Investigation of a Ball Screw Feed Drive System Based on Dynamic Modeling for Motion Control

Transcription

1 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 1 Feruary 216; received in revised form 28 April 216; accepted 2 May 216 Astract This paper examines the frequency response relationship etween the all screw nut preload, all screw torsional stiffness variations and tale mass effect for a single-axis feed drive system. Identification for the frequency response of an industrial all screw drive system is very important for the precision motion when the viration modes of the system are critical for controller design. In this study, there is translation and rotation modes of a all screw feed drive system when positioning tale is actuated y a servo motor. A lumped dynamic model to study the all nut preload variation and torsional stiffness of the all screw drive system is derived first. The mathematical modeling and numerical simulation provide the information of peak frequency response as the different levels of all nut preload, all screw torsional stiffness and tale mass. The trend of increasing preload will indicate the arupt peak change in frequency response spectrum analysis in some mode shapes. This study provides an approach to investigate the dynamic frequency response of a all screw drive system, which provides significant information fo r etter control performance when precise motion control is concerned. Keywords: Ball nut preload, all screw drive system, dynamic modeling 1. Introduction Precision computer numerical control (CNC) machines are widely used in modern industry for mass production. The all screws are widely applied in the linear actuators of machinery and equipment ecause of the high efficiency, less acklash, easy lurication, and easy maintenance. Since all screw play a significant role in converting rotary motion into linear mot ion Preloading is effective to eliminate acklash and increase the stiffness of all screw for precision motion concerns. The dynamic frequency responses of the feed drive system depends on the stiffness cominations of the all screw, all nut, fixed support earings, the flexile coupler and the stiffness etween the all screw and the working tale. Such frequency response results from the axial mode shapes and torsional mode shape of the all screw drive system when it is actuated y servo motor. Each mode shape affects and determines the motion control frequency response andwidth when the control speed is limited and ecomes a critical issue. The working tale mass and the olt stiffness etween the machine ed ase and attached ground floor also plays an important role for the frequency response when the viration mode of the CNC machine is concerned with precision accuracy. As in intelligent control field, the control signals that fed into the controlled plant are ased on the feedack control error that should e learnale. Therefore, some control efforts [1-2] are focused the design of the andwidth of the filter that can filter out the un-learnale errors contents and can e get ack to control system for ettering control history. Since the andwidth of controlled system determines the motion speed response and its performance. The lumped dynamic model derivation is significant in determining the andwidth for motion control law that can e used for controller design. Since such frequency contents of compensated error are suggested to e within the andwidth when control signals are actuating. This paper will derive the dynamic model of the all screw feed drive system first and examine the relationship etween the all screw preload variation, all screw torsional stiffness and effect of tale mass. * Corresponding author, ychuang@cc.ncue.edu.tw 29

2 Simulation results will unveil the frequency response of the translational and torsional peak modes. Numerical simulation shows stale convergence y using hyrid particle swarm optimization of iterative learning control [2] on this developed all screw drive system. 2. Mathematical Model and Numerical Simulation 2.1. Dynamic Model of the Ball Screw Drive System To model the feed drive system, this paper set differnt stiffness of the all nut stiffness for different preload etween the all screw shaft and the all nut. The presetting preload value can e deployed y inserting different all size for single all nut design or using disk spring that applied to the all screw when doule all nut is the preference. Fig. 1 shows the picture of the in-la single-axis feed drive platform. To analyze the dynamic characteristic of the all screw system under different preload and varying tale mass, the feed drive system is modeled y a lumped parameter system shown in Fig. 2. Fig. 2 is the schematic illustration for the single-axis all screw feed drive system. In general, mechanical systems have three passive linear components. The spring and the mass are energy-storage elements, while the viscous damper is the dissipated energy. Both of the rotational and translation mechanical system modeled elow are actuated y the servo motor torque, indicated as T. As the same derivation in [3], the overall stiffness of a all screw feed drive system can e determined y the stiffness of the all screw itself, which is comprised of the all screw shaft, the all nut, supporting earings of the all screw, and the stiffness etween the all screw and the working tale. M t X t B t ( X t ) K n ( R X X t ) Rearranging Eqs (1)-(4), we have M t X t M [ X ] + J θ J m [ θ m] B t X t B [ X ] + Q θ Q m [ θ m] K n K n K n R K n K e + K n K n R K n R K n R K g + K n R 2 K g [ K g K g ] X t X [ θ ] = [ ] θ m T where the stiffness matrix is different from [3]. Fig. 1 The in-house single axis platform (4) (5) J m m Q m m K g ( m ) T J θ m + Q m θ + R [K n (R θ + X X t )] = K g (θ m θ ) M X + B (X ) + K e (X ) + K n (R θ + X X t ) = (1) (2) (3) Fig. 2 Illustration for the schematic diagram of the single-axis lumped parameters all screw drive system Copyright TAETI 3

3 2.2. Numerical Simulation Tale 1 list the simulated parameters and associated values used in Eqs (1)-(4). The dynamic equation of the single-axis feed drive system model with varied preload and tale mass can e expressed in a compact form: [M]{u } + [C]{u } + [K]{u} = f (6) where [M], [C], and [K] are the 4x4 square matrices, referred as the mass (or the moment of inertia), the viscous damping, and the stiffness matrices, respectively. {u} represents a four degree of freedom model. It consists with the X t, X, θ, and θ m for the displacement of the working tale, axial displacement of the all screw, rotation angle of the all screw, and the rotation angle of the motor, respectively. Tale 1 Important parameters of all screw drive system Parameters value working tale mass (Mt) 47.9Kg all screw mass (M) 9Kg inertia moment of the motor (Jm) kgm 2 inertia moment of the all screw (J) kgm 2 equivalent axial stiffness of all screw shaft (Ke) N/m stiffness of the all nut (Kn) N/m torsional stiffness of the all screw (Kg) N/m viscous damping coefficient of the guide way of the working 1N s/m tale (Bt) viscous damping coefficient of the supporting earing of the all 1N s/m screw (B) rotational viscous damping coefficient of the motor (Qm) N ms rotational viscous damping coefficient of the support earing of N ms all screw (Q) angle conversion axial displacement of the constant (R).25 motor torque (T) Nm/s 2 displacement of the working tale (Xt) State Variale (m) axial displacement of the all screw (X) State Variale (m) rotation angle of the motor (θm) State Variale (rad) rotation angle of the all screw (θ) State Variale (rad) The homogeneous solution of Eqs (5) represents the transient response of the lumped system whereas the forcing function of applied motor torque renders the tale positioning. Homogeneous solution of the Equation (2.1.5) results in four eigenvectors V1, V2, V3, and V4 associated with each eigenvalue ( λ = ω 2 ) of , , , (rad 2 /s 2 ) V1={ }V2={ } V3 ={ }V4={ } (7) The four eigenvectors corresponding to the Eigen frequencies of Hz, Hz, 5259 Hz, 93.9 Hz w is calculated y 2% of the rated dynamic load. These eigenvectors represent the mode shapes of the all screw feed drive system. The first three significant Eigen frequencies are related to the three resonant frequencies. The first and second modes are from the axial viration. The third mode is from the torsional viration. In Eqs (7), the first mode of the working tale and the all screw is moving in phase while the second mode is moving out of phase. Fig. 3 shows the ode plot of the dynamic system ased on different K n values. The preload variation is simulated from 2%, 4%, 6%, 8% to 1% of the rated dynamic loading of the all screw. Fig. 3 Bode plot of the all screw drive system ased on the preload value of 2%(lue), 4%(green), 6%(red), 8%(cyan) and 1%(purple) of the rated dynamic loading 31 Copyright TAETI

4 Enlargement of the first, second and third modes of the ode plot, the three frequencies are ranging from 3.5 Hz to 3.8 Hz, 294 Hz to 38 Hz and 52 Hz to 55 Hz respectively. It is ovious that the second translational mode will e affected more than the first mode when the all nut preload is varying. The solution of the fourth mode is aout 93 Hz, the contriution of this mode is not significant even though the preload is varied. the all screw is investigated y calculating K g y πd r 4 G/32L. Fig. 7 shows the variations of Kg in the range of ± 1%. As shown, the third mode demonstrates large frequency shift with increasing the torsional stiffness of the all screw. Fig. 8 indicates the frequency shift is range from 57 Hz to 58 Hz, while the first and second modes are not changed noticealy. Fig. 4 The enlargement of the first mode of the ode plot in Fig. 3 ased on different all nut stiffness Fig. 7 Bode plot of the all screw torsional stiffness 1%( lue), 15%(green), 11%(red), 95%(cyan), 9%( purple) Fig. 5 The enlargement of the second mode of the ode plot in Fig. 3 ased on different all ut stiffness Fig. 6 The enlargement of the third mode of ode plot in Fig. 3 ased on different all nut stiffness As stated, the third eigenvector indicates the torsional viration. Since the servo motor drives the all screw through a coupler providing damping and stiffness. The torsional stiffness of Fig. 8 The enlargement of the torsional mode of the ode plot in Fig. 7 ased on different all screw torsional stiffness Figs detail the effect of the tale mass. The characteristic frequency shifts from the 24Hz to 3.5Hz when the tale mass increases from 47.9 kg to kg. As predicted, the tale mass preserves the effect on the first andwidth in the translational mode and some effect on the second mode. Increasing tale mass does not affect the andwidth of the third torsional mode. Fig. 12 shows the numerical simulation plot of the convergence error when the all screw drive system is controlled y hyrid particle swarm optimization for the dynamic andwidth tuning of an iterative learning control. Copyright TAETI 32

5 3. Conclusions Fig. 9 Bode plot of the working tale mass 47.9Kg(lue), 7.63Kg(green), 94.18Kg(red) Fig. 1 The enlargement of the first mode of the ode plot in Fig. 9 ased on different working tale mass A lumped dynamic model for descriing different all nut preload level, all screw torsional stiffnesss and the tale mass effects of the all screw feed drive system is derived and numerically simulated. Based on the different percent of the preload and tale mass, the frequency spectrum analysis of the nu merical simulation provides the limits and constraints of andwidth tuning for motion control applications. The preload variation can e diagnosed y the peak frequency change and the magnitude of the peak frequency in a specific frequency range when the axial mode or the torsional mode is excited. The derivation and numerical simulation results of the lumped dynamic model provides significant information in determining the zero phase andwidth tuning. Application of a hyrid particle swarm optimization iterative learning control law deploys successfully when the frequency contents of the compensated error was constrained in every control actuation. Acknowledgement This work was supported y MOST Grant E for which the authors are very much grateful. References Fig. 11 The enlargement of the second mode of the ode plot in Fig. 9 ased on different working tale mass Fig. 12 Plot of a convergence error of the numerical s imulation y using HPSO-ILC [2] for all screw drive system [1] M. S. Tsai, C. L. Yen, and H. T. Yau, Integration of an empirical mode decomposition algorith m with iterative learn ing control for high-precision machining, IEEE/ASME Transaction on Mechatronic, vol. 18, no. 3, pp , 213. [2] Y. C. Huang, Y. W. Su, and P. C. Chuo, Iterative learning control andwidth tuning using the particle s warm optimization technique for high precis ion motion, Microsystem Technologies. DOI: 1.17/ s , 215. [3] G. H. Feng and Y. L. Pan, Investigation of all screw preload variation ased on dynamic modeling of a preload adjustale feed-drive system and spectrum analysis of all-nuts sensed viration s ignals, International Journal of Machine Tools & Manufacture, vol. 52, no. 1, pp , Copyright TAETI