Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 3, pp. 1001-1012, Warsaw 2016 DOI: 10.15632/jtam-pl.54.3.1001 RESEARCH ON THE DISTURBANCE GENERATED BY A SOLAR ARRAY DRIVE ASSEMBLY DRIVING A FLEXIBLE SYSTEM Jiangpan Chen, Wei Cheng School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing, China e-mail: chenjiangpan@hotmail.com; cheng wei@buaa.edu.cn The present work is aimed at presenting the disturbance generated by a solar array drive assembly (SADA) driving a flexible system. Firstly, the vibration equation of SADA is obtainedbysimplifyingand linearizing the electromagnetic torque.Secondly, thedisturbance model of SADAdriving adiscrete flexible system is achievedbased on the vibration equation established. Taking a two-dof flexible system as the study object, this disturbance model is simulated and analyzed. Lastly, a continuous flexible system, which is designed to simulate the solar array, is used to illustrate the simulation method of the disturbance emitted by SADAdriving a continuous flexible system.All the achievements obtained from this project will provide a theoretical basis for the prediction of the disturbance emitted by the SADA driving solar array on the orbit. Keywords: solar array drive assembly, disturbance, flexible system, electromagnetic torque, vibration equation 1. Introduction Micro-vibrations of a satellite, which are characterized by a low amplitude and a wide frequen- cy band, can significantly degrade its pointing accuracy and imaging quality. There are many possible disturbance sources, such as reaction/momentum wheel assembly, control moment gy- roscope and SADA (Luo et al., 2013). Due to a longer exposure time than in a low orbit remote sensing satellite, the performance of a high orbit remote sensing satellite is more sensitive to micro-vibrations in the low frequency band. Three reasons make the disturbance emitted by the SADA driving solar array on the orbit extremely complex andmainly in the low frequency band,which are: (1) distribution of the natural frequency of the solar array in the low frequency band is dense (Cui, 2006); (2) exciting energy of SADA in the low frequency band is large, which makes the natural vibration of the solar array easily excited out; (3) vibration attenuation of the solar array in the low frequency band is slow without air damping in space. Therefore, to study the disturbance produced by the SADAdriving solar array is imperative. Because of the gravity, the solar array, which has characteristics of low stiffness, largemass and large size (Chen, 2010), is difficult to unfold on the Earth. Testing of the disturbance generated by the SADA driving solar array in the atmospheric environment is hard to carry out. A disturbance model used to predict the disturbance emitted by the SADA driving solar array on the orbit is necessary. From researches carried out thus far, lots of studies have focused on the disturbance proper- ties of SADA.Themechanism of electromagnetic vibration of SADAwas studied byXia (1994), and the methods to lower the amplitude of the electromagnetic vibration were also proposed, such as using damping (including mechanical damping and electrical damping) and choosing a more appropriate drive circuit. The expression for electromagnetic vibration frequency of SADA was presented in Bodson et al. (2006). In Yang et al. (2010) and Zhu et al. (2015), argued that the electromagnetic field between the stator and rotor of SADA could be equivalent to an 1002 J. Chen,W. Cheng electromagnetic spring-viscous damping system. Stiffness of the electromagnetic spring, named as electromagnetic stiffness, was obtained by experimental testing by Yang et al. (2010). Zhu et al. (2015) validated feasibility of the equivalent electromagnetic field to an electromagnetic spring-viscous damping system, and also investigated the coupling effect between the solar array and the electromagnetic spring-viscous damping system. Meanwhile, Si et al. (2010), Elsodany et al. (2011) and Szolc et al. (2012) proposed dynamic models of SADA. Among them, the dy- namic model proposed by Si et al. (2010) was more complete, in which the friction torque and fluctuation torque of SADAwere taken into account. The disturbance emitted by SADAdriving a rigid system could be obtained by the dynamicmodels easily. Chen et al. (2014) modeled and simulated the disturbance produced by SADA driving a rigid system through a flexible trans- mission shaft based on the dynamic model of SADA. In this paper, the coupling effect between the electromagnetic stiffness and the flexible transmission shaft was studied by changing the stiffness of the transmission shaft. However, the electromagnetic stiffness of the electromagnetic spring has not appeared in the dynamic models as an explicit expression up to now, and just has been hidden in the electromagnetic torque, which prevented the establishment of the coupling effect between the electromagnetic stiffness and the flexible system. As the solar array is an extremely complex ty- pical continuous flexible system, the dynamicmodels existing do not have the ability to describe the disturbance caused by the SADA driving solar array on the orbit precisely and easily. As a consequence, in this paper, the first part is to establish the vibration equation of SADA; the second part is tomodel, simulate and analyze the disturbance produced by SADAdriving a discrete flexible system; the third part is to introduce the simulationmethod of the disturbance generated by SADA driving a continuous flexible system; the last part summarizes the paper and states conclusions drawn from this work. 2. Modeling of the vibration equation of SADA The SADAdiscussed in this paper is a two-phase hybrid steppermotor. Steppermotors aremo- tors which translate digital pulse sequences to precisemovements of the rotor shaft, rather than rotating continuously as conventional motors do (Anish et al., 2012). Because of the advantages of high resolution, high torque and low noise, the stepper motor is widely used in the field of spacecraft (Letchmanan et al., 2005). Both solar array and digital antenna use stepper motors as the driver. Themost frequently used dynamic model of SADA at present is given by J0θ̈=Te−C0θ̇−Tl (2.1) where J0 is the moment of inertia of the rotor, θ is the angular displacement of the rotor, Te is the electromagnetic torque,C0 is the viscous damping coefficient and Tl is the load torque. As to a two-phase hybrid stepper motor, when the linear flux model without saturation effects is assumed, the drivingmodel presented by Kapun et al. (2007) can be used in the form ϕ=LI Te = 1 2 I T∂ϕ ∂θ = 1 2 I T∂L ∂θ I (2.2) where ϕ is the flux linkage vector, L is the symmetric inductance matrix and I is the current vector. The expressions of I andL are given by I=    IA IB If    L=    LAA LAB LAf LBA LBB LBf LfA LfB Lff    (2.3) Research on the disturbance generated by a solar array... 1003 where IA and IB signifies the current of phase A and phase B respectively, If is the fictitious constant rotor current, Lii (i=A,B) is the self-inductance of the stator winding, LAB =LBA is themutual inductance between the stator windings,Lff is the self-inductance of the fictitious rotor winding, Lif = Lfi (i = A,B) is the mutual inductance between the i-th stator winding and the fictitious rotor winding, see Yang et al. (2010) and Kapun et al. (2007). Assuming that the mutual inductance between the stator windings is negligible and igno- ring the second and more harmonics of the cyclical permeance function, the elements of the inductance matrixL can be defined as LAA =LBB =L0 LAB =LBA =0 LAf =LfA =Lm0+Lm1cos(zθ) LBf =LfB =Lm0+Lm1 sin(zθ) Lff =Lf0+Lf1cos(4zθ) (2.4) where z is the rotor teeth number. Substituting equations (2.3) and (2.4) into equation (2.2), the electromagnetic torque can be obtained as Te =Tm−Td =Km[IB cos(zθ)− IA sin(zθ)]−Kd sin(4zθ) (2.5) where Tm is the motor torque, Td is the detent torque,Km = zIfLm1 andKd =2zI 2 fLf1 is the motor torque constant and detent torque constant, respectively (Letchmanan et al., 2005). In general, the detent torque constant Kd is so small that the detent torque Td is far less than themotor torque Tm, and can be ignored (Yang et al., 2010). Thus, equation (2.5) can be simplified as Te =Km[IB cos(zθ)− IA sin(zθ)] (2.6) In order to improve the resolution and running smoothness of the steppermotor, a subdivi- sion driver is frequently used.Yang et al. (2007) pointed out that the ladder sine curve current is frequently used as the subdivisiondriving current in the subdivisiondriver.Thus, the expression for the two-phase current IA and IB after subdivision can be shown as IA = I cos(γi) IB = I sin(γi) (2.7) where I is the amplitude of the two-phase current, γ is the electrical step angle after subdivi- sion and i = 1,2, . . . is the step number. Substituting equation (2.7) into equation (2.6), the electromagnetic torque can be rewritten as Te =KmI sin(γi−zθ)=KmI sin [ z (γi z −θ )] =KmI sin(z∆θ) (2.8) In equation (2.8), the physical meaning of∆θ is that:∆θ is the included angle between the actual position and theoretical equilibrium position of the rotor during the time period that after the input of the i-th digital pulse signal completed and before the input of (i+1)-th digital pulse signal, where γi/z and θ is the theoretical equilibrium position and actual position of the rotor in this time period, respectively. To ensure SADA running without losing step, the range of ∆θ is given by −α¬∆θ¬α (2.9) where α is the micro-step angle after subdivision. The expression for α is given by α= 2π zpn (2.10) where p is the beat number and n is the subdivision number. The value of p and n of the SADA studied in this paper is 4 and 256, respectively, thus, we can obtain that − π 512 ¬ z∆θ¬ π 512 (2.11) 1004 J. Chen,W. Cheng A significant result can be obtained from equation (2.11) that: z∆θ is so small that equation (2.8) can be linearized as Te =KmI sin(z∆θ)=KmIz∆θ=KmIγi−KmIzθ (2.12) Substituting equation (2.12) into equation (2.1), the dynamic model of SADA can be rew- ritten as J0θ̈+C0θ̇+K0θ=KmIγi−Tl (2.13) where K0 = KmIz and KmIγi is called the electromagnetic stiffness and exciting torque, re- spectively. From equation (2.13) we can get that: 1) the electromagnetic field between the stator and rotor of SADA can be equivalent to an electromagnetic spring-viscous damping system; 2) the rotor of SADA coupled with the load vibrates with viscous damping under the action of the exciting torqueKmIγi. Thus, the equivalent system diagram of SADA can be shown as Fig. 1. Fig. 1. Equivalent system of SADA Assuming that the stator of SADA is a rigid body, the disturbance torque generated by SADA acting on the satellite can be defined as Tdis =KmIγi−C0θ̇−K0θ (2.14) Thus, the disturbance model of SADA can be defined as J0θ̈+C0θ̇+K0θ=KmIγi−Tl Tdis =KmIγi−C0θ̇−K0θ (2.15) From equation (2.15) we can obtain that the dynamic model of SADA can be redefined as a vibration equation, and the electromagnetic stiffness appears in this vibration equation as an explicit expression which leads to the coupling relationship between the electromagnetic spring-viscous damping system and the solar array easily achieved by establishing the vibration equation of the coupling system. Thus, the disturbance torque aroused by the SADA driving solar array acting on the satellite can be facilely obtained. 3. Disturbance model of SADA driving a discrete flexible system 3.1. Modeling Assuming that the discrete flexible system is a torsion spring-damping-moment of inertia systemwithn-dofs, the equivalent systemof SADAdriving this discrete flexible system is shown in Fig. 2. Research on the disturbance generated by a solar array... 1005 Fig. 2. Equivalent system diagram of SADA driving a discrete flexible systemwith n-dofs Thus, the disturbance model of SADA driving this flexible system can be defined as Jβ̈+Cβ̇+Kβ=F Tdis =KmIγi−C0θ̇−K0θ (3.1) whereJ,C,K,F andβ is themassmatrix, dampingmatrix, stiffnessmatrix, excitation vector and angular displacement vector, respectively. All the expressions are shown in equation (3.2) J=       J0 0 · · · 0 0 J1 · · · 0 ... ... ... ... 0 0 · · · Jn       C=       C0+C1 −C1 · · · 0 −C1 C1+C2 · · · 0 ... ... ... ... 0 0 · · · Cn       K=       K0+K1 −K1 · · · 0 −K1 K1+K2 · · · 0 ... ... ... ... 0 0 · · · Kn       F=       KmIγi 0 ... 0       β=       θ θ1 ... θn       (3.2) 3.2. Simulation and analysis Taking a two-dof torsion spring-damping-moment of inertia system as the simulation object, Matlab/Simulink toolbox isused to simulate thedisturbancemodelof SADAdrivingthisdiscrete flexible system. Table 1 shows the simulation parameters of SADA and the two-dof flexible system. The simulation time and time step size is 100s and 1/2048s, respectively. Figures 3 to 5 illustrate the simulation results of the disturbance model of SADA driving this two-dof flexible system. It can be seen from Figs. 3a and 3b that the two-phase currents IA and IB are ladder cosine/sine curves, and the amplitude I is 0.3 A. 1006 J. Chen,W. Cheng Table 1. Simulation parameters of SADA and the two-dof flexible system Parameter Symbol Value Unit Rotor teeth number z 300 – Beat number p 4 − Subdivision number n 256 – Given angular velocity of rotor ω0 0.012 ◦/s Amplitude of two-phase current I 0.3 A Moment of inertia of rotor J0 0.0005 kg·m 2 Viscous damping coefficient C0 0.01 N·m·s/rad Motor torque constant Km 10 N·m/(A·rad) Electromagnetic stiffness K0 900 N·m/rad Moment of inertia of flexible system J1/J2 1/4 kg·m 2 Viscous damping coefficient C1/C2 0.02/0.01 N·m·s/rad Stiffness of torsion spring K1/K2 1000/2000 N·m/rad Fig. 3. Simulation results of the two-phase current (a) full view (b) partially enlarged view As shown in Fig. 4, the simulation result of the rotor angular displacement indicates that after 100 seconds of running, the angular displacement of the rotor is 1.2◦, which is equal to the given angular velocity ω0 (0.012 ◦/s) times the running time (100s). Fig. 4. Simulation result of the angular displacement of the rotor The simulation result of thedisturbance torque shown inFig. 5 indicates that thedisturbance frequencies in the frequency band of 0∼40Hz are 1.438Hz, 8.563Hz, 10.25Hz, 20.5Hz and 30.75Hz.Substituting theparameters shown inTable 1 into equation (3.2), thenatural frequency of the equivalent system of SADA driving the two-dof flexible system can be calculated easily, which is demonstrated as f1 =1.440Hz f2 =8.558Hz f3 =310.272Hz (3.3) Therefore, the disturbance frequencies 1.438Hz and 8.563Hz are the natural frequencies of the equivalent coupling system of SADA driving the flexible system. Research on the disturbance generated by a solar array... 1007 Fig. 5. Simulation result of the disturbance torque As mentioned above, SADA is to translate digital pulse sequences to precise movements of the rotor shaft. The input frequency of the digital pulse signal is given by fd = πω0 180 α = πω0 180 2π zpn =10.24Hz (3.4) Thus, the disturbance frequencies 10.25Hz, 20.5Hz and 30.75Hz are the input frequency of the digital pulse signal and its harmonics. As a consequence,we can obtain that the disturbance frequencies of SADAdriving a discrete flexible systemaremainly consists of twoparts: 1) natural frequencies of the equivalent systemof SADAdrivingtheflexible system;2) input frequencyof thedigital pulse signal and itsharmonics. These disturbance frequencies are consistentwith those obtained byYang et al. (2010) andChen et al. (2014), which confirmedly validates the effectiveness of the disturbancemodel proposed in this paper. 4. Simulation method of SADA driving a continuous flexible system Since the solar array is a typical continuous flexible system, equation (3.1) used to describe vibration of the equivalent system of the SADA driving solar array is difficult to establish. Taking a continuous flexible system,which is designed to simulate the solar array, as the studied object, the simulation method of the disturbance generated by SADA driving a continuous flexible system is demonstrated in this part. 4.1. Continuous flexible system The continuous flexible system is made of a thin aluminum plate, steel counterweight beam and connecting bracket. The size of the thin aluminum plate is 5m×1.6m×0.001m. The steel counterweight beam is arranged along the four sides of the thin aluminum plate continuously, whose section size is 0.03m×0.03m. The connection bracket is composed of two identical steel beams. The length of each steel beam is 1m, and its section size is 0.03m×0.03m as well. A schematic diagram of this continuous flexible system is shown in Fig. 6. ANSYS 12.1 is used to create the finite element model of this continuous flexible system. In the finite element model, element SHELL 181 is used to mesh the thin aluminum plate and element BEAM 4 is used to mesh the steel counterweight beam and the connecting bracket. What is more, a node should be laid in the installation point which is named as nodea. The modal shapeof thisfinite elementmodel is determined for theboundarycondition of constraining all-dofs of nodea. The calculation results of the modal analysis are listed in Table 2. 1008 J. Chen,W. Cheng Fig. 6. Schematic diagram of the continuous flexible system Table 2.Calculation results of the modal Modal order Frequency [Hz] Modal shape 1 0.503 first-order out-of-plane bending 2 1.364 first-order in-plane rocking 3 2.110 first-order torsional 4 2.222 second-order out-of-plane bending As the former four order modal shapes of the continuous flexible system designed are the same as the actual solar array mentioned in Zhu et al. (2014), this continuous flexible system has the ability to simulate the actual solar array. 4.2. Continuous flexible system coupling with SADA As discussed above, the electromagnetic field between the stator and rotor of SADA can be equivalent to an electromagnetic spring-viscous damping system, thus, the equivalent system of the continuous flexible system coupling with SADA can be shown as in Fig. 7. Fig. 7. Equivalent system of the continuous flexible system coupling with SADA Research on the disturbance generated by a solar array... 1009 ANSYS12.1 is used to create the finite elementmodel of this equivalent system. In this finite element model, themoment of inertia of the rotor J0 is meshed by using elementMASS 21; the electromagnetic torsion spring-viscous damping system ismeshedbyusing elementCOMBIN14; the finite element model of the continuous flexible system in the equivalent system is the same as the finite element model created above. Element MASS 21, element COMBIN 14 and the finite element model of the continuous flexible system should share one node, which is nodea. As the rotor of SADA has only one rotation dof, nodea has only this dof (Y -rotation) as well. Constraining all the other dofs of nodea and all-dofs of the other node of element COMBIN 14, then the creation process of the finite element model is accomplished. ANSYS 12.1 is used to find the modal shapes of this coupling finite element model. The calculation results of the modal analysis are presented in Table 3. Table 3.Calculation results of the modal analysis Modal order Frequency [Hz] Modal shape 1 0.503 first-order out-of-plane bending 2 0.606 first-order torsional 3 1.364 first-order in-plane bending 4 2.222 second-order out-of-plane bending Comparing with the calculation results shown in Table 2, the natural frequency of the first- order torsional modes moves forward and the other natural frequency remains the same, which is caused by the coupling effect between the continuous flexible system and the electromagnetic spring of SADA. 4.3. Simulation and analysis ANSYS 12.1/Transient is used to simulate the disturbance emitted by SADA driving the continuous flexible system designed. The simulation process is done on the basis of the coupling finite element model created in Section 4.2. In the simulation processing, firstly, the exciting torque KmIγi acts on node a; secondly, the simulation time and time step size is 20s and 1/2048s respectively; thirdly, each substep of the angular displacement and angular velocity of nodea are written down; lastly, substituting the angular displacement and angular velocity of nodea into equation (3.1), the disturbance torque of SADA driving this continuous flexible system is obtained. The simulation results are shown in Figs. 8 and 9. Fig. 8. Simulation result of the angular displacement of the rotor (node a) It can be seen from Fig. 8 that due to huge torsional vibration of the continuous flexible system coupling with SADA in the low frequency band (0.606Hz), the angular displacement of the rotor is no longer a straight line. However, after 20 seconds of running, the angular displacement of rotor is very close to 0.24◦, which equals to the given angular velocity of the rotor ω0 (0.012 ◦/s) times the running time (20s). What can be obtained from Fig. 9 are the disturbance frequencies in the frequency band of 0 ∼ 40Hz are 0.625Hz, 10.25Hz, 20.5Hz and 30.75Hz. Among them, 0.625Hz is the natural 1010 J. Chen,W. Cheng Fig. 9. Simulation result of the disturbance torque frequency of the first-order torsionalmode of the continuous flexible system coupledwith SADA shown in Table 3; 10.25Hz, 20.5Hz and 30.75Hz is the input frequency of the digital pulse signal and its harmonics. Due to low exciting energy of the exciting torque KmIγi beyond the low frequency band and the existence of the viscous damping inner SADA, the natural frequencies of high-order torsional modes of the continuous flexible system coupled with SADA do not appear in the disturbance frequencies. Therefore, thedisturbance frequencies ofSADAdrivingthe continuousflexible systemmainly consist of twoparts: 1) natural frequencies of the torsionalmodeof the continuous flexible system coupled with SADA in the low frequency band; 2) input frequency of the digital pulse signal and its harmonics. These conclusions are the same as those obtained for SADA driving athe discrete flexible system in Section 3.2. This proves that the simulationmethod of SADAdriving the continuous flexible system demonstrated in this part is right. 4.4. Summarizing The simulation method of the disturbance aroused by SADA driving the continuous flexible system can be summarized as follows: • Before the start of simulation, the finite element model of the continuous flexible system coupled with SADA, which can be equivalent to an electromagnetic spring-viscous dam- ping system, should be created firstly, and the coupling node named nodea. Appropriate boundary conditions should be formulated as well. • In the process of simulation, the exciting torque KmIγi should act on nodea, and each substepof theangular displacement andangular velocity ofnodea shouldbewrittendown. What is more, an appropriate simulation time and time step size should be chosen. • After the end of simulation, substituting the angular displacement and angular veloci- ty of node a into equation (3.1), the disturbance torque produced by SADA driving the continuous flexible system could be obtained. 5. Conclusions According to the above discussions, the following conclusions can be drawn: • By simplifying and linearizing the electromagnetic torque of SADA, the electromagnetic field between the stator and rotor of SADA can be found as equivalent to an electro- magnetic spring-viscous damping system, and the vibration equation of SADA can be established. Research on the disturbance generated by a solar array... 1011 • The disturbance model of SADA driving a discrete flexible system is achieved based on the vibration equation of SADA.A two-dof flexible system is used to simulate and analyze the disturbance model. The simulation results show that the disturbance frequencies of SADAmainly consist of two parts: 1) natural frequencies of the equivalent system of SADA driving the flexible system; 2) input frequency of the digital pulse signal and its harmonics. • Themethod to simulate the disturbance emitted by SADAdriving the continuous flexible system, which is designed to simulate the solar array, is proposed. And the disturbance frequencies of SADAmainly consist of two parts: 1) natural frequencies of the low-order torsional mode of the continuous flexible system coupling with SADA in the low frequency band; 2) input frequency of the digital pulse signal and its harmonics. In conclusion, all the achievements of this project will provide a theoretical basis for the prediction of disturbances emitted by the SADA driving solar array on the orbit. References 1. AnishN.K., KrishnanD.,Moorthi S., SelvanM.P., 2012, FPGAbasedmicrostepping sche- me for stepper motor in space-based solar power systems, Industrial and Information Systems, IEEE Conference, 1-5 2. BodsonM., Sato J.S., Silver S.R., 2006, Spontaneous speed reversals in steppermotors, IEEE Transactions on Control Systems Technology, 14, 2, 369-373 3. Chen J.P., Cheng W., Han W., 2014, Analysis and simulation of stepper motor disturbance considering structural coupling,Applied Mechanics and Materials, 526, 103-108 4. Chen Y.M., 2010, Application of shape memory alloys to control the vibration of solar panels, Mechanical Engineering and Automation, 1, 173-175 5. 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Zhu S.Y., Lei Y.J., Wu X.F., Zhang D.P., 2015, Influence analysis on dynamic characteristic parameters of solar array drive system, Journal of Vibration Engineering, 28, 2, 183-189 16. Zhu S.Y., Xie Y., Lei Y.J., 2014, Characteristic analysis of disturbance aroused by solar array tracking drive, Journal of National University of Defense Technology, 36, 1, 27-33 Manuscript received July 16, 2015; accepted for print January 28, 2016