Finite element modal analysis of an HDD considering the flexibility of spinning disk spindle, head suspension actuator and supporting structure

Transcription

1 DOI /s y TECHNICAL PAPER Finite element modal analysis of an HDD considering the flexibility of spinning disk spindle, head suspension actuator and supporting structure G. H. Jang Æ C. H. Seo Æ Ho Seong Lee Received: 15 July 2006 / Accepted: 2 October 2006 Ó Springer-Verlag 2006 Abstract This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk spindle system with fluid dynamic bearings (FDBs), the head suspension actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to G. H. Jang (&) C. H. Seo PREM, Department of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul , South Korea ghjang@hanyang.ac.kr C. H. Seo 4hee2@paran.com H. S. Lee Servo & Mechanical Group, Storage System Division, Samsung Electronics Co. Ltd, 416 Maetan-3dong, Yeongtong-gu, Suwon, Gyeonggi-do , South Korea hoseong.lee@samsung.com determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head suspension actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock. 1 Introduction Vibration analysis of an HDD (hard disk drive) has been important issue in the HDD industry because the vibration between read/write head and rotating disk in an HDD plays the major role in determining the disk memory capacity. Servo system cannot position the recording head back to the data track if the in-plane vibration exceeds the allowable track misregistration, approximately 5% of data track pitch. Severe vibration due to shock results in the head crash a failure of the HDD in which the head scrapes across the platter surface, often grinding away the thin magnetic film. However, a complete simulation model of the HDD is not yet developed to analyze the free and forced vibration. One of the difficulties is to develop the theory to connect the spinning disk spindle system with the flexible supporting structure. The other is to calculate the stiffness and damping coefficients of the

2 several bearings of an HDD fluid dynamic bearings (FDBs) in the spindle motor, ball bearings in the actuator and air bearings between head and disk interface, as shown in Fig. 1. Many researchers have developed the simulation model to investigate the free and forced vibration of an HDD. Tseng et al. (2003) proposed the analytical model of the HDD spindle system with FDBs by including the flexibility of housing. They extracted several lower natural frequencies and mode shapes of the supporting structure of the HDD spindle system by using the finite element method (FEM). The extracted mode shapes were applied to the HDD spindle system with rotating flexible disks to determine a set of equations of motion by using the Lagrange equation. Recently, Jang et al. (2005) proposed a consistent method to predict the natural frequencies and mode shape of a rotating disk spindle system in an HDD with FDBs considering the flexibility of a complicated supporting structure by using the FEM and substructure synthesis. They derived the finite element equations of each substructure of an HDD spindle system from the spinning flexible disk to the flexible base plate by satisfying the geometric compatibility in the internal boundary between each substructure. They also imposed the rigid link constraints between the sleeve and FDBs to describe the physical motion at this interface. However, their model did not include the head suspension actuator and the air bearing between spinning disk and head. On the other hand, many researchers investigate the dynamics of the head suspension actuator system. Xu and Guo (2003) performed the modal analysis of the actuator with pivot bearings by using FEM without considering disk spindle system. Jayson et al. (2003) and Gao et al. (2005) analyzed the coupled effect of the actuator and the stationary disk using the FEM, but they did not consider the effect of the flexible supporting structure and the spinning disk spindle system with the FDBs. This paper extends Jang s method (Jang et al. 2005) to analyze the free vibration of a flexible HDD composed of the spinning disk spindle system with FDBs, the head suspension actuator with pivot bearings, and the base plate with complicated geometry. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Effect of suspension preload is replaced by introducing the equivalent stiffness of linear spring. The proposed method is applied to a 2.5-in. HDD and the validity of this research is verified by comparing the numerical results with the experimental ones. 2 Method of analysis 2.1 Finite element analysis of an HDD spindle system This research follows the Jang s method (Jang et al. 2005) to analyze the free vibration of a rotating disk spindle system of an HDD in Fig. 1. Finite element equations of each substructure in the HDD disk spindle system are derived with the introduction of consistent variables to satisfy the geometric compatibility at the internal boundaries. Motion of the rotating spindle and shaft can be described by Timoshenko beam including the axial motion. Motion of the spinning disk is superposed by both its rigid body motion measured from the fixed coordinate and its elastic deformation, i.e., in-plane and transverse elastic displacement, measured from the rotating coordinate Fig. 1 Mechanical structure of a 2.5-in. HDD

3 system. Complicated supporting structures are modeled by the tetrahedral element with rotational degrees of freedom. Two journal bearings and one thrust bearing support the HDD spindle system of this research. The herringbone grooves of the journal bearing and spiral grooves of the thrust bearing are inscribed in the stationary sleeve. The stiffness and damping coefficients of FDBs are calculated in the five degrees of freedom, i.e., the displacement in x, y and z directions and the rotation with respect to x and y axes by using the in-house computer program named HYBAP (Hydrodynamic Bearing Analysis Program; Jang and Kim 1999). The FDBs can be considered as bearing elements with stiffness and damping coefficients in FEM. Rigid link constraints are introduced to connect the bearing element to the tetrahedral element of the stationary part. 2.2 Quadrilateral 4-node shell element for the suspension of an actuator Figure 2 shows the mechanical structure of a head suspension actuator of a 2.5-in. HDD. Enormous number of elements is needed to model the thin and complicated suspension if thin suspension is modeled by the tetrahedral element. This research develops a quadrilateral 4-node shell element with rotational degrees of freedom to efficiently model the suspension, i.e., hinge, flexure and load beam as well as to connect this 2-D shell element to the 3-D tetrahedral element of arm, E-block and fantail of an actuator without violating the geometric compatibility. Shell element is derived by combining a 4-node membrane element and a 4-node bending plate element in Fig. 3 (Choi and Lee 2001). The displacement and rotation fields of the membrane elements can be expressed with the interpolation function, N i, and nonconforming modes, Nj : u ¼ X4 v ¼ X4 h z ¼ X4 N i u i þ X4 j¼1 N i v i þ X4 N i h zi j¼1 N j u j N j v j ð1þ This research uses four non-conforming modes to achieve the high accuracy and suppress membrane locking. Strain displacement relation are obtained by using the functional proposed by Hughes and Brezzi (1989). Element stiffness matrix for the membrane is derived from strain displacement and general stress strain relation. The displacement and rotation fields of the bending plate elements can be expressed with the interpolation function, N i, and non-conforming modes, N j : Fig. 2 Mechanical structure of the actuator of a 2.5-in. HDD

4 Fig. 3 Quadrilateral 4-node shell element combined a bending plate and a membrane w ¼ X4 h x ¼ X4 h y ¼ X4 N i w i N i h xi þ X4 j¼1 N i h yi þ X4 j¼1 N j hxj N j hyj ð2þ This research also uses four non-conforming modes so that Mindlin thick plate theory can be applicable to both thin and thick model. Stiffness matrix for the bending plate is derived from strain displacement relation obtained by using the Mindlin thick plate theory. The substitute shear strain fields proposed by Donea and Lamain (1987) are used to suppress the shear locking phenomenon. The consistent mass matrix is formed by using the interpolation functions of both translation and rotational degree of freedom. 2.3 Connection between the spinning disk and the read/write head Air bearing between the spinning disk and the read/ write head is mechanically a spring element with very small damping. The heads are connected to the spinning disk with the spring elements of axial stiffness as shown in Fig. 4. The local reference frame, x 2 y 2 z 2; is located at the center of the disk which has the infinitesimal rigid body translation of X D ; Y D and Z D, and the infinitesimal rigid body tilting motion of h x and h y with respect to the inertial reference frame, x 1 y 1 z 1 : The elastic deformation of the disk is observed with respect to the local reference frame. The deformation of the air bearing is determined by R h, the displacement of the head, and R p, the displacement of the corresponding point p on the disk: DR ¼ R p R h And R p can be written as follows: ð3þ R p ¼ X D i 1 þ Y D j 1 þ Z D k 1 þ rðcos h i 2 þ sin h j 2 Þþw r k 2 ð4þ where i 1 j 1 k 1; i 2 j 2 k 2; r; h and w r are the unit vectors of the inertial reference frame and the local reference frame, and the radius, the phase angle and the axial elastic deformation of a point p on the disk. The displacement of the head R h can be expressed as follows: R h ¼ u h i 1 þ v h j 1 þ w h k 1 ð5þ where u h ; v h and w h are the elastic deformation of a head in x, y and z directions. The potential energy U due to the axial deformation of air bearing can be approximated by neglecting the radial displacement: UðZ D ; h x; h y; w r; w h Þ ¼ 1 2 DRT k air DR ð6þ ffi 1 2 k airðz D þ r sin hh x r cos hh y þ w r w h Þ 2 Fig. 4 Connection of the air bearing between spinning disk and head where k air is the axial stiffness of air bearing. The reaction forces of air bearing with respect to the rigid body displacements Z D ; h x and h y, and the elastic deformation w r and w h can be determined with the derivative of the potential energy (Cook et al. 1989):

5 8 h 9 2 >= ¼ k air 6 4 >; 38 1 r sin h r cos h 1 1 r sin h r 2 sin 2 h r 2 cos h sin h r sin h r sin h >< r cos h r 2 sin h cos h r 2 cos 2 h r cos h r cos h 7 1 r sin h r cos h >: 1 r sin h r cos h 1 1 Z D h x h y w r w h 9 >= >; ð7þ Then, the stiffness matrix of an air bearing K air in terms of Z D ; h x; h y; w r and w h can be determined as follows: This research uses the restarted Arnoldi iteration method with the deflation technique to solve the eigenvalue problem with large asymmetric matrix as r sin h r cos h 1 1 r sin h r 2 sin 2 h r 2 cos h sin h r sin h r sin h K air ¼ k air r cos h r 2 sin h cos h r 2 cos 2 h r cos h r cos h r sin h r cos h r sin h r cos h 1 1 ð8þ 3 Numerical analysis of an associated eigenvalue problem The global finite element equation of an HDD composed of the spinning disk spindle system with FDBs, the head suspension actuator with pivot bearings, and the base plate of an HDD is a very large and asymmetric matrix due to the complicated geometry of base plate and suspension actuator, the gyroscopic effect of the rotating substructures and damping coefficients of FDBs. The global finite element equation can be expressed as follows: M x þðc þ GÞ_x þ Kx ¼ 0 ð9þ where M, G, C and K are the mass, gyroscopic, damping and stiffness matrices of the global finite element equation. Equation 9 is transformed to a statespace form as follows: G C k M M 0 or kay ¼ By u ku ¼ K 0 0 M u ku ð10þ shown in Eq. 10 (Lehoucq and Sorensen 1996). The Arnoldi iteration method is a well-known technique to approximate a few eigenvalues and the corresponding eigenvectors of a general asymmetric matrix by reducing it to an upper Hessenberg form. This research uses the sparse algorithm for the matrix multiplication (Saad 1995) and the frontal technique as a linear solver (Hinton and Owen 1977) in the Arnoldi iteration method in order to save computation time and memory space. 4 Results and discussion 4.1 Finite element model As shown in Figs. 5 and 6, a finite element model of a 2.5-in. HDD is developed. The head is assumed to be located in the middle of the spinning disk rotating at 5,400 rpm. The disk is discretized by 360 annular sector elements. Rotating hub, clamp, spacer, yoke and permanent magnet are discretized by Timoshenko beam elements including gyroscopic effect. Stator core, sleeve and base plate are discretized by 4-node tetrahedron elements with rotational degrees of freedom. Fig. 5 Finite element model of the spinning disk spindle system

6 Table 1 Major stiffness coefficients of journal bearing K xx (N/m) K xy (N/m) K yy (N/m) K hxh x (Nm/rad) K hxh y (N s/m) K hyh y (N s/m) Upper journal bearing Lower journal bearing Table 2 Major damping coefficients of journal bearing C xx (N s/m) C xy (N s/m) C yy (N s/m) C hxh x (Nm s/rad) C hxh y (Nm s/rad) C hyh y (Nm s/rad) Upper journal bearing Lower journal bearing Fig. 6 Finite element model of a 2.5-in. HDD composed of the spinning disk spindle, the head suspension actuator, and the base plate The stiffness and damping coefficients of the FDBs are calculated by using the HYBAP. Tables 1, 2 and 3 show the major components of the stiffness and damping coefficients of journal and thrust bearings. The rigid link constraints are included to constrain the physical motion between the sleeve and shaft. Hinge, flexure and load beam of the actuator are discretized by 4-node shell elements with rotational degrees of freedoms. The stiffness coefficients of the pivot bearings are calculated in the five degrees of freedom by using the in-house computer program named the WinBAP (Window Bearing Analysis Program; Jang and Jeong 2003). Table 4 shows the major stiffness coefficients of the pivot bearing. The rigid link constraints are also included to constrain the physical motion between the inner and the outer race. The coil of the actuator is not isotropic material, so that equivalent Young s modulus is determined by applying the modal testing of the coil. The first natural frequency of the coil component is measured, and the equivalent Young s modulus is calculated by matching the measured frequency with the simulated one, which is determined by the finite element analysis. Effect of preload is replaced by a linear spring. When a preload is applied to suspension, the displacement of the suspension is calculated through the finite element analysis of suspension. And then the equivalent stiffness of suspension is determined after the applied preload is divided by the calculated displacement. The stiffness element of air bearing is included between spinning disk and head. Table 5 shows the type and the number of the elements used in the finite element model of a 2.5-in. HDD. The total numbers of elements and degrees of freedom are 60,373 and 132,398, respectively. 4.2 Numerical results Table 6 shows the calculated damped natural frequencies and modal damping ratios of the 2.5-in. HDD. Jang et al. (2005) classified the vibration modes of an HDD spindle system without an actuator into three Table 3 Major stiffness and damping coefficients of thrust bearing Upper thrust bearing K zz (N/m) C zz (N s/m) Table 4 Major stiffness coefficients of pivot bearing K xx (N/m) K yy (N/m) K zz (N/m) K hxh x (Nm/rad) K hyh y (Nm/rad) K hxh y (N/rad) K hyh x (N/rad)

7 Table 5 Element type and element number of each components of an HDD Component Element number Element type Supporting structure Base plate 34,532 Tetrahedron element Stator 674 Sleeve 724 Thrust cap 147 Thrust yoke 521 Disk spindle Disk 360 Annular sector element Shaft 18 Rotating Timoshenko Hub 12 Spacer and clamp 4 beam element Actuator Overmold 4,615 Tetrahedron element Latch-pin 48 Coil 1,146 Arm and E-block 4,160 Pivot shaft 3,371 Dummy 483 Suspension 5,375 Shell element Yoke Yoke 3,414 Tetrahedron element Magnet of yoke 760 Bearing FDB 5 Spring and damper element Air bearing 2 Spring element Pivot bearing 2 Spring element Table 6 Comparison between numerical and experimental damped natural frequencies of a 2.5-in. HDD at 5,400 rpm Mode number Simulation Experiment Error (%) Modal damping ratio (f) Damped natural frequency (Hz) Damped natural frequency (Hz) Half-speed whirl Half-speed whirl Mode 1b Mode Mode 1f Mode Mode 4b ,229 1, Mode ,259 1, Mode ,273 1, Mode ,331 1, Mode ,427 1, Mode ,445 1, Mode 4f ,589 1, Mode ,895 1, Mode 11b ,066 2, Mode ,147 2, Mode ,213 2, Mode ,372 2, Mode ,530 2, Mode 11f ,606 2, Mode ,941 2, groups, i.e., half-speed whirl modes, the pure disk modes and the modes coupled with the motion of the base structure. First, the vibration modes of an HDD with an actuator can be similarly classified into two groups, i.e., the half-speed whirl mode and the elastic deformation mode. The half-speed whirl mode is the rigid body mode of the spinning disk spindle. However, head suspension actuator has elastic deformation as shown in Fig. 7, because the rigid body motion of the spinning disk results in the elastic motion of the head suspension actuator through air bearing. The elastic deformation mode of an HDD can be classified into three groups, i.e., the pure disk mode, the (0,0) disk mode and the (0,1) disk mode. The vibration modes 4 and 11 correspond to the pure disk modes with two and three nodal diameters, respectively, and they generate the elastic bending motion of head suspension and arm of the actuator. They split

8 Fig. 7 Simulated half-speed whirl mode Fig. 10 Simulated disk (0,1) mode coupled with the actuator and base plate (mode 15) Fig. 8 Simulated pure disk mode with two nodal diameters (mode 4) into the forward (4f, 11f) and backward (4b, 11b) frequencies as the spindle rotates. In modes 4b, 4f and 11f, fantail has bending motion, and in mode 11b, fantail has torsional motion. Figure 8 shows the vibration mode 4b. In the disk spindle system without an actuator, the vibration modes 4 and 11 correspond to the pure disk modes and the base structure has no motion (Jang et al. 2005). But in the disk spindle system with Fig. 9 Simulated disk (0,0) mode coupled with the actuator and base plate (mode 2) an actuator, disk motion is transferred to the actuator and base plate through air bearing and pivot bearing. Vibration modes 2, 3, 5, 6, 7, 8 and 9 are the (0,0) disk mode coupled with the motion of actuator and base structures. Figure 9 shows the vibration mode 2 that is coupled with the bending motion of the arms 1, 2, 3 and fantail. Vibration modes 1, 10, 12, 13, 14, 15 and 16 are the (0,1) disk modes coupled with the motion of actuator and base structures. Figure 10 shows the vibration mode 15, i.e., the (0,1) disk mode coupled with the bending motion of the arms 1, 2, 3 and torsional motion of the fantail. Table 7 shows the description of modes of a 2.5-in. HDD rotating at 5,400 rpm. 4.3 Experimental verification Experimental modal testing is performed to verify the proposed numerical method. Figure 11 shows the experimental setup for the modal testing. A 2.5-in. HDD is fixed to the jig by screws. An impact hammer is used to excite the HDD and the frequency response functions are measured with a laser Doppler vibrometer. Figures 12 and 13 show the frequency response functions of the outer rim of the disk and the head with the excitation of the outer rim of the disk, respectively. The disk modes are dominantly observed in the frequency response function of the outer rim of the disk when the outer rim of the disk is excited. However, the natural frequencies corresponding to the motion of the suspension, arm, fantail as well as the disk are observed in the frequency response function of the head even when the outer rim of the disk is excited. It can be explained that the motion of the disk is transferred to the head suspension actuator through air bearing. Table 6 shows the comparison between the simulated and experimental damped natural frequencies in the HDD at 5,400 rpm. Figures 14, 15 and 16 are

9 Table 7 Description of the vibration modes for a 2.5-in. HDD at 5,400 rpm Mode number Disk spindle base plate a Actuator b Disk Hub Shaft Base plate Suspension Arm Fantail Half-speed whirl Rigid body motion (+) 1,2 Bending 2,3 Bending No motion Half-speed whirl Rigid body motion (+) 1,2 Bending 2,3 Bending No motion Mode 1b (0,1) (+) Bending(+) (+) 1,2 Bending(+) 1,2,3 Bending(+) Bending(+) Mode 2 (0,0) (+) Axial(+) (+) 1,2 Bending( ) 1,2,3 Bending(+) Bending(+) Mode 1f (0,1) (+) Bending(+) (+) 1,2 Bending( ) 1,2,3 Bending(+) Bending(+) Mode 3 (0,0) (+) Axial(+) (+) 1,2 Bending( ) 1,2,3 Bending(+) Bending(+) Mode 4b (0,2) No motion No motion c 1,2 Bending( ) 1 Bending(+); Bending( ) 2,3 bending( ) Mode 5 (0,0) ( ) Axial( ) ( ) 1 Bending(+); 1,3 Bending(+); Bending( ) 2 bending( ) 2 bending( ) Mode 6 (0,0) (+) Axial(+) (+) 1 Bending(+); 1,3 Bending(+); Bending(+) 2 bending( ) 2 bending( ) Mode 7 (0,0) ( ) Axial( ) ( ) 1 Bending( ); 1,2 Bending(+); Bending(+) 2 bending(+) 3 bending( ) Mode 8 (0,0) ( ) Axial( ) ( ) 1,2 Bending( ) 1,2,3 Bending(+) Bending(+) Mode 9 (0,0) (+) Axial(+) (+) 1,2 Bending 3 Bending Bending(+) Mode 4f (0,2) No motion No motion c 1,2 Bending(+) 1 Bending(+); Bending( ) 2,3 bending( ) Mode 10 (0,1) ( ) Bending( ) ( ) 1,2 Bending( ) 1,2,3 Bending(+) Bending(+) Mode 11b (0,3) No motion No motion d 1,2 Bending(+) 1 Bending(+); Torsion 2,3 bending( ) Mode 12 (0,1) ( ) Bending( ) ( ) 1,2 Bending( ) 1,2,3 Bending(+) Torsion Mode 13 (0,1) ( ) Bending( ) (+) 1,2 Bending( ) 1,2,3 Bending(+) Torsion Mode 14 (0,1) ( ) Bending( ) (+) 1,2 Torsion 1,2,3 Torsion Torsion Mode 15 (0,1) ( ) Bending( ) (+) 1,2 Bending( ) 1,2,3 Bending(+) Torsion Mode 11f (0,3) No motion No motion d 1,2 Bending(+) 1 Bending(+); Torsion 2,3 bending( ) Mode 16 (0,1) ( ) Bending( ) (+) 1,2 Bending( ) 1,2,3 Bending(+) Torsion a Phase of the spindle and base is explained with respect to the motion of disk. In-phase phase and out-of-phase motion are denoted as + and, respectively b Phase of the arm 2, 3 and fantail is explained with respect to the motion of arm 1. Arms 1, 2, 3 and suspensions 1, 2 are denoted from the top component c Concave shape around the spindle d Bending shape which has one nodal line parallel to the short side of the base plate Fig. 11 Experimental setup for modal testing LDV Amp. Star-modal system Impact hammer Charge amp. PULSE 3560C the measured mode shapes corresponding to the simulated mode shapes in Figs. 8, 9 and 10. The damped natural frequencies and mode shapes of the proposed numerical method match well with those of the experimental modal testing, and they verify the validity of the proposed method except the half-speed whirl.

10 Fig. 12 Frequency response function of the outer rim of the disk with the excitation of the outer rim of the disk Fig. 15 Measured disk (0,0) mode coupled with the actuator and base plate (mode 2) Fig. 16 Measured disk (0,1) mode coupled with the actuator and base plate (mode 15) Fig. 13 Frequency response function of the head with the excitation of the outer rim of the disk actuator with pivot bearings and the base plate with complicated geometry. Experimental modal testing shows that the proposed method well predicts the natural frequencies and corresponding mode shapes of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode results in elastic deformation mode of the head suspension actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock. References Fig. 14 Measured pure disk mode with two nodal diameters (mode 4) 5 Conclusion This paper presents a FEM to analyze the free vibration of a flexible HDD composed of the spinning disk spindle system with FDBs, the head suspension Choi CK, Lee TY (2001) Development of flat shell elements with directly modified non-conforming modes and analysis of shell structures. Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Daejeon Cook RD, Malkus DS, Plesha ME (1989) Concepts and applications of finite element analysis, 3rd edn. Wiley, New York Donea J, Lamain LG (1987) A modified representation of transverse shear in quadrilateral plate elements. Comput Methods Appl Mech Eng 63: Gao F, Yap FF, Yan Y (2005) Modeling of hard disk drives for vibration analysis using a flexible multibody dynamics formulation. IEEE Trans Magn 41(2):

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