Analysis of Friction-Induced Instabilities in a Simplified Aircraft Brake

Transcription

1 his is a pre-publication version of a paper submitted to SAE 1999 Brake Colloquium: Analysis of Friction-Induced Instabilities in a Simplified Aircraft Brake Osama N. Hamzeh, W. Woytek worzydlo, Hsien J. Chang Computational Mechanics Company. Inc. Slawomir. Fryska AlliedSignal Inc. Abstract he stability and dynamic characteristics of frictioninduced vibrations in a simplified aircraft brake model are investigated. A finite element model equipped with nonlinear frictional contact algorithm is used. he constitutive model of the interface is based on an extended version of the Oden-Martins law [1]. he interface material constants are obtained via asperitybased homogenization methodology from the profilometric information on the surface. Initial uncoupled analyses are performed to identify the basic dynamic modes of the model. Frequencies of normal vibrations of the model are found to be dependent on the interface stiffness and the piston pressure. o study the dynamic behavior of the system, its transient response is computed after a perturbation of the steady-state sliding position. It is found that, while the vibrations are subdued in some cases categorized as stable, they grow in other, unstable cases. It is also shown that the triggering mechanism of instability can be either the velocity-dependent coefficient of friction or the dynamic coupling of certain vibration modes of the system (even without velocity-dependent friction). hese two unstable modes exhibit different dynamic characteristics. rotor disks. he rotors (distinguishable by a larger diameter) interlock, via notches on the outside perimeter, with the wheel. he stators (smaller diameter) are connected, via similar notches on the inner perimeter, to the torque tube and a wheel hub. When the brakes are applied, the entire stack is compressed by an arrangement of pistons, thus engaging a multitude of braking surfaces. he aerospace industry distinguishes five basic types of aircraft brake vibration: squeal, whirl, chatter, gear walk, and shimmy. INRODUCION Friction-induced vibrations are a major concern in a wide variety of mechanical systems. his is especially the case in braking systems, where friction is both the principal performance factor and a potential cause of detrimental vibrations, noise, and excessive wear. In aircraft braking systems friction induced oscillations can lead to excessively high loads in the landing gear and the brake structure, with consequences ranging from noticeable human discomfort to structural failure of the brake components. A typical modern aircraft brake (see Figure 1) consists of a stacked sequence of carbon disks mounted on a mandrel and housed inside a wheel (not shown). he stack includes alternating stator and Figure 1. A typical aircraft brake configuration. Squeal vibration is characterized by torsional motion of stators, torque tube, and piston housing. ypical frequency range is between 100 Hz and 20 khz. Squeal usually occurs during landing stops, but it can also be observed during taxis. Higher contact pressures and higher energies increase the severity of this type of

2 vibration. Squeal can be excited by the characteristics of the friction material and by modal coupling between axial and tangential degrees of freedom of the brake system. Whirl vibration shows up as out of plane "wobble" motion involving the brake disks, torque tube and piston housing. Whirl frequency range is 100 Hz to 300 Hz, roughly the same as the first squeal mode. Indeed, coupling between squeal and whirl modes is often observed. Whirl typically occurs during high velocity landing stops, especially if there are bumps in the runway. Chatter mode of vibration involves torsional motion of rotors and wheel, typically coupled with fore-aft motion of the landing gear. Chatter is largely controlled by tire stiffness and it can occur at the end of taxi stops. he frequency of chatter vibration is low, in the range between 10 Hz and 100 Hz. Gear walk is defined as fore-aft motion of the landing gear, typically coupled with torsional motion of rotors and wheel. Gear walk oscillations can build up to significant levels, creating passenger discomfort and potential for structural failures. ypical gear walk vibration frequency is also very low, between 10 Hz and 50 Hz. Shimmy vibration mode involves torsional and lateral motion of the wheel and the landing gear. Because of low frequencies (10 Hz - 50 Hz) and high energies involved it can be very destructive, especially in two wheel main gears with off-center bracing. Because of inherent nonlinearity and dynamic complexity of friction-induced vibrations, their understanding, modeling and prediction technology has been lagging behind the needs of the industry and the engineering applications. For decades, the source of friction-induced vibrations was attributed to the specific characteristic of the frictional interface, in particular to velocity-weakening and slip-weakening of friction. An overview of these approaches can be found in the treatise by Bowden and abor [2,3] and in a review article by Ibrahim [4]. While these models accounted for instabilities observed in some systems, in many others they could not produce realistic results or explain the observed behavior. An additional insight into the source of dynamic instabilities came from recognition of the importance of the dynamics of the entire system and its vibrations on the occurrence of friction-induced vibrations. olstoi and co-workers [5,6] observed in their early experimental work that normal vibrations strongly contribute to the difference between static and dynamic coefficients of friction. Following these observations, Oden and Martins [1] introduced a constitutive model for frictional interfaces that represents both the nonlinear normal compliance and sliding resistance of the interface. his model, when combined with detailed transient analysis, can reproduce many of the observed phenomena of friction, such as stick-slip motion. he dependence of friction-induced oscillations upon the dynamic characteristics of the system has been clearly shown in experimental investigations of D'souza and coworkers [7,8,9,10] and in numerical simulations of Oden and Martins [1] and worzydlo et.al. [11,12,13,14,15]. he corresponding mechanism of friction-induced oscillations has been identified as frictional coupling between specific dynamic modes of the system. A strong sensitivity was shown of this coupling to the dynamic characteristics of the system and to the contact nonlinearities. Presently, the importance of system dynamics in prediction of friction-induced instabilities is broadly recognized, and there has been a growing number of applications of these ideas to more complex mechanical systems, including brakes. Liu et.al. [16,17] and Gordon [18] expanded the concepts developed in the earlier works and applied them to modeling of aircraft brake assemblies, represented as interacting rigid bodies. Somewhat similar approach was applied to automotive brakes by Youan [19]. Hulten [20] applied finite element method to modeling of drum brake squeal, considering both friction-velocity slope and the system dynamics. Finite element modeling of low-frequency brake squeal was presented by Matsushima et.al. [21], and application of finite elements to studies of brake groan was studied by Brecht et.al. [22]. he present paper is dedicated to further advancement of understanding and numerical modeling of frictioninduced vibrations in brake systems. In particular, we investigate the influence of nonlinear behavior of the contact interface on the stability of vibrations in aircraft brakes. he primary factors of interest are: nonlinear normal compliance of the surface, and velocity-dependent friction (also referred to as velocity-weakening of friction), at various levels of normal pressure and roughness of the surface. A finite element model of a simplified aircraft brake is the primary object of this study. A nonlinear interface constitutive law (namely, a modified version of Oden- Martins model) with velocity-dependent friction is incorporated into a dynamic representation of the problem. he nonlinear response of the interface in the normal and frictional directions is computed from a surface profile using an asperity-based homogenization method [24,25]. he resulting nonlinear computational model is the basis for steady-state and transient solutions aimed at determining the stability of the brake operation. It is found that under some conditions the sliding is stable, while in other cases a dynamic instability occurs. wo different mechanisms of instability are identified: (i) negative damping due to the negative slope of friction-velocity curve and (ii) friction-induced dynamic coupling of certain vibration modes of the system.

3 In the following sections, the approach used in the analysis of dynamic stability of frictional systems is briefly explained. he model used in the study is illustrated. Boundary conditions, material properties, and loading cases are chosen to approximate those of the actual brake system. he response of the interface in the normal and frictional directions is then computed using an asperity-based method and surface profiles. For these models, dynamic vibration modes without friction (uncoupled) are first determined. his is followed by case studies of stable and unstable sliding. A parametric study of the effects of varying selected system parameters on the stability and frequencies of the vibrations is presented. normal velocity of the stator. As a cross-check, a perturbation in the plane parallel to the contact surface was often performed, usually by slightly varying the coefficient of friction. In a dynamically stable case, the vibrations introduced by perturbation will die out. If the case is unstable, on the other hand, these vibrations will grow. ANALYSIS OF DYNAMIC INSABILIY OF SYSEMS WIH FRICION he overall approach to modeling dynamic stability and behavior of frictional systems follows the techniques developed by Oden and Martins [1] and worzydlo et.al. [12,13,14,15]. he main steps of this approach are summarized below. Let the system with frictional interfaces be discretized using a finite element approximation. he set of equations describing the dynamic equilibrium of the system can be written as: M u&& + Cu& + Ku+ P ( u,u& ) + P ( u,u& ) = F( t) N (1) where: u, u&,& u& column vectors of discrete displacements, velocities and accelerations; M, C, K mass, damping and stiffness matrices; F consistent load vector corresponding to external forces; and P N, P vector of consistent forces due to normal and frictional response of the interface, respectively. he frictional interface between the stator and the rotating disk is modeled using a modified version of Oden-Martins model [1], which was extended to include velocity-dependent friction. he specific nonlinear response of the interface was computed using an asperity-based method, as shown later. Figure 2 illustrates the approach followed in the analysis of stability of frictional systems. First, a steady-state sliding position { u &, u&, u} = { 0, 0, ur} is calculated by solving equation (1) with zero velocities and accelerations: Ku + P N (u,0) + P (u,0) = F (2) Starting from this position, a precise study of the fully nonlinear frictional behavior of the system is performed via the transient solution of the equations of motion (1). he steady-state position is intentionally and minutely perturbed and the dynamic response is computed. he most typical perturbation is to impose some minimal Figure 2. he approach followed in modeling of friction-induced vibrations. he dynamic equations represent a highly nonlinear set of second order differential equations for the unknown displacements u as a function of time that are solved using a Newmark method. he set of Newmark parameters β and γ was selected to slightly dampen high-frequency vibrations (such as compressive waves and numerical noise) while preserving the essential modes that might affect stability of the case at hand. It can be expected that in the actual brake system some level of mechanical damping is typically present and has to be overcome by the unstable effects to produce actual instability. A FINIE ELEMEN MODEL FOR HE BRAKE ASSEMBLY he brake vibration problem was solved using an hpadaptive finite element code PHLEX [23], which is capable of modeling continuum nonlinear dynamic problems, with nonlinear frictional contact on a moving rigid surface or assembly of surfaces.

4 on the pistons are considered: 690, 3447, and 8274 kpa (100, 500, and 1200 psi). he spinning velocities selected for the analyses are: 20, 80, and sometimes 800 rpm. he above values correspond to typical taxi and landing conditions. ASPERIY-BASED INERFACE MODEL Figure 3. A schematic drawing of the simplified brake model. An essential component in simulation and prediction of friction-induced vibrations is a sound constitutive model of the frictional interface. he model should incorporate both normal compliance of the interface and sliding resistance. It has been shown in our previous research that these factors can strongly affect the occurrence and characteristics of friction-induced vibrations, such as squeak and chatter. he actual significance of these parameters depends, to a large extent, on the dynamic characteristics of the system under consideration. In this work, the constitutive characteristics of the contact interface are derived through asperity-based statistical homogenization approach [24,25]. he raw data needed were obtained experimentally and included: a) profile shapes on the surface (from two samples), and b) elasto-plastic compression test results for the composite material. hese parameters were provided for a series of profiles taken on the surface of the disk for 1x1 mm and 8x8 mm sample areas. Based on these results, the following was performed for selected profiles: Figure 4. he mesh used in the analysis of the brake problem. Shade indicated the order of approximation. For this study, a simplified model of an arcraft brake was defined, which includes the stator disk, the pistons, the torque tube, and the spinning rotor - see Figure 3. hen, a finite element model was developed (Figure 4) that includes finite element representation of the torque tube, the pistons, and the stator disk. he rotor disc (not shown in Figure 4) is represented as a prescribed surface, which rotates at a constant angular velocity about the axis of symmetry of the model. A torque tube, attached to the inner side of the disk at one end and fixed in the xy-plane at the other, is added to provide the required torsional stiffness. In the current model, the z- axis coincides with the normal direction, and the contact surface is parallel to the xy-plane. he material properties correspond to typical components of carbontype aircraft brakes. For better dynamic representation, the mass density of the pistons is increased to account for the mass of the piston housing assembly that is not included in the model. Loading conditions are chosen to represent the actual operating modes of the brakes. hree levels of pressure profile sampling; calculation of statistics of the profile, the surface, and asperity peaks; determination of elasto-plastic deformations of representative surface asperities; statistical homogenization of interface properties; and determination of the constants of the Oden-Martins interface constitutive law. For a more detailed description of this procedure the reader can refer to [24,25]. Figure 5. A fragment of the profile on the surface. Vertical scale is exaggerated. As an example, consider a representative profile obtained from a 1x1 mm sample. he bulk material data recovered from compression test results are: Young's modulus E = 4500 MPa, Poisson's ratio ν = 0.2, and the yield stress σ y = 138 MPa. A plot of a short fragment of the profile is shown in exaggerated vertical scale in Figure 5. Using the profile data and a profilometric sampling code, the following values of the standard

5 deviation of profile heights, slopes an curvatures were obtained: σ = µm, σ' = , and σ'' = /µm. he density of summits on the surface is equal to /µm 2. he statistical information of the surface and the material properties of the bulk material are then used to define the response of the surface to normal and tangential loading. First, a representative asperity is defined as having the tip curvature radius of R = 5 µm and height h =.85 µm. For this asperity, elasto-plastic solution is obtained using an analytical solution to a spherical contact problem. he calculated response of a single asperity is then used in conjunction with a statistical homogenization procedure to produce constitutive characteristics of the interface, including: normal pressure versus approach, real contact area versus approach, and friction force versus approach. Here, by approach is meant the coming together of the two surfaces from the point of initial contact of the tips of the tallest asperities (due to asperity deformation, the base surfaces do approach each other under pressure). wo of these curves, namely normal pressure and frictional resistance, are shown in Figure 6. where σ N is the normal stress (compression is positive), a is the normal approach (penetration) on the interface, and a& is its time derivative. he coefficients c N, b N, m N, and l N depend on the properties of the contacting surfaces and the materials of the two bodies. he second constitutive equation of the interface is a friction law of the form: a < 0 σ a 0 σ σ σ = 0 c < c = c a a a m m m and d& = 0 d& = λσ ( λ 0) (4) where d & is the sliding velocity calculated as the time derivative of sliding distance, and the index indicates a direction tangential to the contact surface. he friction force is a function of the normal approach of the two surfaces, which in turn depends on the normal force. he friction law is usually regularized by a continuous function around the zero velocity point [1,13,14]. For the surface profile presented above, fitting the computed response curves (Figure 6) to the Oden- Martins law yields the following parameters: c N = 2.6 x N cm 5.5 m N = 3.5 c = 7.8 x N cm 5.5 m = 3.5 Figure 6. Normal pressure and frictional resistance as a function of approach. he statistical homogenization procedure determines response of the surface in a form of series of discrete data or curves. In order to use this data in numerical analyses, it is convenient to represent it in a closed-form equation, such as Oden-Martins contact and friction law [1]. he constitutive equations in this law consist of two terms: a normal interface law and a friction law. For the normal response, instead of assuming a typical Signorini-type rigid contact condition, the normal compliance of the interface is modeled as: he above parameters correspond to a static coefficient of friction of 0.3. his value represents static coefficient of friction for clean surfaces. In our numerical simulations with frictional sliding we used a lower value of the kinetic coefficient of friction, equal to Additionally, to study the effects of velocity-dependent friction, in some cases the friction law was modified to model velocity-dependence. he coefficient of friction, namely the ratio of frictional to normal stresses, is given as a function of the relative tangential velocity at the interface; namely, σ σ µ = = N µ ( v ) (5) σ σ N N = c N a mn + b a N = 0 for a < 0, ln a& for a 0, and (3) he specific velocity-dependent curve used in this study has three different slopes as shown in Figure 7. As a part of parametric sensitivity studies, we also considered a "soft" version of the interface,

6 corresponding to about ten-fold higher asperities than the above data. he Oden-Martins parameters for this surface are: able 1. orsional (F t ) and normal (F n ) frequencies of uncoupled vibrations. he spectral analysis produces two main frequencies: Normal: his mode of vibrations is due to the compliance of the interface. It has a frequency of about 2650 Hz. Figure 7. Coefficient of friction curves used in the analysis: a) constant, b) velocity-dependent. c N = 8.22 x N cm 5.5 m N = 3.5 orsional: Since the model is fixed at the top of the torque tube, the stator rotates back and forth around the z-axis at a frequency of about 600 Hz. his mode is affected mainly by the torsional stiffness of the tube and the rotational mass moment of inertia of the model. An additional, weaker mode is also contained in the tangential signal with a frequency of about 2050 Hz. his mode represents the rotation of the piston and the neighboring segment of the disk about the radius that passes through that piston ("swaying" of the pistons). his mode, as shown later, becomes the prominent mode in the unstable case. c = 2.46 x N cm 5.5 m = 3.5 MODELING OF UNCOUPLED VIBRAIONS (WIHOU FRICION) In order to define basic dynamic characteristics of the system, the oscillations of the model with no friction are first examined. First, steady-state sliding with friction is solved for. From the steady-state sliding position, the friction is set to zero, thus providing perturbation of the equilibrium position. he model is let free to vibrate, and the transient response is computed. he modal frequencies are calculated using Fast Fourier ransform (FF). A brief parametric study of the effect of the interface compliance and the normal load on the uncoupled modes of the model was performed. he results are summarized in able 1. For the purpose of this study, both the "hard" and the "soft" interface compliances are used. he level of piston pressure is set to the three values of: 690, 3447, and 8274 kpa (100, 500, and 1200 psi). For each of the cases in the table, the frequencies were obtained from FF analysis. Figure 8. Frequency of normal vibrations as a function of normal load. In Figure 8, the normal frequency is plotted as a function of normal pressure for both surfaces. It is clear that increasing the stiffness of the interface increases the frequency of the normal vibrations of the model. Moreover, due to the nonlinear response of the interface, increasing the normal load increases the normal frequency by increasing the stiffness in that direction. he effect of the normal load is more evident on the softer interface.

7 able 2. Summary of parametric case studies of friction-induced vibrations. SUDIES OF FRICION-INDUCED INSABILIIES In this section, we perform studies of friction-induced dynamic instability of the system under various conditions. A compilation of all the cases with indication of the nature of the results is included in the section "Parametric Study", particularly in able 2. In general, three types of behavior were observed: Stable sliding, unstable sliding triggered by velocity-dependent friction, and unstable sliding triggered by coupling between different dynamic modes of the system with or without contribution from velocity-dependent friction. A useful way of measuring the growth (or decay) of vibrations is via the ratio of the amplitude change within one period of oscillations: G = A t+ /A t (4)

8 where G is the growth (or decay) ratio, A is the amplitude, t is the time, and is the period of oscillations for a given frequency. In the following subsections, three case studies are described in detail. While the first exhibits stable behavior, the other two are unstable. In the stable case, vibrations introduced by perturbing the steady-state sliding position decay over time. In the remaining cases, the vibrations tend to grow and the sliding is considered unstable. he cause of instability, however, is different in the two unstable cases: One is unstable due to the negative slope of the friction-velocity curve, and the other is unstable due to dynamic coupling of vibration modes induced by the presence of friction. on bottom of the stator at the same position. Due to the relatively small thickness of the stator and to the fact that its orthotropic material is more flexible in the xy-plane, contact pressure is concentrated under the pistons. Actually, portions of the stator's face are totally separated from the rotor surface. It is noteworthy that the experimental measurements of contact pressure on the stator also showed higher concentration of pressure under the pistons; Figure 10 shows the measured trace of contact pressure under one of the pistons. SABLE SLIDING CASE In this case the pistons are subjected to a pressure of 690 kpa (100 psi), and the contact surface spins at an angular velocity of 80 rpm. he hard interface model with velocity-dependent friction was used. he velocity of 80 rpm falls into the less steep part of the friction-velocity curve shown in Figure 7-b. Figure 10. race of experimentally measured contact pressure under one of the pistons. Figure 9. he deformed shape (displacements scaled 6000 times) and the distribution of contact pressure (kpa) on the bottom of the stator. Plots represent the steady-state sliding configuration. Figure 9 shows the deformed shape (displacements exaggerated) at the steady-state sliding position. he same figure also shows the contact pressure distribution Figure 11. ime histories of the contact area, normal load, and torque moment on the stator during stable oscillations.

9 Figure 12. ime histories of tangential and normal displacements of node 3 and their FF during stable oscillations. Starting from the steady-state sliding position, the model is perturbed and the transient response is computed. Figure 11 shows the time histories of the contact area, resultant normal force, and torsional torque integrated over the contact surface. he oscillations introduced by perturbing the steady-state sliding position decay over time and the system returns to steady-state sliding. Regarding the frequency spectrum, the overall torsional frequency of 600 Hz is present in this system. However, the spectral analysis of the time history of the tangential displacement of a selected node - node 3 (which lies on the face of the stator in contact with the spinning surface directly underneath the inner edge of a piston, see Figure 3) yields a dominant frequency of 2150 Hz (see Figure 12). As in the uncoupled case, the normal oscillations still occur at a frequency of 2650 Hz. Animation of transient response reveals that the frequency of 2150 Hz corresponds to rotation of the piston and the neighboring parts of the disk about a radial axis that passes through the center of the disk and the piston; we call this mode "swaying" of the pistons. Although this frequency existed earlier in the uncoupled vibrations, it was rather a secondary mode. However, it became the primary mode in the present case due to introduction of friction. he energy supplied into the system by the negative damping of friction is a function of both the slope of the friction-velocity curve and the applied normal load. Since both of these factors are at a low level in this case, the energy supplied was too small to initiate unstable behavior. UNSABLE CASE DUE O VELOCIY-DEPENDEN FRICION In this case the piston pressure is increased to 8274 kpa (1200 psi), and the contact surface spins at a low angular velocity of 20 rpm. Similarly as in the previous example, the hard interface model with velocitydependent friction is used. he rotational velocity of 20 rpm falls into the steeper part of the friction-velocity curve shown in Figure 7-b. Figure 13 shows the initial time histories of the contact area, resultant normal force, and torsional torque on the contact surface. In Figure 14, the deformed shape and the distribution of contact pressure are shown for a specific time step during unstable oscillations (time = sec). Figure 15 represents the time histories of the tangential and normal displacements of node 3 (located under one of the pistons) and their FF. he oscillations grow over time, which is characteristic of unstable behavior. Again, the major mode in the tangential motion has a frequency of 2150 Hz. In this mode, the pistons rotate (sway) about their radial axis; all pistons vibrate in sync. In addition, the basic torsional mode of 600 Hz is also present. he amplitude of the oscillations in the x-

10 direction increases from one cycle to the next by a ratio of about below, when instability is triggered by dynamic coupling, both tangential and normal oscillations are excited. Figure 14. he deformed shape (displacements scaled 600 times) and the distribution of contact pressure (kpa) on the bottom of the stator. he plot represents a specific time step during unstable oscillations triggered by velocity-dependent friction (time = sec). Figure 13. ime histories of the contact area, normal load, and torque moment on the stator for the case that is unstable due to velocitydependent friction. In order to identify the source of instability, an example with a constant, velocity-independent coefficient of friction was solved. o further increase the potential of instability, a high value of µ=0.8 was used. Still, the sliding in this case was stable. herefore, it is concluded that the negative slope of the friction-velocity curve, not the amount of friction, is the definite cause of instability here. Another evidence in support of this conclusion is that when the spinning velocity was increased to 80 rpm, the strength of instability, measured by the amplitude growth ratio, decreased from 1.27 to about his can be explained by the fact that the velocity of 80 rpm falls onto the less steep part of the friction-velocity curve shown in Figure 7-b. Another interesting observation is that, unlike in the case shown in the next section, the normal oscillations do not grow over time. he energy provided to the system by the velocity-dependent friction excites mainly the tangential vibrations (at least initially). As will be shown UNSABLE CASE DUE O DYNAMIC COUPLING he case presented in this section exhibits instability triggered purely by dynamic coupling between vibration modes of the system, in absence of velocity-weakening of friction. his case is similar to the stable example presented above (80 rpm, 690 kpa), except that: 1. the soft interface is used, and 2. the coefficient of friction is assumed to be constant at Despite the relatively low coefficient of friction and the non-existence of negative friction slope, the oscillations also grow and the case is unstable. Figure 16 shows time histories of the displacements of node 3 in the x- and z-directions and their FF. Figure 17 shows the phase plots of motion of node 3 in the x- and z- directions. It is important to note that both normal and tangential oscillations are exited and grow in this case. Moreover, the strength of instability, as measured by the growth ratio, is stronger than in the case induced by velocity-dependent friction. Due to nonlinear effects inherent in this problem, the oscillations do not grow infinitely, but develop into a limit cycle oscillation.

11 Figure 15. ime histories of tangential and normal displacements of node 3 and their FF during unstable oscillations due to velocity-dependent friction. Figure 16. ime histories of tangential and normal displacements of node 3 and their FF during unstable oscillations due to dynamic coupling.

12 Figure 18 shows the deformed shape (displacements exaggerated) at a selected time step during unstable oscillations. Clearly, the deformation of the system is more complex than in the previous cases, with undulations visible in the disk. he figure also shows contact pressure distribution on the stator at the same time step. he contact pressure is unevenly distributed under the pistons due to their rotation. Figure 18. he deformed shape (displacements scaled 600 times) and the distribution of contact pressure (kpa) on the bottom of the stator. he plot represents a specific time step during unstable oscillations triggered by dynamic coupling (time = sec). Figure 17. Phase plots of tangential and normal motions of node 3 during unstable oscillations due to dynamic coupling. Since no velocity-dependent friction is used here, another mechanism is responsible for triggering instability. Due to the presence of friction, selected vibration modes of the system are dynamically coupled. his follows the same principle, although at a more complex form, as the examples shown in our previous work [12,13,14,15]. he coupled modes grow quickly, and may become violent. Investigation of the animated motion reveals rather complex modes of vibration. he pistons start wobbling about their axes in an out-ofphase fashion, following a deflection wave which circles the stator. he stator, in addition to its basic symmetric torsional mode, engages in in-plane unsymmetric vibrations in the radial direction. he normal vibrations also grow, so that the stator momentarily separates from the spinning surface; this is clear from the time histories of the total contact area and the normal load on the interface plotted in Figure 19. Figure 19. ime histories of the contact area, normal load, and torque moment for the case that is unstable due to dynamic coupling.

13 As an additional test, the same case was re-run at a much higher angular velocity of 800 rpm. he overall behavior was similar, namely the coupling of dynamic modes of the system and the resulting unstable oscillations. his example indicates that the above coupling mechanism can be triggered at much higher velocities, where the friction-velocity curve does not produce any negative damping (typically, the frictionvelocity curve has the highest slope at low velocities and levels off at higher sliding speeds). It should be noted here that, although this is a relatively simple qualitative study, it confirms that unstable oscillations due to pure dynamic coupling can be present in actual brake systems. In fact, because of the larger number of interacting components in a complete brake system, the possibility of existence of dynamic modes that can be coupled by friction is even greater than in this simplified model. PARAMERIC SUDY In this section we summarize the results of a parametric study of the effect of various characteristics of the system on the stability of sliding. In particular, the following parameters are considered: piston pressure of 690, 3447, and 8274 kpa (100, 500, and 1200 psi), spinning velocities of 20 and 80 rpm (and in some cases 800 rpm), "hard" and "soft" contact interfaces, and velocity-dependent as well as constant coefficients of friction. Case studies that use velocity-dependent friction are summarized in able 2, shown earlier. he table shows for each case whether it is stable or not, the mechanism triggering the instability, the dominant frequencies in normal and tangential oscillations, and the corresponding amplitude growth ratios as defined in equation (6). Some additional results obtained with both constant and velocity-dependent coefficients of friction are summarized in able 3. able 3. Additional case studies. Analyzing these cases, the following observations can be made: he dynamic instability can be triggered by the negative slope of friction-velocity curve or by frictioninduced coupling of dynamic modes of the system (even in absence of velocity-dependent friction). he strength of instabilities (measured by amplitude growth ratios) due to velocity-dependent friction increases with the negative slope of the frictionvelocity curve and with increasing normal pressure. hat explains why in our studies the instability is stronger at 20 rpm than at 80 rpm. he instability due to dynamic coupling is more likely to occur when certain modes within the system have frequencies close enough to be coupled by friction. Also, it can be said that the occurrence of this mode of instability is less systematic than of the frictionvelocity case, and is strongly affected by various dynamic parameters and nonlinearities in the system, such as the normal compliance of the interface. he nonlinear compliance of the interface affects the oscillation frequencies of the system by making them dependent upon the piston pressure. he strength of this influence varies with the the overall hardness of the surface, stiffness of the disk, etc. In the present model, the softer interface produced more unstable cases. While the instability due to velocity-dependence of friction can occur at rather lower sliding speeds (where the friction-velocity curve exhibits a steeper negative slope), the instability due to dynamic coupling can occur at a wide range of velocities. CONCLUSION he paper presented an analytical investigation of the stability of vibrations in a simplified aircraft brake model. he model incorporated three main components: a disk (stator) with one of its faces in frictional contact with a spinning rigid surface (rotor), six pistons with applied normal pressure, and a torque tube that provided the torsional stiffness of the system. A nonlinear constitutive model was used to represent the interface. he parameters of this models were obtained from profilometric data of the surface and asperity-based homogenization techniques. Velocity-dependent coefficient of friction was also incorporated in the model. his model was used to analyze stability of transient behavior following a perturbation of the steady-state sliding position. In these analyses, both stable and unstable modes were observed, depending upon the parameters of the system and the loading conditions. In particular, two major instability modes were detected: 1. Unstable mode caused by the negative slope of the friction-velocity curve. In this mode, primarily the tangential vibrations were excited. Moreover, stronger instability was observed under higher

14 normal loads and higher slopes of the frictionvelocity curve. 2. Unstable mode caused by friction-induced coupling of certain dynamic modes of the system. In this mode, both normal and tangential oscillations were excited. he occurrence of this mode was highly sensitive to various dynamic parameters of the system and nonlinearities inherent in the contact interface. Although the brake model analyzed in this study was quite simplified, the material properties and dimensions were representative of the components of the actual brake system. hus, it can be expected that the conclusions of this study will extend to more complete models. It can also be expected that the techniques presented here and general observations will be applicable to automotive brakes and other similar systems. ACKNOWLEDGMENS he support of this work by AlliedSignal Inc., Aerospace Equipment Systems, is greatly appreciated. REFERENCES 1. Oden, J.., and Martins, J. A. C., "Models and Computational Methods for Dynamic Friction Phenomena," Comp. Meth. Appl. Mech. and Engng., 52, pp , Bowden, F.P. and abor, D., he Friction and Lubrication of Solids, Clarendon Press, Oxford, Bowden, F.P. and abor, D., he Friction and Lubrication of Solids - Part II, Clarendon Press, Oxford, Ibrahim, R.A., "Friction-induced vibration, chatter, squeal and chaos; Part I: Mechanics of Contact and Friction; Part II: Dynamics and Modeling", in Applied Mechanics Reviews, 47, pp , olstoi, D. M., "Significance of the Normal Degree of Freedom and Natural Normal Vibrations in Contact Friction," Wear, 10, pp , olstoi, D. M., Borisova, G. A., and Grigorova, S. R., "Role of Intrinsic Contact Oscillations in Normal Direction During Friction," Nature of the Friction of Solids, Nauka i iechnika, p.116, Minsk, Aronov, V., D'Souza, A. F., Kallpakijan, S., and Shareef, I., "Experimental Investigation on the Effect of System Rigidity on Wear and Friction-Induced Vibrations,"Journ. Lubr. echnol., 105, pp , Aronov, V., D'Souza, A. F., Kallpakijan, S., and Shareef, I., "Interaction Among Friction, Wear and System Stiffness - Part1: Effect of Normal Load and System Stiffness; Part 2: Vibrations Induced by Dry Friction; Part 3: Wear Model,"Journ. Lubr. echnol., 106, pp , D'Souza, A.F. and Dweib, A.H., "Self-Excited Vibrations Induced by Dry Friction, Part 2: Stability and Limit Cycle Analysis," J. Sound and Vibration, Vol. 137, 2, pp , Dweib, A.H. and D'Souza, A.F., "Self-Excited Vibrations Induced by Dry Friction, Part 1: Experimental Study," J. Sound and Vibration, Vol. 137, 2, pp , worzydlo, W. W., and Becker, E., "Influence of Forced Vibrations on the Static Coefficient of Friction-Numerical Analysis," Wear, 43, pp. 1-22, worzydlo, W. W., Becker, E. B. and Oden, J.., "Numerical Modeling of Friction-Induced Vibrations and Dynamic Instabilities", in Ibrahim, R. A. and Soom, A., Editors, Friction-Induced Vibration, Chatter, Squeal and Chaos, ASME, De-Vol. 49, New York, pp , worzydlo, W. W., Becker, E. B. and Oden, J.., "Numerical Modeling of Friction-Induced Vibrations and Dynamic Instabilities", in Applied Mechanics Reviews, 47, pp , worzydlo, W. W. and Hamzeh, O.N. "On the Importance of Normal Vibrations in Modeling of Stick-Slip in Rock Sliding," J. of Geophysical Research, 102, No. B7, pp. 15,091-15,103, July worzydlo W.W., Hamzeh, O.N., Zaton, W., Judek,. and Loftus, H., "Experimental and Numerical Studies of Friction-Induced Oscillations of a Pin-on- Disk Apparatus", Proc. of ASME Design Engineering echnical Conferences, Sixteenth Biennial Conference on Mechanical Vibration and Noise, ASME, Sacramento, September 14-17, Liu, S.Y., Gordon, J.. and Özbek, M.A., "A Nonlinear Model for Aircraft Brake Squeal Analysis; Part I: Model Description and Solution Methodology", AIAA paper AIAA CP, pp , Liu, S.Y., Gordon, J.. and Özbek, M.A., "A Nonlinear Model for Aircraft Brake Squeal Analysis; Part II: Stability Analysis and Parametric Studies", AIAA paper AIAA CP, pp , Gordon, J..,"A Perturbation Analysis of Nonlinear Squeal Vibrations in Aircraft Braking Systems", Proc. of ASME Design Engineering echnical Conferences, Sixteenth Biennial Conference on Mechanical Vibration and Noise, ASME, Sacramento, September 14-17, Youan, Y., "An Eigenvalue Analysis Approach to Brake Squeal Problems", Proc. of Dedicated Conference on Automotive Braking Systems, 29th ISAA, Florence, Italy, June 1996, pp. 1-9.

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