Modeling Friction Phenomena in Flexible Multibody Dynamics

Transcription

1 Modeing Friction Phenomena in Fexibe Mutibody Dynamics Oivier A. Bauchau and Changuan Ju Danie Guggenheim Schoo of Aerospace Engineering, Georgia Institute of Technoogy, 270 Ferst Dr., Atanta, GA 30332, USA Abstract The dynamic response of fexibe systems of arbitrary topoogy and compexity are now readiy modeed using mutibody dynamics anaysis concepts. In most formuations, the joints connecting the fexibe bodies of the system are not modeed per se: ony the effect of joints is modeed through a set of inematic constraints. This paper focuses on the deveopment of methodoogies for the anaysis of uniatera contact conditions in joints and of the resuting norma and friction forces. This invoves a host of phenomena such as: contact inematics, contact conditions, and the modeing of the norma and tangentia contact forces. Furthermore, the high eve of noninearity associated with a number of these phenomena impies chaenging numerica issues. Athough friction forces can be readiy evauated using Couomb friction aw, the resuting accuracy is questionabe: sticing and sipping can co-exist in different parts of the contact area, a phenomenon nown as micro-sip. Numerous modes of friction have been deveoped and presented in the iterature; appication of the LuGre mode wi be discussed in this paper. 1 Introduction Mutibody dynamics anaysis is a powerfu too for the comprehensive simuation of the dynamic response of fexibe systems of arbitrary topoogy and compexity. In present formuations, the joints connecting the various fexibe bodies are rarey modeed per se. Rather, the effect of joints, i.e. the constraints they impose on the behavior of the entire system are modeed through a set of inematic constraints; the piece of hardware that actuay constitutes the joint is not modeed. As mutibody formuations become more widey accepted, the need to mode a wider array of phenomena increases. In particuar, it is necessary to deveop methodoogies for the anaysis of uniatera contact conditions in joints and of the resuting norma and friction forces. For reaistic simuations, the actua piece of hardware and contact mechanics shoud be modeed more precisey. This invoves a host of phenomena such as: contact inematics, noninear norma contact forces, tangentia oading, and siding contact. Furthermore, the high eve of noninearity associated with a number of these phenomena impies chaenging numerica issues. Athough friction forces can be readiy evauated from Couomb friction aws, the resuting accuracy is questionabe. Sticing and sipping can co-exist in different parts of the contact area, a phenomenon nown as micro-sip. The actua reationship between the friction force, the norma pressure and the sip motion is not we understood. A further compication is that the accurate evauation of the norma or pressure forces across the sip pane requires modeing of system dynamics in both norma and sip panes. Hence, joint behavior and system dynamics are intimatey couped. Finay, friction forces depend on the motion in a highy noninear way, compicating anaytica and numerica soution procedures. Numerous modes of friction have been deveoped and presented in the iterature, each addressing some of the above issues. This paper focuses on the deveopment, impementation and vaidation of modes that capture the behavior of joints in a reaistic manner. These modes wi be presented within the framewor of a finite eement based, noninear mutibody dynamics formuations that ensure unconditiona noninear stabiity of the computation for compex systems of arbitrary topoogy. The proposed approach can be divided into three parts. First, the modeing of the joint configuration: this purey inematic part of the probem deas with the Computer Methods in Appied Mechanics and Engineering, 195(50-51), pp ,

2 description of the configuration of the joint and the evauation of the reative distance, q, and the reative tangentia veocity, v, between the contacting bodies. Second, the enforcement of the contact conditions: in most cases, contact at the joint wi be of an intermittent nature. This uniatera contact condition is readiy expressed in terms of the reative distance as q 0. The enforcement of the uniatera contact condition is a critica aspect of the computationa procedure. Finay, the evauation of the contact forces, when contact occurs: this ast part of the probems deas with the computation of the norma and tangentia forces that arise at the interface between contacting bodies. The contact forces must be computed based on suitabe phenomenoogica aws. A survey of the mutibody iterature reveas that very simpe modes have been used thus far. For instance, the norma contact forces have been modeed using a quadratic potentia that corresponds to a inear force-approach reationship, or the potentia corresponding to Hertz s probem [1]. The cassica form of Couomb s aw has been the basis for the modeing of frictiona phenomena [2]. Increasing the versatiity and accuracy of uniatera contact modes in mutibody systems is the focus of this paper. To achieve this goa, the three parts of the mode must be consideraby expanded. The inematics of the contacting bodies must be generaized so as to aow a variety of joint configurations to be considered. The proper enforcement of the contact condition is centra to the numerica stabiity of the proposed procedure. Robust schemes must be used to treat this chaenging numerica probem. Finay, contact and frictiona forces must be modeed in an accurate manner. Note that the probem is highy couped because contact and frictiona forces depend on the overa dynamic response of the system; joint behavior cannot be investigated without modeing the entire system. The advantage of the proposed approach is that the three parts of the probem can be formuated and impemented independenty. For instance, once a given friction aw has been impemented, it can be used for various types of joints and for systems of arbitrary configurations. It is aso possibe to evauate the performance of various friction aws for a given joint configuration by comparing their predictions with experimenta measurements. Because the formuation is deveoped within the framewor of finite eement based mutibody dynamics, it can dea with systems of arbitrary configurations. This paper is aid out as foows: section 2 presents the inematic description of two joints with cearance and the uniatera contact condition associated with the cearance is discussed in section 3. The modeing of the frictiona process is presented in sections 4 and 5, and numerica exampes of the proposed procedure appear in section 6. 2 Kinematic Description of Joints with Cearance The inematic description of joints with cearance wi be divided into two- and three-dimensiona modes. For two-dimensiona probems, a revoute joint with cearance, caed the panar cearance joint, is viewed as a panar joint with a uniatera constraint. The cearance, or distance between the inner and outer races, can be evauated from the inematic variabes associated with the panar joint. A simiar approach can be appied to spherica joints with cearance. In the three-dimensiona case, a spatia cearance joint is deveoped; the cearance can sti be reated to the inematic variabes of the joint, athough this reationship is more compex. 2.1 The Panar Cearance Joint A revoute joint with cearance can be modeed as a panar joint with the appropriate addition of a uniatera contact condition. Consider the panar joint depicted in fig. 1; the outer and inner races of the joint are modeed as bodies K and L, respectivey. In the reference configuration, the position of body K is defined by its position vector u 0 and its orientation is determined by a body attached orthonorma basis B0 = ( ē 10, ē 20, ē30), with ē 30 norma to the pane of joint and the over bar indicates a unit vector. The radius of the outer race is denoted ρ. In the deformed configuration, body K undergoes a dispacement u and its orientation is defined by an orthonorma basis B = ( ē 1, ē 2, ē3). The inematic variabes associated with body L, which represents the inner race of the revoute joint, are defined in a simiar manner. The panar joint is associated with the foowing constraint conditions C 1 = ē T 1 ē 3 = 0; C 2 = ē T 2 ē 3 = 0; C 3 = ē T 3 (u 0 + u) = 0, (1) where u 0 = u 0 u 0 and u = u u. Kinematic condition C 3 = 0 impies that body L remains in the pane norma to ē 3. Conditions C 1 = C 2 = 0 impy that ē 3 remains norma to that same pane. The impementation 2

3 of the hoonomic constraints, eqs. (1), is discussed in ref. [3]. Contact may occur between the inner and outer races of the joint. As shown in fig. 1, the candidate contact points [2] are readiy found as Z = u 0 +u +ρ n and Z = u 0 +u +ρ n, where n = (u 0 +u)/ u 0 +u is the unit vector joining the centers of the two races. The reative distance between the races then becomes q = n T (Z Z ) = ρ ρ u 0 + u. (2) The virtua wor done by the norma contact force is δw = f n n T δ(z Z ) = f n n T δ(u u ) = f n δq, where f n is the magnitude of the norma contact force. Expanding this resut yieds [ ] δu T [ ] δw = δu F n δu T [ ] = δu f n n. (3) n The present formuation is deveoped within the framewor of the time discontinuous Gaerin procedure [4] that provides agorithms for the integration of mutibody systems featuring unconditiona noninear stabiity [3]. In this approach, the discretization of the system is based on its configurations at the beginning and end times of a typica time step, denoted t i and t f, respectivey, and the subscripts ( ) i and ( ) f denote the vaues of a quantity at t i and t f, respectivey. In addition, the subscript ( ) j indicates quantities at time t j, immediatey after the initia discontinuity. The norma contact force is taen to be constant over a time step and is discretized as F n m = f n m [ nm n m ], (4) where u m = (u f + u i )/2, n m = 2(u 0 + u m )/( u 0 + u f + u 0 + u i ), and f n m the constant magnitude of the contact force over the time step. Note that the discretization of the contact force does not invove discretized quantities at time t j ; this contrasts with the discretization of the eastic forces in the fexibe eements of the mode that does invove the discretized quantities at time t j [3], resuting in a inear in time approximation for these forces. For contact probems, it was found to be preferabe to use constant in time approximations for the norma and tangentia components of the contact force. It is readiy verified that this discretization impies that the wor done by the norma contact force over a time step is W = f n m(q f q i ). Next, the friction forces wi be evauated. The time derivatives of the candidate contact point positions are Ż = u + ρ ω n and Ż = u + ρ ω n, where () denotes a derivative with respect to time, and ω and ω are the anguar veocity vectors of bodies K and L, respectivey. The reative veocity is V r = Ż Ż, and the reative tangentia veocity, V t, then becomes V t = P V r, (5) where P = U n n T is the operator that projects the reative veocity vector onto the common tangentia pane at the point of contact; U is the identity matrix. The unit vector aong the reative tangentia veocity is denoted ē = V t /V t, where V t = V t. The virtua wor done by the friction force is δw = f t ē T (δz δz ), where f t is the magnitude of the frictiona force. Expanding this expression yieds δu T δu T ē δw = δψ δu F t = δψ δu f t ρ ñē ē. (6) δψ δψ ρ ñē The discretized friction force is seected as F t m = fm t ē m ρ ñ m ē m ē m ρ ñ m ē m, (7) where f t m is the constant magnitude of the tangentia contact force over the time step. The wor done by the friction forces over a singe time step then becomes W = f t m tv t m, provided the foowing definitions are made: V t m = (U n m n T m)v r m, ē m = V t m/v t m and tv r m = (u f u i )+ρ ñ T mr ρ ñ T mr, where r and r are the incrementa rotations of bodies K and L, respectivey. If the friction is dissipative, f t tv t 0, and the above discretization guarantees that W = f t m tv t m 0, i.e. the discretization of the frictiona process is dissipative. Note that the formuation presented here is aso vaid for a spherica joint with cearance. 3

4 2.2 The Spatia Cearance Joint When the motion of the joint cannot be assumed to remain panar, a more compex, three-dimensiona configuration must be considered, such as that presented in fig. 2. The outer race of the joint, denoted body K, is ideaized as a cyinder of radius ρ. The inner race, denoted body L, is ideaized as a thin dis of radius ρ. In the reference configuration, the position of the outer race is defined by the position vector u 0 of its center and its orientation is determined by a body attached orthonorma basis B 0 = ( ē 10, ē 20, ē 30), with ē 30 aong the axis of the cyinder. In the deformed configuration, the outer race center undergoes a dispacement u and its orientation is defined by an orthonorma basis B = ( ē 1, ē 2, ē 3). The inematic variabes associated with body L are defined in a simiar manner, with ē 30 norma to the pane of the dis. An orthonorma basis d 1, d 2, d 3 is now defined in the foowing manner: d 3 = ē 3 is aong the axis of the cyinder, d 1 maes an arbitrary ange ϕ with ē 1, and d 2 competes the basis d 1 = cos ϕ ē 1 + sin ϕ ē 2; d 2 = sin ϕ ē 1 + cos ϕ ē 2; d 3 = ē 3. (8) Consider now the pane tangent to the cyinder, defined by vectors d 2 and d 3, as depicted in fig. 3. Point P, of position vector u 0 + u + ρ d 1, beongs to this pane. The reative distance q between dis L and this tangent pane is now evauated. The candidate contact points on the pane and dis are denoted Z and Z, respectivey, see fig. 3. The tangent to the dis at the candidate contact point must be in the pane of the dis and parae to the contacting pane, i.e. norma to ē 3 and d 1, respectivey. The foowing basis is now defined d 2 = d 1 ē 3 h 1 ; d 3 = ē 3; d 1 = d 2 ē 3, (9) where h 1 = d 1 ē 3. Ceary, d 2 is parae to the tangent at the candidate contact point, d 1 points toward the candidate contact point, and d 3 is norma to the pane of the dis. The vector from point P to point Z is P Z = [ (u 0 + u ) + ρ d 1 ] [ (u 0 + u ) + ρ d 1 ] = u0 + u + ρ d 1 ρ d 1, (10) where u 0 = u 0 u 0, and u = u u. The reative distance q is found by projecting P Z aong the unit vector d 1, see fig. 3, to find T q = d 1 P Z T = d 1 (u 0 + u) ρ dt 1 d 1 + ρ. (11) As shown in fig. 3, the candidate contact point Z is in the tangent pane, but not necessariy on the cyindrica surface defining the outer race of the joint. The reative distance q defined by eq. (11) is ceary a function of the ange ϕ that defines the ocation of the tangent pane around the cyinder. The reative distance q between the cyinder and dis is found by minimizing q with respect to the choice of ϕ, i.e. by setting d q/dϕ = 0 to find d T 2 [ρ g 1 ē 3 (u h 0 + u)] = 0, (12) 1 where g 1 = ē T 3 d 1. Note that for sma anges g 1 0 and h 1 1. The same resut coud be obtained by imposing the condition that P Z be orthogona to d 2. In summary, the reative distance between the inner and outer races of the joint is q = ρ ρ h 1 d T 1 (u 0 + u), (13) where h 1 = d 1ē T 3 = d 1 d 1, and ange ϕ is impicity defined by eq. (12). n T If contact occurs, the virtua wor done by the norma contact force is δw = f d 1 δ(z Z ) = n T f d 1 δ(q d 1) = f n δq, where f n is the magnitude of the norma contact force. Expanding this resut yieds δw = δu δψ δu δψ T F n = δu δψ δu δψ T f n d 1 (ρ g 1 /h 1 ) h 1 d 1(u 0 + u) d 1 (ρ g 1 /h 1 ) h 1, (14) 4

5 where h 1 = d 1ē 3. The proposed discretization of the norma contact force, taen to be constant over a time step, is d 1m F n m = fm n (ρ g 1m /h 1m ) h 1m d 1m(u 0 + u m ) d, (15) 1m (ρ g 1m /h 1m ) h 1m where d 1m = ( d 1f + d 1i )/2, e 3m = (ē 3f + ē 3i )/2, h 1m = d 1me 3m, h 1m = (h 1f + h 1i )/2, g 1m = (g 1f + g 1i )/2, and f n m is the constant magnitude of the contact force over the time step. It is readiy verified that this discretization impies that the wor done by the norma contact force over a time step is W = f n m(q f q i ). Next, the friction forces wi be evauated. The time derivatives of the candidate contact point positions are Ż = u + ω λ and Ż = u + ρ ω d 1, where λ = (u 0 + u) + ρ d 1 + q d 1. The reative veocity is V r = Ż Ż, and the reative tangentia veocity, V t, then becomes V t = P V r, (16) where P = U d T 1 d 1 is the operator that projects the reative veocity vector onto the common tangentia pane at the point of contact. The unit vector aong the reative tangentia veocity is denoted ē = V t /V t, where V t = V t. The virtua wor done by the friction force is δw = f t ē T (δz δz ), where f t is the magnitude of the frictiona force. Expanding this expression yieds δw = δu δψ δu δψ T F t = δu δψ δu δψ T f t ē λē ē ρ d 1ē. (17) The discretized friction force is seected as F t m = fm t ē m λ m ē m ē m ρ d 1mē m, (18) where fm t the constant magnitude of the contact force over the time step. The wor done by the friction forces over a singe time step then becomes W = fm tv t m, t provided the foowing definitions are made: V t m = (U d 1md T 1m)V r m, ē m = V t m/vm t and tv r T m = (u f u i ) + ρ d 1mr λ T mr, where r and r are the incrementa rotations of bodies K and L, respectivey. If the friction is dissipative, f t tv t 0, and the above discretization guarantees that W = fm tv t m t 0, i.e. the discretization of the frictiona process is dissipative. A reaistic mode of a journa bearing with cearance is obtained by using two spatia cearance joints connected by a rigid body, as depicted in fig The Uniatera Contact Condition The approaches to the modeing of uniatera contact conditions can be categorized in two main casses. The first one considers impact to be an impusive phenomenon of nu duration [5, 6, 7]. The configuration of the system is frozen during the impact, and an appropriate mode is used to reate the states of the system immediatey before and immediatey after the event. There are two aternative formuations of this theory: Newton s and Poisson s methods. The first reates the reative norma veocities of the contacting bodies through the use of an appropriate restitution coefficient. The second divides the impact into two phases. At first, a compression phase brings the reative norma veocity of the bodies to zero through the appication of an impuse at the contact ocation. Then, an expansion phase appies an impuse of opposite sign. The restitution coefficient reates the magnitudes of these two impuses. Athough these methods have been used with success for mutibody contact/impact simuations, it is cear that their accuracy is inherenty imited by the assumption of a vanishing impact duration. 5

6 In the second approach, contact/impact events are of finite duration, and the time history of the resuting interaction forces is computed as a by-product of the simuation [8, 9, 10]. This is achieved by introducing a suitabe phenomenoogica aw for the contact forces, usuay expressed as a function of the inter-penetration, or approach, between the contacting bodies. This approach is obtained at each instant of the simuation by soving a set of inematic equations that aso express the minimum distance between the bodies when they are not in contact, such as eqs. (2) or (13). When the bodies are in contact, the reationship between the norma contact force and the approach is given by a constitutive aw; various aws can be used, but the cassica soution of the static contact probem presented by Hertz [1] has been impemented by many investigators. Energy dissipation can be added in an appropriate manner, as proposed by Hunt and Crossey [11]. If the norma contact forces can be derived from a potentia, V (q), the wor done by these forces over a time step of the simuation is W = f n m(q f q i ), as impied by the time discretizations proposed in eqs. (4) and (15). The wor done by these forces then becomes W = V (q f ) V (q i ), as expected, if the norma forces are discretized as f n m = (V (q f ) V (q i ))/(q f q i ) To be successfu, the approach described above must be compemented with a time step size seection procedure. When contact between the two bodies is about to tae pace, the contact mode wi dictate the time step for the anaysis. Let q 0 and q 1 be the reative distances between the bodies for two consecutive time steps of size t 0 and t 1, respectivey. To avoid arge penetration distances and the ensuing arge norma contact forces at the first time step after contact, the time step size wi be seected so that the change in reative distance, q = q 1 q 0, be of the order of a user defined characteristic penetration distance, ε p. To achieve this goa, the desired change in reative distance is seected as q = ε p { 1 if κ 1 κ α if κ > 1, (19) where the quantity κ, defined as κ = (q 1 /ε p )/ q min, measures the proximity to contact. The desired time step size is then estimated as t new = q v m, (20) where v m is the average reative veocity during the previous time step, v m = 2 (q 1 q 0 )/( t 0 + t 1 ). The foowing vaues were found to give good resuts for a wide range of probems: q min = 5 and α = 1.2. It is important to note that the success of the present approach hinges upon two features of the mode: the oca fexibiity of the contacting bodies and time adaptivity. If the contacting bodies are assumed to be rigid, the contact forces present a discontinuity at the instant of contact that causes numerica probems during the simuation. Taing oca fexibiity into account transforms the discontinuous force into a force with steep time gradients that are then resoved using time adaptivity. This contrasts with event driven computationa strategies that first invove the determination of the exact time of contact, then a different set of governing equations is used when the bodies are in contact. If severa contacts occur simutaneousy, the compementarity principe is then a very effective soution strategy [2]. The choice between these two contrasting approaches is one based on computationa considerations. For instance, the even driven approach is effective when deaing with systems of rigid bodies featuring mutipe contacts; the compementarity principe then gives an eegant soution to an otherwise untractabe probem. On the other hand, when deaing with systems modeed with finite eement techniques, the approach proposed here seems to be more effective. 4 Modeing the Frictiona Process The detaied modeing of frictiona forces poses unique computationa chaenges that wi be iustrated using Couomb s aw as an exampe. When siding taes pace, Couomb s aw states that the friction force, F f, is proportiona to the magnitude of the norma contact force, f n, F f = µ (v) f n v/v, where µ (v) is the coefficient of inetic friction and v the magnitude of the reative veocity vector tangent to the friction pane, v. If the reative veocity vanishes, sticing taes pace. In this case, the frictiona force is F f µ s f n, where µ s is the coefficient of static friction. Appication of Couomb s aw invoves discrete transitions from sticing to siding and vice-versa, as dictated by the vanishing of the reative veocity or the magnitude of the friction force. These discrete 6

7 transitions can cause numerica difficuties that are we documented, and numerous authors have advocated the use of a continuous friction aw [12, 13, 14, 10], typicay written as F f = µ (v) f n v v (1 e v /v 0 ), (21) where (1 e v /v 0 ) is a reguarization factor that smoothes out the friction force discontinuity and v 0 a characteristic veocity usuay chosen to be sma compared to the maximum reative veocity encountered during the simuation. The continuous friction aw describes both siding and sticing behavior, i.e. it competey repaces Couomb s aw. Sticing is repaced by creeping between the contacting bodies with a sma reative veocity. Various forms of the reguarizing factor have appeared in the iterature; a comparison between these various modes appears in [15]. Repacing Couomb s friction aw by a continuous friction aw is a practice widey advocated in the iterature; however, this practice presents a number of shortcomings [16]. First, it aters the physica behavior of the system and can ead to the oss of important information such as abrupt variations in frictiona forces; second, it negativey impacts the computationa process by requiring very sma time step sizes when the reative veocity is sma; and finay, it does not appear to be abe to dea with systems presenting different vaues for the static and inetic coefficients of friction. In reaity, frictiona forces do not present the discontinuity described by Couomb s aw but rather, a veocity dependent, rapid variation. The reguarization factor discussed above smoothes the discontinuity through a purey mathematica artifact that maes no attempt to more accuratey represent the physica processes associated with the friction phenomenon. On the other hand, physics based modes address the behavior of the frictiona interface under sma reative veocity. Typicay, these micro-sip modes aow sma reative dispacements to tae pace during sticing; the frictiona interface then behaves ie a very stiff viscous damper. Of course, such modes are incapabe of capturing the tangentia stiffness of the joint during micro-sip. The most common micro-sip mode is the Iwan mode [17], aso named eastic-perfecty pastic mode [18]. In this approach, the frictiona interface is modeed as a spring in series with a Couomb friction eement featuring a friction force of magnitude µf n. When the force in the spring reaches this magnitude, the force in the Iwan mode saturates unti the direction of sip reverses. An aternative approach to the treatment of micro-sip invoves modification of the friction aw itsef. Over the years, severa friction modes have been proposed that more accuratey mode various physica aspects of the friction process, such as the Vaanis mode [19]. Dah [20] proposed a differentia mode abe to emuate the hysteretic force-dispacement behavior that characterizes micro-sip. The primary shortcoming of the Dah mode is that it does not incude the dependence of the friction coefficient on sip veocity. In particuar, it does not capture the rapid decrease in the friction coefficient as the interface begins to sip. Since this drop is a major contributor to stic-sip osciations, its omission coud be a major drawbac in its appication to joint dynamics. More recenty, Canudas de Wit et a. have proposed the LuGre mode [21] that is based on a phenomenoogica description of friction. This mode is abe to capture experimentay observed phenomena such as pre-siding dispacements, the hysteretic reationship between the friction force and the reative veocity, the variation of the brea-away force as a function of force rate, and stic-sip motion associated with the Stribec effect. The LuGre mode has further been refined by Swevers et a. [22] and Lampaert et a. [23]. 5 The LuGre Friction Mode The state of the art in friction modeing was advanced using the paradigm of intermeshing bristes to expain the friction forces between two contacting bodies [24]. The briste mode captures micro-sip and aso accounts for the drop in friction force as the siding speed is increased. However, according to its authors, the mode is computationay burdensome. One of the more promising friction modes deveoped from the briste paradigm is the LuGre mode [21], which captures the variation in friction force with sip veocity, maing it a good candidate for studies invoving stic-sip osciations. Aso, when inearized for very sma motions, the LuGre mode is shown to be equivaent to a inear spring/damper arrangement. The LuGre mode is an anaytica friction mode summarized by the foowing two equations. µ = σ 0 z + σ 1 dz dt + σ 2v; (22) 7

8 dz dt = v σ 0 v z. (23) µ + (µ s µ )e v/vs γ The first equation predicts the instantaneous friction coefficient µ as a function of the reative veocity, v, of the two contacting bodies and an interna state of the mode, z, that represents the average defection of eastic bristes whose interactions resut in equa and opposite friction forces on the two bodies. The second equation is an evoution equation for the average briste defection. The coefficients σ 0, σ 1, and σ 2 are parameters of the mode; µ s and µ are the static and inetic friction coefficients, respectivey; v s the Stribec veocity; and γ a fina mode parameter which is often seected as γ = 2. The friction force acting between the bodies is then f f = µf n, (24) where f n is the norma contact force. For convenience, the mode is now rewritten in nondimensiona form as ˆµ = (1 ˆβˆσ 1 )ẑ + (ˆσ 1 + ˆσ 2 )ˆv; (25) ẑ = ˆv ˆβẑ, (26) where ˆµ = µ/µ, ˆv = v/v s, ẑ = σ 0 z/µ, ˆσ 1 = σ 1 v s /µ and ˆσ 2 = σ 2 v s /µ. The nondimensiona time is τ = Ωt, where Ω = σ 0v s µ (27) is the inverse of the time constant of the LuGre mode. The notation () indicates a derivative with respect to this nondimensiona time. Finay, ˆβ = ˆv /ĝ(ˆv), where ĝ(ˆv) = 1 + (µ s /µ 1)e ˆv γ. (28) The evoution equation of the LuGre mode, eq. (26), wi be discretized in the foowing manner, based on a time discontinuous Gaerin procedure [4], z f z i τ + β g z f + z j 2 = v g, z j z i τ β g z f z j 6 = 0, (29) where, for simpicity, the hat was dropped from a symbos. The subscript () g indicates the foowing average () g = 1/2(() f +() j ). Note the presence of quantities at time t j that wi bring about the numerica dissipation required for the simuation of the frictiona process. 5.1 Properties of the Proposed Discretization For steady state soutions z f = z j = z i = z ss, eq. (29 b ) is identicay satisfied, whereas eq. (29 a ) impies β ss z ss = v ss, which can be written as v ss (s vss s zss z ss /g(v ss )) = 0, where s vss = sign(v ss ) and s zss = sign(z ss ). The first soution of this equation is v ss = 0, which corresponds to sticing. The second is z ss = g(v ss ) with s vss s zss = 1, corresponding to steady state siding. For arge siding veocity, g(v ss ) 1, and hence, z ss 1. The discretization aso impies an important evoution aw for the strain energy of the bristes. This evoution aw is obtained by summing up eqs. (29 a ) and (29 b ) mutipied by (z f + z j )/2 and (z f z j )/2, respectivey, to yied V f V i τ = z g v g ( z g g(v g ) s v g s zg ) 1 2 τ (1 + 6 β g τ )(z j z i ) 2, (30) where V = 1/2 z 2 is the nondimensiona strain energy stored in the bristes. The second term of the evoution aw, eq. (30), is a numerica dissipation term that is aways negative. As the time step decreases, z j z i as impied by the time discontinuous Gaerin approximation, and the numerica dissipation vanishes. The first term is negative whenever s vg s zg = 1, or when s vg s zg = 1 and z g /g(v g ) > 1. In summary, (V f V i )/ τ < 0 when z g > g(v g ). This impies the decreasing of the strain energy of the eastic bristes whenever z g > g(v g ). Since the strain energy is a quadratic form of the briste average defection, this impies 8

9 the decreasing of the briste defection under the same conditions. Consequenty, the briste defection must remain smaer than the upper bound of g(v g ), which, in view of eq. (28), is equa to µ s /µ. It foows that z g < µ s /µ. This inequaity impies the finiteness of the briste defection and of its strain energy. The discretizations of the friction forces for the cearance joints proposed in this wor, eqs. (7) and (18), impy that the wor done by the friction force over one time step is W = f t m tv t m = µ g f n g tv g ; introducing eq. (25) then yieds σ 0 W µ 2 τfn g = [(1 β g σ 1 )z g + (σ 1 + σ 2 )v g ] v g. (31) With the hep of eq. (30), this wor can be expressed as σ 0 W µ 2 τfn g = σ 2 vg 2 z2 g g(v g ) v g V ( f V i σ 1 1 z ) g τ g(v g ) s v g s zg vg τ (1 + 6 β g τ )(z j z i ) 2. (32) The first two terms are aways dissipative, with the second term corresponding to the main energy dissipation during siding. The third term corresponds to the change in the potentia of the eastic deformation of the bristes. Athough this term can be positive or negative, it is finite since the potentia is, itsef, finite. As discussed earier, when z g > g(v g ), (V f V i )/ τ < 0 and the potentia energy of the briste defection is reeased to the system. Under a simiar condition, s vg s zg = 1 and z g > g(v g ), the fourth term aso becomes positive. Ceary, the third and fourth terms are non-dissipative ony when the potentia of the eastic bristes is reeased. As discussed earier, this potentia is finite and increases in its vaue stem from wor done against friction forces; it is this very wor that coud be reeased at a ater time. Finay, the ast term is a numerica dissipation term. Note that as the time step size is decreased, (z j z i ) 2 rapidy decreases, and the numerica dissipation vanishes. In summary, the proposed discretization of LuGre mode guarantees the dissipative nature of the friction forces, when combined with the proposed discretizations for the friction forces, eqs. (7) and (18). To be successfu, the approach described above must be compemented with a time step size seection procedure. When friction occurs, the friction mode wi dictate the time step for the anaysis. In view of the rapid variation of the function g(v) for sma reative veocities, the time step size must be reduced when the reative veocity is of the order of the Stribec veocity v s. To achieve this goa, the time step size for the next time step is seected as Ω t new = τ min { 1 if ν 1 ν α if ν > 1, (33) where Ω, given by eq. (27), is the inverse of the time constant of the LuGre mode and the quantity ν, defined as ν = (V r /v s )/(ˆv min ), measures the smaness of the reative veocity. The existence of a time constant, 1/Ω, associated with the friction process as described by the LuGre mode, enabe a rationa time adaptivity strategy. The foowing vaues of the parameters give good resuts for a wide range of probems: τ min = 0.02, ˆv min = 5, and α = 1.2. Here again, the success of the present approach hinges upon two features of the mode: the physics based mode of the friction force between the contacting bodies and time adaptivity. The discontinuous friction force impied by Couomb s aw is repaced by a force with steep time gradients that are then resoved using time adaptivity. This contrasts with event driven computationa strategies that first invove the determination of transition times (from stic to sip or sip to stic), then different sets of governing equations are used depending on the specific friction regime. The compementarity principe can aso be used to formuate friction probems [2]. When deaing with systems modeed with finite eement techniques, the approach proposed here seems to be more effective because it invoves a singe set of governing equations for a friction regimes and furthermore, the unconditiona stabiity of the integration process can be guaranteed based on energy arguments. 9

10 6 Numerica Exampes Two exampes wi be studied in this section. The same contact and friction modes were used for both exampes. A inear spring of stiffness constant = 15 MN/m was used for the contact mode. The parameters for the time adaptivity agorithm, eq. (20), are: ε p = m, ˆq min = 5, and α = 1.2. The LuGre mode was used to mode the friction phenomena with the foowing parameters: σ 0 = 10 5 m 1, σ 1 = σ 2 = 0 s/m, v s = 10 3 m/s, µ = µ s = 0.30, and γ = The Spatia Mechanism The spatia mechanism depicted in fig. 5 consists of cran of ength L a = 0.2 m connected to the ground at point S by means of a revoute joint that aows rotation about an axis parae to ī 1. The motion of the cran is prescribed as θ = Ωt, where Ω = 20 rad/s. At point P, the cran connects to a fexibe in of ength L b = 1 m through a universa joint that aows rotations about axes ī 2 and ī 3. The other end of the in attaches to a spherica joint at point Q. In turns, this joint connects a to prismatic joint of mass m Q = 5 g that aows reative dispacements aong axis ī 1. Finay, this prismatic joint is attached to a fexibe beam, cantievered at point O. The physica properties of the fexibe beam are: bending stiffnesses, I 22 = I 33 = 23 N m 2, torsiona stiffness, GJ = 18 N m 2, and mass per unit span, m = 1.6 g/m; those of the in are: bending stiffnesses, I 22 = I 33 = 12 N m 2, torsiona stiffness, GJ = 9 N m 2, and mass per unit span, m = 0.85 g/m; finay, the sectiona properties of the cran are: bending stiffnesses, I 22 = 23.2, I 33 = 29.8 N m 2, torsiona stiffness, GJ = 28 N m 2, and mass per unit span, m = 1.6 g/m. Two cases wi be contrasted in this exampe; for case 1, the spherica joint at point Q is treated as a inematic constraint, whereas in case 2, the same spherica joint features the cearance mode described in section 2.1 with ρ = 50 and ρ = 49.5 mm. For case 1, the simuation was run at a constant time step of t = 10 3 s; for case 2, the time adaptivity agorithms were used. Simuations were run for a tota of six revoutions of the cran to obtain a periodic soution, and resuts wi be presented for the fifth revoution of the cran. The reative tangentia veocity at the cearance joint is shown in fig. 6 for case 2. Note the severa occurrences of neary vanishing reative veocities at cran anguar positions from 150 to 250 degrees. The resuting frictiona force is shown in fig. 7. Note that at cran ange of about 275 degrees, contact in the joint is ost and the frictiona force vanishes. This intermittent contact behavior, couped with the eastic response of the system, creates rapid variations in the norma contact force that are refected in the friction force. Fig. 8 shows the time history of the root forces in the beam at point O. Whereas the overa responses for cases 1 and 2 are quaitativey simiar, it is cear that the strong variations in both norma and tangentia contact forces excite the eastic modes of the system, resuting in fu couping between the dynamic response of the system and the behavior of the contact forces. The same comments can be made concerning the in mid-span forces shown in fig. 9; the osciatory component of the stresses woud strongy impact the fatigue ife of these structura components. Finay, the impication of the varying reative veocity on the time step size used in the simuation is evident in fig. 10; ceary, for this probem, the time step size for the simuation is driven by the friction mode. The parameters associated with the time step size contro agorithm for friction, eq. (33), are: τ min = , ˆv min = 5, and α = The Supercritica Rotor The simpe rotor system depicted in fig. 11 features a fexibe shaft of ength L s = 6 m with a mid-span rigid dis of mass m d = 5 g and radius R d = 15 mm. The shaft is a thin-waed, circuar tube of mean radius R m = 50 mm and thicness t = 5 mm; its sectiona properties are: bending stiffnesses, I 22 = 28.2 and I 33 = 28.7 N m 2, torsiona stiffness, GJ = 22.1 N m 2, and mass per unit span, m = g/m. The center of mass of the shaft is ocated at a 1 mm offset from its geometric center. At point R, the shaft is connected to the ground by means of a revoute joint; at point T, it is supported by a spatia cearance joint. The radius of the cyinder is ρ = 80.8 mm and that of the dis is ρ = 80 mm. At first, the natura frequencies of the shaft were computed and the first critica speed zone was found to correspond to shaft anguar speeds Ω [44.069, ] rad/s. 10

11 The system is initiay at rest and a torque is appied at point R with the foowing schedue { (1 cos 2πt) t ts Q(t) = Q 0, (1 cos 2πt s ) t > t s where Q 0 = N m and t s = 0.93 s. The time history of the resuting anguar veocity of the shaft is depicted in fig. 12 which aso indicates the unstabe region. After 2 s, the rotor has crossed the unstabe region of operation and stabiizes at a supercritica speed of about 50 rad/sec. The trajectory of the midspan point M is shown in fig. 13. Since the shaft must first cross the unstabe operation zone, the trajectory first spiras away from the axis of rotation of the shaft, as expected. Once the unstabe zone is crossed, the shaft regains equiibrium and due to the friction in the spatia cearance joint, the ampitude of the motion decreases. Since in supercritica operation the shaft is sef centering, a dispacement of about 1 mm (corresponding to the center of mass offset) is expected for point M. The time history of the norma contact force is depicted in fig. 14; arge contact forces are generated as the shaft crosses the unstabe zone, however, whie regaining stabiity, intermittent contact episodes are observed, 2.1 and 2.6 s into the simuation. When contact is restored, arge impact forces are experienced, up to about 100 N, i.e. an order of magnitude arger than those observed during the continuous contact regime. In supercritica operation, the contact force decreases to smaer eves. Simiar behavior is observed in fig. 15 that depicts the bending moments at the root of the shaft. The parameters associated with the time step size contro for friction, eq. (33), were seected as: τ min = 0.1, ˆv min = 500, and α = 1.2. In this exampe, the reative veocity at the spatia cearance joint aways remains much arger than the Stribec veocity, and hence, appication of Couomb s aw woud probaby give satisfactory resuts. On the other hand, the first exampe presents numerous stic-sip transitions: the reative veocity at the joint vanishes numerous times at each revoution of the cran. These two exampes show that the proposed approach to the modeing of friction forces by means of the LuGre mode is capabe of deaing the various regimes of friction. In the case of high reative veocity, the computationa cost associated with the LuGre mode is minimum, because its use has no impact on the required time step size, on the other hand, when ow reative veocities impy stic sip events, the proposed approach sti perform we, athough sma time step sizes are required. 7 Concusions The present paper has proposed an approach to increase the versatiity and accuracy of uniatera contact modes in mutibody systems. Two joint configurations were deveoped, the panar and spatia cearance joints that can dea with typica configurations where contact and cearance are iey to occur. More genera configurations coud be deveoped based on the same principes. The inematic anaysis of the joint yieds two important quantities: the reative distance between the bodies that drives the intermittent contact mode and the reative tangentia veocity that drives the friction mode. An arbitrary contact force-approach reationship can be used for the contact mode. For the friction mode, the use of the LuGre mode was proposed in this wor. This physics based mode is capabe of capturing a number of experimentay observed phenomena associated with friction. From a numerica stand point, it eiminates the discontinuity associated with Couomb s friction aw. Discretizations were proposed for both norma contact and friction forces that impy an energy baance for the former and energy dissipation for the atter. When combined with the energy decaying schemes used in this effort, these properties of the discretizations guarantee the noninear stabiity of the overa numerica process. The numerica simuations rey on time step adaptivity; simpe, yet effective strategies were given to evauate the required time step size when contact and friction are occurring. The efficiency of the proposed approach was demonstrated by reaistic numerica exampes that demonstrate the couping between contact and friction forces and the overa dynamic response of the system. 8 Acnowedgments This wor was supported by the Air Force Office of Scientific Research under Contract # F ; Capt. Car Ared is the contract monitor. 11

12 References [1] S.P. Timosheno and J.M. Gere. Theory of Eastic Stabiity. McGraw-Hi Boo Company, New Yor, [2] F. Pfeiffer and C. Gocer. Muti-Body Dynamics with Uniatera Contacts. John Wiey & Sons, Inc, New Yor, [3] O.A. Bauchau. Computationa schemes for fexibe, noninear muti-body systems. Mutibody System Dynamics, 2(2): , [4] C. Johnson. Numerica Soutions of Partia Differentia Equations by the Finite Eement Method. Cambridge University Press, Cambridge, [5] T.R. Kane. Impusive motions. Journa of Appied Mechanics, 15: , [6] E.J. Haug, R.A. Wehage, and N.C. Barman. Design sensitivity anaysis of panar mechanisms and machine dynamics. ASME Journa of Mechanica Design, 103(3): , Juy [7] Y.A. Khuief and A.A. Shabana. Dynamic anaysis of constrained systems of rigid and fexibe bodies with intermittent motion. ASME Journa of Mechanisms, Transmissions, and Automation in Design, 108:38 44, [8] Y.A. Khuief and A.A. Shabana. A continuous force mode for the impact anaysis of fexibe muti-body systems. Mechanism and Machine Theory, 22: , [9] H.M. Lanarani and P.E. Niravesh. A contact force mode with hysteresis damping for impact anaysis of muti-body systems. Journa of Mechanica Design, 112: , [10] A. Cardona and M. Géradin. Kinematic and dynamic anaysis of mechanisms with cams. Computer Methods in Appied Mechanics and Engineering, 103: , [11] K.H. Hunt and F.R.E. Crossey. Coefficient of restitution interpreted as damping in vibroimpact. Journa of Appied Mechanics, 112: , [12] J.C. Oden and J.A.C. Martins. Modes and computationa methods for dynamic friction phenomena. Computer Methods in Appied Mechanics and Engineering, 52: , [13] T. Baumeister, E.A. Avaone, and T. Baumeister III (eds.). Mars Mechanica Engineers Handboo. McGraw-Hi Boo Company, New-Yor, [14] J.E. Shigey and C.R. Mische. Mechanica Engineering Design. McGraw-Hi Boo Company, New Yor, [15] A.K. Banerjee and T.R. Kane. Modeing and simuation of rotor bearing friction. Journa of Guidance, Contro and Dynamics, 17: , [16] O.A. Bauchau and J. Rodriguez. Modeing of joints with cearance in fexibe mutibody systems. Internationa Journa of Soids and Structures, 39:41 63, [17] W.D. Iwan. On a cass of modes for the yieding behavior of continuous and composite systems. ASME Journa of Appied Mechanics, 89: , [18] A.A. Ferri. Friction damping and isoation systems. ASME Journa of Vibration and Acoustics, 117B: , [19] K.C. Vaanis. A theory of viscopasticity without a yied surface. Archives of Mechanics, 23(4): , [20] P.R. Dah. Soid friction damping of mechanica vibrations. AIAA Journa, 14: ,

13 [21] C. Canudas de Wit, H. Osson, K.J. Astrom, and P. Lischinsy. A new mode for contro of systems with friction. IEEE Transactions on Automatic Contro, 40: , [22] J. Swevers, F. A-Bender, C.G. Ganesman, and T. Prajogo. An integrated friction mode structure with improved presiding behavior for accurate friction compensation. IEEE Transactions on Automatic Contro, 45(4): , [23] V. Lampaert, J. Swevers, and F. A-Bender. Modification of the Leuven integrated friction mode structure. IEEE Transactions on Automatic Contro, 47(4): , [24] D.A. Haessig and B. Friedand. On the modeing and simuation of friction. ASME Journa of Dynamic Systems, Measurement, and Contro, 113: ,

14 List of Figures 1 Configuration of the panar cearance joint Configuration of the spatia cearance joint Reative distance between the candidate contact points Mode of a journa bearing using two spatia cearance joints Configuration of the spatia mechanism Reative tangentia veocity at the spherica joint for case Friction force at the spherica joint for case Beam root forces at point O. Case 1: dashed ine; case 2: soid ine. Force component F 2 :, F 3 : Lin mid-span forces. Case 1: dashed ine; case 2: soid ine Time step size used in the simuation for case Configuration of the supercritica rotor system Time history of the anguar veocity of the shaft. The dashed ines indicate the unstabe zone of operation: Ω [44.069, ] rad/s Trajectory of the shaft mid-span point M. For 0 < t < s: soid ine; for < t < s: dashed ine; for < t < s: dashed-dotted ine; for < t < s: dotted ine Time history of the norma contact force for the spatia cearance joint at point T Time history of root bending moments in the shaft at point R

15 e 20 Reference configuration e 20 e 10 u u, R, R Deformed configuration e 2 e 2 e 1, R u 0 0, R u 0 0 e 10 q Z Z e 1 i 2 n i 1 i 3 Figure 1: Configuration of the panar cearance joint. 15

16 Deformed configuration Reference configuration e20 e e30 r e20 u0, R0 u, R 10 e2 e3 u, R e10 r e1 e2 e1 e30 u0, R0 e3 i3 i2 i1 Figure 2: Configuration of the spatia cearance joint. 16

17 e 2 d 2 d 2 e 2 Z q Z P d 1 d 1 e 1 e 1 Figure 3: Reative distance between the candidate contact points. 17

18 Spatia cearance joints Rigid body Outer race Figure 4: Mode of a journa bearing using two spatia cearance joints. 18

19 Revoute joint h L a S i 3 Cran P Lin Universa joint L b i 2 O Beam Prismatic joint Q T Spherica joint i 1 Figure 5: Configuration of the spatia mechanism. 19

20 1 Reative Veocity [m/sec] Cran Ange θ [deg] Figure 6: Reative tangentia veocity at the spherica joint for case 2. 20

21 Friction Force [N] Cran Ange θ [deg] Figure 7: Friction force at the spherica joint for case 2. 21

22 50 0 Forces [N] Cran Ange θ [deg] Figure 8: Beam root forces at point O. Case 1: dashed ine; case 2: soid ine. Force component F 2 :, F 3 : 22

23 Forces [N] Cran Ange θ [deg] Figure 9: Lin mid-span forces. Case 1: dashed ine; case 2: soid ine. 23