 Advances in Technology Innovation , vol. 2, no. 2, 2017, pp. 29 - 33 29 Investigation of a Ball Screw Feed Drive System Based on Dynamic Modeling for Motion Control Yi-Cheng Huang * , Xiang-Yuan Chen Department of Mechatronics Engineering, National Changhua University of Education, Changhua, Taiwan. Received 01 February 2016; received in revised form 28 April 2016; accept ed 02 May 2016 Abstract This paper e xa mines the frequency response relationship between the ball screw nut preload, ball screw torsional stiffness variations and table mass effect for a single-a xis feed drive system. Identificat ion for the frequency response of an industrial ball screw drive system is very i m- portant for the precision motion when the v i- bration modes of the system are critica l for controller design. In this study, there is transla- tion and rotation modes of a ball screw feed drive system when positioning table is actuated by a servo motor. A lu mped dynamic model to study the ball nut preload variation and torsional stiffness of the ball screw drive system is d e- rived first. The mathe matica l mode ling and numerical simulat ion provide the informat ion of peak frequency response as the diffe rent levels of ball nut preload, ball screw torsional stiffness and table mass . The trend of increasing preload will indicate the abrupt peak change in fre - quency response spectrum ana lysis in some mode shapes . This study provides an approach to investigate the dynamic frequency response of a ball screw drive system, which provides sig- nificant information fo r better control perfor- mance when precise motion control is con- cerned. Keywor ds : Ball nut preload, ball screw drive system, dynamic modeling 1. Introduction Precision co mputer nu merica l control (CNC) mach ines are widely used in modern industry for mass production. The ball screws a re widely applied in the linear actuators of machinery and equipment because of the high effic iency, less backlash, easy lubrication, and easy maintenance. Since ba ll screw play a significant role in co n- verting rotary motion into linear mot ion Pre - loading is effective to eliminate backlash and increase the stiffness of ball screw for precision motion concerns. The dynamic frequency re- sponses of the feed drive system depends on the stiffness comb inations of the ball screw, ball nut, fixed support bearings, the fle xible coupler and the stiffness between the ball screw and the working table. Such frequency response results fro m the a xial mode shapes and torsional mode shape of the ball screw drive system when it is actuated by servo motor. Each mode shape af- fects and determines the motion control fre - quency response bandwidth when the control speed is limited and becomes a critica l issue. The working table mass and the bolt stiffness between the machine bed base and attached ground floor also plays an important role for the frequency response when the vibration mode of the CNC machine is concerned with precision accuracy. As in intelligent control fie ld, the control signals that fed into the controlled plant are based on the feedback control error that should be learnable. There fore, so me control efforts [1-2] are focused the design of the bandwidth of the filter that can filter out the un -learnable er- rors contents and can be get back to control system for bettering control h istory. Since the bandwidth of controlled system determines the motion speed response and its performance. The lu mped dynamic model derivation is significant in determin ing the bandwidth for motion control law that can be used for controller design. Since such frequency contents of compensated error are suggested to be within the bandwidth when control signals are actuating. Th is paper will derive the dynamic model of the ball screw feed drive system first and e xa mine the re lationship between the ball screw preload variation , ball screw torsional stiffness and effect of table mass. * Corresponding aut hor, Email: ychuang@cc.ncue.edu.t w Advances in Technology Innovation , vol. 2, no. 2, 2017, pp. 29 - 33 30 Copyright © TAETI Simu lation results will unveil the frequency response of the translational and torsional peak modes. Nu me rical simulat ion shows stable convergence by using hybrid partic le swarm optimization of iterative lea rning control [2] on this developed ball screw drive system. 2. Mathematical Model and Numer- ical Simulation 2.1. Dynamic Model of the Ball Screw Drive System To model the feed drive system, th is paper set differnt stiffness of the ball nut stiffness for diffe rent preload between the ba ll screw shaft and the ball nut. The presetting preload value can be deployed by inserting different ball size for single ball nut design or using disk spring that applied to the ball screw when double ball nut is the preference. Fig. 1 shows the picture of the in-lab single-a xis feed drive p latform. To analyze the dynamic characteristic of the ball screw system under different preload and vary- ing table mass, the feed drive system is modeled by a lumped parameter system shown in Fig . 2. Fig. 2 is the schematic illustration for the sin- gle-a xis ball screw feed drive system. In general, mechanica l systems have three passive linear components. The spring and the mass are ener- gy-storage elements, while the viscous damper is the dissipated energy. Both of the rotational and translation mechanical system modeled below are actuated by the servo motor torque, indicated as T. As the same derivation in [3], the overall stiffness of a ball screw feed drive sys- tem can be determined by the stiffness of the ball screw itself, which is comprised of the ball screw shaft, the ball nut, supporting bearings of the ball screw, and the stiffness between the ball screw and the working table. )( T bm gKmmQmmJ      (1) 𝐽𝑏 × �̈�𝑚 + 𝑄𝑚 × �̇�𝑏 + 𝑅 × [𝐾𝑛 × (𝑅 × 𝜃𝑏 + 𝑋𝑏 − 𝑋𝑡)] = 𝐾𝑔 × (𝜃𝑚 − 𝜃𝑏 ) (2) 𝑀𝑏 × �̈�𝑏 + 𝐵𝑏 × (�̇�𝑏 − 0) + 𝐾𝑒 × (𝑋𝑏 − 0) + 𝐾𝑛 × (𝑅 × 𝜃𝑏 + 𝑋𝑏 − 𝑋𝑡 ) = 0 (3) tXb X b RnK tXtBtXtM )( )0(     (4) Rearranging Eqs (1)-(4), we have [ 𝑀𝑡 0 0 0 0 𝑀𝑏 0 0 0 0 𝐽𝑏 0 0 0 0 𝐽𝑚 ] × [ �̈�𝑡 �̈�𝑏 �̈�𝑏 �̈�𝑚] + [ 𝐵𝑡 0 0 0 0 𝐵𝑏 0 0 0 0 𝑄𝑏 0 0 0 0 𝑄𝑚 ] × [ �̇�𝑡 �̇�𝑏 �̇�𝑏 �̇�𝑚] + [ 𝐾𝑛 −𝐾𝑛 −𝐾𝑛𝑅 0 −𝐾𝑛 𝐾𝑒 + 𝐾𝑛 𝐾𝑛 𝑅 0 −𝐾𝑛 𝑅 𝐾𝑛 𝑅 𝐾𝑔 + 𝐾𝑛 𝑅 2 −𝐾𝑔 0 0 −𝐾𝑔 𝐾𝑔 ] × [ 𝑋𝑡 𝑋𝑏 𝜃𝑏 𝜃𝑚 ] = [ 0 0 0 𝑇 ] (5) where the stiffness matrix is different from [3]. Fig. 1 The in-house single axis platform Fig. 2 Illustration for the schematic d iagra m of the single-a xis lu mped para meters ball screw drive system Advances in Technology Innovation , vol. 2, no. 2, 2017, pp. 29 - 33 31 Copyright © TAETI 2.2. Numerical Simulation Table 1 list the simu lated para meters and associated values used in Eqs (1) -(4). The dy- namic equation of the single-a xis feed drive system model with varied p reload and table mass can be expressed in a compact form: [𝑀]{�̈�} + [𝐶]{�̇�} + [𝐾]{𝑢} = 𝑓 (6) where [M], [C], and [K] a re the 4x4 square ma - trices, refe rred as the mass (or the mo ment of inertia ), the viscous damping, and the stiffness matrices, respectively. {u} represents a four degree of freedom model. It consists with the Xt , Xb , θb , and θm for the displace ment of the working table, a xia l displace ment of the ball screw, rotation angle of the ball screw, and the rotation angle of the motor, respectively. Table 1 Important parameters of ball screw drive system Parameters value working table mass (M t) 47.09Kg ball screw mass (M b) 9Kg inertia moment of the motor (Jm) 4.45× 10−4kgm2 inertia moment of the ball screw (Jb) 1.3× 10−3kgm2 equivalent axial stiffness of ball screw shaft (Ke) 1.8663× 107N/m stiffness of the ball nut (Kn) 2.3345× 108N/m torsional stiffness of the ball screw (Kg) 3.49× 108N/m viscous damp ing coefficient of the guide way of the working table (Bt) 10N s/m viscous damp ing coefficient of the supp orting bearin g of the ball screw (Bb) 10N s/m rotational viscous damp ing co- efficient of the motor (Qm) 0N ms rotational viscous damp ing co- efficient of the sup p ort bearing of ball screw (Qb) 0N ms angle conversion axial dis- p lacement of the constant (R) 0.0025 motor torque (T) 1.8× 10−3Nm/𝑠2 disp lacement of the working table (Xt) State Variable (m) axial disp lacement of the ball screw (Xb) State Variable (m) rotation angle of the motor (θm) State Variable (rad) rotation angle of the ball screw (θb) State Variable (rad) The homogeneous solution of Eq s (5) rep- resents the transient response of the lumped system whereas the forc ing function of applied motor torque renders the table positioning. Ho mogeneous solution of the Equation (2.1.5) results in four e igenvectors V1, V2, V3, and V4 associated with each e igenvalue ( λ = ω2 ) of 3.2124 × 104 , 3.3322 × 106 , 1.0699 × 1011 ,3.421 × 107(rad2 /s 2). V1={ −0.7299 −0.6836 0 0 }V2={ 0.1762 −0.9844 0 0 } V3 ={ 0 0 0.9482 −0.3176 }V4={ 0 0 0.7067 0.7076 } (7) The four eigenvectors corresponding to the Eigen frequencies of 28.52 Hz, 290.52 Hz, 52059 Hz, 930.9 Hz w is calculated by 2% of the rated dynamic load. These eigenvectors repre- sent the mode shapes of the ball screw feed drive system. The first three significant Eigen fre - quencies are re lated to the three resonant fre- quencies. The first and second modes are fro m the a xia l vibration. The third mode is fro m the torsional vibration. In Eqs (7), the first mode of the working table and the ball scre w is moving in phase while the second mode is moving out of phase. Fig. 3 shows the bode plot of the dynamic system based on diffe rent Kn values. The pre- load variation is simulated fro m 2% , 4%, 6% , 8% to 10% of the rated dynamic loading of the ball screw. Fig. 3 Bode plot of the ball screw drive system based on the preload value of 2% (b lue), 4%(green), 6%(red), 8%(cyan) and 10%(purple) of the rated dynamic loading Advances in Technology Innovation , vol. 2, no. 2, 2017, pp. 29 - 33 32 Copyright © TAETI En large ment of the first, second and third modes of the bode plot, the three frequencies are ranging fro m 30.5 Hz to 30.8 Hz, 294 Hz to 380 Hz and 52000 Hz to 55000 Hz respectively. It is obvious that the second translational mode will be affected more than the first mode when the ball nut preload is vary ing. The solution of the fourth mode is about 930 Hz, the contribution of this mode is not significant even though the preload is varied. Fig. 4 The enla rge ment of the first mode of the bode plot in Fig. 3 based on different ball nut stiffness Fig. 5 The enlarge ment o f the second mode of the bode plot in Fig. 3 based on diffe rent ball but stiffness Fig. 6 The en large ment of the third mode of bode plot in Fig. 3 based on different ba ll nut stiffness As stated, the third eigenvector indicates the torsional vibration. Since the servo motor drives the ball screw through a coupler providing damping and stiffness. The torsional stiffness of the ball screw is investigated by calculating Kg by πdr 4G/32L . Fig. 7 shows the variations of Kg in the range of ± 10%. As shown, the third mode de monstrates large frequency shift with increasing the torsional stiffness of the ball screw. Fig. 8 indicates the frequency shift is range fro m 50700 Hz to 58000 Hz, wh ile the first and second modes are not changed notice- ably. Fig. 7 Bode plot of the ball screw torsional stiffness 100% ( blue), 105% (green), 110%(red), 95%(cyan), 90%( purple) Fig. 8 The enla rge ment of the torsional mode of the bode plot in Fig. 7 based on different ball screw torsional stiffness Figs. 9-11 detail the effect of the table mass. The characteristic frequency shifts from the 24Hz to 30.5Hz when the table mass increases fro m 47.09 kg to 94.18 kg. As predicted, the table mass preserves the effect on the first bandwidth in the translational mode and some effect on the second mode. Increasing table mass does not affect the bandwidth of the third to r- sional mode. Fig. 12 shows the numerical si m- ulation plot of the convergence error when the ball scre w drive system is controlled by hybrid particle swa rm optimization for the dynamic bandwidth tuning of an iterative learning con- trol. Advances in Technology Innovation , vol. 2, no. 2, 2017, pp. 29 - 33 33 Copyright © TAETI Fig. 9 Bode plot of the working table mass 47.09Kg(blue), 70.63Kg(green), 94.18Kg(red) Fig. 10 The en large ment of the first mode of the bode plot in Fig. 9 based on dif- ferent working table mass Fig. 11 The enla rge ment of the second mode of the bode plot in Fig. 9 based on different working table mass Fig. 12 Plot of a convergence error of the nu- me ric a l s imu lat ion by using HPSO-ILC [2] for ball screw drive system 3. Conclusions A lu mped dynamic model fo r describing diffe rent ball nut preload level, ba ll screw tor- sional stiffnesss and the table mass effects of the ball screw feed drive system is derived and numerically simulated. Based on the different percent of the preload and table mass , the fre- quency spectrum analysis of the nu merical si m- ulation provides the limits and constraints of bandwidth tuning for motion control applications. The preload variation can be diagnosed by the peak frequency change and the magnitude of the peak frequency in a specific frequency range when the a xial mode or the torsional mode is excited. The derivation and numerical simulation results of the lumped dynamic mode l provides significant informat ion in determining the zero phase bandwidth tuning. Application of a hybrid particle swarm optimization iterat ive learning control law deploys successfully when the fre- quency contents of the compensated error was constrained in every control actuation. Acknowledgement This work was supported by MOST Grant 104-2221-E-018-015 for which the authors are very much grateful. References [1] M. S. Tsai, C. L. Yen, and H. T . Yau, “In- tegration of an empirica l mode decompos i- tion algorith m with iterative learn ing con- trol for h igh-precision machining,” IEEE/ASM E Transaction on Mechatronic, vol. 18, no. 3, pp. 878-886, 2013. [2] Y. C. Huang, Y. W. Su, and P. C. Chuo, “Iterative learn ing control bandwidth tuning using th e pa rt ic le s wa rm opt imiza t ion techn ique fo r h igh p rec is ion mot ion,” Microsystem Technologies. DOI: 10.1007/ s00542-015-2649-6, 2015. [3] G. H. Feng and Y. L. Pan, “Investigation of ball screw p reload variation based on d y- namic modeling of a preload adjustable feed-drive system and spectrum analysis of b a ll-n uts sense d v ib rat ion s igna l s,” In- ternational Journal of Machine Too ls & Manufacture, vol. 52, no. 1, pp. 85-96, 2012.