Frequency Response of Vibration Control with Linear Actuator and Permanent Magnet Phaisarn SUDWILAI, Koichi OKA and Yuta HIROKAWA

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1 The th MAGDA Conference in Pacific Asia, Noeber 4-6,, Kaohsiung, Taiwan 3B5 Frequency Resonse of Vibration Control with Linear Actuator and Peranent Magnet Phaisarn SUDWILAI, Koichi OKA and Yuta HIROKAWA This aer resents the frequency resonse of the ibration control syste. The syste rototye corises a linear actuator, a controller, sensors and eranent agnets. The reduction of ibrations and deforations of thin steel sheets was accolished by controlling an air-ga between the eranent agnets and the steel sheets. The feedbac control systes were designed using the Linear Quadratic Regulator (LQR) ethod and robust control ( H ). The design of the controller and the frequency resonses of the roosed ibration suression echanis hae been erfored through both nuerical analyses and exerients. A sinusoidal signal was eloyed as a continuous disturbance to the syste rototye where different conditions of disturbance were carried out by arying the signal frequency fro 3 to 6 rad/sec. The frequency resonses shown in the Bode diagras show that the agnitude of ibration body dislaceent in the case without feedbac control is higher than the case with feedbac control. In conclusion, the feedbac control has been alied successfully, and the roosed ibration suression echanis is effectie for this ibration control syste. Keywords: ibration control, frequency resonse, eranent agnet, linear actuator, linear quadratic regulator, H.. Introduction Corresondence: Koichi OKA, Deartent of Intelligent Mechanical Systes Engineering, Kochi Uniersity of Technology, Miyanoochi 85, Tosayaada-cho, Kai city, Kochi 78-85, Jaan eail: oa.oichi@ochi-tech.ac.j In the rocess of lating, coating and rolling of steel sheets, ibration in coneyances often leads to robles of deforation, eeling and non-unifor roducts, because the sheets are ery flexible. As the results, a ibration suressor with echanical contacts is not suitable as a countereasure. Objects are easily daaged due to their aterial aeu when they hae just been rolled, coated or lated. Therefore, a noncontact suression echanis is ore suitable for controlling the iniiation of deforation of the steel sheets and can iniie such robles. Noncontact ibration control ethods which use attractie force of electroagnets hae already been roosed ()-(4). The rincial weaness of these ethods is that the control range is ery constricted, because the attractie force of the agnet aries in inerse roortion to the square of airga length. If the ibration alitude of the object is large, iniiing deforation by eans of electroagnets is ineffectie for the control of the object. On the other hand, a control ethod using eranent agnets located at the desired iron sheet assage has also been roosed (5). Howeer, this ethod uses assie control and can not be successfully suressed under large disturbances. A noncontact susension syste using eranent agnets and linear actuators has been roosed (6). The ey eleent of the design is the force control echanis. A linear actuator dries a eranent agnet and aries the air-ga between the agnet and the object. Variation of the sie of the air-ga changes the attractie force. Here the ibration control syste uses this tye of force control echanis for ibration suression, since the control range is alost the sae as of the actuator stroe length. In this syste ibration control range is exected to be corresondingly wide. In this aer, ibration suression echanis and the frequency resonse of the ibration control systes of the roosed were studied. The structure of the ibration suression syste is introduced and the ai of the syste is shown. The rincile of control echanis is exlained and exeriental exainations are carried out to deonstrate the frequency resonse and feasibility of the control ethod.. Vibration Suression Mechanis A scheatic diagra of a ibration suression echanis for lating or coating rocess is shown in Fig.. The steel sheet is fed fro the right side of the figure and is directed uwards by a roller. While the steel sheet is being fed into the solution bath, lating or coating is carried out. After the lating rocess is coleted, the steel is seasoned or cooled in a ertical feed. During the seasoning rocess, the steel is esecially sensitie to deforation. Consequently ibration control in the seasoning rocess is ery iortant. In this aer, a ibration suression echanis, as shown in Fig. is roosed. The echanis consists of two eranent agnets, two linear actuators, sensors which easure the steel sheet dislaceent, and a controller. The sensors and controller are oitted in the figure. The intention of the syste is to reduce ibration caused by the roller feed echanis, air wiers, and

2 The th MAGDA Conference in Pacific Asia, Noeber 4-6,, Kaohsiung, Taiwan high seed lating. Air wiers, reresented by the green eleent in the figure, adjust the thicness of the lating. The design of the rototye of ibration suression echanis is illustrated in Fig.. In this figure, the ertical sheet in the center reresents the steel sheet and is labeled. The agnets on both sides are drien in the horiontal direction by actuators, which are labeled. The rincile of the ibration suression ethod is as follows. When the ibration body swings to the right of the equilibriu osition, as shown in Fig. (a), the left actuator () dries the left agnet to the right, the dislaceent between the left agnet and ibration body is reduced, and the left attractie force becoes larger. Consequently, the agnetic force caused the ibration body return to the original equilibriu osition. Siilarly, when the ibration body swings to the left, as shown in Fig. (b), the right actuator dries the right agnet to the left, the attractie force on the right side of the ibration body becoes larger and this force caused the object to return to the equilibriu osition. The attractie force of the eranent agnet can suress ibration and deforation of the object. Steel sheet Linear actuator Fig. Vibration suression echanis Peranent agnet. ROLLER Linear actuator Attractie force 3. Exeriental Deice An exeriental syste to exaine the erforance of the roosed ibration control echanis was anufactured. The hotograh of the syste is shown in Fig. 3. The syste consists of a ibration body, two eranent agnets installed on a slider, a oice coil otor () and sensors. The ibration body is structured as a arallel sring ade by hoshor brone and installs steel lates on the both end-sides facing to the eranent agnets. Two eranent agnets are installed on the slider drien by the. Two sensors are installed for easuring the dislaceents of the ibration body and the slider. A laser sensor is used for the otion of the ibration body and an eddy current sensor is used for the otion of agnets which is the sae as the oeent. The control syste bloc diagra is shown in Fig. 3. As seen in the figure, the controller is using a DSP. It calculates the generating force of the fro two sensor signals and the outut is current signal for the filter. The outut of filter is current signal for alifier. The alifier changes the signal to the current for the and the changes the currents to the force roortionally. For DSP rograing, MATLAB with SIMULINK and Pass/C67 were used. In this aer, the syste was odeled in order to analye and adot the linear control theory, to synthesie the control syste and frequency resonse. For odeling, the secifications should be figured out. The relationshi between attractie force of eranent agnet, force of ibration body and resultant force with dislaceent of a eranent agnet and a steel late on the ibration body set to is shown in Fig. 5. As shown in the figure, the relationshi of attractie force of eranent agnet and dislaceent is nonlinear. Disturbance force was set to sinusoidal function by dc otor drie as shown in dash line in Fig. 3. The relationshi between oltage of dc otor and disturbance frequency as shown in Fig. 6. (a) (a) (a3) Peranent agnet Attractie force Disturbance (b) (b) (b3) Fig. Method of controlling ibration (a)-(a3) is swing to right (b)-(b3) is swing to left Slider Fig. 3 Photograh of syste rototye

3 The th MAGDA Conference in Pacific Asia, Noeber 4-6,, Kaohsiung, Taiwan.5.5 Dislaceent of ibration body Alifier Filter DSP BOARD MATLAB PC Fig. 4 Bloc-diagra of a syste Attractie force (N) Resultant force Dislaceent of eranent agnet 4. Syste Modeling In the odel, the otion of the ibration body, as suorted by a arallel sring is assued to be translational. An illustration of the odel is shown in Fig. 7. Other secification alues of the rototye are indicated in Table The sybols used in the odel are:, : dislaceent of the ibration body and eranent agnet, d : air-ga length when the ibration body is centered between the agnets, f : resultant force of agnets, fc, f d : actuator generating force and the disturbance force. The resultant force is assued as inerse roortion to the square of ga length. f. () ( d ) ( d ) The otion equation of the ibration body is c f fd. () The otion equation of the slider is c f f. (3) c Dislaceent () Fig. 5 Attractie force of eranent agnet ibration body and resultant force, air-ga of the equilibriu osition is 6 Frequency (rad/sec) Disturbance d d f f V dc (V) Fig. 6 Relationshi between dc-oltage suly to dc otor and frequency of disturbance force Fig. 7 Syste odelling f c

4 The th MAGDA Conference in Pacific Asia, Noeber 4-6,, Kaohsiung, Taiwan Table Paraeters of the ibration control syste. c : daing coefficient of the. Ns/ ibration body c : daing coefficient of the Ns/ eranent agnet : constant of ibration body 5 N/ : constant of eranent agnet N/ : ass of ibration body.47 g : ass of eranent agnet.9 g : constant of lineariation The inuts of the syste are defined as the forces of the actuator () f c and the disturbance f d. The oututs are the dislaceent of the ibration body and the slider. The odel can be reresented by Eqs. (), () and (3). By lineariation of the attractie force of the agnets, the resultant force can be reresented by f ( ). (4) Equations () and (3) becoe as ( ), (5) c fd c f ( ). (6) c The syste can be considered as a bloc diagra shown in Fig. 6. There are PD controllers in the loos of ibration body and the agnets. The feedbac gains are calculated by ean of the LQR control theory. Using the LQR ethod, a state sace odel can be deried fro Eqs. (4) (6) as; x Ax Bu, (7) y Cx. (8) Where, u fc, y [ ] T,state ector x [ ' ] T. f d f c N/ s cs c f - s cs K K s PP Fig. 8 Bloc-diagra of a ibration control syste DP K K s PV DV PD Controller 5. Siulation Results Siulation results were carried out with nonlinear force. The results of the frequency resonse of ibration suression were recorded when the frequency of disturbance are set at 3 and 3 rad/sec. The conditions of the siulations were; ) without feedbac control ) with feedbac control when the controller was designed by LQR ethod and 3) with feedbac control when the controller was designed by robust control ( H ), the weighting atrices for calculating the otial gains by LQR ethod were equal Dislaceent (). sec Fig. 9 Dislaceent signal of ibration body, the feedbac control (LQR ethod), disturbance frequency set to 3 rad/sec Dislaceent ().4 sec. sec Fig. Dislaceent signal of ibration body, the feedbac control (LQR ethod), disturbance frequency set to 3 rad/sec. According to the siulation results in case of frequency of disturbance are set to 3 and 3 rad/sec as shown in Figs. 9 and. The red line is dislaceent of ibration body the syste without feedbac control. The blue line is the syste with feedbac control. It should be noted that the alitude of ibration body when the syste with feedbac control and without feedbac control are sall different. Siulation results shows that the feedbac control designed by LQR ethod cannot suress disturbance

5 The th MAGDA Conference in Pacific Asia, Noeber 4-6,, Kaohsiung, Taiwan at low and high frequency. Howeer, in case the controller was designed by robust control ( H ) the syste can suress disturbance ore than when the controller was designed by LQR ethod, as shown in Figs. and Dislaceent ().. sec Gain (db) H LQR 3 Frequency (rad/sec) Fig. 3 Bode diagra of ibration body, the syste conditions are without feedbac control and with feedbac control (LQR ethod and H ) -.. sec Fig. Dislaceent signal of ibration body, the feedbac control ( H ), disturbance frequency set to 3 rad/sec.... Dislaceent () 6. Exeriental Results Vibration control exerients were also conducted using the exeriental rototye shown in Fig. 3. Trial under arious conditions for studied in frequency resonses such as frequency of disturbance are 3, 36 and 57 rad/sec are shows in Figs. 4, 5 and 6, the feedbac control was designed by LQR ethod. Bode diagra shown in Fig Dislaceent ().sec sec Fig. Dislaceent signal of ibration body, the feedbac control ( H ), disturbance frequency set to 3 rad/sec. According to siulation results in case of feedbac control was designed by robust control ( H ) as shown in Figs. and, all of results roed that the erforance of feedbac control was designed. The bode diagra of ibration control syste when the conditions were; ) without feedbac control ) with feedbac control (LQR ethod) and 3) with feedbac control ( H ), as shown in Fig sec Fig. 4 Dislaceent signal of ibration body, the feedbac control, frequency of disturbance force set to 3 rad/sec Dislaceent (). sec Fig. 5 Dislaceent signal of ibration body, the feedbac control, frequency of disturbance force set to 36 rad/sec

6 The th MAGDA Conference in Pacific Asia, Noeber 4-6,, Kaohsiung, Taiwan Dislaceent () sec ( H ). A odel was set u for feasibility analysis; the syste rototye was anufactured for exeriental confiration. Fro the results of the exerients, according to the odel based on analysis of rototye erforance, feedbac control were achieed by closedloo stability and deonstrated the efficiency of the feedbac control. Further research is required to iroe the exeriental design by alying robust control ( H ). References Fig. 6 Dislaceent signal of ibration body, the feedbac control, frequency of disturbance force set to 57 rad/sec. According to the exeriental results in the case of disturbance are set to 3, 36 and 57 rad/sec as shown in Figs. 4, 5 and 6, the blue lines are dislaceent of ibration body in case of without feedbac control. The red lines are dislaceent of ibration body in case of with feedbac control. As the results, in case without feedbac control alitude of ibration body higher than with feedbac control. Howeer, the syste that design by LQR ethod cannot suress disturbance dislaceent. 3 4 Gain (db) LQR Frequency (rad/sec) Fig. 7 Bode diagra of ibration body, the syste condition is with feedbac control. Figure 7 shows bode diagra of ibration body in case of without feedbac control and with feedbac control was designed by LQR ethod. Fro these results, it can be obsered that the gains of ibration body in case of with feedbac control are lower than without feedbac control. 7. Conclusions The frequency resonse of ibration suression control schee based on linear actuator and eranent agnet is roosed in this aer. The feedbac control was designed using the LQR ethod and robust control [] Matsuda K., Yoshihashi M., Oada Y. and Tan ACC: Self-Sensing Actie Suression of Vibration of Flexible Steel Sheet, Trans. ASME, J. Vib. Acoust. Vol.8, (996). [] Fujita M., Matsuura F. and Naeriawa T.: - Analysis and Synthesis of a Flexible Bea Magnetic Susension Syste, Proc. 3 rd Int. Sy. Magnetic Bearings., (99). [3] Stehens L.S., Tieran M.A. and Caseore, M.A. :Gain Scheduled PID Control for Electroagnetic Vibration Absorbers, Proc. 6 th Int. Sy. Magnetic Bearings., 33-34(998). [4] Betschon F. and Schob R., On-Line Adated Vibration Control, Proc. 6 th Int. Sy. Magnetic Bearings, (998). [5] Oa K. and Higuchi T.: Magnetic Leitation Syste by Reluctance Control: Leitation by Motion Control of Peranent Magnet Int. J. Alied Electroagnetics in Materials, Vol.4, (994). [6] Oa K. and Inoue Y.: Vibration Control With Linear Actuated Peranent Magnet: Proc. 6 th Int. Conf. Motion and Vibration Control,ol., (). [7] McFarlane D.C. and K. Gloer, A loo shaing design rocedure using H synthesis, IEEE Trans. On Autoatic Control AC-37(6), ,99 [8] McFarlane D.C. and K. Gloer, AN H Design Procedure using Robust Stabiliation of Noralied Corie Factors, Proc. 7 th on Decision and Control, ,988 [9] D.-W. Gu, P. Hr. Peto and M.M. Konstatino,: Robust Control Design with MATLAB:,Sringer, London, 5 [] Shino Electric co, LTD,: Electroagnet Vibration Control Syste for Steel Sheet (in Jaanese), htt://