APVC2009. Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations

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1 Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations Kiyong Park, Gunhee Jang* and Chanhee Seo Department of Mechanical Engineering, Hanyang University Seoul, , Korea *Corresponding Author, Abstract This paper proposes an effective numerical method to determine the forced response of the disk-spindle system represented by asymmetric finite element equations. The asymmetric global finite element matrices are transformed to the modal domain by using the mode vectors from the standard eigenvalue problem. Even though the asymmetric equations of motion in modal domain are coupled, the degrees of freedom in modal domain are much reduced so that they are numerically integrated. Then the results are transformed to determine the forced response of the system. The proposed method is verified by comparing with the response from the other methods. It shows that the proposed method is accurate and computationally effective to determine the forced vibration of a spinning flexible disk-spindle system supported by bearings. Key words: eigen value problem, adjoint eigen value problem, biorthogonality 1. Introduction Rotating machines have been applied to many industries such as turbomachines in large machines and computer hard disk drives (HDD) in small machines. In most cases, their dynamics determine the performance of the whole system. One of the examples is the dynamics of flexible spinning disk-spindle system in a HDD supported by complicated base structure and fluid dynamic bearings (FDBs), because it plays the major role in determining the shock characteristics and the disk memory capacity of a HDD. Fig.1 shows the photograph of the motion between spinning disk and a head due to external shock. The flying height of a head is around 10 nm so that it is very difficult not only to measure the motion of a head with respect to spinning disk by using a high-speed camera but also to predict the motion between spinning disk and a head due to various shocks. Many researchers have developed the simulation model to investigate the free and forced vibration of a HDD. Tseng et al. proposed the analytical model of the HDD spindle system with FDBs by including the flexibility of housing (1). 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. proposed a consistent method to predict the natural frequencies and mode shape of a rotating disk-spindle system in a HDD with FDBs considering the flexibility of a complicated supporting structure by using the FEM and substructure synthesis (2), (3), (4). They derived the finite element equations of each 1

2 substructure of a 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, they did not include the asymmetry of finite element equations due to the gyroscopic effect and the stiffness and damping effect of the FDBs when they calculated the forced response. This paper proposes an effective numerical method to determine the forced response of the spinning flexible disk-spindle system represented by asymmetric finite element equation. The asymmetric global finite element matrices are transformed to the modal domain by using the mode vectors from the standard eigenvalue problem. Even though the asymmetric equations of motion in modal domain are coupled, the degrees of freedom in modal domain are much reduced so that they are numerically integrated. Then the results are transformed to determine the forced response of the system. The proposed method is verified by comparing with the response from the other methods, i.e. (1) the mode superposition and modal approximation with standard eigenvalue problem, and (2) the mode superposition with standard and adjoint eigen value problems. Fig. 1 Shock test of a head-disk 2. Method of Analysis 2.1 Finite element formulation of flexible spinning disk-spindle system This research follows the Jang s method to model the flexible spinning disk-spindle system of a HDD in Fig. 2 and 3 (2). 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 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) (5). The FDBs can be considered as bearing elements with stiffness and damping coefficients in finite element method. Rigid link constraints are introduced to connect the bearing element to the tetrahedral element of the stationary part. 2

3 Fig.2 Finite element model for supporting structures (flange, stator, housing and fluid dynamic bearing) Fig.3 Finite element model for rotating shaft, spindle and disk 2.2 Free vibration analysis of a flexible spinning disk-spindle system The global finite element equation of a HDD composed of the spinning disk-spindle system with FDBs and the base plate of a HDD is a very large and asymmetric matrix due to the complicated geometry of base plate, the gyroscopic effect of the rotating substructures and the stiffness and damping coefficients of FDBs. The global finite element equation can be expressed as follows: ( C + G ) u + Ku 0 M u& + & = (1) where M, G, C and K are the mass, gyroscopic, damping and stiffness matrices of the global finite element equation. The asymmentric eigenvalue problem can be represented as follows: 2 λ Mx + λ( C + G) x + Kx = 0 (2) Equation (2) is transformed to a state-space form as follows: λ G C M M x = K 0 λx 0 0 x M λx or λ Ay = By (3) 3

4 This research uses the restarted Arnoldi iteration method with the deflation technique to solve the eigenvalue problem with large asymmetric matrix as shown in Eq. 3 (6), (7). 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 and the frontal technique as a linear solver in the Arnoldi iteration method in order to save computation time and memory space. 2.3 Forced vibration analysis by using the adjoint eigenvalue problem The finite element equation of a spinning disk-spindle system with excitational force can be represented as follows: C + M G M u + K 0 u& & 0 0 u = Q M u& 0 or M * r& + K * r = Y (4) The eigen vectors from Eq. (4), Φ are complex conjugate, and the response can be approximated by using the lower 2n eigen vectors. r ~ ( t) = Φz ( t) = 2 2 The adjoint eigen vectors can be obtained from the adjoint eigen value problems in order to decouple the asymmetric finite element equation of the motion in modal domain. The adjoint eigen value problem of Eq. (3) can be represented as follows: T T [{ φ } { φ } L { φ } ] ( ) 1 n z t λ A y = B y (6) (5) The biorthogonality relationship between the standard eigen vector and the adjoint eigen vector is utilized to decouple the equations of motion in modal domain: ~ T * ~ ~ T * ~ ~ T Φ M Φz + Φ K Φz = Φ Y ~ ~ or M z& ( t) + K z( t) = F (7) Equation (7) can be represented with 2n decoupled equations of motion: m z& + k z = f ( i 1, 2,..., 2n) (8) i i i i i = The above equations are decoupled ones. The displacement due to the excitation in Eq. (5) can be determined once Eq. (8) is integrated directly or numerically. 2.4 Forced vibration analysis by using the standard eigenvalue problem The disadvantage of the method in 2.3 is that the adjoint eigen value problem has to be solved in addition to the standard eigen value problem. It takes long computation time when the equations of motion have large degrees of freedom. Substituting Eq. (5) into Eq.(4) and multiplying the eigen vector from the standard eigen value problem, the following equations can be obtained: 4

5 ~ T * ~ ~ T * ~ ~ T ~ ~ Φ M Φz + Φ K Φz = Φ Y or M z( t) + Kz( t) = F & (9) Even though the above equations are coupled ones, the degrees of freedom are much reduced so that the displacement due to the excitation in Eq. (5) can be determined once Eq. (9) is integrated numerically. 3. Result and Discussion 3.1 Finite element model and experimental verification This research develops the finite element model of a spinning disk-spindle system of a 3.5 inch HDD at 7,200 rpm by using the proposed method. It has 4,120 elements. The natural frequencies and the mode shapes are compared with the experimental results. Table 1 shows the comparison between numerical and experimental natural frequencies, and it shows that the developed finite element model predicts the natural frequencies well within 5% error. Table 1 Numerical and experimental natural frequencies for a HDD spindle system at 7,200 rpm Modde number Mode shape Natural frequency(hz) Experimental Analysis Mode 1b Rocking Mode 1f Rocking Mode 2 Axial Mode 3b Disk(0,2) Mode 3f Disk(0,2) Mode 4b Disk(0,3) Mode 4f Disk(0,3) Forced vibration analysis This research includes a tetrahedron element as mounting table to attach the HDD spindle system because the HDD is usually fixed to a computer. Fig. 4 shows a finite element model of a 3.5 inch HDD on the mounting table for shock simulation. A half-sinusoidal shock with a period of 2ms and a magnitude of 300G in Fig.5 is applied to the mounting table. The response of the HDD spindle system depends on the number of superposed modes. This research investigates the convergence of the Jang s method (4), the conventional method in 2.3 and the proposed method in 2.4 by calculating the maximum displacement at the end of the spinning disk with the increase of the superposed modes a shown in Fig.5. Sixty modes are required in Jang s method (4), which neglects the off-diagonal elements of the equations of motion in modal domain. However, forty modes are sufficient to guarantee the convergence in the conventional and proposed method. 5

6 Fig. 4 Finite element model of a 2.5 HDD on the mounting table for shock simulation acceleration[g] Time[s] Fig. 5 Half-sinusoidal shock -1.4 x 10-6 Convergence Jang's method Adjoint method Standard method Displacement(m) Number of mode superposition Fig. 6 Maximum displacement of a disk by increasing the number of superposed modes Fig.7 shows the displacement at the outer rim of the disk in three different methods. The transient response from the proposed method in Fig.7 (b) exactly matches with that from the conventional method in Fig.7(c). However, the Jang s method in Fig.7 (a) is very slightly different from the other methods. Table 2 shows the computation time for three different methods. The computer has Intel core2 Duo CPU 3.00GHz, 3.5GB RAM. The conventional method takes twice computation time than the other methods because it solves the adjoint eigen value problem as well as the standard eigen value problem. Table 2 and Fig. 7 show that the proposed method, which uses the standard eigen value problem and the numerical integration in modal domain, guarantee not only the accuracy but also short computation time. 6

7 1.5 x Magnitude(m) Time(s) (a) Jang s method 1.5 x Magnitude(m) Time(s) (b) Proposed method 1.5 x Magnitude(m) Time(s) (c) Conventional method Fig. 7 Radial displacement of the disk Table 2 Compuation time (second) Jang s method Standard eigenvalue problem Adjoint eigenvalue problem Free vibration analysis Forced vibration analysis

8 4. Conclusion This paper proposes an effective numerical method to determine the forced response of the flexible spinning disk-spindle system represented by asymmetric finite element equation. It only solves the standard eigen value problem once and then integrates the coupled equations of motion in modal domain. The proposed method has the same level of accuracy and less computational time than the conventional method. This research can be effectively applied to design a robust HDD spindle system. 5. References (1) Tseng, C. W., Shen, J. Y. and Shen, I. Y., "Vibration of rotating-shaft HDD spindle motors with flexible stationary parts", IEEE Transactions on Magnetics, vol. 39, 2003, pp (2) Jang, G. H., Han, J. H. and Seo, C. H., "Finite element modal analysis of a spinning flexible disk-spindle system in a HDD considering the flexibility of complicated supporting Structure", Microsys. Technol., vol. 11, 2005, pp (3) Jang, G. H., Seo, C. H. and Lee, H. S., Finite element modal analysis of an HDD considering the flexibility of spinning disk spindle, head suspension actuator and supporting structure, Microsys. Technol., vol. 13, 2007, pp (4) Jang, G. H. and Seo, C. H., Finite element shock analysis of an operating HDD considering the flexibility of a spinning disk-spindle, a head-suspension-actuator and a supporting structure, IEEE Transactions on Magnetics, Vol. 11, No. 9, 2007,pp (5) G. H. Jang and S. H. Lee, Determination of the dynamic coefficients of the coupled journal and thrust bearing by the perturbation method, Tribology Letters, vol.22, 2006, pp.239~246. (6) Lehoucq, R. B. and Sorensen, D. C., Deflation techniques for an implicitly restarted Arnoldi iteration, J. Matrix anal. Appl. SIAM, 1996,pp (7) Leonard Meirovitch, 2001, "Fundamentals of vibrations", International ed., McGraw-Hill, pp (8) Anil K. Chopra, "Dynamics of structures", 3rd ed., Prentice-Hall, 2001, pp Acknowledgements This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF D00040). 8