Analytical Model Development and Model Reduction for Electromechanical

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1 Copyright © 2004 by ASME ANALYTICAL MODEL DEVELOPMENT AND MODEL REDUCTION FOR ELECTROMECHANICAL BRAKE SYSTEM Jaeho Kwak, Bin Yao*, and Anil Bajaj School of Mechanical Engineering Purdue University West Lafayette, IN, 47907, USA *Email: byao@purdue.edu ABSTRACT This paper presents a detailed analytical model development which can describe the dynamic behavior of the electromechanical brake-by-wire (BBW) system over the entire operating range. The complete model has 10 degree-of-freedom (DOF) and includes essential nonlinearities such as gear backlashes, Coulomb frictions, and disk gap clearance. Such a full model is reduced to 6 degree-of-freedom model in SIMULINK for simulation study of the effectiveness and the achievable performance of different hardware and controller designs, an invaluable tool in the early design stage of a product.

Simulation results show that the model is able to reproduce various nonlinear characteristics including typical structural hysteresis as shown in real brake assembly. The linearized version of the full nonlinear model is then obtained for its modal properties to understand the modes that are critical to the low frequency dynamics of the overall system. The results of the modal analyses are subsequently utilized to obtain two simplified models, one for non-contact mode and the other for the contact mode ... more. less.

of operation. The concepts of two simplified models well capture the dynamic characteristics of the system over the frequencies of interest and are being used in the controller (e.g., clamping force control) and estimator (e.g., gap clearance estimation) designs that are under investigation. 1. INTRODUCTION With regards to future vehicle concepts, the automotive industry is developing a fully electromechanical brake-by-wire (BBW) system as it allows reduced number of components, improvement in response times, and possibility of adding safety features such as anti-skidding through coordinated individual control of each brake. Especially, integration of ABS, Brake Assist, Traction, and Stability control becomes largely a matter of computer code, not of complex layers of addition hardware. However, the BBW system demands more precise control than a conventional electrohydraulic brake system such as clamping force regulation to generate consistent braking force and disk gap management to avoid residual brake torque and pad wear. As a result, rigorous mathematical modeling plays a significant role in the development of the BBW system consisting of brushless DC motor, planetary gear train, roller screw drive, and floating brake caliper. Unfortunately, such a rigorous mathematical modeling has been lacking in the previous studies of BBW systems [1, 2, 5-7]. The aim of this paper is to develop in early design stage a reference model with relevant degrees of freedom. This model must describe the dynamic behavior of electromechanical brake components for the entire instrument range. Also, since there is a wide range of efficiency on the brake actuator due to changing environmental conditions (thermal, friction, and pad wear) and the nonlinearities (Coulomb friction, gear backlash, and brake gap clearance) of its components, the each wheel brake need to be operated in a clamping force feedback control and in a disk gap management [1, 3-5]. So, simplified models for future controller design and parameter estimator design are necessary. The rest of this paper is organized as follows: Section 2 derives the complete nonlinear analytical model for an electromechanical brake system. From the developed full degrees of freedom model, natural frequencies and vibration modes are compared to capture the dominant frequency mode for the simplified model developed in Section 3. Then two simplified models for clamping mode and gapping mode defined by contacting condition (braking status) are developed in Section 4. In Section 5 the simulation results for the characteristics of complete nonlinear model and the validity of two simplified models are presented. Proceedings of IMECE04 2004 ASME International Mechanical Engineering Congress and Exposition November 13-20, 2004, Anaheim, California USA IMECE2004-61955 DSC TOC 2 Copyright © 2004 by ASME 2. ANALYTICAL MODEL DEVELOPMENT For the dynamic model development of electromechanical brake (Brake-by-wire) on a floating disk brake caliper, the structure [1, 2] consisting of brushless DC motor, planetary gear unit, roller screw drive and disk brake caliper is selected among various brake types and components [1-7]. This type of brake is made structurally stiff by fixing the nut carrier to the caliper housing through the use of the central bearing and by choosing roller screw drive instead of ball screw drive as shown in Fig. 1. The aim of the model development is to provide a reference model which describes the behavior of the real brake for the entire instrument range. This reference model serves in initial design studies for feasibility and for aid in design. Also, it should possess sufficient accuracy to serve as a realistic nonlinear simulation test bed for design of a clamping force estimator and design of a controller. 2.1 Structure of the Benchmarked Brake [1] Figure 1 . Layout for Electromechanical Disk Brake The electromechanical brake is implemented based on the conventional floating caliper principle as shown in Fig. 1. With this principle, the anchor holder is firmly installed on the knuckle of the vehicle. The caliper housing with both pads floats through sliding pin with two degree-of-freedom in the interlocking of the clamping force. As an electromechanical converter, a brushless DC motor is used. The angular position of motor for the electrical commutation and for the regulation function is measured via an encoder. The cylinder shaped motor rotor is interlocked within the caliper housing and it is rigidly connected with the sun gear of the planetary gear unit. So, the moment of inertia of the rotor and the sun gear will be lumped together in the following modeling. The planet gear of the planetary gear unit is in mesh with the sun gear and with the internal ring gear that is bolted in the brake caliper housing and propels the nut carrier. To convert the rotation of the planetary gear into a linear movement for the necessary clamping force of the brake caliper assembly, a roller screw drive is used. In the operation of the brake, the actuator-side brake pad is shifted by the spindle of the roller screw drive in the moving direction of the disk brake. As soon as the actuator-side pad comes in contact with the brake disk rotor, the entire brake housing floats until the required clamping force is generated. Here, the central bearing of the nut carrier supports a radial force as well as a axial force to react against the clamping force. So the large axial clamping force is immediately absorbed by the nut-screw mechanism without transferring down to the motor and gear unit. A combination of ball bearings and roller bearings makes possible the relative rotation between the motor rotor and the screw nut, as shown in Fig. 1 and Fig. 2. For the further radial support of the drive unit, a roller bearing is intended between the encoder rotor and the encoder stator. 2.2 Dynamic Model 2.2.1 Planetary Gear Train (PGT) To meet the required power and high torque needs within space constraints, planetary gear trains have been widely used in aerospace and automotive applications. They help amplify the force/torque that can be produced by an electronic motor. In addition to being compact in size and physical construction, high efficiency and low sensitivity to shock loading also make the planetary gear train (PGT) attractive for electromechanical brakes. However, the dynamic behavior of a PGT is very complex due to the nonlinear effects such as backlash, the variation of gear mesh stiffness and damping, and various frictions as well as the complex motion of the planetary gear which generates revolution and rotation movement at the same time. For example, the modeling of dynamic tooth forces still remains as important issue that has not been resolved even for single-mesh PGT. Dynamic analysis including the prediction of natural modes and gear mesh forces has been one of the major research area for planetary gear trains [8-13]. For simplified PGT dynamic analysis in most researches, however, an equivalent conventional gear train concept or a carrier fixed PGT is used to remove the relative motion caused by nut carrier rotation [9, 10], or a purely torsional model is used ignoring the bearing stiffness between planetary gear and nut carrier pin [12, 13]. This section details the dynamic model development a PGT including the movement of relative motion and the bearing stiffness aimed at exploring the dynamic behavior of power transmissions. Formulation In the electromechanical brake, the single stage PGT is considered. It consists of a sun gear connected with motor rotor, an internal ring gear fixed with caliper housing, and 3 identical planetary gears coupling the sun gear with the internal ring gear, as shown in Fig. 2. The planetary gears are mounted on the rigid nut carrier through bearings and pins. In the present usage of the PGT, the sun gear is the input and the nut carrier thrusting the spindle is the output of the PGT unit. In the technical model development, the planetary gear dynamics can be simplified for the purpose of this paper. However, the contacting dynamics (the stiffness of bearing or gear mesh) lying in the power train cannot be neglected in relation to the stiffness of the other brake components during the development stage of the electromechanical brake system. For this reason, in the following, the PGT unit will be modeled as an eight degree- of-freedom (DOF) system without translational spindle Sun Gea r Nut Carrie r J s J n m b J p m sp z b ¸ n ¸ s ¸ p z s Brake Calipe r Roto r Bearings Calipe r Side Pad Spindle Slider Pin Actuator Side Pad Motor Rotor Planetary Rolle r Internal Ring Gear Planetary Gear Central Bearing Brake Disk Anchor Holde r 3 Copyright © 2004 by ASME movement as shown in Fig. 2. The following assumptions are also made in developing the dynamic model of the PGT: 1) The gear wheels and the nut carrier are assumed to be rigid bodies. 2) The gear flexibilities are modeled as averaged linear gear mesh springs and dampers acting on the plane of the motion. 3) XY-axis is a fixed coordinate, i.e., the lateral movement of the caliper will be ignored. 4) x i y i -axis is the moving coordinate system attached to the center of the planetary gear i (i=1~3) and moving with it. Figure 2 . PGT Model for Rotational Dynamics Basic Equation of Motion for PGT The angular displacements of central members s ¸ , ci ¸ (i=1~3), and n ¸ are defined as absolute displacements about a fixed coordinate frame while the angular displacements of planets pi ¸ are defined relative to the carrier with positive direction along the negative z i -axis of the moving coordinate systems x i y i z i which are related to the absolute angular displacement of planetary gear pci ¸ as pcicipi ¸ ¸¸ = 2 or picipci ¸ ¸¸ = 2 (1) To derive the basic equations of motion for PGT only, the Lagrange 9s method neglecting damping is employed. Thus 33 2222 11 333 2222 111 1 [()()] 2 1 [()(())()] 2 sspicipipincinn ii spissscipipipripipinpicipnincin iii LJJmrJ krrrkrrrkr ¸¸¸¸¸ ¸¸¸¸¸¸¸ == === =+ 2++ 2 2 2+ 2++ 2 1 1 1 1 1 (2) The Lagrange 9s equations of motion for PGT only are given as - j jj dLL Q dt ¸¸ ?? \x2\x2 = ?? ?? \x2\x2 (3) where ,1,1,2,2,3,3, j spcpcpcn = , and only the generalized forces s m QT = (motor torque) and nnL QT = 2 (nut load to next step) exist. Eq. (3) in conjunction with Eq. (2) yields eight ordinary differential equations given below: (i=1~3) 333 22 111 s ssspisspispipisspicim iii J rkrrkrkT ¸¸¸¸ === + 2 2= 1 1 1 (4) 2 ()() (())0 pipicispispisspipripipi spispipripinpici Jkrrkkr krrkrrr ¸¸¸¸ ¸ 2 2++ + 2+= (5) 22 2222 ()(()) +(())0 pipincipipispissspispipripinpipi spispninprinpicipninn JmrJkrkrrkrrr krkrkrrkr ¸¸¸¸ ¸¸ + 2 2+ 2+ + ++ 2= (6) 33 22 11 nnnpnicinpninnL ii J rkrkT ¸¸¸ == 2 += 2 1 1 (7) In this section, the damping term and nonlinear terms such as Coulomb friction and backlash are not presented for the simple expression. Complete nonlinear models including the terms are derived in Appendix [A]. 2.2.2 Conversion by Roller Screw Drive To convert a rotatational motion into a linear movement, a set of mechanical components are available with different working principles. For electromechanical brakes, a planetary roller screw drive with reduction is particularly suitable due to its high load-carrying capacity, its small building area, its small pitch and high rotating speed. The main components are the nut carrier, the spindle to lead thread rolling, and rollers. With a rotatational motion of the nut, the rollers move in radial and axial direction. Each roller is led back after a circulation. The planetary roller screw drive can be simplified as a two-mass model ignoring the inertia of the roller between nut carrier and spindle as shown in Fig. 1 and Fig. 2. On the assumption of a rigid planet nut carrier for the brake modeling, the rotation angle of the screw nut carrier equals to the rotation angle of the planet nut carrier, and the planet nut carrier load torque is related to the equivalent driving axial force actuating the spindle by 2 nLns p TF À ?? = ?? (8) where p is the pitch of the roller screw, and ns F is the equivalent axial force to actuate the spindle in the actuation direction of the disk brake. With the contacting stiffness ns K and the contacting damping ns C , the axial force ns F can be calculated from the rotation angle of the nut carrier n ¸ , the spindle axial displacement s z , the brake caliper displacement b z and their derivatives as described in Eq. (9). r y 2 ¸ s x 2 ¸ n y 1 J n r p J s r s X x 1 Y J p2 r r T m 1 pn k 1 p r k 1 s p k x 3 J p3 J p1 y 3 2 s p k 2 p r k 2 pn k 3 p n k 3 s p k 3 p r k ¸ c2 ¸ p2 ¸ p1 ¸ c1 ¸ p3 T nL ¸ c3 r p 4 Copyright © 2004 by ASME Figure 3. Friction Assembly for Translational Movement 2.2.3 Floating Disk Brake Assembly The floating saddle friction assembly is modeled as a two- degree-of-freedom lumped mass-spring-damper system as shown in Fig. 3, in which different model equations and model parameters should be used according to the gaps between the two pads and the rotor. In this paper, the brake caliper dynamics are considered as a 2-phase model because the gap between the caliper side pad and the disk rotor is relatively very small when compared with the gap between the actuator side pad and the brake disk rotor (seeFig.1 and Fig. 3). Also, the following conditions are assumed in the theoretical model development: 1) Brake disk rotor is assumed to be a fixed (rigid) body because its stiffness is very high in relation to the stiffness of the other brake components. 2) The axial stiffness of central bearing is ignored in relation to the stiffness of the other brake components during clamping mode. 3) The brake caliper is assumed as a lumped mass b m attached to the fixed disk rotor through a spring with a lumped stiffness of c K and a lumped damping c C . Equation of Motion for Spindle Dynamics ( s zBG e ) From Fig. 3, the spindle dynamics is presented as 1 ()0 spssbsbspns mzCzzFF + 2+ 2= (9) where 111 ()() s pspssps F KzBGCz = 2+ ()() 22 nsnsnbsnsnbs pp FKzzCzz ¸¸ ÀÀ =+ 2++ 2 Equation of Motion for Brake Caliper ( s zBG e ) The brake caliper mass is actuated by the transmission forces described in Fig. 3. Then, ()0 bbsbbscbcbbtns mzCzzKzCzFF + 2++ 2+= (10) The caliper friction force bt F is mainly due to the lateral force between the caliper and sliding pin caused by the brake friction force between the rotor and the brake pad. As such, it consists of a static component and a component that is proportional to the total clamping force, and is assumed to be modeled by () 01 sgn() btbbtbcsp FzFF µµ = 2+ÅÅ (11) where 0 bt F is the static friction between the caliper and slider pin, b µ is the friction coefficient between the pad and the disk rotor, and c µ is the friction coefficient between the caliper and the sliding pin. The complete analytical models including damping, Coulomb friction, backlash, and brake gap clearance are derived in Appendix [A] and implemented in SIMULINK as shown in Fig. 11. Their complete nomenclature and values are listed in Appendix [B]. 3. NATURAL FREQUENCY AND VIBRATION MODE In this section, several modal analyses are developed based on natural frequency and vibration mode shapes without nonlinear effects. These analyses have a useful geometric interpretation in early design stage of electromechanical brake system for tuning resonances away from operating speeds and optimizing the structural design. Also, through these analyses, a reduced order planetary gear train (PGT) model will be derived by removing modes of higher frequencies so as to avoid using a model with unnecessary a large number of degrees of freedom. We begin with the determination of the natural frequencies and mode shapes for only the 8-DOF PGT model without translational brake assembly. The 8-DOF PGT model is subsequently reduced into the 4-DOF model of PGT through natural frequency and mode shape analysis. In section 4, this reduced PGT model will be then integrated with the spindle dynamics and the caliper dynamics to form the 6-DOF brake model to investigate the natural frequencies and mode shapes of the entire brake system during the two distinct operating conditions respectively: gapping mode before the pad makes contact with the rotor and clamping mode thereafter. 3.1 Full PGT Model The natural frequencies and mode shapes of the 8 3DOF PGT model when the planet nut carrier is not constrained (i.e. 0 nL T = ), are obtained by solving the free vibration eigenvalue problem from the derived equations, Eq. (4) to Eq. (7) ignoring damping terms. All planetary gears are assumed identical and equally spaced with equal sun-planet mesh stiffness, ring-planet mesh stiffness, and planetary gear-nut bearing stiffness. The obtained natural frequencies and mode shapes given in Fig. 4 reveal several characteristics that can be classified into three groups: a rigid body mode, planet modes, and an overall mode [10] as follows: Rigid Body Mode: So long as the input and output members are not constrained, the stiffness matrix is semi-definite resulting in a crigid body d mode at zero frequency, as is shown in Fig. 3(a). In this mode, all gears rotate as rigid bodies without any gear or bearing mesh deflections. Hence, the X n ¸ BG s z b z c K c C bt F ns C ns K s b C 1 s p C 1 s p K s p m b m Y 5 Copyright © 2004 by ASME following kinematic relationships are obtained when only rigid body mode is considered: cin ¸ ¸ = where (i=1~3) (12) (2) s pici pi pi rr r ¸ ¸ + = (13) (14) From the above equations, the overall gear ratio of the PGT is calculated as 2() spi s sr tot nss rr rr N rr ¸ ¸ + + === (15) Figure 4 . Natural Mode Shapes for Full PGT Model Planet Modes: These are the asymmetric modes at the double mode frequencies w n3 * and w n5 *, as shown in Fig. 4 (e-h), as well as the axi-symmetric modes at the modal frequencies w n4 and w n6 as shown in Fig. 4 (c, d) characterized by motions of planets relative to each other. At these modes, only the planetary gears move with respect to each other such that the input sun gear and the output nut carrier do not move. Also, at the multiple double mode frequencies (asymmetric), the summation of modal amplitude is zero. Over All Mode: Besides the rigid body mode and the planet modes, there is a third mode at the frequency, w n2 as shown in Fig. 4(b). At this mode, all planets move exactly the same way forming an axi-symmetric mode shape to generate the non-zero displacements of the input and output members. This type of mode is the so-called coverall mode d. It is characterized by existence of vibrations of all PGT members in contribution to the overall motion. 3.2 Reduced PGT Model In general, the PGT with equally spaced three identical planetary gears (i.e., 123 ppp J JJ = = , 123 p pp mmm = = ), has both the symmetric modes that contribute to the overall input- output motion and that generate relative internal motion of planets only. Thus, if only the input-output motion of the PGT unit is of concern, as is the case in subsequent study of the overall brake dynamics, the axi-asymmetric double modes can be ignored for simplicity. As a result, all planets have the same motion, i.e., 123 pppp ¸ ¸¸¸ = == , and 123 cccc ¸ ¸¸¸ === , and can be simplified as a lumped planet having inertia and stiffness property model by relations: 3 1 p pi i J J = = 1 and 3 1 p pi i mm = = 1 (16) 3 1 s pspi i kk = = 1 , 3 1 p rpri i kk = = 1 and 3 1 p npni i kk = = 1 (17) Subsequently, the number of DOF of the overall PGT unit reduces to four: ,,,and, s pcn ¸ ¸¸¸ and the governing linear dynamic equations are simplified. By solving the free vibration eigenvalue problem for the reduced PGT, natural frequencies and mode shapes of the reduced 4 3DOF PGT unit are obtained as shown in Fig. 5. Figure 5. Natural Mode Shapes for Reduced PGT Model (a) Rigid Body Mode (w n1 = 0 [Hz]) (b) Over All Mode (w n2 = 2,764 [Hz]) (c) Planet Mode (w n3 = 28,978 [Hz]) (d) Planet Mode (w n4 = 37,868 [Hz]) (a) Rigid Body Mode (w n1 = 0 [Hz]) (b) Over All Mode (w n2 = 2,764 [Hz]) (c) Planet Mode (w n4 = 28,978 [Hz]) (d) Planet Mode (w n6 = 37,868 [Hz]) (e) Planet Mode (w n3 = 28,731* [Hz]) (g) Planet Mode (w n5 = 37,831* [Hz]) (f) Planet Mode (w n3 = 28,731* [Hz]) (h) Planet Mode (w n5 = 37,831* [Hz]) 2() spi s ci s rr r ¸ ¸ + = 6 Copyright © 2004 by ASME The obtained natural frequencies and mode shapes for the reduced PGT model reveal the same characteristics with the full PGT model that can also be classified into three groups: a rigid body mode, planet modes, and an overall mode. As expected, the natural frequencies and mode shapes of the two planet modes of the reduced PGT model are exactly the same as the two axi-symmetric modes of the full PGT model. This reduced 4-DOF PGT model is especially useful to simplify the simulation program. 4. SIMPLIFIED MODEL In section 3 and Appendix [A], several analytical models including nonlinearities were developed which describe the dynamic behavior of an electromechanical disk brake with high accuracy for simulation based analysis and design purposes. For the analytical design of estimators and feedback controllers, it is however not meaningful to use the model in its full complexity. Rather, the model structure and the number of parameters to be dealt with can be further reduced based on the dominant factors analyzed in previous sections. According to status of braking system, simplified models for gapping mode and for clamping mode are respectively developed in this section. The derived simplified clamping mode and gapping mode models are then used to construct estimation algorithms for clamping force and spindle position in respect based on the measurement of motor position via encoder and the motor current. Note that in the ideal case, the contacting position between the disk rotor and the pad is known. In addition, an analytical formula that relates the dominant mode of overall modes is developed. This formula provides estimate to the lowest natural frequency, from which the major parameters influencing the lowest natural frequency in clamping mode is clearly identified. This information is very useful in the early design stage of the prototype brake system. 4.1 Clamping Model During the clamping mode the spindle (actuator side pad) makes contact with the brake rotor, disk as shown in Fig. 1 and Fig. 3. In addition to the reduced PGT model shown in Fig. 5, the translational spindle dynamics, Eq. (9), and caliper dynamics Eq. (10), can be incorporated into a 6-DOF clamping model. For the clamping model, the natural frequencies are obtained by solving the free vibration eigenvalue problem as shown in Table 1. Table 1. Natural Frequencies for Clamping Mode Model w n1 w n2 w n3 w n4 w n5 w n6 Hz 5 1080 2764 12894 28978 37868 In this paper, we only have to consider the lowest flexible mode around 5 [Hz] in the estimator and controller design as the frequencies of all other modes are above 1 [kHz]. The lowest mode is concerned with the translational movement of friction assembly. Therefore, the rotational movement of planetary gear train (PGT) can be lumped together as a rigid body motion. Consequently, when ignoring load friction losses, total equivalent mass beq m reflected to translational brake movement can be obtained as: () 22 222 2 2 2222 p n beqbspnpnps ss r r mmmJmrJJ pprpr ÀÀÀ ???? ?????? =+++++ ???? ?????? (18) The corresponding values of Eq. (18) based on the nominal parameter values of brake assembly are presented in Table 2. As presented in Table 2, the equivalent mass of sun gear which is amplified by the square of gear ratio and pitch ratio is most dominant whereas spindle mass and caliper mass are negligible where compare to total equivalent mass. From Fig. 3, therefore, the translational brake assembly can be assumed as a series combination of springs without mass ignoring friction losses, as is shown in Fig. 6. Also, from Eq. (9), the clamping force 1 s p F when ignoring friction losses, is defined as same as the nut- spindle transfer force ns F : 1 nssp F F E (19) Figure 6 Translational Movement of Brake Assembly containing Massless Junction Table 2. Equivalent Mass Reflected to Translational Brake Assembly Movement Mass reflected to brake movement Value [kg] m b (brake caliper) 6.41 m sp (spindle with pad) 0.54 J n (2 À /p) 2 (nut carrier) 5612.95 m p r n 2 (2 À /p) 2 (translational planetary gear) 420.43 J p (2 À /p) 2 (2r p /r s ) 2 (rotational planetary gear) 20.45 J s (2 À /p) 2 (2r n /r s ) 2 (sun gear) 31360.70 m beq 37421.48 From the equilibrium forces at massless junctions Ë A , Ë B , and Ë C , and rigid PGT assumption, the clamping force can be defined by an equivalent spring constant terms the motor angle: 22 s nseqs n pr FK r ¸ À ?? ?? = ?? ?? (20) where . 2 nb p z ¸ À + ns F s z b z Ë A 1 s p K ns K c K Ë B Ë C 1 11 spnsc eq s pnsnscspc KKK K K KKKKK = ++ 7 Copyright © 2004 by ASME So, the lowest mode can be obtained analytically by Eq. (18) and Eq. (20) to have the natural frequency. 1 4.95[] eq n beq K wHz m == (21) Since this frequency is close to the lowest modal frequency of Table 1, the assumption of massless series spring as shown in Fig. 6 is reasonable. Therefore, a simplified clamping model reflected to the sun gear (motor angle) can be obtained by the rigid body assumption of PGT in rotational movement and the series spring assumption of friction assembly in translational movement. Thus, 2 2 0 sgn() 22 s ceqsmceqssceqeqs n pr JTCMK r ¸ ¸¸¸ À ?? ?? = 2 2 2 ?? ?? i (22) 4.2 Gapping Model The situation when the spindle (actuator side pad) makes no contact with the brake disk rotor, as shown in Fig. 1 and Fig. 3, is referred to as the gapping mode in this paper. This equation governing for system behavior during the gapping mode can be obtained by setting 111 0,0,and0 spspsp KCF === in Eq. (9). The natural frequencies for gapping mode operation are obtained by solving free vibration eigenvalue problem for the no constrained system. These frequencies are given as shown in Table 3. Table 3. Natural Frequencies for Gapping Mode Model w n1 w n2 w n3 w n4 w n5 w n6 Hz 0 395 2764 12375 28978 37868 This analysis reveals that aside from the rigid body mode, the lowest frequency of the flexible modes is around 400 [Hz], which is much above the frequencies that are usually dealt with in any estimator and controller design. As such, we will only consider the rigid body mode for the gapping mode (i.e., assume all other frequencies to be much higher). Therefore, the kinematic relationships Eq. (12)-(14) and the dynamic equation (22) for the rigid PGT still remain valid. Furthermore, the rigid contact assumption between the brake caliper and the brake disk rotor indicates that 0 bbb zzz === , and the rigid nut- screw transmission leads to 22 s s s n pr z r ¸ À ?? ?? = ?? ?? (23) As the inertia effect and viscous damping force of the spindle are negligible, from Eq. (9), 1 0 nssp FF H= , which leads to the following gapping mode model from Eq. (22): 0 sgn() geqsmgeqssgeq JTCM ¸¸¸ = 2 2 i (24) Neglecting the gear friction losses, the corresponding values for each of the terms in Eq. (24) for the nominal parameter values can be obtained by using the expression in Eq. (18). As with the results in Table 2, the inertia of sun gear s J is the most dominant due to the square of gear ratio and pitch ratio. Thus we have the assumption: g eqceq J J E (25) A similar relationship for friction losses are obtained by geqceq CC E and (26) Therefore, the simplified gapping model of Eq. (24) can be seen to be identical to the simplified clamping model of Eq. (22) except for the translational massless brake movement which generates clamping force. Note that Eq. (26) is only valid when the load dependent friction losses are ignored. 5. SIMULATION for MODEL VALIATION In this section, simulation results for nonlinear and linear models are presented. The nonlinear simulation results show that the model is able to reproduce various nonlinear characteristics such as gap clearance, Coulomb friction, and hysteresis curve. Then, the clamping force and the spindle position are estimated based on the linearized models assuming that the gap clearance and total stiffness are known. 5.1 Nonlinear Simulation for Full Model For the nonlinear simulation, a step type of open loop motor voltage input is given as 20 [V] to generate and release a typical clamping force, as shown in Fig. 7. Corresponding to a positive voltage input, the spindle position is increased until it contacts the rotor disk. Then, the clamping force also increases following an initial delay to overcome 0.5 [mm] clearance gap between pad and disk rotor. Note that the rate of change in spindle position is decreased due to the clamping load. Corresponding to the subsequent negative voltage input, the spindle position and clamping forces are released. However, the spindle position does not return to its original position due to the retracting force and various nonlinear characteristics of brake assembly. This is one of reasons why the clearance gap management is required with the clamping force feedback control in the future. Fig. 8 shows a hysteresis curve existing between motor angular position and clamping force which is common in brake assembly due to the Coulomb friction losses. This hysteresis is generated after overcoming the clearance gap. An additional hysteresis curve (dotted line) is presented assuming that the brake pad is exposed to the 200% pad stiffness variation through pad wear and thermal exchange [1]. 00 g eqceq MM E 8 Copyright © 2004 by ASME 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0 0.5 1 1.5 2 2.5 x 10 4 Time [sec] Nonlinear Simulation Voltage Input [V]E+2 Spindle Position [m]E+7 Clamping Force [N] Figure 7. Simulation Results for Nonlinear Full Model 0 2 4 6 8 10 12 14 -0.5 0 0.5 1 1.5 2 2.5 x 10 4 Motor Angular Position [rad] Clamping Force [N] Hysteresis Curve Nominal Pad Stiffness 200% Pad Stiffness Figure 8. Hysteresis Curve between Motor Angular Position and Clamping Force. 5.2 Linear Simulation for Simplified Models Based on the simplified models derived in section 4, the clamping force and spindle position can be estimated assuming that the gap clearance and total stiffness are known. To verify the accuracy of Eq. (20) for clamping force estimation, square wave type of open loop motor voltage inputs with 1 [Hz], 5 [Hz], and 10 [Hz] are given to the actuator. Fig. 9 shows that all estimated clamping forces follow the measured clamping forces with small absolute error. Here, by cmeasured d we mean the results of simulations from the complete (8 DOF) model for system. This means that the assumption of a series connected brake spring and rigid PGT is reasonable for the estimation of the clamping force. However, the relative error is increased on small clamping force actuation due to the ignored stiffness and dynamics. Note also that the clamping force is estimated here without gap clearance (BG=0) between the actuator side pad and the rotor disk. By Eq. (23), the spindle position can be obtained from the motor position. However, this relationship is only valid while the spindle makes no contact with the disk rotor. Open loop motor voltage inputs for several frequencies (square types in 1 [Hz], 5 [Hz], and 10 [Hz]) are given to generate spindle position without contacting. As shown in Fig. 10, all cases have similar absolute estimation errors. The estimation errors are mainly caused by the neglected backlash of the gear and nut- screw transmissions. Because of this, the estimation error does not decrease when the command spindle position becomes smaller. Overall, the estimation errors for all test cases are within the same level of tolerance as the assumed backlashes, indicating the practical possibility of estimating the spindle position based on the motor angle measurement. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 x 10 4 Clamping Force Force [N] Time [sec] Measured Estimated Figure 9. Clamping Force Estimation for Open Loop Input 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 3 4 5 x 10 -4 Spindle Position Time [sec] Position [m] Measure Estimated Figure 10. Spindle Position Estimation for Open Loop Input 6. CONCLUSION A detailed analytical model for electromechnical brake is developed in early design stage, including essential nonlinear characteristics such as backlash, Coulomb friction and gap clearance. The linearized version of the full nonlinear model is then obtained for its modal properties to understand the modes that are critical to the low frequency dynamics of the overall system. The results of the modal analyses are subsequently utilized to reduce the order of PGT model for dynamic simulations and to obtain two simplified models, one for non- contact mode and the other for the contact mode of the 9 Copyright © 2004 by ASME operation. The main difference between the two models is characterized by clamping force load. The derived two simplified models well capture the spindle position and clamping force of the system over the frequency range of interest. These model are now being used in the controller (e.g., clamping force control) and estimator (e.g., gap clearance estimation) designs that are under investigation. ACKNOWLEDGEMENT The authors gratefully acknowledge Christopher S. Keeney, Senior Principal Engineer; Dennis A. Kramer, Senior Program Manager; and John Grace, Vice President of Light Vehicle System with ArvinMeritor in Troy, MI for their financial and technical support of this research. REFERENCE [1] R. Schwarz, cReconstruction of the Braking Force in Vehicles with Electromechanically Operated Brakes, d Technical Report VDI, No. 393, Technical University of Darmstadt, Germany, 1999 (German). [2] R. Schwarz, R. Iserman, J. Bohm, J. Nell, and P. Rieth, cModeling and Control of an Electromechanical Disk Brake, d SAE 980600, Detroit, 1998. [3] C. Maron, T. Dieckmann, S. Hauck, and H. Prinzler, cElectromechnical Brake System: Actuator Control Development System, d SAE 970814, Detroit, 1997. [4] E.G.M.Holweg, R.L.Klomp, J.B.Klaassens, and E.A.Lomonova, cModeling and Inverse Model-Based Control of an Electro-Mechanical Brake Actuator, d 1st IFAC Conference on Mechatronic Systems, Darmstadt, Germany, September 18-20, Volume I, pp39-44, 2000. [5] W. D. Jonner, H. Winner, L. Dreilich, and E. Schunck, cElectrohydraulic Brake System-The Approach to Brake- by-Wire Technology, d SAE960991, Detroit, 1996. [6] D. E. Schenk, R. L. Wells, and J. E. Miller, cIntelligent Braking for Current and Future Vehicles, d SAE950762. [7] B. Breuer, M. Barz, K. Bill, S. Gruber, M. Semsch, T. Strothjohann, and C. Xie, cThe Mechanical Vehicle Corner of Darmstadt University of Technology-Interaction and Cooperation of Sensor Tire, New Low-Energy Disk Brake and Smart Wheel Suspension, d International Journal of Automotive Technology, Vol. 3, No. 2, pp63-70, 2002. [8] R. Kasuba and R. August, cGear Mesh Stiffness and Load Sharing in Planetary Gearing, d ASME, 84DET229, pp348- 353. [9] R. August, R. Kasuba, J. L. Frater, and A. Pintz, cDynamics of Planetary Gear Train, d NASA CR 3793, June 1984. [10] A. Kahraman, cPlanetary Gear Train Dynamics, d ASME Journal of Mechanical Design, Vol. 116, pp713-720, Sep. 1994. [11] Kahraman, cNatural Modes of Planetary Gear Trains, d Journal of Sound and Vibration, Vol. 173, pp125-130, 1994. [12] A. Kahraman, cFree Torsional Vibration Characteristics of Compound Planetary Gear Sets, d Mechanism and Machine Theory, Volume 36, Issue 8, pp953-971, August 2001. [13] J. Lin and R. G. Parker, cAnalytical Characterization of the Unique Properties of Planetary Gear Free Vibration, d ASME Journal of Mechanical Design, Vol. 121, pp316- 321, July 1999. APPENDIX [A] Complete BBW Model Sun Gear (Motor Rotor) ( ) ( ) 0112 12 () sssssnsnsssgsspspsspsp s spspsspspm JCCsgnMMrkfrcf rkfrcfT ¸¸¸¸¸ ¡¤ ++ 2+++ £¦ + += Planetary Gear (Rotation) () 0112 1212 ()() 0 ppcpppppgpprprpprpr pprprpprprpspsppspsp JCsgnMMrkfrcf rkfrcfrkfrcf ¸¸¸¸ ¡¤ 2++++ £¦ + + 2 2= Planetary Gear (Revolution) ( ) 2 120112 121 ()()[] ()() ppncppnpnpnnpnpnpppppgpprprpprpr sspspsspspnpprprnppr JmrJrkfrcfCsgnMMrkfrcf rkfrcfrrkfrrc ¸¸¸¸ + 2++ 2 2++ 2 2 2+ 2+ 2 0 pr f = Nut Carrier ( ) 0112 1212 ()() ()0 2 nnnnsnnsnnnnsnsnsns nsnsnsnsnpnpnnpnpn JCCsgnMMKfCf p KfCfrkfrcf ¸¸¸¸¸ À ¡¤ ++ 2+++ £¦ ?? + + 2 2= ?? Spindle 111212 ()0 spssbsbspspspspnsnsnsns mzCzzKffCffKfCf + 2++ 2 2= Brake Caliper () 01 ()sgn()0 bbsbbscbcbbbtbcspns mzCzzKzCzzFFF µµ + 2+++++= 1 2 0 () 2 () () 22 () () 22 0 nbsns nsnbsnsnbsns nbsnsnbsns ns p forzzB pp fzzBforzzB pp z zBforzzB f ¸ À ¸¸ ÀÀ ¸¸ ÀÀ § + 2< ª ª ª =+ 2 2+ 2e ¨ ª ª + 2++ 2d 2 ª © = () 2 () () 22 nbsns nbsnbsns p forzzB pp z zforzzB ¸ À ¸¸ ÀÀ § + 2< ª ª ¨ ª + 2+ 2e ª © 1 2 0 () () () () () 0 s sscppsp s pssscppspssscppsp s sscppspssscppsp sp f orrrrB f rrrBforrrrB rrrBforrrrB f ¸¸¸ ¸¸¸¸¸¸ ¸¸¸¸¸¸ § 2 2< ª = 2 2 2 2 2e ¨ ª 2 2+ 2 2d 2 © = () () () s sscppsp s sscppssscppsp f orrrrB rrrforrrrB ¸¸¸ ¸¸¸¸¸¸ § 2 2< ª ¨ 2 2 2 2e ª © 1 2 0 (()) (()) (()) (()) (()) 0 ppnpcpr prppnpcprppnpcpr ppnpcprppnpcpr pr f orrrrB frrrBforrrrB rrrBforrrrB f ¸¸ ¸¸¸¸ ¸¸¸¸ § 2+< ª = 2+ 2 2+e ¨ ª 2++ 2+d 2 © = (()) (()) (()) ppnpcpr ppnpcppnpcpr f orrrrB rrrforrrrB ¸¸ ¸¸¸¸ § 2+< ª ¨ 2+ 2+e ª © 1 2 0 () () () () () 0 () () ( ncnpn pnncnpnncnpn ncnpnncnpn ncnpn pn ncnnc f orrB frBforrB rBforrB f orrB f rforr ¸¸ ¸¸¸¸ ¸¸¸¸ ¸¸ ¸¸¸¸ § 2< ª = 2 2 2e ¨ ª 2+ 2d 2 © 2< = 2 2 ) npn B § ¨ e © { { 1 2 0 0 s sp ss s sp ss f orzBG ff zBGforzBG f orzBG ff zforzBG < = 2e < = e 111 2 nnbnng M MrM p À ?? =+ ?? 10 Copyright © 2004 by ASME APPENDIX [B] Nomenclature and Parameters Inertia and Mass (i=1~3) Value Rotor Inertia (Sun Gear) J s [kgm 2 ] 2.112E-4 Planetary Gear Inertia (J pi = J p /3) J p [kgm 2 ] 1.410E-6 Planetary Gear Mass (m pi = m p /3) m p [kg] 2.340E-2 Nut Inertia J n [kgm 2 ] 3.199E-4 Spindle Mass with Pad m sp [kg] 5.380E-1 Brake Caliper Mass m b [kg] 6.410 Contact & Structural Stiffness (i=1~3) Value Sun-Planetary Gear (k spi = k sp /3) k sp [N/m] 3.000E+8 Planetary-Ring Gear (k pri = k pr /3) k pr [N/m] 4.500E+8 Planetary Gear-Nut (k pni = k pn /3) k pn [N/m] 9.000E+7 Roller Screw Stiffness K ns [N/m] 3.000E+9 Pad Stiffness (per piece) K p [N/m] 3.000E+8 Caliper Stiffness K c [N/m] 4.286E+7 Backlash and Gap Value Sun-Planetary B sp [m] 0.0001 Planetary-Ring B pr [m] 0.0001 Planetary-Nut B pn [m] 0.0001 Nut-Screw B ns [m] 0.0001 Disk Gap BG [m] 0.0005 Radius (i=1~3) Value Sun Gear r s [m] 0.022 Planetary Gear r pi [m] 0.010 Nut Carrier r n [m] 0.032 Bearing friction C s [Nms/rad] 0.001 Bearing friction between sun gear and nut C sn [Nms/rad] 0.001 Static friction M s0 [Nm] 0.100 Sun Load friction M s1 [-] 0.010 Bearing friction C p [Nms/rad] 0.001 Static friction M p0 [Nm] 0.100 Planetary Load friction M p1 [-] 0.010 Bearing friction C n [Nms/rad] 0.002 Static friction M n0 [Nm] 0.200 Nut Load friction M n1 [-] 0.020 Contact & Structural Damping (i=1~3) Value Sun-Planetary Gear(c spi = c sp /3) c sp [Ns/m] 204.57 Planetary-Ring Gear (c pri = c pr /3) c pri [Ns/m] 204.57 Planetary Gear-Nut(c pni = c pn /3) c pn [Ns/m] 111.84 Nut-Roller Screw C ns [Ns/m] 1270 Pad (per piece) C sp [Ns/m] 3000 Spindle and Caliper C sb [Ns/m] 7000 Brake Caliper Damping C c [Ns/m] 3500 Fi g ure 11 . SIMULINK Block for Com p lete Electromechanical Brake S y ste m

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