Journal home page: S. Lignon*, J-J. Sinou, and L. Jézéquel. ABSTRACT

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1 Journal home page: Stability analysis and µ-synthesis control o brake systems Journal o Sound and Vibration, Volume 98, Issues 4-5, December 006, Pages S. Lignon, J-J. Sinou and L. Jézéquel STABILITY ANALYSIS AND µ-synthesis CONTROL OF BRAKE SYSTEMS S. Lignon*, J-J. Sinou, and L. Jézéquel. Laboratoire de Tribologie et Dynamique des Systèmes UMR CNRS 553 Ecole Centrale de Lyon, 36 avenue Guy de Collongues, 6934 Ecully, France. ABSTRACT The concept o riction-induced brake vibrations, commonly known as judder, is investigated. Judder vibration is based on the class o geometrically induced or kinematic constraint instability. Ater presenting the modal coupling mechanism and the associated dynamic model, a stability analysis as well as a sensitivity analysis have been conducted in order to identiy physical parameters or a brake design avoiding riction-induced judder instability. Next, in order to reduce the size o the instability regions in relation to possible system parameter combinations, robust stability via µ-synthesis is applied. By comparing the unstable regions between the initial and controlled brake system, some general indications emerge and it appears that robust stability via µ-synthesis has some eect on the instability o the brake system. NOMENCLATURE C damping matrix G controller K stiness matrix M mass matrix N normal load P initial system T tangential load x scalar x vector x vector o velocity x vector o acceleration set structured uncertainties set λ eigenvalue o the nominal system λ eigenvalue o the controlled system µ brake riction coeicient µ structured singular value

2 INTRODUCTION Friction-induced vibration and instability are complicated phenomena that have been studied in detail by many researchers [-3]. However, they are still a major concern in a wide range o mechanical systems due to the diiculty in resolving the problem. This is especially the case or brake systems, where riction-induced vibration due to coupling modes can cause severe damage or/and noise. So the prevention and prediction o unstable vibrations are actually very complex and important problems or the vehicle brake industry. In order to avoid these problems, the eects o some speciic system parameters (typically mass, stiness, damping, ) need to be studied in order to detect the stable and unstable zones o the mechanical system subject to riction-induced instability. Though many studies have been conducted and some o them have been successully applied to particular brake systems and running conditions, it can be very diicult to ind suitable values o the system parameters in order to obtain stable brake systems or all operating conditions. In these cases, the engineer thereore needs to ind suitable devices to control instability in the brake system. In recent decades, riction-induced vibration has received considerable attention rom a number o researchers: Ibrahim [-], Bowden and Tabor [3], Rabinowitz [4], Armstrong Hélouvry [5], and Oden and Martins [6]. Their investigations were conducted in order to ind dierent mechanisms o riction-induced system instability. This type o analysis was then introduced in the context o brake noise to predict the dynamic behaviour o brake systems and to prevent instability (Ouyang et al. [], North [7-8], Kinkaid et al. [0], etc). In order to ind the most suitable mechanism to describe riction-induced vibration in brake systems, these dierent mechanisms have to be examined. They all into our classes: stickslip, variable dynamic riction coeicient, sprag-slip [] and geometric coupling o degrees o reedom [9, 5-]. The sprag-slip action was described by Spurr [] and does not depend on a riction coeicient varying with the relative rotation speed o the brake disc. Next, a number o investigations have been developed by considering kinematic constraint or geometric instability. This mechanism involves the coupling o the dierent degrees o reedom. It can be seen as an extension o the idea o the sprag-slip model []. Earles et al. [5, 6, 8, 0] and North [7-8] conducted extensive studies o kinematical constrained instability models. They demonstrated that instability may occur even i the riction coeicient is constant. In this study, a modal coupling mechanism involving two system modes coupled together due to the riction interace will be considered. This instability may be deined as a geometrical coupling where two system modes move closer in requency as the riction coeicient increases. In this study, the application o robust control via µ-synthesis or a brake system is tested in order to avoid instability or in order to reduce the instability regions. In the irst section, some basic concepts o µ-synthesis will be introduced. In the second section, the modal coupling mechanism used in this study will be briely presented and the application o µ-synthesis or judder instability will be investigated. Next, a stability analysis and some interesting studies o possible system parameter combinations or the initial and controlled brake systems will be undertaken in order to examine the varying eects o robust control analysis on the size o the instability regions. This sensitivity analysis will be conducted in order to ind the physical parameters or a brake design which avoid riction-induced instability in the case o controlled and uncontrolled brake systems. Finally, some natural extensions and possible applications o this methodology will be briely described in the conclusions.

3 µ-synthesis. INTRODUCTION The robustness o a system P with uncertainties represented by a set set o block-diagonal matrices is studied with the non singularity o the matrix I P (where I is the identity matrix), or set. In order to treat this problem, the structured singular value µ is introduced; this parameter µ will be deined below. This theory was introduced by Doyle [4] in 98 and has become a standard tool in the robustness analysis o linear systems. It directly considers the problem o robust stability or a known plant subject to a block-diagonal structured uncertainty connected in eedback. The utility o µ lies in the act that essentially any block diagram interconnection o systems and uncertainties may easily be rearranged into this standard orm, i.e. where the uncertainty structure is block-diagonal.. STRUCTURED SINGULAR VALUE n n This section deines the structured singular value µ (.). We consider matrices P R and introduce a structure set to deine µ ( P ). This structure set is a prescribed set o blockdiagonal matrices and may be deined dierently or each problem depending on the uncertainty o the problem. n n By deinition, µ ( P ) is deined or P C by: µ ( P) = () min { σ ( ) : set, det ( I P ) = 0} unless no set makes I P singular, in which case µ ( P ) = 0. σ ( ) corresponds to the maximum singular value o the matrix. µ is then a unction o two variables: the complex matrix P and the structure set. Considering the loop shown in Figure, µ ( P ) can be interpreted as a measure o the smallest uncertainty (represented by the matrix ) that causes instability o the constant matrix eedback loop. The norm o this destabilizing is exactly. It means that the weaker µ ( P ) µ ( P ) is, the more robust the system. Details concerning the calculation o the structured singular value are given in Packard and Doyle [5]..3 µ-synthesis The deinition o µ allows an extension o the stability analysis o systems by considering the system illustrated in Figure. This system is composed o three blocks, P and G that deine the perturbation matrix, the initial system which should be controlled, and the controller, respectively. The input / output couples are ( u0, Y 0), ( u, Y ) and ( u, Y ) which deine respectively the perturbation variables associated with the perturbation matrix, the measurable output (with 3

4 the input control u ) and the perormance variables where u includes the commands and the excitation and Y represents the errors and the results. Then, µ-synthesis consists o determining the controller G allowing the stability o the system in the presence o the uncertainty. The resolution is conducted by applying successive iterations. As the controller is oten denoted by K (notation already used here or the stiness matrix), this resolution is called the D-K iterations. These iterations are repeated unless µ <. I µ <, the robust stability o the system is assured or the given uncertainties. The theory o µ-synthesis is developed in Packard et al. [6], Venini [7], Balas et al. [8] and Markerink et al. [9]. Some applications can be ound in Lanzon [30], Wu and Lin [3]. 3 BRAKE SYSTEMS In order to demonstrate the suitability o µ-synthesis to brake systems and in order to link the eect o speciic parameter variation with stability o the design eatures, a parameter model including riction orces at the rubbing surace and mechanisms or riction-induced system instability is established and the equations o motion are determined. The problem considered in this study deals with a modal coupling mechanism [-] that results rom the coupling o two system modes due to the riction interace. The irst mode corresponds to the suspension mode o the ront axle assembly and the second mode corresponds to the normal mode o the brake piston elements. This phenomenological model was established through experimental investigations [3] and the riction-induced vibration was observed in the Hz range without variation o the brake riction coeicient. The act that instability may occur even i the coeicient o riction is constant is a very common phenomenon that has been observed by many researchers [9,, 5-]. In the ollowing sections, two analytical models (the initial and controlled systems) will irst be presented. Second, a stability analysis or each system will be undertaken and the initial and controlled systems will be compared in order to demonstrate the suitability o robust control or brake systems. 3. INITIAL SYSTEM The initial system studied here is modelled as a three-degrees-o-reedom system, as illustrated in Figure 3(A): translational and normal displacement in the y-direction o the mass m deined by Y () t and X () t, respectively, and the translational displacement in the x- direction o the mass m deined by X () t. As previously explained, each mode is linked to a single vibration mode o the brake system: ( k, m ) and ( k ), m deine the dynamic o the brake piston elements and the dynamic o the suspension mode o the ront axle assembly, respectively. The modal coupling mechanism involves the two modes coupled together due to the riction interace. This mechanism may induce a classic lutter instability where two solutions or the dynamic behaviour o the mechanical system exist. The irst solution is an unstable equilibrium whereas the second is a periodic solution. Then, any perturbation o the equilibrium point implies sel-excited vibrations. In order to simulate the modal coupling mechanism due to the riction interace, this riction interace slopes with an angle θ. This assumption may be seen as a geometric coupling with the braking system. This slope couples the normal and tangential degree-o-reedom induced by the brake riction coeicient only. 4

5 By considering this system composed o two masses m and m interconnected by stinesses k and k (Figure 3), the dynamic equilibrium around its static equilibrium position is expressed by the ollowing system o equations mx + c( X X ) + k( X X) = 0 my + cy + ky = Nsinθ + Tcosθ () mx + c( X X ) + k( X X) = Ncosθ + Tsinθ By applying the hypothesis o maintained contact between the mass m and the moving belt, the geometric constraint imposes X = Y tanθ (3) By eliminating x in equations (3) and considering Coulomb s riction law T =µ N, the - degrees-o-reedom system has the orm Mx + Cx + Kx = 0 (4) x = X Y where { } T. x, x and x are the acceleration, velocity, and displacement response - dimensional vectors o the degrees-o-reedom, respectively. The mass matrix M, the damping matrix C and the stiness matrix K o the system are given by m 0 M = (5) 0 m ( tan θ + ) c ctanθ C = c( tanθ + µ ) c( tan θ µ tanθ ) + c( + µ tanθ) (6) k ktanθ K = (7) k( tanθ + µ ) k( + µ tanθ) + k( tan θ µ tanθ) Finally, the dynamic system may be rewritten in state variables: z = Az (8) where x z = (9) x and 0 I A = (0) M K M C 3. CONTROLLED SYSTEM BY APPLYING µ-synthesis µ-synthesis is applied to the brake system by assuming that the riction coeicient µ is uncertain. This uncertainty corresponds to possible variations o the riction with time. This 5

6 controlled system is illustrated in Figure 3(B): the controller U is placed in parallel with the suspension. 3.. Deinition o the controlled system By considering section and equations (5-7) o the initial brake system, the nominal system is represented by: z = Az+ Bu () y = Cz + Du where B 0 0 = 0 + µ tanθ m ( tan + ) θ () [ ] C = (3) D = 0 (4) In the presence o uncertainties, the nominal system is modiied to introduce the variables corresponding to the uncertain parameters: z = Az+ B0u0 + Bu y0 = Cz 0 + D00u0 + D0u (5) y = Cz + D0u0 + Du In the case under consideration, the uncertainty is introduced on the riction coeicient µ. As there is only one uncertainty, the set set is reduced to scalar variables, which elements are noted δ. We have then µ = µ ( + δ), whereδ is the degree o uncertainty. The matrix A is then transormed to A = A+δ. A, where A is the previous matrix and A is deined by: A = µ k µ tanθ ( k k) µ c µ tanθ( c c) m( tan + ) m( tan + ) m( tan + ) m( tan + ) θ θ θ θ (6) By deinition, we have 0 = δ. 0 allows us to determine the matrices B 0, C 0 and A= B. D. C. This relation u y,which results in: δ ( δ ) D ij : 6

7 0 0 B 0 0 = m ( tan + ) θ (7) C 0 = k tan ( k k) c tan ( c c) (8) D 00 = 0 (9) D 0 = 0 (0) D = () 0 0 The relations between z, y 0, y, u 0 and u determined in this section are the basis o the µ- synthesis resolution. 3.. Resolution As explained previously, equations (6-) correspond to the complete description o the controlled system and contain the nominal system and the perturbations linked to the uncertainties rom a general standpoint. The D-K iterations, allowing us to determine the robust controller by µ-synthesis, are conducted using Matlab sotware [8]. The controller is determined where the structured singular value µ o the system is less than unity and we obtain the Bode diagram o the controller allowing robust stability o the system. The structured singular value µ obtained or the brake system is plotted in Figure 4. We observe that µ is less than unity or all requencies ω, which means that robust stability is assured. This result is obtained ater two D-K iterations. The controller determined by the algorithm and corresponding to this result is illustrated in G ω which is sought in the orm: Figure 5. The controller is approximated by a unction ( ) G ( ω) α β ω = G + G () where α G and β G are constants, depending on the values o the parameters o the system. - - For the coniguration ω = 387 rad s, ω = 36. rad s, ζ = 0.008, ζ = (i.e m = kg, m = kg, c = 5 N m s, c = 5 N m s, k =.5 0 N m, 5-5 k = 0 N m ), θ = 0. rad, µ = 0.3, the numerical results give α G = 0 and β G = This approximation o G ( ω ) allows a good representation o the controller, as we can see in Figure 6, and it will be useul in the stability analysis o the controlled system. 3.3 STABILITY ANALYSIS OF THE INITIAL AND CONTROLLED SYSTEMS In this section, the stability o the initial and controlled brake systems will be compared. To examine the stability o the initial system, the eigenvalues λ o the matrix A (deined in equation (7)) need to be determined. As long as the real part o all the eigenvalues λ remains negative, the system is stable. When at least one o the eigenvalues has a positive real part, the system is unstable. Moreover, the imaginary part o the eigenvalue having a positive real part represents the requency o the unstable mode. 7

8 G ω α β ω For the controlled system, the orm ( ) = G + G means that G α is equivalent to a stiness and β G is equivalent to a mass. It enables us to take this into account directly in the mass and stiness matrices. By noting K and M the new mass and stiness matrices 0 0 K = K+ 0 + tan. (3) µ θαg 0 0 M = M+ 0 + tan. (4) µ θβg a new matrix A may be deined or the controlled system by 0 I A = M K M C (5) The advantage o these notations is that the stability analysis is similar or the initial and the controlled systems: the sign o the real part o the eigenvalues λ o A gives the result concerning the stability o the system. First, the evolutions o requencies in relation to the brake riction coeicient or the initial and controlled brake systems are given in Figure 7. The evolutions o the associated real parts and the representation in the complex plan are given in Figures 8-9. As illustrated in Figure 8, Hop biurcation points occur at µ 0 = 0.35 and µ 0 = 0.46 or the initial and controlled systems respectively. A Hop biurcation point is deined by the ollowing conditions Re( λ center ( µ )) = 0 µ = µ 0 Re( λ non center ( µ )) 0 µ = µ 0 (6) d ( Re( λ ( µ ))) 0 dµ µ = µ 0 where λ center deines a pair o purely imaginary eigenvalues while all o the other eigenvalues λnon center have nonzero real parts at µ = µ 0. The last condition o equation (34), called a transversal condition, implies a transversal or nonzero speed crossing o the imaginary axis. I µ < µ 0 the initial system is stable; it has two stable modes at dierent requencies, as illustrated in Figure 7. As the brake riction coeicient increases, these two modes move closer until they reach the biurcation zone. We obtain the coalescence or µ = µ 0 o two imaginary parts o the eigenvalues. Finally, the initial system becomes unstable or µ > µ 0. In the case o the controlled brake system, the stable and unstable regions are obtained or µ < µ 0 and µ > µ 0, respectively. In Figure 8, it may be observed that the instability region versus the riction coeicient µ is smaller or the controlled system than or the initial system ( µ 0 > µ 0). This illustrates the suitability o robust control via µ-synthesis. 8

9 An interesting observation is that the mode that becomes unstable and reaches the biurcation zone is dierent or the initial and controlled systems (Figure ). Then, in order to demonstrate the suitability o µ-synthesis and in order to compare the stability analysis o the initial and controlled brake systems, dierent sets o two combinations o physical parameters k, k, c, c and θ are tested. Figures 0-4 show the zones o instability or the initial and controlled systems: the dashed line corresponds to the initial system and the solid line corresponds to the controlled system. Figures 5-6 show the evolutions o the requencies and the associated real parts in the complex plane. For all tested combinations o the dierent parameters with the brake riction coeicient (Figures 0-4), the controller provided by µ-synthesis allows an increased stable zone or the brake system. However, or some values o these parameters which correspond more or less to the nominal setting, this improvement is very weak. Another interesting result o µ-synthesis is that the intervals o instability requencies are reduced or the controlled brake system in comparison with the initial system, as illustrated in Figures 5-6 and Table. Finally, dynamical responses o the system are presented in Figures 7-8 in order to illustrate the advantages o the controlled brake system versus the initial brake system. In this case, we consider a combination o parameters corresponding to an unstable zone or the initial system - - and a stable zone or the controlled system ( m = kg, m = kg, c = 5 N m s, c = 5 N m s, k =.5 0 N m, k = 0 N m, θ = 0. rad, µ = 0.4 ). In such a case, the temporal response o the initial system grows exponentially while that o the controlled system is sotened (Fig. 7-8). Figure 9 illustrates the oscillations o the initial and controlled systems. It illustrates the dierence between the behaviour o the two systems. The instability is maniested by an exponentially increasing curve. On the other hand, the response o the controlled system is more complex but is limited in amplitude. 4 CONCLUSION This study presents an application o µ-synthesis in order to eliminate riction-induced vibration or a brake system. A model or judder instability analysis and an associated stability analysis or the initial and controlled systems are developed. For urther understanding o the eects caused by variations in some parameters and the suitability o µ-synthesis, a stability analysis using two parameter evolutions has been conducted. Robust stability via µ-synthesis or brake systems appears interesting or reducing the size o the instability regions in relation to the possible system parameter combinations. This procedure can be applied to a brake design avoiding riction-induced judder instability. REFERENCES [] R.A. Ibrahim, Friction-Induced Vibration, Chatter, Squeal and Chaos: Part I - Mechanics o Contact and Friction, ASME Applied Mechanics Review 47(7) (994) [] R.A. Ibrahim, Friction-Induced Vibration, Chatter, Squeal and Chaos: Part II Dynamics and Modeling, Applied Mechanics Review, ASME Applied Mechanics Review, 47(7) (994) [3] F.P. Bowden, D. Tabor, The Friction and Lubrication o Solids, Oxord University Press, Clarendon Press, Oxord, (second corrected edition, 00. Oxord Classic Text in the Physical Sciences). 9

10 [4] E. Rabinowicz, Friction and Wear o Materials, Wiley, New York, 965. [5] B. Armstrong-Hélouvry, P. DuPont, C. Canudas de Wit, A survey o models, analysis tools and compensation methods or the control o machines with riction, Automatica 30 (7) (994) [6] J.T. Oden, J.A.C. Martins, Models and computational methods or dynamic riction phenomena, Computer Methods in Applied Mechanics and Engineering 5 (985) [7] M.R. North, 4 th FISITA congress, Paper /9. A Mechanism o disc brake squeal, 97. [8] M.R. North, ImechE, C38,. Disc brake squeal, 976. [9] D.A. Crolla, A.M. Lang, Tribologie, Vehicle Tribology, Brake Noise and Vibration State o Art 8 (99) [0] N.M. Kinkaid, O.M. O Reilly, P. Papadopoulos, Automotive disc brake squeal, Journal o Sound and Vibration 67 (003) [] H. Ouyang, J. E. Mottershead, M. P. Cartmell, M. I. Friswell, Friction-induced parametric resonances in discs: eect o a negative riction velocity relationship, Journal o Sound and Vibration 09) (998) [] H. Ouyang, J. E. Mottershead, D. J. Brookields, S. James, M. P. Cartmell, A methodology or the determination o dynamic instabilities in a car disc brake. International Journal o Vehicle Design 3 (000) 4-6. [3] H. Ouyang, J. E. Mottershead, M. P. Cartmell, D. J. Brookield, Friction-induced vibration o an elastic slider on a vibrating disc, International Journal o Mechanical Sciences 4(3) (999) [4] H. Ouyang, J. E. Mottershead, Unstable travelling waves in the riction-induced vibration o discs, Journal o Sound and Vibration 48(4) (00) [5] S.W.E Earles, G.B. Soar, Proc.I.Mech.E.Con. on Vibration and Noise in Motor Vehicles, Paper C00/7. Squeal noise in disc brakes, 97. [6] S.W.E Earles, M. N. M. Badi, SAE 78033, On the interaction o a two-pin-disc system with reerence to the generation o disk-brake squeal, 978. [7] N. Millner, SAE Paper An Analysis o Disc Brake Squeal, 978. [8] S.W.E Earles, C.K. Lee. Trans ASME, Instabilities arising rom the rictional interaction o a pin-disc system resulting in noise generation, J. Engng Ind, 98, Series B, n, [9] H.R. Mills, 938/939 Research Report n 9000B and Research Report n 96B o the Institution o Automobile Engineers. Brake Squeal, 938/939. [0] S.W.E Earles, P.W. Chambers, Disc Brake Squeal Noise generation: Predicting its Dependency on System parameters Including Damping, Int. J. o vehicle Design 8(4/5/6) (987) [] J.J. Sinou, F. Thouverez, L. Jezequel, Analysis o riction and instability by the centre maniold theory or a non-linear sprag-slip model, Journal o Sound and Vibration 65 (003) [] R.T. Spurr, A theory o brake squeal, Proc. Auto. Div., Instn. Mech. Engrs, (96) [3] R.P. Jarvis, B. Mills, Vibrations induced by dry riction, Proc. Instn. Mech. Engrs, 78(3) (963/964) [4] J.C. Doyle, Analysis o eedback systems with structured uncertainty, IEE Proc., Part D 9 (98) [5] A. Packard, J. Doyle, The Complex Structured Singular Value, Automatica 9() (993)

11 [6] A. Packard, J. Doyle, G.J. Balas, Linear, Multivariable Robust Control With a µ Perspective, Journal o Dynamic Systems, Measurement and Control, 5 (993) [7] P. Venini, Robust control o uncertain structures, Computers & Structures 67 (998) [8] G.J. Balas, J. Doyle, K. Glover, A. Packard, R. Smith, µ-analysis and Synthesis Toolbox (or use with MATLAB), The Math Works, Natick, MA. [9] J. Markerink, S. Bennani, B. Mulder, Design o a Robust Controller or the HIRM using µ-synthesis (997) Report, GARTEUR (Group or Aeronautical Research and Technology in Europe). [30] A. Lanzon, Pointwise in requency perormance weight optimization in µ-synthesis, Int. J. Robust Nonlinear Control 5 (005) [3] J.D. Wu, J.H. Lin, Implementation o an active vibration controller or gear-set shat using µ-synthesis, Journal o Sound and Vibration 8 (005) [3] J.P. Boudot, Modélisation des bruits de reinage des véhicules Industriels, PhD Thesis, Ecole Centrale de Lyon, 995. Figure : P eedback connection Figure : General structure o the problem

12 Figure 3 : Braking model (A) initial system (B) controlled system Figure 4 : Evolution o the structured singular value µ

13 Figure 5 : Bode diagram o the controller Figure 6 : Approximation o the controller ( Result o the µ -synthesis, Approximation) 3

14 Figure 7 : Evolution o the requency o two coupling modes ( Initial system, Controlled system) Figure 8 : Evolution o the real part o two coupling modes ( Initial system, Controlled system) 4

15 Figure 9 : Evolution o the requency versus real part o two coupling modes ( Initial system, Controlled system) Figure 0 : Stability as a unction o brake riction coeicient µ and stiness k ( Initial system, Controlled system) 5

16 Figure : Stability as a unction o brake riction coeicient µ and stiness k ( Initial system, Controlled system) Figure : Stability as a unction o brake riction coeicient µ and mass m ( Initial system, Controlled system) 6

17 Figure 3 : Stability as a unction o brake riction coeicient µ and mass m ( Initial system, Controlled system) Figure 4 : Stability as a unction o brake riction coeicient µ and angle θ ( Initial system, Controlled system) 7

18 Figure 5 : Frequency and real part o eigenvalue o the initial system or various riction coeicient µ and stiness k Figure 6 : Frequency and real part o eigenvalue o the controlled system or various riction coeicient µ and stiness k 8

19 Figure 7 : Temporal response o the initial system or µ = 0.4 Figure 8 : Temporal response o the controlled system or µ = 0.4 9

20 Figure 9 : Divergent oscillations or 0.4 µ = ( µ 0 < µ < µ 0) Parameter Initial System Controlled System k Hz 49-6 Hz k 4-65 Hz Hz m Hz Hz m Hz Hz θ 5 Hz 49-5 Hz Table : Comparison o the instability regions or the initial and controlled brake system 5 6 0