DESIGN AND ANALYSIS OF FLEXIBLE STRUCTURES WITH PARAMETRIC UNCERTAINTIES FOR ACTIVE VIBRATION CONTROL USING ANSYS

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1 INTERNATIONAL JOURNAL OF R&D IN ENGINEERING, SCIENCE AND MANAGEMENT Vol.1, Iue I, AUG.14 ISSN 9-85X Reearch Paer DESIGN AND ANALYSIS OF FLEXIBLE STRUCTURES WITH PARAMETRIC UNCERTAINTIES FOR ACTIVE VIBRATION CONTROL USING ANSYS Sunny Bhatia a, Sueh C Babu b a At. Profeor Manav Rachna International Univerity, Faridabad b Aociate Profeor School of Engineering,Aeejay Stya Univerity, Gurgaon ABSTRACT The flexible beam are the common tructural element of many engineering alication which are ubjected to vibration. Thee vibration can everely affect the fatigue life of the tructure and any arametric uncertainty can even woren the ituation. Thee arametric uncertaintie are caued by factor uch a joint clearance, friction, material non-uniformitie and manufacturing and aembly error. Thi wor deal with the tudy of uncontrolled and controlled dynamic reone of a cantilever beam lie tructure with arametric uncertaintie ubjected to an imule load at the ti of the beam. An integrated analyi rocedure ha been develoed for the control of tructure. ANSYS arametric deign language (APDL) wa ued to incororate the cloed loo control law into the finite element (FE) model. Simulation were erformed by varying the height and width of a beam. Simulation reult how that active vibration ureion i achieved even in the reence of uncertaintie eyword: arametric uncertainty, FEM, PID, flexible tructure 1. Introduction Vibration i an imortant aect of many engineering ytem, from machine tool to tructure-borne noie in aircraft. In mot cae, uch vibration i undeirable and require attenuation or vibration control. The flexible beam are the common tructural element of many engineering alication which are ubjected to vibration. Thee vibration can everely affect the fatigue life of the tructure and any arametric uncertainty can even woren the ituation. Thee arametric uncertaintie are caued by factor uch a joint clearance, friction, material nonuniformitie and manufacturing and aembly error during the alication. Active vibration control can be ued to eliminate undeired vibration in engineering tructure with arametric uncertaintie. Thi wor deal with the active vibration control of beam lie tructure with arametric uncertaintie. A finite element model baed on Euler-Bernoulli beam theory i utilized to tudy the dynamic characteritic of tructure with uncertaintie. Flexible tructure are generally characterized by the undamed natural frequencie and daming ratio of their flexible mode. Thee arameter are ubject to error when they are etimated. Such uncertaintie are imortant and hould be taen into account in a controller deign. The roer cature of modal arameter uncertaintie in dynamic model of flexible tructure for control ha been the ubject of ongoing effort. In thi tudy, the cloed loo control law are incororated into the finite element (FE) model by uing ANSYS arametric deign language (APDL).. Firt, the rocedure i teted on the active vibration control roblem with three degree of freedom ytem to validate the method. The analytical reult obtained by the Lalace tranform method and by ANSYS are comared. It i oberved that the analytical reult were well matched with the reult obtained through ANSYS. The convergence tudy for finalizing the meh ize i erformed and modal analyi of beam tructure i carried to find out the mode hae and natural frequencie of the beam. Then the harmonic and tranient analyi of a cantilever beam i done Correonding Author unny.fet@mriu.edu.in Available at: R&D Publication Page 17

2 with the variation in the arameter lie height and width to account for the uncertainty in the tructure and finally the active vibration control i alied to ure the vibration uing ANSYS.The active vibration concet i demontrated through imulation tudy in ANSYS for roortional integral derivative (PID) controller aragulle et al. (4) introduced the active vibration control uing APDL (ANSYS Parametric Deign language) in ANSYS. They analyzed the reult obtained from the APDL for the two-degree of freedom ytem and found ueriority with the Lalace tranform method. Samuel da Silva and V. Loe Júnior tudied Robut control to Parametric Uncertaintie in Smart Structure Uing Linear Matrix Inequalitie. Shu-Xin Wang *, Yan-Hui Wang, Bai-Yan He Dynamic modeling of flexible multibody ytem with arameter uncertainty. Nabil Aouf and Benoit Boulet dicued Uncertainty Model and Robut Comlex-Rational Controller Deign for Flexible Structure.Rade Matušů, Roman Proo, and Libor Peař tudied Parametric and untructured aroach to uncertainty modelling and robut tability analyi. The main motivation of thi wor i to develo general deign and analyi cheme by incororating the control law directly into the FE rogram. The cloed loo control law i incororated into the FE model by uing APDL. The FE analyi with cloed loo control action are carried out by ANSYS. Cloed loo-fe imulation of flexible tructure lie cantilever beam with arametric uncertaintie are erformed. The control gain are determined by the imulation.. Three-degree of freedom ring ma ytem In thi, firt the reult are obtained for the active vibration control of a three degree of freedom ytem analytically & then reult are comared with ANSYS modeling & imulation. It wa oberved that the ANSYS reult were in cloe roximity with the analytical reult. Figure 1. (a) The three-degree of freedom ytem((m1=m=m=5 g, 1===5 N/m) and (b) bloc diagram of the cloed loo control ytem.1 Analytical Solution The ytem conidered and a bloc digram of the cloed loo control ytem are hown in figure 1. f i the vibration generating force, and f 1 i the controlling force. X r, X, F 1 and F are the Lalace tranform of the reference inut (x r ), outut dilacement (x ), the force f 1 and f, reectively. G 1 i the tranfer function of the control action, and i taen a i G1 ( ) D ( ) for the PID (roortional-integral-derivative)control., i, D are the roortional, integral, derivative contant, reectively. H 1 i the tranfer function from F 1 to X, and H i the tranfer function from F to X. The reference inut, X r, i taen a zero for the vibration control. X e (t) i defined a the error ignal, where X e (t) =X r (t)-x (t). The vibration generating force i taen a a unit imule in the tudy, and thu F() =1 The equation of motion for undamed multi-degree of freedom vibrating ytem i given a where M, and f are the ma, tiffne and force matrice, reectively. Alying Lagrange equation, the mathematical model of the ytem in figure 1(a) can be found a (1) () Page 18

3 Page 19 DESIGN AND ANALYSIS OF FLEXIBLE STRUCTURES WITH PARAMETRIC UNCERTAINTIES FOR ACTIVE VIBRATION CONTROL USING ANSYS 1 m m m x x x = t z 1 () Then the tranfer function can be written a the following, after ubtituting the value of the mae and tiffne: (4) (5) () Where (7) Subtituting X r ()=1 and F () =1, the tranfer function of the cloed loo ytem in figure 1(b) i found to be x ) ( 7x1 ) ( X i d d i d x1 7 1x1 1x1 x1 54 i + i 7x1 7x1 x1 1 (8) x (t) can be found by taing the invere Lalace tranform of X (). The time hitorie of x (t) are given in figure for the uncontrolled and controlled cae. Fig Analytical Solution for controlled & uncontrolled cae

4 . Solution by ANSYS Cloed loo imulation i erformed by utilizing comuter-aided engineering rogram ANSYS. Control action are incororated into the FE model of multi-degree of freedom vibratory ytem uing APDL. The element MASS1 and COMBIN14 are ued in order to contruct the FE model of -DOF ytem. MASS1 i the lumed ma while COMBIN14 i the ring-damer. The value of damer i taen zero ince daming i ignored. Modal analyi i erformed to find the undamed natural frequencie of the ytem. The cloed loo control i realized with the following macro : um= err= *do,t,*dt,t,dt *get,x1,node,1,u,x *get,x,node,,u,x *get,x,node,,u,x err=1-(x1+x+x)/ um=um+err*dt diff=(err-err)/dt ucon=*err+i*um+d*diff d,4,ux,ucon err=err time,t olve *enddo The variable dt, t and err are the time te, the time at teady tate and the error ignal, reectively. The reference value i taen a 1 to calculate the error ignal. The time te i taen a dt=1/f, where f i the highet undamed natural frequency ince the differential control require maller time te for higher accuracy. So, dt=.5. The uncontrolled and controlled reone obtained by cloed loo imulation are hown in figure. It i oberved that the analytical and ANSYS olution are in agreement. Thi agreement of the two method validate the method adoted in the reearch. Fig ANSYS Solution for controlled and uncontrolled cae Effect of uncertainty in tiffne To introduce uncertainty in the -DOF ytem, 1 ercent variation in the tiffne of the ring i done and the reone i oberved uing ANSYS. It wa oberved that although there i not much variation in the amlitude of the vibration but the ytem Page

5 vibrate in a different hae a hown in figure 4. So, there i a requirement of vibration control ince any uncertainty in the ytem can everely affect the vibration attern of the ytem which can lead to failure of the ytem. Fig. 4 Effect of uncertainty in tiffne It conclude that there i a need to aly control on the vibration to revent the damage to the ytem. Again the active control i carried out with *do-*enddo loo and the reult of the control action are comared with the uncontrolled cae with varying tiffne. It i clear from the Fig. 5 that ame controller Fig. 5 ANSYS Solution for Controlled Cae i uitable for controlling the vibration for all the three cae. A it i clear from the figure that the ettling time a well a magnitude of the vibration i reduced remarably by uing the active control method o, it rove to be an effective tool in controlling vibration. Thu if we can aly thi method of control to the flexible tructure which are ued more often than not in the mechanical equiment then the damage to the equiment which i caued by the vibration can be reduced to a good extent. Flexible tructure In thi ection, figure how cantilever beam which i fixed at one end and free at the other. Page 1

6 Fig. Configuration of beam tye tructure Numerical Examle and Dicuion A an illutrative examle, conider a cantilever beam having the following characteritic: L= Length of the beam = m, b= width of the beam =. m, h = thicne of the beam =.95 m, E = Young Modulu of the beam.5e1 Pa and ρ = denity = 78 g/m. Tranient analyi (a) Uncontrolled Vibration ( Oen Loo Sytem) The firt mode i conidered to calculate the time te and Δt i.8.in the tranient analyi, the coefficient of Rayleigh daming (α and β) are defined..α=β i taen in thi tudy. Fe=F (imulive force) for t=δt and Fe= at the ubequent time te. Oen loo reult i hown in fig 7. Fig.7 Ti Dilacement of cantilever beam under tranient loading. Effect of Uncertainty on tranient reone Fig. 8 how the effect of change in the width on the attern of vibration. It i clear from the figure that a the width i increaed the amlitude of vibration decreae and on decreaing the width amlitude increae. Page

7 Fig.8 Effect on vibration due to change in width Thu with change in the dimenion of a tructure it natural frequency change and hence there i a change in the reone of a tructure to the imule load. (b) Controlled Vibration (Cloed Loo imulation) The control loo imulation i obtained by dilacement feedbac control. Fig.9 Control and uncontrolled imulation of cantilever beam uing ANSYS The figure 9 how the comarion between the cloed loo and oen loo imulation. It i clear from the figure that on the alication of the control rocedure the amlitude of vibration reduce. Active vibration control in reence of uncertainty For controlling the vibration when uncertainty i involved in the ytem a robut PID controller i deigned having the gain value a =4, i =1, d =.A deicted from the figure 1 the amlitude of the vibration of the beam i reduced ignificantly on the alication of a PID controller. Page

8 Reult & Concluion Fig.1 Cloed and Oen loo Simulation by ANSYS The active control of flexible beam under free vibration i imulated and exerimentally tudied. By incororating the control law directly into the FE rogram, the cloed loo control roblem of flexible tructure in reence of uncertaintie can be analyzed more eaily. Modeling of flexible tructure, determining the feedbac gain and evaluating the erformance of the deign are the main te in the active vibration control. Thi can be achieved by the rocedure develoed in thi wor. The reult how that the energy of the controlled reone i maller than that of uncontrolled reone for higher gain. Better vibration ureion are achieved for = in dilacement feedbac. It i oberved that exerimental reult erformed by other cientit and imulation reult are in good agreement. Uer friendly FE rogram uch a ANSYS, ABAQUS, MSC/NASTRAN and control rogram uch a MATLAB are currently available. It i neceary to develo macro to integrate control action to the FE rogram a rooed in thi thei. Grahical uer interface (GUI) rocedure can be roduced in the FE rogram, o the uer can imulate control roblem without develoing macro. The following iue may be invetigated in the future: Active control of tructure having comlicated geometrie can be analyzed with the rocedure rooed. Otimal control law can be incororated into the FE model of tructure uing the rooed technique. The control of tructure having different tye of actuator uch a fluid ower and ervo motor can be imulated by the rooed technique The FE matrice can be exorted to other comuter rogram uch a MATLAB to erform cloed loo vibration control analye REFERENCES [1] Dafang Wu, Liang Huang, Bing Pan, Yuewu Wang, Shuang Wu. (11). Exerimental Study and Numerical Simulation of Active Vibration Control on High Order Mode of Piezoelectric Flexible Beam. IEEE, [] Jingjun Zhang, Lili He, Ercheng Wang, Ruizhen Gao. (8). Active Vibration Control of Flexible Structure Uing Piezoelectric Material. International Conference on Advanced Comuter Control ( ). China: IEEE. [] T. A. Zahidi Rahman, I. Z. Mat Daru. (11). Active Vibration Control of a flexible late. 4th International Conference on Mechatronic (ICOM 11). uala Lumur, Malayia: IEEE. Page 4

9 [4] Dong Jinghi, Guo ang Zheng Wei,Shen,Ding Ji. (1). Study on active vibration control technique baed on cantilever beam. International Conference on Comuter, Mechatronic, Control and Electronic Engineering (CMCE) ( ). IEEE. [5] HU Chao, NI Bo, GUO Wang. (1). Modal Vibration Control of Cantilever Plate. Proceeding of the 9th Chinee Control Conference ( ). Beijing, China: IEEE. [] Mohd Sazli Saad, Hihamuddin Jamaluddin, Intan Zaurah Mat Daru. (11). Active Vibration Control of Flexible Beam Sytem Uing Proortional Control Scheme in Finite Difference Simulation Platform. IEEE. [7] Vibration Control of Active Structure Uing Hybrid PD/Reetitive Control. (11). International Conference on Mechatronic and Automation (. 9-91). Beijing, China: IEEE. [8] ANSYS, 4. ANSYS uer manual, ANSYS, Inc., Canonburg, PA, USA, [9] MATLAB,The MathWor, Inc., United State, [1] H. aragulle, Levent Malgaca(7).Integration Of Active Vibration Control Method With Finite Element Model Of Smart Structure Graduate School Of Natural And Alied Science Douz Eylül Univerity [11] Hatch R. Michael (1).Vibration Simulation Uing MATLAB and ANSYS.CHAPMAN & HALL/CRC Boca Raton London New Yor Wahington, D.C.IEEE td, 1988, IEEE Standard on Piezoelectricity ANSI/IEEE Std ,. Page 5