. the value of this spring is assumed to be very superior to the local stiffness). Forthe reader comprehension, it can be noticed that these nodes have the same geometric position in the modelbut they are dissociated on Figure 4 for a better illustration of the contact reduction.

Then, the contact forces (but not the moment) computed at the reduced contact interface are distributedon the coupling nodes of the part surface. Indeed, this reduction method is interesting only if the reducedcontact interface computation is relevant and does not generate error on the non-linear static equilibrium. Inthis study it was interesting to exploit the *DISTRIBUTING and *COUPLING cards in Abaqus in order todetermine the best set of parameters [1] and decrease the error induced by this reduction and thus improvethe previous reduction method.

One of the main contribution of this study on simplified contact modelling is to create one SE reducing onlythe disc/pad interface. All the other interfaces are then linearized during the SE creation. This choice allowedus to reduce the error involved by contact reduction and better understand its effect. Indeed increasing thenumber of reduced contact interfaces sums up the error involved by reduction. The major drawback is thatindividual damping can not be considered anymore for each component during SE assembly.

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(a) (b) (c) (d)

(e)

Figure 3: Unstable modes shape on the 0-6kHz range. (a)1900Hz (b)3800Hz (c)4700Hz (d)5300Hz(e)5900Hz

Figure 4: The reduced contact model

2.2 Super-element creation and super-element assembly

The first step consists in reducing the number of nodes at the disc/pad interface with the strategy explainedearlier. Once the contact is reduced and the static equilibrium found, Craig and Bampton reduction [3] iscomputed and the super-element is generated. It requires the deactivation of the contacts but the displace-ments on the future condensation nodes (i.e the reduced contact interface nodes) are still applied. Dynamicmodes on the 0-12khz range are kept to ensure SE validity on the 0-6kHz range.

The next step concerns the super-element assembly which is done thank to Matlab as shown on Figure 5.This subsection details the computation method to compute the non-linear static step of the SE assembly atthe exact static equilibrium.

The non-linear static system solved by Abaqus is described as follows.

Knl(Us)Us + Fnl(Us) = Fext (8)

As the SE is created in a perturbation step, the system linearized around the static equilibrium can be writtenas follows.

Knl,UsUs + Fnl(Us) = Fext (9)

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Where Knl,Us defines the linearized stiffness matrix at the vicinity of the static equilibrium Us.

The local stability is studied introducing a perturbation ∆U around the static equilibrium:

Knl,Us(Us + ∆U) + Fnl(Us + ∆U) = Fext + ∆F (10)

The difference between Eq.9 and 10 considering the exact static equilibrium (i.e. ∆U is null) needs to satisfythe following relation:

Fnl(Us + ∆U) = Fnl(Us). (11)

Another enhancement, based on experimental data is to define a cubic stiffness for the reduced contact onSE assembly whereas this contact was computed using a linear penalty method in Abaqus. An equivalentcontact penetration has to be found in order to match with our SE assembly the normal contact force valueobtained with Abaqus, which justifies the contact output explained earlier. Knowing the value of Abaqusnormal contact force for each reduced node and the cubic stiffness parameters, it is easily done in solvingfollowing relation.

FcMatlab(δ + ξ) = FcAbaqus(δ) (12)

Then the corrective term ξ in node to node contact penetration can be easily found. Finally, the stabilityanalysis (i.e. the determination of the non-linear static equilibrium and the complex eigenvalue analysisaround this equilibrium point) can be performed as previously explained in Section 1.3.

Figure 5 sums up all the steps of the defined Abaqus plus Matlab simulation for the global strategy.

Figure 5: Global strategy

3 Application for the reduced industrial model

This section presents the application of the reduction strategy defined above to the industrial brake model.The finite element model was reduced while keeping a good prediction concerning stability regarding thereference, non-reduced, Abaqus model (see Section 1).

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For now one problem remains concerning contact reduction: the *CONTACT PAIR card used for contactis not compatible with our reduced contact modelization. The point is our contact reduction involves node-based surface creation which can not be defined as the master surface of a contact pair in Abaqus. Onesolution could be to directly define node to node contact between reduced node of both side directly inAbaqus with *GAP elements with friction. However, this strategy does not allow us to generate unsymmet-rical terms in matrices which is the key point for stability analysis. So we decided to create reduced contactnodes only for the pad and impose coincident nodes on the disc volume mesh and keep a node to surfaceformulation with *CONTACT PAIR card. The reduced contact nodes and their coincident nodes on discmesh will then be used for condensation and node to node contact will be defined later in Matlab betweenthem.

With this reduction strategy well defined we created several Abaqus models with several number of reducedcontact nodes at the disc/pad interface. Table 2 sums up the characteristics of these models and Figure 6 givesan insight of the reduced contact nodes layout over the pad surface. As point of comparison, the industrialAbaqus model contains 1370 nodes on the surface of each pad.

Reduced contact node per pad 6 22 52 106Total coincident nodes on disc volume mesh 12 44 104 212

Total references nodes for SE 26 90 210 526Number of node to node contact elements 12 44 104 212

Table 2: Reduced models characteristics

(a) 12 nodes (b) 44 nodes

(c) 104 nodes (d) 212 nodes

Figure 6: Reduced contact nodes layout over the pad surface for several reduced model

From now and for a better comprehension the several Abaqus models with reduced contact interface will becalled Abaqus 12, Abaqus 44, Abaqus 104 and Abaqus 212 in reference to the number of reduced contactelements they contain.

3.1 Stability analysis on Abaqus reduced models

Stability analysis results are here compared between the four reduced models presented above and the ref-erence Abaqus model. Complex eigenvalue analysis was computed on each of these models varying the

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friction coefficient at the disc/pad from 0.1 to 0.9. Then we followed, for the two first instabalities, theevolution of their associated real part and frequency on each of the models. Results are given on Figure 7.

Figure 7: Evolution of the frequencies (a,b) and the real part (c,d) of the eigenvalues against the frictioncoefficient µ for the (a,c) 1.9kHz and (b,d) 3.8kHz instabilities.

The weakness of Abaqus 12 model is highlighted here. Even if the 1.9kHz instability is pretty well repre-sented this is not so obvious concerning the 3.8kHz instability. For this second instability, the frequenciesthat couple are not the same considering a friction coefficient value below or above 0.4. For the particularvalue of µ = 0.4 the instability totally disappear and the three frequencies are distinct. It suggests that achange in contact status can completely modify the characteristics of the brake system.

All the other models are able to predict these two instabilities considered both for real part and frequencyestimation. Mode coupling at 3.8kHz is slightly delayed for Abaqus 44 model, unstable mode appears froma friction coefficient of 0.4. Increasing the number of nodes from 44 to 104 reduces significantly the gapon frequency estimation between reduced models and the reference. This improvement is not so distinctbetween Abaqus 104 and 212.

These results showed that Abaqus 104 is the good compromise between model size reduction and instabilitiesrepresentation. It will be used in the next section to create super-elements.

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3.2 Stability analysis on SE assembly

Now we propose to validate the use of SE assembly for the stability analysis (as previously explained inSection 2 and Figure 5).

Based on Abaqus 104 model, we propose to generate a SE for each step of the friction coefficient. Actually,super-element only remains available at the vicinity of the non-linear static equilibrium point and it responseis considered to be a linearization of this equilibrium. Indeed in can easily be imagined that this equilibriumcan change as the friction coefficient varies.

The generated SE size is 756 DOF which is way less than the initial model with more than one million DOF.This gives an indication about the interface reduction and SE creation interest.

Results on stability, obtained thanks to Matlab computation on SE assembly, fits good enough with the resultson the corresponding Abaqus 104 model as shown in Figure 8. The maximum error committed on frequencyestimation never exceeds half a percent.

Figure 8: Evolution of the frequencies (a,b) and the real part (c,d) of the eigenvalues against the frictioncoefficient µ for the (a,c) 1.9kHz and (b,d) 3.8kHz. A new super-element is generated at each µ for complexeigenvalue analysis on SE assembly.

In conclusion, these results clearly demonstrate the efficiency of the global strategy based on SE generationand nodal interface reduction.

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4 Acknowledgement

This work was achieved within Stellab program. The authors would like to thank the PSA Peugeot Citroenstaff for their assistance.

5 Conclusion

An efficient reduction method of a non-linear industrial brake system model is proposed. A convergence onstability analysis results, on Abaqus models with reduced contact interface, has been found in accordancewith the industrial reference. This contact reduction allows us to generate Craig & Bampton super-elementto compute a reliable stability analysis.

References

[1] Abaqus analysis user’s manual, ch. 31 Constraints.

[2] D. Brizard, O. Chiello, and J.-J. Sinou X. Lorang, Performances of some reduced bases for the stabilityanalysis of a disc/pads system in sliding contact, Journal of Sound and Vibration 330 (2011), no. 4,703–720.

[3] N. Coudeyras, J.-J. Sinou, and S. Nacivet, A new treatment for predicting the self-excited vibrations ofnonlinear systems with frictional interfaces: The constrained harmonic balance method, with applica-tion to disc brake squeal, Journal of Sound and Vibration 319 (2009), no. 3, 1175–1199.

[4] B. Herve, J.-J. Sinou, H. Mahe, and L. Jezequel, Extension of the destabilization paradox to limit cycleamplitudes for a nonlinear self-excited system subject to gyroscopic and circulatory actions, Journal ofSound and Vibration 323 (2009), no. 3, 944–973.

[5] W. V. Nack, Brake squeal analysis by finite elements, International Journal of Vehicle Design 23 (2000),no. 3, 263–275.

[6] J.-J. Sinou and L. Jezequel, Mode coupling instability in friction-induced vibrations and its dependencyon system parameters including damping, European Journal of Mechanics-A/Solids 26 (2007), no. 1,106–122.

[7] P. Villard, S. Nacivet, and J.-J. Sinou, Superelement reduction of industrial finite element brake system fora constrained harmonic balance method, INTER-NOISE and NOISE-CON Congress and ConferenceProceedings, 2012, pp. 3142–3151.

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