Finite element analysis of unbalance response of high-speed polygon mirror scanner motor considering the flexibility of the structure

Transcription

1 Microsyst Technol (2) 15: DOI 1.17/s TECHNICAL PAPER Finite element analysis of unbalance response of high-speed polygon mirror scanner motor considering the flexibility of the structure K. M. Jung Æ G. H. Jang Æ C. H. Seo Æ M. G. Kim Received: 2 July 27 / Accepted: 2 April 2 / Published online: 15 May 2 Ó Springer-Verlag 2 Abstract This paper presents the finite element method and the mode superposition method to analyze the unbalance response of a high speed polygon mirror scanner motor supported by sintered bearing and flexible supporting structures. Finite element equations of each component of the polygon mirror scanner motor and the flexible supporting structures are consistently derived by satisfying the geometric compatibility in the internal boundary between each component. The rotating polygon mirror is modeled by annular sector element, and its rigid body motion is also considered. The rotating components except for the polygon mirror are modeled by Timoshenko beam element including the gyroscopic effect. The flexible supporting structures are modeled by using a 4-node tetrahedron element and 4-node shell element with rotational degrees of freedom. The rigid link constraints are imposed at the interface between sleeve and sintered bearing to describe the physical motion at this interface. A global matrix equation obtained by assembling the finite element equations of each substructure is transformed to a state-space matrix-vector equation, and both damped natural frequencies and modal damping ratios are calculated by solving the associated eigenvalue problem by using the restarted Arnoldi iteration method. Unbalance responses are calculated by superposing the eigenvalues and eigenvectors of the free vibration analysis. The validity of the proposed method is verified by comparing the simulated unbalance response with the experimental results. This research also shows that the flexibility of supporting structures plays an important K. M. Jung G. H. Jang (&) C. H. Seo M. G. Kim PREM, Department of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul , Republic of Korea ghjang@hanyang.ac.kr role in determining the unbalance response of the polygon mirror scanner motor. 1 Introduction A polygon mirror scanner motor is a driving device of a polygon mirror to reflect the laser beam to print image information in a laser beam printer. It is the key component to determine the printing speed and quality of a laser beam printer. Nowadays, the operating speed of the polygon mirror scanner motor has been increased up to 5, rpm for fast printing speed, and it may be increased in the near future. The polygon mirror scanner motor also requires the low vibration for high printing quality at the same time. One of the major excitation forces is the centrifugal force due to the mass unbalance of the rotor of the polygon mirror scanner motor. Even though mass unbalance is reduced below the certain value in the manufacturing stage, residual unbalanced mass still plays the important role on generating the vibration and noise in high speed operation because the centrifugal force is proportional to the square of the operating speed. Therefore, it is important to investigate the dynamic characteristics of the polygon mirror scanner motor due to the mass unbalance of its rotor to achieve high printing speed as well as the high printing quality of a laser beam printer. Unbalance response has been one of the interested areas in the rotor dynamics. Many researchers have investigated the unbalance response of the rotating disk, shaft and bearings by using several different numerical methods. Recently, Zirkelback and Ginsberg (22) summarized the prior researches on unbalance response, and they studied the unbalance response of rotating shaft system by using

2 163 Microsyst Technol (2) 15: the Ritz series. Shiau et al. (24) studied the unbalance response of a hydrodynamic thrust bearing-mounted rotor using the finite element method. They did not include the flexibility of the stationary supporting structure. However, the flexibility of the stationary supporting structure determines the unbalance response of the rotating system in some applications, and a polygon mirror scanner motor is one of the examples. Several researchers have investigated the dynamics of the polygon mirror scanner motor. Kawamoto (17) studied unbalance response of the polygon mirror scanner motor by assuming a rotating part of a polygon mirror scanner motor as a rigid body supported by air bearing, and he investigated the unbalance response due to bearing length. He also studied the transient contact phenomenon of a polygon mirror scanner motor during start stop operation experimentally (Kawamoto 21). This paper presents the finite element method and the mode superposition method to analyze the unbalance response of a high speed polygon mirror scanner motor supported by sintered bearing and flexible supporting structures. The validity of the proposed method is verified by comparing the simulation results of the natural frequencies and the unbalance responses with the experimental ones. This research also proposes a possible solution of the polygon mirror scanner motor to reduce the vibration due to mass unbalance. 2 Method of analysis 2.1 Free vibration analysis of a polygon mirror scanner motor The polygon mirror scanner motor of this research is operated from 2, to 3, rpm in nominal condition and the mass unbalance is limited less than 4. mg/cm in manufacturing stage by the balancing machine. Figure 1 shows the mechanical structure of a polygon mirror scanner motor. The rotating parts consist of polygon mirror, clamp, hub, shaft and permanent magnet. Rotating parts are supported by the sintered bearing and the supporting structures which consist of housing, sleeve, stator, condenser and PCB plate. This research followed Jang s method to analyze the free vibration characteristics of a rotating polygon mirror scanner motor supported by the flexible supporting structure (Jang et al. 25). The rotating polygon mirror is modeled by annular sector element, and its rigid body motion is also considered. The rotating components except for the polygon mirror are modeled by Timoshenko beam element including the axial deformation. Figure 2 shows the finite element model of rotating parts of the polygon mirror scanner motor. The sintered bearing is modeled by the stiffness and Condenser damping element with five degrees of freedom. The flexible supporting structures are modeled by using a 4-node tetrahedron element and 4-node shell element with rotational degrees of freedom. Figure 3 shows the interface between the rotating part and stationary part. The rigid link constraints are imposed at the interface between sleeve and sintered bearing to describe the physical motion at this interface. A global matrix equation obtained by assembling the finite element equations of each substructure is transformed to a state-space matrix-vector equation, and both damped natural frequencies and modal damping ratios are calculated by solving the associated eigenvalue problem by using the restarted Arnoldi iteration method. 2.2 Unbalance response analysis of the polygon mirror scanner motor using the mode superposition method The rotating polygon mirror can be approximated by the flexible rotating disk, and its finite element equation can be obtained by using the Lagrange equation as follows: d dt ot P o_x P Clamp PCB plate Sleeve Shaft ot P ox P þ ou P ox P ¼ Q P Hub Housing Polygon mirror Stator PM Fig. 1 Mechanical structure of a polygon mirror scanner motor Shaft Clamp PM Hub Polygon mirror Fig. 2 Finite element model of rotating parts Annular sector element Rotating Timoshenko beam element ð1þ X P ¼ X; Y; Z; h x ; h y ; ðwþ e ð2þ where T P, U P, X P and Q P are the kinetic energy, the strain energy, the nodal displacement vector and the generalized excitation force of the polygon mirror, respectively. X, Y, Z, h x, h y are the rigid body displacements and (w) e is the

3 Microsyst Technol (2) 15: Fig. 3 Interface between the rotating part and stationary part Fig. 4 Centrifugal force due to mass unbalance of a polygon mirror scanner motor transverse elastic deformation of the polygon mirror (Jang et al. 25). Figure 4 shows the centrifugal force due to mass unbalance, the inertial reference frame XYZ, and the local reference frame xyz located at the center of the rotating polygon mirror after it has the tilting motion h x and h y.in the local reference frame, the centrifugal force can be expressed as follows: F U ¼ mex 2 cos Xt^i þ sin Xt^j ð3þ where m, e, ^iand ^j are mass unbalance, eccentricity and unit vectors of the local reference, respectively. Equation 3 can be represented in the inertial reference frame: F U ¼ mex 2 cos Xt cos h y ^I þ mex 2 cos Xt sin h x sin h y þ sin Xt cos h x ^J ð4þ þ mex 2 sin Xt sin h x cos Xt cos h x sin h y ^K where ^I, ^J and ^K are the unit vectors of the inertial reference frame, and they are in the direction of the rigid body displacements of the polygon mirror, i.e., X, Y, Z, respectively. Then, the following finite element equation of the polygon mirror due to the centrifugal force can be obtained as follows: M P X Y >< Z h x h >: y ð w Þ e >= >< þ G P X _ Y _ Z h x _h y ð_w Þ e >= >< þ K P X Y Z h x >; >: >; >: >; ðwþ e mex 2 ðcos Xt cos h y Þ mex 2 ðcos Xt sin h x sin h y þ sin Xt cos h x Þ >< >= mex ¼ 2 ðsin Xt sin h x cos Xt cos h x sin h y Þ >: >; h y >= ð5þ where M P, G P and K P are the mass, gyroscopic and stiffness matrices of the rotating polygon mirror, respectively. The components of M P, G P and K P are explained in Jang et al. (25). If the tilting motions of the polygon mirror, h x and h y are small, Eq. 5 can be approximated to the following Eq. 6: M P X X _ Y Y _ >< >= >< Z Z _ þ G P h x _h x h >: y _h >; >: y ð w Þ e ð_w Þ e mex 2 cos Xt mex 2 sin Xt >< >= ¼ >: >; >= >< þ K P >; >: X Y Z h x h y ðwþ e >= >; ð6þ or M P X P þ G P _X P þ K P X P ¼ Q P The global finite element equation of the polygon mirror scanner motor is assembled from the finite element equations of each components of the polygon mirror scanner motor, i.e., annular sector elements for the rotating polygon mirror, beam elements for the shaft and hub, stiffness and damping elements for the sintered bearings, and 4-node tetrahedron elements and 4-node shell elements for the flexible supporting structure. The global finite element equation can be expressed as follows: M xðtþþðcþgþ_xðtþþkxðtþ ¼QðtÞ ð7þ where M, G, C, K and Q are the mass, gyroscopic, damping, stiffness matrices and excitation force vector due to mass unbalance of the global finite element equation, respectively. It is composed of very large matrices, and G

4 1632 Microsyst Technol (2) 15: are obtained by multiplying the Eq. with ~U T : m i _z i þ k i z i ¼ f i ði ¼ 1; 2;...; 2nÞ ð1þ The solution z i at time t þ Dt can be determined that as follows (Jang and Seo 27): Fig. 5 Finite element model of a polygon mirror scanner motor and C are the asymmetric matrices due to the gyroscopic effect of the rotating parts and damping of the sintered bearings. The Eq. 7 can be represented by the following state space equation. ðc þ GÞ M _x M x or M _r þ K r ¼ Y þ K M x ¼ Q _x ðþ 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. (Lehoucq and Sorensen 16). Once the natural frequencies and mode vectors are determined, the unbalance response r(t) can be approximated with the lowest 2n modes with complex number by using mode superposition method: rðtþ ~rðtþ ¼ X2n i¼1 z i ðtþ/ i ¼ ~ U~z ðþ where / i is the eigenvector which is calculated by the eigenvalue problem. The decoupled 2n equations of motion Table 1 Element number and type of each component of a polygon mirror scanner motor Component Element number Element type Rotating part Polygon mirror 27 Annular sector element Shaft 13 Rotating Timoshenko Clamp and PM 2 beam element Hub 14 Stationary part PCB plate 77 Shell element Sleeve 1,22 Tetrahedron element Stator 1,566 Housing 2,56 Condenser 2 Bearing Sintered bearing 1 Stiffness and damping element z i ðt þ DtÞ ¼ z i ðþe t k i m Dt i þ Af i ðþþbf t i ðt þ DtÞ ð11þ A ¼ 1 m 2 i m i e k i m Dt i 1 m 2 i e k i m Dt i ð12þ Dt k i k i Dt k i B ¼ m i 1 m 2 i þ 1 m 2 i e k i m Dt i ð13þ k i Dt k i Dt k i The time response z i (t) can be determined by solving the above Eq. 11 repetitively respect to the time. Then, the unbalance response r(t) due to mass unbalance can be obtained from the Eq.. 3 Result and discussion 3.1 Free vibration analysis and experiment Figure 5 shows the finite element model of a polygon mirror scanner motor and Table 1 shows the element number and type of each component in the finite element model. The finite element model is developed by using total 7,11 elements. Table 2 shows the major design parameter of a PCB plate as important part in this paper. In this model, the polygon mirror is modeled by the annular sector elements, and the hub, clamp and permanent magnet are modeled by the Timoshenko beam elements including the gyroscopic effect. The supporting structures, which are composed of the PCB plate, sleeve, housing, stator and condenser, are modeled by the 4-node shell and tetrahedron elements with rotational degrees of freedom. The sintered bearing is modeled by the stiffness and damping element with five degrees of freedom. This research assumes that the oil of sintered bearing are fully filled in the clearance at the operating speed and then the stiffness and damping coefficients of sintered bearing are calculated by using HYBAP (fluid bearing analysis program) (Jang and Lee 26). It calculates the pressure distribution by using the Reynolds equation and the finite element method. And stiffness and damping coefficients are calculated by using the perturbation method. Displacements and rotations of four nodes, at which the PCB plate is fixed by screws, are assumed to be as fixed boundary condition. Experimental modal testing is performed to verify the proposed simulated results. Figure 6 shows the experimental setup for the modal testing. The PCB plate is fixed to the jig by screws and an impact hammer is used to excite

5 Microsyst Technol (2) 15: Table 2 Major design parameter of a PCB plate Length (mm) Width (mm) Thickness (mm) Poisson s ratio Density (kg/m 3 ) Young s modulus (N/m 2 ) Table 3 Simulated and measured natural frequencies (f d ) of a polygon mirror scanner motor at rpm Mode number Simulation Experiment Error (%) f d (Hz) f d (Hz) Value ,53 225e a polygon mirror scanner motor. The response is measured by using a laser doppler vibrometer (sensor) and PULSE 356C (signal analyzer). The mode shapes are measured by using the STAR modal system. Table 3 shows the comparison between the simulated and experimental damped natural frequencies of the polygon mirror scanner motor at rpm. Figure 7 shows the comparison between simulated and experimental mode shape of the vibration mode 1 of the polygon mirror scanner motor. Table 3 and Fig. 7 show that the simulated damped natural frequencies and mode shapes match well with those of the experimental modal testing. Table 4 shows the simulated vibration modes of a polygon mirror scanner motor at rpm. The vibration modes of a polygon mirror scanner motor can be classified into three groups, i.e., the vibration mode in which the motion of supporting structures is dominant, the vibration mode in which the motion of the polygon mirror is dominant and the vibration mode in which both the motions of supporting structures and the polygon mirror are dominant. In vibration modes 1, 4, 7, and, the motion of supporting structures is dominant. In these modes, supporting structures have large elastic deformation due to the flexibility of the PCB plate. In vibration modes 2 and 3, the motion of the polygon mirror is dominant. In these modes, the polygon mirror has the rocking motion and the elastic deformation motion of supporting structures is small as compared to the motion of the polygon mirror. In vibration modes 5 and 6, both the motion of supporting structures and the polygon mirror are dominant. In these modes, both Mode Mode Mode Mode 4 1,45 1, Mode 5 1,72 1, Mode 6 1,47 1,4 5.4 Mode 7 2,34 2, Mode 2,53 2, Mode 2,715 2,71.17 the motion of supporting structures and the polygon mirror are large. The free vibration analysis of the polygon mirror scanner motor is performed due to the change of rotating speed. Figure shows the simulated damped natural frequencies of the polygon mirror scanner motor from 14, to 34, rpm by the increment of 2, rpm. As the rotating speed increases, the damped natural frequencies of vibration modes 1 and 2 decrease and the damped natural frequencies of vibration modes 3 and 6 increase due to the gyroscopic effect of the rotating part and the coupled effect between the rotating and stationary parts. The damped natural frequency of vibration mode 1 decreases from 56 Hz at rpm to 53 Hz at 32, rpm. The resonance may occur when the polygon mirror scanner motor operates near 32, rpm. 3.2 Unbalance response analysis and experiment The unbalance response of the polygon mirror scanner motor is determined by using the proposed mode superposition. The measured mass unbalance and the measured Fig. 6 Experimental setup for modal testing LDV Amp. Star-modal system Polygon mirror scanner motor Impact hammer Charge amp. PULSE 356C

6 1634 Microsyst Technol (2) 15: Fig. 7 Comparison of the first vibration mode between the simulation and the experiment Table 4 Vibration modes of a polygon mirror scanner motor at rpm Mode number Mode shape PCB plate Polygon mirror Mode 1 Bending Rocking along Y-axis (?) Mode 2 X-axis (1,) Rocking along X-axis (?) Mode 3 Y-axis (1,) Rocking along Y-axis (?) Mode 4 X-axis (1,) Rocking along X-axis (-) Mode 5 X-axis (1,) Rocking along X-axis (-) Mode 6 Y-axis (1,) Rocking along Y-axis (-) Mode 7 X-axis (1,) No motion Mode (2,) No motion Mode (2,) No motion The phase of the polygon mirror is explained with respect to the motion of the PCB plate. In-phase and out-of-phase motion are denoted as? and -, respectively. The major axis and the minor axis of the PCB plate are denoted as X-axis and Y-axis. The difference between mode 4 and mode 5 is due to the phase of the condenser eccentricity are applied to the simulation. In this paper, the unbalance response is the axial displacements of the top center of polygon mirror scanner motor. The convergence Damped natrualfrequency [Hz] Rotating speed [rpm] Mode Mode Mode 7 Mode 6 Mode 5 Mode 4 Mode 3 Mode 2 Mode 1 Fig. Simulated damped natural frequencies due to operating speed Mode number Fig. Convergence of unbalance responses due to the increase of mode number of unbalance response is verified by increasing the mode number. Figure shows the convergence of the unbalance responses of the polygon mirror scanner motor at 3, rpm due to increase of mode number. It shows that the lowest 11 modes may be sufficient to approximate the unbalance response in this model. Experiment is performed to verify the proposed numerical method. Figure 1 shows the experimental setup to measure the run-out, i.e., the axial displacement by using a capacitance sensor. Figure 11 shows the comparison between the simulated axial displacement and the experimental axial repeatable run-out (RRO) at 3, rpm. RRO is the repeatable vibration component subtracted nonrepeatable run-out from the measured total run-out. RRO is compared with the simulated result, because RRO is the displacement dominantly generated by mass unbalance. Figure 12 shows the simulated axial displacement and the experimental axial RRO from 22, to 34, rpm by the increment of 2, rpm. The simulated axial displacement due to the change of rotating speed matches with that of the experimental ones. Unbalance response has peak value at 32, rpm because the rotating speed of Hz is very close to the first natural frequency of 53 Hz. In the speed region faster than 3, rpm, the simulation result is smaller than the experiment. It is due to the fact that the oil in the sintered bearing is assumed to be

7 Microsyst Technol (2) 15: Fig. 1 Experimental setup for run-out testing Gap sensor Amplifier Polygon mirror scanner motor Optic sensor Computer Fig. 11 Comparison of the axial displacement at 3, rpm between the simulation and the experiment Time [sec] Time [sec] (a) Measured axial displacement at 3, rpm (b) Simulated axial displacement at 3, rpm Simulation Experiment T =.3 [mm] T = [mm] Rotating speed [RPM] Rotating speed [rpm] Fig. 12 Comparison of the axial displacement from 22, to 34, rpm between the simulation and the experiment Fig. 13 Comparison of the axial displacement between the original design (T =.3 mm) and the modified design (T = mm) fully filled so that the stiffness and damping coefficients are overestimated in this research. 3.3 Reduction of unbalance response Unbalance response of the polygon mirror scanner motor increases rapidly near 32, rpm due to the resonance between the first natural frequency and the operating speed. The unbalance response may be reduced by shifting the first natural frequency to high frequency. The first vibration mode has dominant bending motion of the PCB. This research increases the thickness of PCB plate, which is dominantly related to the bending stiffness, from.3 to mm by 2%. The free vibration and the unbalance response analysis of the modified design are performed from 22, to 34, rpm by the increment of 2, rpm.

8 1636 Microsyst Technol (2) 15: Table 5 Damped natural frequencies (f d ) of the original and the modified design at 3, rpm Mode number Table 5 shows the comparison of the damped natural frequencies of the polygon mirror scanner motor between the original design and the modified design at 3, rpm. Thick PCB increases the natural frequencies of all the vibration modes which have dominant bending motion of the PCB. For example, the natural frequency of the first vibration mode increases from 54 to 61 Hz. Figure 13 shows the comparison of the axial displacement of the original design and the modified design from 22, to 34, rpm by the increment of 2, rpm. The resonance phenomenon occurs at 32, rpm (533 Hz) for the original design, but there is no resonance in the operating speed range from 22, to 34, rpm. It decreases overall unbalance response and it decreases the unbalance response at the operating speed of 3, rpm from.5 to.27 lm by 71%. 4 Conclusion T =.3 mm T = mm Increasing f d (Hz) f d (Hz) rate (%) Mode Mode Mode 3 1,165 1, Mode 4 1,453 1, Mode 5 1,2 2, Mode 6 2,33 2, Mode 7 2,61 2, Mode 2,715 3, Mode 2,755 3, This paper presents the finite element method and the mode superposition method to analyze the unbalance response of a high speed polygon mirror scanner motor supported by sintered bearing and flexible supporting structures. The validity of the proposed method is verified by comparing the simulated natural frequencies and run-out with the experimental ones. This research also shows that a rapid increase of the vibration of the polygon mirror scanner motor at 32, rpm is the resonance in which the natural frequency of the first vibration mode matches with the operating speed. And this research also proposes a modified design of polygon mirror scanner motor to reduce the vibration due to centrifugal force. The proposed method can well predict the unbalance response due to mass unbalance. And it also can be utilized to develop the robust polygon mirror scanner motor. References Jang GH, Lee SH (26) Determination of the dynamic coefficients of the coupled journal and thrust bearings by the perturbation method. Tribol Lett 22(3): doi:1.17/s Jang GH, Seo CH (27) Finite element shock analysis of an operating HDD considering the flexibility of a spinning diskspindle, a head-suspension-actuator and a supporting structure. IEEE Trans Magn Jang GH, Han JH, Seo CH (25) Finite element modal analysis of a spinning flexible disk-spindle system in a HDD considering the flexibility of complicated supporting structure. Microsyst Technol 11(7):4 4. doi:1.17/s Kawamoto H (17) Rotor dynamics of polygonal mirror scanner motor supported by air bearings in digital electrophotography. J Imaging Sci Technol 41: Kawamoto H (21) Contact characteristics of a laser scanner motor in a laser printer in the low speed region. J Imaging Sci Technol 45:4 44 Lehoucq RB, Sorensen DC (16) Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J Matrix Anal Appl 17:7 21 Shiau TN, Hsu WC, Chang JR (24) Dynamic response of a hydrodynamic thrust bearing-mounted rotor. J Eng Gas Turbines Power 126: doi:1.1115/ Zirkelback NL, Ginsberg JH (22) Ritz series analysis of rotating shaft system: validation, convergence, mode functions and unbalance response. J Vib Acoust 124: doi:1.1115/1.1511