Modeling of Cubic-Stewart vibration isolation platform with hybrid vibration isolator

Transcription

1 he 4th IFoMM World Congress, aipei, aiwan, October 5-3, 5 DOI Number:.6567/IFoMM.4.WC.O9.9 Modeling of Cubic-tewart vibration isolation platform with hbrid vibration isolator Y. Zhang Z.-B. Chen Y.-. Jiao 3 arbin Institute of echnolog arbin Institute of echnolog arbin Institute of echnolog arbin, China arbin, China arbin, China Abstract In order to protect the instrument and equipment for space applications, vibration control is necessar. In this paper, a Cubic-tewart vibration isolation platform emploing a novel hbrid vibration isolator (I) is presented. he I is composed of the active piezoelectric stack actuator and the passive rubber isolator, which features compact structure and high reliabilit. he simulation results the active piezoelectric stack actuator can eliminate the resonance peak significantl and the passive rubber isolator is effective to isolate a part of vibration once active control fails. Finall, the simulation analsis of the Cubic-tewart platform is established based on the transmissibilit and single input single out (IO) control algorithm. he results verif the effectiveness of our proposed vibration isolation sstem. Kewords: vibration isolation, piezoelectric-stack actuator, hbrid vibration isolator, Cubic-tewart platform I. Introduction ibration isolation and control have become a widespread and significant challenge in spacecraft. he various vibration source of the spacecraft impacts the sensitive instrument seriousl, such as vibration generated b flwheel attitude adjustment, solar arra drives, refrigeration equipment and shock loads and nonlinear vibrations generated b completing a particular action[]. At present, most of the sensitive instruments which required vibration isolation mainl used passive damping isolation or fleible structure. he passive vibration isolation cannot meet the higher demand because of its small vibration isolation frequenc range, small vibration attenuation, and high frequenc vibration suppression with low frequenc vibration amplification [, 3]. As the stricter requirements in the aerospace Engineering, the passive vibration isolation sstems alone cannot often be satisfied. In contrast, the active vibration isolation can overcome these insufficiencies, but it has high requirement to the control sstem design. Once an the control problems will lead to the failure of the whole isolation sstem [4], so the sstem stabilit of the active vibration isolation is poor. z@hit.edu.cn chenzb@hit.edu.cn 3 jiaoh@hit.edu.cn With the development of the material technolog and advance control technolog, the hbrid vibration isolation sstems have attracted a lot of attention which emplo the passive isolation and the active control isolation together, because the can combine with the advantages of active vibration isolation and passive vibration isolation. Nakamura et al. developed a hbrid actuator, which was composed of an air actuator and a giant magnetostrictive actuator in series configuration [5, 6]. Nguen proposed a hbrid active mount featuring piezoelectric stack and a rubber element that can provide high performance in terms of vibration control [7]. Jang et al. developed a hbrid mount sstem with air springs and piezoelectric stack actuators for microvibration control [8]. As a popular parallel mechanism invented b Gough and summarized b tewart[9], stewart platforms have wide-range application such as flight simulators, vehicle simulators, vibration tables, and space dock mechanisms []. One of such sstems called Cubic Configuration was invented b Geng and used b Intelligent Automation Inc.[]. he cubic configuration has several unique characteristics, making such stewart platform capable of decoupled control, since adjacent legs are orthogonal to each other, as well as using the smmetr of cube to provide identical legs []. he features of the Cubic- tewart would be useful in vibration isolation control. In this paper, aiming to utilize the advantages of the passive vibration and the active control, a hbrid vibration isolator is developed. In our design, the vulcanized rubber and the piezoelectric stack are combined in the series configuration. In that manner, the advantages of the passive and active vibration sstems can be utilized simultaneousl. In addition, our proposed I has simple and compact structure, small qualit and size, small energ consumption, which are distinct features and advantages when the I is used in the aerospace engineering. Because of its compactness, the I is suitable for application on multi-dimensional platform, the I is applied on the Cubic-tewart vibration isolation platform in this paper. he joints which connect platform and the I adopt omnidirectional tpe fleible hinge to avoid the clearance/ friction/ crawling problems of the traditional hinge, improve the control precision, realize multi degree of freedom vibration control of sensitive instruments.

2 uipment linear bearing output rod II. brid vibration isolator A. Mechanism design of the hbrid isolator o achieve the active and passive vibration isolation together, our proposed hbrid isolator shown in Fig. consists of an active part and a passive part, which are combined in serial connection. As the active part of the hbrid sstem, a piezoelectric stack actuator is selected. Piezoelectric stack actuators have several suitable properties for microvibration control, such as a fast response time, high dnamic force generation, and easiness of control due to the linearit between the input and output. Of the various passive devices, the rubber is used in this stud because of its several features, such as sample structure, high damper, eas to metal bonding. hole preloaded spring head board transmission rod rubber tailgate regulation equipment linear bearing output rod (I) (I) outer casing bolt Piezo-stack wire hole preloaded spring head board b) B Fig.. he structure of I: he structure of I. he structure parameters of the vulcanized rubber. o describe the mechanical design of the I clearl, the I structure is divided into two subsstems: the active and passive vibration isolation structures. he two parts are connected in series configuration in order to make full use of their isolation performances, i.e. the leveling function of the vulcanized rubber and the fast response characteristics of the piezoelectric stack actuator. he active vibration isolation structure contains the piezoelectric stack actuator, the adjustment parts, force output rod, and the pre-loaded spring. he linear bearing outside of the force output rod is used to guarantee the aial displacement of the piezoelectric stack. With the pre-loaded spring, the piezoelectric stack is alwas compressed to avoid the radial displacement and the structure gap. he passive structure is composed of the vulcanized rubber and the force transmission rod. he specifications of the I are summarized in able I. Element Item alue ubber ize (, B,, B, ) (ardness JI) G (tatic shear modulus) 9mm,8mm, 3mm,.5mm,4.4mm 4 A.45 N/mm Piezoelectric Maimum stroke length 36um stack actuator Maimum force output 4N Maimum tension 3N Input voltage -3~+5v brid I mm vibration isolator Effective length ( I) 33mm ABLE. I. pecifications of the I B. Dnamic model of the hbrid isolator he displacement of the piezoelectric stack actuator can be formulated b, FL z d33 () EA where the F z, E, A, L, and d 33 are the driving force, the modulus of elasticit, the action area, the length, the driving voltage and the piezoelectric stress constant of the piezoelectric stack, respectivel. Based on the above analsis, we can build the dnamic model of the I with the mass-spring model illustrated in Fig.. According to the Newton s second law, the sstem equilibrium equation can be formulated b, B m k ( ) Fz a) b) ( ) ( ) ( ) m c k k Fz where m is the mass of load and m is middle mass, namel, the mass of the piezoelectric stack and the adjustment, etc. and are the displacements of the eternal load and the middle mass. k and k are the stiffness of the piezoelectric stack actuator and the rubber. is the displacement of the disturbance. he c is the damping coefficient of the rubber. F z is the output force of the piezoelectric stack actuator. k m f z m k c ensor Controller Fig.. he theoretical model of the I he Eq. () can be rewritten in the form of the equation of state. k k m m m Fz k c k k k c m m m m m m Based on the dnamic model, the sstem can be formulated b the state equation and output equation, where the acceleration of the load platform is measured as the sstem output, the state equation (3) and the output equation can be written to simplif the notation b, ( t) A( t) Bu( t) Gw( t) (4) ( t) C( t) Du( t) where (t) is the time-varing state variables. u(t), w(t) and (t) are the control variables, the disturbance and the output. A, B, G are the () (3)

3 Phase (deg) Magnitude (db) parameter matrices of the Eq. (3). C and D can be k k epressed as C m m, D. m C. Controller of the hbrid isolator A suitable control strateg plas a ver important role in the active vibration control. he linear quadratic regulator (LQ) is emploed to simulate the effect of our isolator in this paper, because the LQ control method can provide inherent robustness and simplicit in design and implementation. he vibration isolation objective is to decrease the acceleration response of the load platform. With the consideration of the vibration isolation performance and the stabilit of the sstem, the objective function is formulated as follows, J X QX dt t (5) where Q and are the state weight matri and the control weight matri. We determine Q and according to the empirical value. he state feedback controller gain matri L can be solved b LQ method, and LX (6) where the L is the feedback gain matri. he P can be solved form the iccati algebra equation of the controller. PA A P PB B P Q (7) Let L=[k c,k c,k c3,k c4 ] represent the feedback gain of the LQ controller. he active control force of the piezoelectric stack actuator can be formulated b, Fz kc kc kc3 kc4 (8) ubstitute Eq.(9) into Eq.(4) and use Laplace transform to formulate the sstem transfer functions of the passive control (open-loop, ) and the integrated passive and active vibration control (IPAC) (closed-loop, ). () s k cs k k 4 3 us () mm s mcs as kcs kk () s ck s a s ( k k k k ) () ( ) c4 c3 4 3 us mm s a4s a3s a5s kkc kk () where a mk mk ; a ckc3 ck kc4k ; a m k m k m k m k m k ck ; 3 c3 c c a4 kc4m cm mkc. D. imulation for the hbrid isolator In order to test the vibration isolation capabilit of the I sstem, a simulation model based on LQ control is established in MALAB/IMLINK. wo sinusoidal vibration signals with the frequenc of z, um and z, um are acted on the base platform, respectivel. he open-loop responses and the closed-loop responses of the load platform are shown in Fig.3. he disturbances can be eliminated b about 8% in z, and almost % in z. he vibration transmission characteristic curves are shown in Fig. 4. he IPAC can basicall eliminate the fist-order vibration peak and passive control plas the main (9) role at the high frequenc Passive IPAC (s) a) (s) b) Drive signal Passive IPAC Fig. 3. he open-loop and closed-loop responses of the load platform with different disturbances: a) he disturbance of um, z sine signal. b) he disturbance of um, z sine signal disturbance Frequenc (z) Passive IPAC Fig. 4. he open-loop and closed-loop vibration transmission characteristic. III. Cubic-tewart vibration isolation sstem A. ransfer characteristics of the vibration isolation sstem Generalized coordinates of the vibration isolation sstem based on Cubic-tewart platform are shown in Fig. 5. Our proposed hbrid vibration isolator is emploed as the support leg of Cubic-tewart platform, which is arranged in a mutuall orthogonal configuration

4 connecting the corners of a cube. An two legs are orthogonal or parallel, each direction of the leg is a set of generalized coordinates. he topolog provides a uniform control capabilit and a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators and sensors of different support legs (being orthogonal to each other) [3]. Based on those features, the IO control method can be adopted for our proposed Cubic-tewart vibration isolation sstem. W O' z O Fig. 5 chematic illustrating the coordinates of the Cubic-tewart vibration isolation sstem As depicted in Fig. 5, the node,, and,, W is the platform joint respectivel. he base frame {,, z} has its origin at node O and the paload frame {,, z } has its origin at node O. he legs and W are parallel with ais. W and are parallel with ais. and are parallel with z ais. It is assumed that each actuator is actuated along the corresponding ais direction when the vibrations are acted on the base platform. Let ', ', ', ', z ', z ' and,,, W W, z, z epress the generalized coordinates of the joints in the base platform and the paload platform, respectivel. In the O-z coordinates, the position and the pose of the base platform can be epressed b generalized coordinates of the joints in the base platform, ' ( W) / ( W ) / z ( z z) / ( z z ( W ( W ( z z z ( W ( W () where l is the length of each leg, and Eq. () can be re-written in matri form, z z z W W M W W l l l l z z z l l l l l l l l () where M is defined as the coordinate transformation matri. In the same manner, the position and pose of the paload platform which are described in the O- z coordinates, can be epressed b the generalized coordinates of the joints in the paload platform. ' ( ' ' ) / ' ( ' ' ) / z ' ( z ' z ' ) / ' ( z ' z ' ( ' ' ( ' ' ( z ' z ' ( ' ' ( ' ' Eq. (3) can be re-written in matri form. (3) ' ' ' ' z ' ' ' ' M ' (4) l l l l l l l l l l l l For each leg, based on the transfer characteristics ((s)) among the input and the output of the actuator established in Eq. (), the transfer characteristics of the Cubic-tewart vibration isolation sstem can be derived as shown in Fig. 6. () ' Base ecitations z () Load platform,, z, θ, θ, θ z ',,, θ ', θ, θ W () ' Coordinate transformation matri M - W z () () () Coordinate transformation matri M Fig. 6 he transfer characteristics of the vibration isolation sstem B. Numerical simulation According to the transfer characteristics, the simulation analsis of the Cubic-tewart platform is established in the MALAB/IMLINK, in which the vibration isolation performance under different ecitation signals are analzed. wo sets of ecitation signals are acted on the base platform, i.e., the sinusoidal ecitation signal and the step ecitation signal. Firstl, the vertical and horizontal sinusoidal vibration signals with the frequenc of z, um are acted on the base platform respectivel. he displacements of the paload platform are obtained to validate the effectiveness of our proposed vibration isolation sstem. he results show that our proposed sstem presents ecellent vibration

5 isolation effect for horizontal and vertical direction low frequenc harmonic ecitation signals, as shown in Fig.7 and Fig.8. Compared to the passive control, the vibration amplitude of the active control is attenuated b 8% in vertical direction and 5% in horizontal direction. he attenuations of the angular displacements are much more obvious (s) (s) Fig. 7 he vibration isolation effect for the vertical harmonic signal ecitation: Linear displacement. Angular displacement (s) (s) Fig. 8 he vibration isolation effect for the horizontal harmonic signal ecitation: Linear displacement. Angular displacement. When the vertical and horizontal step signals with the amplitude of um are acted on the base platform, the corresponding displacements of the paload platform are shown in Fig. 8 and Fig. 9 respectivel. he results show that the platform proposed can reduce the vibration displacement and angular displacement and greatl reduce the number of the oscillation. herefore the stead state of the sstem can be realized quickl (s) (s) Fig. 9 he vibration isolation effect for the vertical step signal ecitation: Linear displacement. Angular displacement (s) (s) Fig. he vibration isolation effect for horizontal step signal ecitation: Linear displacement. Angular displacement.

6 Ⅳ. Conclusions In this paper, we developed a Cubic-tewart vibration isolation platform emploing a hbrid vibration isolator. In our design, the hbrid vibration isolator is composed of the rubber isolator and the piezoelectric stack actuator. LQ control strateg is developed for the active vibration isolation of the I. he simulation results have shown that I can isolate 8% vibration. he piezoelectric stack actuator can eliminate the resonance peak significantl and the rubber isolator can isolate the high frequenc vibration and ensure the reliabilit of the I. sing the feature of the Cubic-tewart that each coordinate direction of motion is linearl independent and the coupling factors greatl reduce, the transmissibilit and distributed control model is built up, then the vibration isolation performance is simulated in MALAB/IMLINK. he simulation results show that the vibrations on the load platform can be significantl reduced b the Cubic-tewart platform emploing the I. owever, our proposed works in this paper are preliminar, much more works will be carried out. In follow-up stud, the eperiments will be conducted to verif the dnamic model and vibration isolation performance of the I and Cubic-tewart vibration isolation sstem. Acknowledgments his research is funded b the National Natural cience Foundation of China (Grant No.775). eferences [] Yao J, an X and u. tructural design of a frog-like hopping robot with consideration of vibration and its kinematic analsis. Journal of vibration and control, 9(): 39-3, 3. [] Yilmaz C, Kikuchi N. Analsis and design of passive band-stop filter-tpe vibration isolators for low-frequenc applications. Journal of ound and ibration, 9(3): 4-8, 6. [3] Liu C, Xu D, and Ji J. heoretical design and eperimental verification of a tunable floating vibration isolation sstem. Journal of ound and ibration, 33(): ,. [4] Dale, et al. Active vibration control for marine applications. Control Engineering Practice, (4): , 4. [5] Nakamura Y, Nakaama M, Masuda K, Yasuda M, suchia M and Fujita. Development of a uniaial hbrid actuator using the combination of an air actuator and a giant magnetostrictive actuator. mart Materials & tructures, : ,. [6] Nakamura Y, Nakaama M, Yasuda M, and Fujita. Development of active si-degrees of freedom micro-vibration control sstem using hbrid actuators comprising air actuators and giant magnetostrictive actuators. mart Materials& tructures, 5(4): 33-4, 6. [7] Nguen. Q., Choi. M., Choi. B., et al. liding mode control of a vibrating sstem using a hbrid active mount. Proceedings of IMechE Part C: Mechanical Engineering cience, 3: ,. [8] Jang DD. et al. Feasibilit stud on a hbrid mount sstem with air springs and piezostack actuators for microvibration control. Journal of Intelligent Material stems and tructures, 3(5): 55-56,. [9] tewart D. A platform with si degrees of freedom, Proceedings of the institution of mechanical engineers, 8(): , 965. [] Dasgupta B. and Mruthunjaa. he stewart platform manipulator: a review, Mechanism and machine theor, 35(): 5-4,. [] Geng Z. J. and anes L.. i degree-of-freedom active vibration control using the stewart platforms, IEEE ransactions on Control stems echnolog, (): 45-53, 994. [] han J., Xue X., and Kang Z. Dnamics and control of piezoelectric-based stewart platform for vibration isolation. IC International Congress on ound and ibration, in press, 4. [3] Preumont A., orodinca M., omanescu I., et al. A i-ais ingle-stage Active ibration Isolator Based on tewart Platform. Journal of ound and ibration, 3(4): , 7.