ACTIVE VIBRATION CONTROL FOR STRUCTURE HAVING NON- LINEAR BEHAVIOR UNDER EARTHQUAKE EXCITATION

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1 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 ATIVE VIBRATION ONTROL FOR TRUTURE HAVING NON- LINEAR BEHAVIOR UNDER EARTHQUAE EXITATION Herlien D. etio 1 and angriyadi etio 1 Departent of ivil Engineering, Institut Tenologi Bandung Departent of echanical Engineering, Institut Tenologi Bandung ABTRAT: A study on actively controlled responses of non-linear syste using Artificial Neural Networ (ANN) is presented in this paper. The capability to learn, the ability to generalize to new inputs and the ability to perfor fast coputations are the aor advantages in application of neural networ. The responses of the non-linear syste, which is governed by Duffing s equation and driven by a haronic force, are deterined through the utilization of equivalent linearization ethod. The active force at any step of aplitude is deterined by using an optial control algorith which is then used as input data for neural networ training. The advantage of utilizing the neural networ as a neuro-controller is that it can considerably reduce coputing steps and tie. 1. INTRODUTION It is coon nowledge that buildings and structures ust withstand ever-changing environental loads, such as wind, earthquaes and waves, over the span of their useful lives. Yet, until very recently, buildings, bridges, and other constructed facilities have been built as passive structures that rely on their ass and solidity to resist outside forces, while being incapable of adapting to the dynaics of an ever-changing environent. In order to design adaptive structures, the concept of structural control was introduced. The control concept has been used for a long tie for airplanes, ships, and space structures. However, its application to civil structures has only recently been taen into consideration. tructural control involves basically the regulating of structural characteristics as to ensure desirable response of the structure under the effect of its loading. These echaniss provide in general a syste of auxiliary forces, the so-called control forces. These forces are designed so that they regulate continuously the structural response. Using the conventional control algorith in order to get the control forces required to control the structure, one ust go through several coplicated steps that usually taes a long coputational tie. By using the bloc of neural networ for coputing the control forces instead of using the conventional control algorith, the coputation tie can be reduced significantly. Active control of linear structures is well developed and is a powerful technique to control vibrations in civil engineering structures against natural dynaic loads. Unfortunately ost of the civil engineering structures exhibit non-linearity in certain local regions or possibly in the whole of the structure. The non-linear systes of routine engineering practice, particularly those subected to dynaic loads, have continued to be solved by linear ethods. For soe types of non-linearity, a linear odel for the structure is no longer appropriate in the analysis of real structures. It is possible to obtain odal properties with sufficient accuracy at low level of excitation, but it ay be copletely distorted at higher levels. In any theoretical studies, it has been found that the period of vibration decreases rapidly with increasing aplitude of vibration. In this study a cubic hardening non-linearity in the for of Duffing s equation is used as non-linear syste odel. The study presents a siple and rapid solution, in regard to coputational cost and the atheatical coplexities of active control of structures having non-linear stiffness. The solution is based on the linear odal analysis. At any step of aplitude, the non-linear ode is used for transforing the set of n coupled equations fro a physical base otion of an n degree-offreedo syste into a set of n uncoupled equations in a odal base syste and then the active Bac to Table of ontents 697

2 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 force is deterined as a function of odal aplitude by using an optial control algorith. The control force and the easured acceleration response of the structure are feed to the artificial neural networ syste as a set of training data. The reliability of the active control for non-linear systes is tested by using various sinusoidal excitation and Flores earthquae base acceleration.. VIBRATION ONTROLLER The ain purpose of vibration control is to reduce and to stabilize the responses of structures under dynaic loading such as wind, earthquae and a high reliability is required in vibration control devices. This is the reason why a passive vibration control device has been used widely as it does not require energy feed. This type of controller has no sensors and can not adapt to various type of external loading. To cope this proble, an active control device which is equipped with sensors used to easured the structure responses, is introduced. The vibration controller can be classified into passive type, sei-active type, active type and cobination of these. Table 1 shows a scheatic view of the vibration control ethods (eto, 3). Table 1 lassification of vibration controller (eto, 3). Passive Type ei active Type Active Type Hybrid Type Fixed point A A Auxiliary structure Auxiliary ass c c c c where o : ontroller; : ensor A : Actuator In the passive type, auxiliary ass, spring or daper is ounted to the ain structure. The auxiliary structures are chosen so that the response of the structure can be reduced. In the active type, a control force is generated with an actuator ounted on the structure to control the vibration of the structure. The vibration control forces are generated by an actuator and fed to the structure through reaction forces. These reaction forces can be obtained by using reaction of a fixed point, auxiliary ass and auxiliary structure. An active structural control syste has the basic configuration as shown scheatically in Figure 1. It consists of: ensors located on the structure to easure either external excitations, or structural response variables, or both. Devices to process the easured inforation and to copute necessary control forces needed based on a given control algorith. Actuators, usually powered by external energy sources, to produce the required forces. A c A A A c Bac to Table of ontents 698

3 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 EXTERNAL FORE TRUTURE TRUTURAL REPONE ENOR ONTROL FORE ENOR ATUATOR ALULATION OF ONTROL FORE Figure 1 onfiguration of structural control syste. 3. NON-LINEAR ODAL ANALYI onsider a non-linear n-degree-of freedo syste subected to a haronic excitation of frequencyω. The second order differential equation corresponding to the otion of this syste can be expressed by X& + X& + X + f X&, X = P cos Ω (1) ( ) t,,, f( X &,X) and P are the ass atrix, the daping atrix, the stiffness atrix, the vector or non-linear force which depends on the spatial displaceents X and their derivatives, and the vector of the aplitude haronic force, respectively. 3.1 Non-Linear Frequency and ode The principal approach of the ethod is the assuption that the ode of vibration in resonance conditions is assued to be the sae as the noral ode of the corresponding non-linear syste and that all the non-resonant co-ordinates can be neglected except the single resonant coordinate. The stationary solution of the ulti-degree-of-freedo syste of Eq. (1) is reduced to singledegree-of-freedo syste described by the single resonant noral co-ordinates at Ω = Ω. Thus the stationary solution of the autonoous conservative syste of Eq. (1) in the resonance condition can be approached as X ( t) = φ q sin Ω t () Here q is the odal aplitude, φ and Ω are the unnown non-linear ode shape and the unnown non-linear resonance frequency of ode. There exist any nuerical procedures for solving non-linear probles. The Newton-Raphson procedures are used for this study (etio et al., 199). The unnown non-linear paraeters can be obtained by inserting Eq. () into the autonoous conservative syste of Eq. (1) and neglecting all the higher haronic ters; one obtains [ D ]( q ){ φ } = λ [ ]{ φ }, where D ]( q ) = + ( q ), [ nl λ = Ω (3, 4) q ) is the non-linear stiffness atrix which depends on the odal aplitude q. nl ( In accordance with the Newton-Raphson procedure, the non-linear Eq. (3) can be rewritten in the for [ + ( q ) λ ] φ = g( λ, φ ) (5) nl The non-linear frequencies and odes of the syste are obtained as a function of odal aplitude q by increasing the odal aplitude progressively: Bac to Table of ontents 699

4 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 ( q ) = λ φ ( q ) = φ ω, φ ( q ) and ( q ) (6, 7) ω are the non-linear noral ode and natural frequency, respectively. This algorith should be rapidly converging for this iterative procedure, based upon the previous values for each given odal aplitude. 3. Equivalent Linearization ethod The principal approach of the ethod is the replaceent of the non-linear differential equation by a linear equation in such a way that an average of the difference between the two systes is iniized. onsider the non-linear second order differential equation of otion (1). The equivalent linear syste ay be expressed in the for X& + X& + X + ' X& + ' X = P cos Ωt (8) and are the equivalent daping atrix and the equivalent stiffness atrix, respectively. The atrices and are deterined by iniizing the difference ε between the non-linear syste (1) and the equivalent linear syste (8) for every x(t). ( X, & X) ' X ' X ε = f & (9) The iniization of ε is perfored according to the criterion T 1 π E( ε' ε) = ε dt = iniu, T = T ω The necessary conditions for the iniization specified by eq. (1) are δe( ε' ε) δ c i δe( ε' ε) = δ i (1) (11, 1) where c i and i are the eleents of the atrices and, respectively. The equivalent linear ters can be constructed according to Eqs. (1), (11) and (1). By assuing a periodic solution, the general solution of Eq. (8) can be approxiated by Fourier series which are truncated to a single haronic function x = cosω t) q (13) ( t) ( φre sin Ω t + φi Re φ and φ are the real and the iaginary parts of the non-linear coplex ode. I 4. TATE EQUATION AND ONTROL TRATEGY 4.1 Active ontrol For any step of aplitude the equivalent state space equation of a non-linear n-degree-of-freedo syste subected to a dynaic excitation p(t) can be expressed by: Z & () t = AZ() t + BU() t + Wp() t, Z ( ) = Z with t (14) X( t) Z( t) =, X & ( t) = O I A 1 ~, 1 ( q ) O =, H B 1 O = E W 1 Bac to Table of ontents 7

5 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 ~ ( q ) = + ( q ), Z(t) is the n state vector of displaceents and velocities, A is the n x n nl atrix of structural paraeters, B is the n x atrix specifying the location of the control forces, U(t) is the vector of control forces, W is the n x 1 atrix specifying external loading location atrix. Optial control of this structure can be deterined by iniizing quadratic perforance t T T [ () t Q Z() t U ( t) R () t ] dt f J = Z + U (15) Q is a n x n syetrical sei definite weighting atrix, R is an x syetrical positive weighting atrix and t f is a tie chosen to be longer than earthquae excitation. The deterination of the Q and R is discussed by oong et al. (1988). The optial control law is given by: U 1 T () t G Z() t = 1 R B P Z() t = (16) The proble of optial control is to deterine a gain vector G in such a way that a perforance index J is iniized subect to the constraining Eq. (14). The state vector Z(t) is accessible through easureent. The n x n syetrical atrix P is deterined by solving the following Riccati equation: PA 1 1 PBR = T T B P + A P + Q (17) Inserting Eq. (16) into Eq. (14), the state space Eq. (8) becoes: () t = ( A + BG ) Z() t Wf () t Z & + (18) The state space Eq. (18) shows that the structural paraeters are changed fro open-loop syste A in Eq. (14) to closed-loop syste A+BG. 4. Artificial Neural Networ An artificial neural networ is a syste with inputs and outputs, coposed of a nuber of siilar processing units (etio et al., 1997b; etio and etio, 3). These processing units operate in parallel and are arranged in a pattern. The units are connected to each other by adustable weights as shown in Figure. hanging the weights will change the input-output behavior of the networ, and weights are chosen in such a way as to achieve the desired input-output relationship. To achieve this goal, systeatic ways of adusting the weights have to be developed, which are called learning algoriths. The accuracy of the networ depends on the nuber of neural in the layer. A neural networ is characterized by the processing units, the networ topology and the learning algorith. The selection of a neural nuber in the networ layer is largely based on experience. The weighting constants are deterined randoly. Through the learning process, those values will change in such a way so that in the end, the whole neural networ will be able to give the expected output. i Input x Output g Input Layer W i Hidden Layers W Output Layer Figure Architecture of artificial neural networ. Bac to Table of ontents 71

6 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, Neural ontrol Force onsider Z(t) is the state vector of the vector of displaceent X(t) and vector of velocity X & (t) of the structure. The control force obtained by optial control procedure is G(Z(t)). Inputs to the neural networ are the acceleration easureents X & (t) and the networ output G( X & (t) ) is the control force G(Z(t)). The neural networ procedure consists of two steps, the learning step and the validation step. If the learning was successful the networ should produce an output sequence very close to the actual syste output. 5. NUERIAL EXAPLE A discrete odel of two-degree-of freedo syste with a local non-linearity was chosen as an exaple to illustrate the advantage and the accuracy of the ethods. A ass-spring syste is depicted in Figure 3, where a cubic non-linear spring and a single haronic excitation of frequency Ω are introduced at ass 1. α 1 1 f 1 f 1 c 1 x 1 c x Figure 3 odel of non-linear two-degree-of-freedo syste. 1 =1 g, =1.5 g, 1 = =1 N/, c 1 =c =.1 Ns/, α 1 =.1 N/ 3. The non-linear natural frequencies ω 1 and ω as function of odal aplitude, shown in Figures 4, are calculated by the non-linear ode ethod of Eq. (6). ω1 ω,8,6,4, 1,8 6,8 11,8 16,8 1,8,5 q1 1,5 1,5 1, 11, 1, 31, 41, 51, Figure 4 The non-linear natural frequencies as a function of odal aplitude q. Figure 5 shows the frequency responses of the structure for three different aplitude forces. oe aspects of the non-linear phenoenon can be seen, such as up phenoena and the dependence of the resonance frequency on the aplitude of vibration. The active force of non-linear syste as a function of aplitude is deterined by using sinusoidal excitation of various aplitudes and frequencies. The control force and the acceleration response of the non-linear structure are feed to the artificial neural networ syste as a set of training data. A three layer neural networ has been used for the neural controller. The input layer consists of 4 neurons with tansigoid and purelin activation functions, the hidden layer has 3 neurons and the output layer has only one neuron that represents the control force. The perforance of the trained neural controller was tested through several nuerical siulation. Figures 6a-b show the coparison between the uncontrolled and controlled structure responses of the non-linear structure deterined fro the optial control and fro the neural controller (ANN). The coparison is ade in the case of the non-linear odel as shown in Figure 3 which was subected to sinusoidal excitation of aplitude F 1 =.8 N, frequencies Ω=.4 and Ω=.8 rad/s at ass 1. q Bac to Table of ontents 7

7 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 Aplitude Excitation Frequency Figure 5 The non-linear frequency response for various level of excitation aplitude. The trained ANN is then used to control structural responses in which the structure is excited by Flores earthquae acceleration on the base. For this purpose, acceleroeters are attached to each ass in order to get the agnitude of acceleration needed by the ANN to copute control forces. Ap litude ( ),,15,1,5 -, ,1 -,15 -, t (sec) Optial ontrol ANN ontrol Uncontrolled (a) A plitude ( ),8,6,4, , -,4 -,6 -,8 t (sec) Optial ontrol ANN ontrol Uncontrolled Figure 6 Displaceent response of ass 1 subected to sinusoidal excitation: (a) F 1 =.8 N and frequency Ω=.4 rad/s at ass 1. (b) F 1 =.8 N and frequency Ω=.8 rad/s at ass 1. The quantitative coparison of R values of displaceent responses at ass 1 is presented in Table. Fro the table, it can be seen clearly that the trained ANN has a good capability in controlling the structure. It can also be seen that the result varies as function of the frequency of the training force. ases Table Reduction of displaceent response at ass 1. Uncontrolled (b) Optial ontrol ANN ontrol ontrolled % Reduction ontrolled % Reduction Training force frequency = Training force frequency = Training force frequency = Training force frequency = Training force frequency = Training force frequency = Test force frequency = Test force frequency = ixed frequency =. and ixed frequency =.3 and Figure 7 is the graph of the displaceent at first floor. The control force given to the structure is presented in Figure 8. Bac to Table of ontents 73

8 International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, Displaceent () t (second) tida Uncontrolled terontrol terontrol:jt ANN ontrolled terontrol: Optial ontrolled ontrol optial Figure 7 Displaceent response of ass 1 subected to Flores earthquae excitation ONLUION Figure 8 ontrol forces for Flores earthquae excitation. lose agreeent between the proposed ethod and the optial control ethod indicates that the approxiate control ethod is adequate for non-linear systes. The advantage of utilizing the neural networ as a neuro-controller is that it can considerably reduce coplicated atheatical probles, the coputation tie and can ensure the stability of control. The results of this study indicate that neural networs are powerful tools in searching solutions for non-linear control probles and can easily be extended for large structures. 7. REFERENE ANN hasil Result JT t (second) Optial hasil Result ontrol optial etio,., etio, H.D. and Jezequel, L. (199). A ethod of non-linear odal identification fro frequency response tests, Journal of ound and Vibration, pp , 158(3). oong, T.T. (1988). tate of the art review: active structural control in civil engineering, Engineering tructure, , etio, H.D., Pagwiwoo,.P., arwoadhi, A. and Andari, Y. (1997a). Active artificial neural networ (ANN) control on cable-stayed bridge pylons under dynaic loading, Proceeding of EP, Bandung. etio, H.D., etio,. and Tioteus (1997b). ontrol of building structures using estiated states, Proceedings of the nd International onference on Active ontrol in echanical Engineering, EL, Lyon, France. etio, H.D. and etio,. (3). Neuro-fuzzy control of building structure using an active ass daper: an experiental study, Proceeding of The Ninth East Asia Pacific onference on tructural Engineering and onstruction, Bali. eto,. (3). Low cost vibration controller, Proceeding of Pan Pacific yposiu for Earthquae Engineering ollaboration, Tsuuba, Japan. Bac to Table of ontents 74