A VIBRATION ISOLATION SYSTEM USING STIFFNESS VARIATION CAPABILITY OF ZERO-POWER CONTROL

Transcription

1 Proceeding of the International Conference on Mechanical Engineering 009 (ICME009) 6-8 December 009, Dhaka, Bangladeh ICME09-AM-0 A VIBRATION ISOLATION SYSTEM USING STIFFNESS VARIATION CAPABILITY OF ZERO-POWER CONTROL M. Emdadul Hoque,, Takehi Mizuno, M. M. Zaglul Shahadat,, Yuji Ihino and Maaya Takaaki Department of Mechanical Engineering, Rajhahi Univerity of Engineering & Technology Rajhahi, Bangladeh. Department of Mechanical Engineering, Saitama Univerity,Japan. ABSTRACT Thi paper preent a vibration iolation ytem combining a poitive tiffne pring in erie with a negative tiffne pring. The negative pring i realized by an active zero-power control. The conventional zero-power control ytem yield contant negative tiffne, and the tiffne depend on the capacity of the permanent magnet or the gap-force coefficient of the magnet. Thi i one of the bottleneck in the field of application of zero-power control where adjutment of tiffne i neceary, uch a developing vibration iolation ytem. To overcome the above problem, the baic zero-power control ytem i modified uch that it can adjut tiffne by adding a minor proportional feedback of diplacement to the zero-power control current. Some experiment have been carried out to meaure the efficacy of the control ytem, a well a the vibration iolation ytem. Keyword: Active Control, Vibration Iolation, Zero-Power Control, Zero-Compliance Sytem, Magnetic Bearing.. INTRODUCTION Vibration iolation ytem i widely ued in many reearche and precie manufacturing ytem, uch a emiconductor indutrie, high preciion meaurement, etc. There are two kind of vibration that may hamper deired operation. They are direct diturbance on the table and ground vibration. Both vibration diturbance can be uppreed by uing paive technique or uing active control. Vibration aborber extract kinetic energy from the vibrating hot ytem and the active ytem introduce oppoing force in the tructure to affect compenation. Vibration iolation of mechanical ytem i achieved with paive technique becaue of it low cot. However, paive technique need the trade-off between the iolation element where high tiffne upenion are thi ytem between bae to iolation table. Poitive tiffne pring i ued between bae to middle table, and a upenion with negative tiffne i ued between middle table to iolation table. In thi reearch, an active zero-power control i ued to realize negative tiffne by uing a hybrid magnet conit of electromagnet and permanent magnet. Thi control achieve the teady tate in which the attractive force produced by the permanent magnet balance the weight of the upended object, and the control current converge to zero. Since there i no teady energy conumption for achieving table levitation, it ha been applied to pace vehicle [0], and to the magnetically levitated carrier ytem in clean room []. A unique characteritic of the zero-power control i that it realize negative tiffne []. Therefore, a very ued for uppreing direct diturbance, and upenion high (theoretically infinite) tiffne of a vibration with oft tiffne are ued for ground vibration iolation iolation Active ytem negative can be achieved by combining a pring k [,]. On the other hand, active technique do not have mechanical pring in erie with a zero-power control. uch limitation, and performance are better a well However, the amplitude of negative tiffne k 3 realized by [3-7]. Recently vibration iolation ytem have been zero-power Active-paive control i olely dependent of developed uing active zero-power controlled magnetic magnetomotive poitive force pring k or gap-force coefficient of the upenion in the view of reducing ytem development cot a well a maintenance cot [8,9]. The above ytem ue a combined ytem with poitive and negative pring in erie. A middle table i introduced in permanent magnet. The amplitude of negative tiffne i fixed for a particular magnet, and the gap between upended object and the permanent magnet, doe not have Fig any. Principle capability of to vibration change iolation tiffne. ytem Hence uing realizing zero-power control ICME009 AM-0

2 i Electromagnet infinite tiffne by combing a mechanical poitive pring with it i a very troubleome tak, becaue a lot of mechanical pring are to be ued to adjut both tiffnee. Thi i one the obtacle to develop vibration iolation ytem uing zero-power control. Moreover, it can be noted that realizing negative tiffne can alo be generalized by uing linear actuator (voice coil motor) intead of hybrid magnet [3]. Thi paper demontrate a control method of zero-power magnetic upenion ytem that ha capability to adjut tiffne. Thi modified zero-power control introduce a proportional feedback of diplacement to the original zero-power controller. The characteritic of the modified control ytem deign are dicued analytically. Finally, a vibration iolation ytem i developed by connecting a mechanical pring in erie with the modified zero-power control. Some experimental reult are preented to how the efficacy of the control ytem, a well a the vibration iolation ytem.. PRINCIPLE OF VIBRATION ISOLATION The vibration iolation ytem i developed to generate infinite (high) tiffne for direct diturbing force and to maintain low tiffne for floor vibration. Infinite tiffne can be realized by connecting a mechanical pring in erie with a magnetic pring that ha negative tiffne [8, 9]. When two pring with pring contant of k and k are connected in erie, the total tiffne k c i given by kk k c =. () k + k The above baic ytem ha been modified by introducing a econdary upenion to avoid ome limitation for ytem deign and upporting heavy payload [4]. The concept i demontrated in Fig.. A pring k 3 i added in parallel with the erial connection of poitive and negative pring. The total tiffne k ~ c i given by ~ kk k c = + k3. () k + k However, if one of the pring ha negative tiffne that atifie k = k, (3) the reultant tiffne become infinite for any finite value of k 3, that i ~ k c =. (4) Thi reearch applie thi principle of generating infinite tiffne againt direct diturbance to the ytem. On the other hand, if low tiffne of mechanical pring for ytem ( k, k 3 ) are ued, it can maintain good ground D 0 vibration iolation performance a well. 3. ZERO-POWER CONTROL SYSTEM Negative tiffne i generated by actively controlled zero-power magnetic upenion. The baic model, controller and the characteritic of the zero-power control ytem i decribed below. 3. Model A baic zero-power controller i deigned for implification baed on linearized equation of motion. It i aumed that the diplacement of the upended ma i very mall and the nonlinear term are neglected. The upended object with ma of m i aumed to move only in the vertical tranlational direction a hown by Fig.. The equation of motion i given by m& x = k x + kii + w, (5) where x : diplacement of the upended object, k : gap-force coefficient of the permanent magnet, k i : current-force coefficient of the electromagnet, i : control current, w : diturbance acting on the upended object. The coefficient k and ki are poitive. When each Laplace-tranform variable i denoted by it capital, and the initial value are aumed to be zero for N S p z pd p v S N m i k i w Permanent magnet Fig. Baic model of zero-power control m k Fig 3. Block diagram of the zero-power controller implicity, the tranfer function repreentation of the dynamic decribed by Eq. (5) become X ( ) = ( k I( ) W ( )) i + m k = ( b0 I( ) + d0w ( )), (6) a0 N S x x ICME009 AM-0

3 where a0 = k / m, b0 = ki / m, and d 0 = / m. 3. Zero-Power Controlled Negative Stiffne Zero-power can be achieved either by feeding back the velocity of the upended object or by introducing a minor feedback of the integral of current in the PD (proportional-derivative) control ytem []. Since PD control i a fundamental control law in magnetic upenion, zero-power control i realized from PD control in thi reearch uing the econd approach. In the current controlled magnetic upenion ytem, PD control can be repreented a I( ) = ( pd + pv) X ( ), (7) where p d : proportional feedback gain, p v : derivative feedback gain. Figure 3 how the block diagram of a current-controlled zero-power controller where a minor integral feedback of current i added to the PD control. The control current of zero-power controller i given by p I( ) = ( p ) X ( ) z d + v + I( ), (8) where p z : integral feedback gain in the minor current loop. From Eq. (6) to (8), it can be written a X ( ) ( ) d0 =, (9) W ( ) 3 + ( b0 pv ) + ( b0 pd a0) + a0 I( ) ( pv + pd ) d0 =. (0) W ( ) 3 + ( b0 pv ) + ( b0 pd a0) + a0 lim i( t) = lim I( ) = 0. t 0 () It indicate that control current alway converge to zero in the zero-power control for any load. The teady diplacement of the upenion, from Eq. (9) and (), i given by d0 F0 lim x( t) = lim X ( ) = F0 =. (3) t 0 a0 k The negative ign in the right-hand ide illutrate that the new equilibrium poition i in the direction oppoite to the applied force. It mean that the ytem realize negative tiffne. Aume that tiffne of any upenion i denoted by k. The tiffne of the zero-power controlled magnetic upenion i, therefore, negative and given by k = k. (4) 3.3 Stiffne Adjutment The tiffne realized by zero-power control i contant, a hown in Eq. (4). However, it i neceary to adjut tiffne of the zero-power control ytem in many application, uch a vibration iolation ytem. There are two approache to adjut tiffne of the zero-power control ytem. The firt one i by adding a minor diplacement feedback gain to the zero-power control current, and the other one i by adding a proportional feedback in the minor current feedback loop [5]. In thi work, tiffne variation capability of zero-power control i realized by the firt approach. Figure 4 how the block diagram of the modified zero-power controller that i capable to adjut tiffne. The control current of the modified zero-power controller i given by p z i k i w m k x Vibration iolation table Leaf pring pd p v Fig 4. Block diagram of the modified zero-power controller that can adjut negative tiffne To etimate the tiffne for direct diturbance, the direct diturbance, W() on the iolation table i conidered to be tepwie, that i F W ( ) = 0, ( F 0 : contant). () From Eq. (0) and () Hybrid magnet for negative tiffne Coil pring for weight upport mechanim Middle table Hybrid magnet for poitive tiffne Coil pring for poitive tiffne Bae Fig 5. Photograph of the vibration iolation ytem I d ( ) = ( + pv + p ) X ( ), (5) ICME009 3 AM-0

4 where : proportional diplacement feedback gain acro the zero-power controller. The tranfer-function repreentation of the dynamic hown in Fig. 4 i given by X ( ) ( ) d = 0. (6) W ( ) 3 + ( b ) 0 pv + ( b0 pd a0) + a0 + b0 p From Eq. (6) and (), the teady diplacement become d0 lim x( t) = lim X ( ) = F0 t 0 a0 + b0 p = F 0 k + ki p F0 =. (7) k + ki p / Therefore, the tiffne of the modified ytem become p k = k ki. (8) It indicate that the tiffne can be increaed or decreaed by changing the feedback gain. 4. EXPERIMENTAL SET UP A ingle-axi vibration iolation ytem combining poitive and negative tiffne upenion in erie i hown in Fig. 5. It conited of a circular bae, a circular middle table and a circular iolation table. The height, diameter and weight of the ytem were 300mm, 00mm middle table. The permanent magnet are made of Neodymium-Iron-Boron (NdFeB). The middle table wa alo upported by three coil pring. The hybrid magnet and the coil pring were ued in tandem to generate poitive tiffne upenion. Another diplacement enor wa ued to meaure the relative diplacement between middle table to iolation table. A different hybrid magnet conited of an electromagnet and ix permanent magnet wa ued to realize negative tiffne. The iolation table wa alo upported by three coil pring a weight upport mechanim. The motion of the iolation table and that of the middle table were retricted to move only in the vertical direction. It wa done by uing a vertical haft which wa fixed to be bae and paed through the center of the iolation table and middle table. The friction between the haft and the iolation table and middle table were reduced to a minimum poible level by uing uitable lubricant and ball bearing. Two additional threaded haft and everal hexagonal nut were employed a topper and limiter for the movement of the iolation table and the middle table. Two leaf pring were ued to confine the rotational motion of the iolation table and middle table. Thee leaf pring were alo behaved a damper for the both table. 5. EXPERIMENTAL RESULTS The experiment have been carried out to meaure the performance both for the modified zero-power controller, a well a for the vibration iolation ytem. Figure 6 how the load-diplacement characteritic of the ytem with the modified zero-power controller (Fig. 4). When the proportional feedback gain, P =0, it can be conidered a a conventional zero-power controller (Fig. 3). The reult how that when the payload were put on the upended object, the table moved in the direction of load application, and gap wa widened. It indicate that the zero-power control realized Fig 6. Load-diplacement characteritic of the modified zero-power controller Fig 7. Load-current characteritic of the modified zero-power controller and 0 kg, repectively. The relative diplacement of the bae to middle table wa meaured by an eddy-current diplacement enor, and poitive tiffne wa realized by a hybrid magnet conited of an electromagnet (80-turn) that wa fixed to the bae, and four permanent magnet (5mm mm) attached to the ICME009 4 AM-0

5 Fig 8. Zero-compliance characteritic of the iolation table to tatic direct diturbance uing modified zero-power controller negative diplacement, and hence it tiffne i negative, a decribed by Eq. (3) and (4). The conventional zero-power controller (P =0) realized fixed negative tiffne of magnitude -9. N/mm. When the proportional feedback gain, P wa changed, the tiffne alo gradually increaed. When P =80A/m, negative tiffne wa increaed to -03. N/mm. It confirm that proportional feedback gain, P can change the tiffne of the zero-power controller, a explained in Eq. (8). Figure 7 how the load-current characteritic of the zero-power controller. The control current neceary for the conventional zero-power controller (P =0,) wa alway zero. It prove the analyi decribed in Eq. (). When the gain P wa increaed, a minor control current wa neceary. The maximum control current for P =80A/m wa ±0.5A up to 0 N load. However, the zero-power characteritic can be retored again by adjuting the weight upport pring, once the ytem i tabilized. Finally, experiment have been carried out with the developed vibration iolation ytem. Zero-compliance characteritic of the iolation table to tatic direct diturbance uing modified zero-power controller i hown in Fig. 8. In thi cae, tepwie tatic direct diturbance in the vertical direction of the table wa generated by putting payload on the center of the table. The diplacement of the iolation table to middle table and that of the middle table to bae were meaured by eddy-current diplacement enor. It i een from the figure that poitive tiffne wa realized between bae to Fig 9. Frequency repone of the iolation table to dynamic direct diturbance middle table, and negative tiffne wa realized between iolation table to middle table. Zero-compliance to direct diturbance of the iolation table to bae wa achieved when Eq. (3) wa atified. In other word, an infinite tiffne of the iolation ytem wa realized that confirm Eq. (4). At higher load, the zero-compliance characteritic may be lot due to the nonlinearity of the zero-power control ytem. A nonlinear compenation to zero-power controller can olve uch problem [6]. Figure 9 how the frequency repone of the iolation table to dynamic direct diturbance. In thi experiment, the iolation table wa excited by an electromagnet which wa fixed to the bae above the iolation table. The diplacement of the table wa conidered a output in thi cae. The frequency repone wa meaured by a dynamic ignal analyzer. The reult how that the table generated very low diplacement at the low frequency region. The diplacement of the table at 0. Hz wa - db [m/n]. It confirm that iolation table generated very high tiffne at low frequency dynamic direct diturbance. Finally, tep repone of the developed vibration iolation ytem wa meaured a hown in Fig. 0. An electromagnet wa employed to generate tepwie diturbance [0 to -0N] at 0.5 Hz. The reult demontrate that the gap between iolation table to middle table wa increaed and the diplacement of the middle table to bae wa decreaed for downward diturbance on the table. However, the iolation table returned to the original poition within 0.5. The diplacement and tranient period to return to the original poition can further be reduced by introducing feedforward control with the zero-power control ytem. Fig 0. Step repone of the iolation table ICME009 5 AM-0

6 6. CONCLUSIONS A vibration iolation ytem ha been developed by combining an active zero-power control with mechanical pring. The zero-power control i modified by introducing a proportional diplacement feedback to the zero-power control current. The modified zero-power controller yielded negative tiffne with the capability of adjuting negative tiffne. The concept wa invetigated by ome experiment. The experimental reult howed that the tiffne of the magnetic upenion ytem depend on the proportional gain of the diplacement feedback. The modified zero-power control had wide tiffne range, and could be applied to develop vibration iolation ytem. The ytem can further be improved by providing minor current feedback to the integral of the zero-power control ytem. A ingle-degree-of-freedom vibration iolation ytem ha been developed uing the fixed poitive upenion and the modified zero-power controller. Infinite tiffne of the iolation ytem wa realized when the negative tiffne wa equal in magnitude with the poitive tiffne. The tatic and dynamic repone of the iolation table howed that the developed iolation ytem could effectively uppre the effect of direct diturbance. 7. ACKNOWLEDGMENT The author gratefully acknowledge the financial upport made available from Japan Society for the Promotion of Science (JSPS) a a Grant-in-Aid, and from the Minitry of Education, Culture, Sport, Science Technology of Japan, a a Grant-in-Aid for Scientific Reearch (B). 8. REFERENCES. Rivin, E. I., 003, "Paive Vibration Iolation," ASME Pre, New York.. Harri, C. M., 996, "Shock and Vibration Handbook," McGraw Hill, 4 th Ed., New York. 3. Yohioka, H., Takahahi, Y., Katayama, K., Imazawa, T., and Murai, N., 00, An Active Microvibration Iolation Sytem for Hi-Tech Manufacturing Facilitie, ASME Journal of Vibration and Acoutic, 3: Benai, L., Elliot, S. J., and Gardonio, P., 004, Active Vibration Iolation Uing an Inertial Actuator with Local Force Feedback Control, Journal of Sound and Vibration, 76(3): Daley, S., Hatonen, J., and Owen, D. H., 006, Active Vibration Iolation in a Smart Spring Mount Uing a Repetitive Control Approach, Control Engineering Practice, 4: Yauda, M., and Ikeda, M., 993, Double-Active Control of Microvibration Iolation Sytem to Improve Performance (Application of Two-Degree-of-Freedom Control). Tranaction of the Japan Society of Mechanical Engineer, Serie C (in Japanee), 59(56): Yauda, M., Oaka, T., and Ikeda, M., 996, Feedforward Control of a Vibration Iolation Sytem for Diturbance Suppreion, Proc. of the 35 th Conf. on Deciion and Control, Japan, pp Mizuno, T., 00, Propoal of a Vibration Iolation Sytem Uing Zero-Power Magnetic Supenion, Proc. of the Aia-Pacific Vibration Conference, pp Hoque, M. E., Takaaki, M., Ihino, Y., and Mizuno, T., 006, Development of a Three-Axi Active Vibration Iolator Uing Zero-Power Control, IEEE/ASME Tranaction on Mechatronic, (4): Sabni, A. V., Dendy, J. B., and Schmit, F. M., 975, A Magnetically Supended Large Momentum Wheel, Journal of Spacecraft, : Morihita, M., Azukizawa, T., Kanda, S., Tamura, N., and Yokoyama, T., 989, A New Maglev Sytem for Magnetically Levitated Carrier Sytem, IEEE Tranaction on Vehicular Technology, 38(4): Mizuno, T., and Takemori, Y., 00, A Tranfer-Function Approach to the Analyi and Deign of Zero-Power Controller for Magnetic Supenion Sytem, Electrical Engineering in Japan, 4(): Mizuno, T., Furuhima, T., Ihino, Y., and Takaaki, M., 007, General Form of Controller Realizing Negative Stiffne, Proc. of the SICE Annual Conference 007, Kagawa Univerity, Japan, Sept. 7-0, 007, pp Mizuno, T., Hoque, M. E., Takaaki, M., and Ihino, Y., 006, Development of A Six-Axi Hybrid Vibration Iolation Sytem Uing Zero-Power Control, Proc. of the 45 th IEEE Conference on Deciion and Control, San Diego, USA, paper no. 400, pp Ihino, Y., Mizuno, T., and Takaaki, M., 009, Stiffne Control of Magnetic Supenion by Local Feedback, Proc. of the European Control Conference 009, Budapet, Hungary, 3-6 Augut, 009, pp Hoque, M. E., Mizuno, T., Takaaki, M., and Ihino, Y., 006, A Nonlinear Compenator of Zero-Power Magnetic Supenion for Zero-Compliance to Direct Diturbance, Tranaction of the Society of Intrument and Control Engineer, 4(9): ICME009 6 AM-0