On the Vibration Behavior Study of a Nonlinear Flexible Composite Beam Under Excitation Forces via Nonlinear Active Vibration Controller

Transcription

1 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: 9 On the Vibration Behavior Study o a Nonlinear Flexible Compoite Beam Under Exitation Fore via Nonlinear Ative Vibration Controller Y. S. Hamed * S. K. Elagan * Abtrat The main objet o thi work i the mathematial tudy o the vibration behavior in a nonlinear lexible ompoite beam deribed by non-linear dierential equation and we ou on the appliation o dierent ontrol algorithm or a lexible ompoite geometrially nonlinear beam like truture with Maro Fiber Compoite (MFC) atuator. Multiple ale perturbation method i applied to obtain the olution o lexible ompoite beam and the nonlinear aturation ontroller (NSC) up to the eond order approximation and to tudy the tability o the teady tate olution near dierent imultaneou reonane ae. The reulting dierent reonane ae are reported and tudied numerially. The tability o the teady tate olution near the eleted reonane ae i tudied applying both requeny repone equation and phae-plane tehnique. The eet o the dierent parameter o the ytem and the ontroller on the ytem behavior are tudied numerially. The eet o NSC ontroller on the ytem are invetigated numerially. Comparion with the available publihed work i reported. Index Term A lexible ompoite beam Stability Vibration ontrol Reonane. I. INTRODUCTION Chao i one o the mot exiting topi in the ield o engineering and phyial iene. Reearher rom variou ield devoted too muh eort in the analyi o haoti behavior a well a the ontrol o both vibration and hao or variou vibrating ytem. Appliation o ompoite material in aeropae and mehanial engineering ha been Y. S. Hamed and S. K. Elagan are with Department o Mathemati and Statiti Faulty o Siene Tai Univerity Kingdom o Saudi Arabia. ( eng_yaer_alah@yahoo.om). Y. S. Hamed i with Department o Engineering Mathemati Faulty o Eletroni Engineering Menoiya Univerity Menou 9 Egypt. S. K. Khalil i with Mathemati Department College o Siene Menoiya Univerity Shebin Elkom Egypt. ( ayed_khalil@yahoo.om). intenively developed in reent year. When ative element e.g. piezo-atuator and enor are embedded into the ompoite together with proper ontrol algorithm then o alled mart or intelligent element are manuatured. They are ued to improve operational parameter or redue the noie and vibration o the truture. One o the mot ommon method o vibration ontrol i the dynami aborber. It ha the advantage o low ot and imple operation at one modal requeny. In the domain o many mehanial vibration ytem the oupled non-linear vibration o uh ytem an be redued to non-linear eond order dierential equation whih are olved analytially and numerially. S. S. Oueini et al. [] tudied a nonlinear aborber baed on aturation phenomena a a vibration aborber or a linear model o a antilever beam. The inluene o damping o the ontroller and loop gain ha been hown uing Multiple Sale Method. Alo the enitivity on the initial ondition ha been heked too. H. A. Sodano et al. [] applied modern Maro Fiber Compoite atuator or vibration redution o inlated toru. They pointed out the advantage o thi kind o atuator in omparion with traditional PZT element. A peial iruit or the uage o Maro Fiber Compoite (MFC) a el-ening atuator ha been preented. O. N. Ahour and A. H. Nayeh [] have ued a nonlinear ontroller a a vibration aborber baed on aturation phenomena working or redution o lexural and torional vibration o the plate. The meaurement o requeny o exitation ha been added to the ytem to inreae the eetivene o the aturation ontrol. The appliation o ompoite material in aeropae and mehanial engineering ha been intenively developed in reent year. When ative element o enor and e.g. piezo-atuator are embedded into the ompoite together with proper ontrol algorithm then o alled intelligent element or mart are manuatured. They are ued to improve operational parameter or redue the vibration and noie o the truture. The Poitive Poition Feedbak (PPF) ontroller applied or a lexible manipulator i preented by J. Shan et al. []. They preented everal vibration mode in the ontrol trategy taking into aount a linear mathematial model o the plant. The (PPF) ontrol ha been ompared with the algorithm o February IJENS

2 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: veloity eedbak. Experimental tudie how that only PPF algorithm i able to work properly while lewing proe i realized. M. K. Kwak and S. Heo [6] preented the eetivene o the Poitive Poition Feedbak (PPF) algorithm applied or a model o a olar panel. They onidered the irt our mode o vibration. The blok-invere tehnique allowed ontrolling more mode than a number o enor and applied atuator. L. Jun et al. [7] applied nonlinear Saturation Controller (NSC) or a nonlinear plant uing tandard PZT pathe. Alo all reult preented or NSC algorithm or ytem are obtained where the ontroller i tritly tuned to the exitation ondition. J. Warminki et al. [8] tudied the experimental and theoretial invetigation o vibration o an autoparametri ytem ompoed o two beam with retangular ro etion. The dierential equation o motion with boundary ondition up to third-order approximation olution are derived uing appliation o the Hamilton priniple o leat ation. The experimental repone o the ytem tuned at : internal reonane ae are perormed or random and harmoni exitation. Mot important vibration mode are extrated rom a real mehanial ytem. It i hown that the ertain mode in the ti and lexible diretion o both beam may interat and intuitively unexpeted out-o-plane motion may appear. Alo the preliminary numerial alulation baed on the mathematial model are preented. J. Warminki et al. [9] preented numerial and experimental tudie o our type ontroller applied to nonlinear beam model. Eetivene o dierent ontrol trategie i evaluated uing numerial imulation in Matlab Simulink otware. The analytial olution or nonlinear NSC are obtained uing Multiple Sale Method. The analytial and numerial analyi or NSC ytem preent the inluene o mot eential parameter on the ontrol quality o a nonlinear plant model or wide range o requeny o exitation and high level o amplitude. The reult or a ingle beam ytem how that NSC and PPF ontroller are the mot eetive or aumed ondition o the plant. M. M. Kamel and Y. A. Amer [] invetigated the behavior o a ingle degree o reedom ytem with dierent quadrati damping and ubi tine non-linearitie imulating the vibration o a antilever beam under multi-parametri exitation ore. M. Eia and Y. A. Amer [] imulated the vibration o a eond order ytem to the irt mode o a antilever beam ubjeted to parametri and external exitation ore at primary and ub-harmoni reonane. They analyzed the ytem uing the multiple time ale perturbation tehnique. They reported that the vibration o the antilever beam an be ontrolled by adding a eedbak ubi non-linear term. M. Eia and M. Sayed [-] preented the eet o dierent ative ontroller on imple and pring pendulum via negative veloity eedbak or it quare or ubi at primary reonane ae. Y. A. Amer [] invetigated the behavior o the oupling o two non-linear oillator o the ytem and aborber repreenting ultraoni utting proe ubjeted to parametri exitation. A threhold value o the linear damping ha been obtained where vibration an be redued dramatially. Thi threhold value an be ued eetively or paive vibration ontrol i it i eonomial. The method o multiple ale perturbation tehnique i applied throughout to determine the olution up to eond order approximation. Guo et al. [6] tudy the non-linear dynami o a our-edge imply upported angle-ply ompoite laminated retangular thin plate exited by both the in-plane and tranvere load. The aymptoti perturbation method i ued to derive the our averaged equation under : internal reonane. Zhang et al. [7] invetigated the haoti dynami and biuration o a imply upported ymmetri ro -ply ompoite laminated piezoeletri retangular plate ubjet to the tranvere in-plane exitation and the exitation loaded by piezoeletri layer. The ae o : internal reonane and primary parametri reonane i onidered. Zhang et al. [8] tudied the haoti dynami o a ix-dimenional non-linear ytem whih repreent the averaged equation o a ompoite laminated piezoeletri retangular plate ubjeted to the tranvere in-plane exitation and the exitation loaded by piezoeletri layer. The ae o :: internal reonane are onidered. Zhang et al. [9] analyzed the haoti dynami and non-linear vibration o a imply upported orthotropi FGM retangular plate ubjeted to the in-plane and tranvere exitation together with thermal loading in the preene o :: internal reonane primary parametri reonane and ub-harmoni reonane o order /. The reonant haoti motion o a imply upported retangular thin plate with parametrially and externally exitation uing exponential dihotomie and an averaging proedure are analyzed by Zhang and Li []. Hamed et al. [-] analyzed the ultraoni mahining model ubjet to multi-external or both multi-external and multi-parametri and both multi-external and tuned exitation ore. The model onit o multi-degree-oreedom ytem oniting o the tool holder and aborber (tool) imulating ultraoni mahining proe. The advantage o uing multi-tool are to mahine dierent material and dierent hape at the ame time. Thi lead to time aving and higher mahining eiieny. Sayed and Hamed [] invetigated the repone o a two-degree-o-reedom ytem with quadrati oupling under parametri and harmoni exitation. The method o multiple ale perturbation method i applied to obtain approximate olution up to and inluding the eond-order approximation. Sayed and Kamel [ 6] tudied the eet o dierent ontroller on the vibrating ytem and the aturation ontrol o a linear aborber to redue vibration due to rotor blade lapping motion. The tability o the numerial olution i invetigated uing both phae plane method and requeny repone equation. The aim o thi work i to tudy eetene o NSC ontrol February IJENS

3 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: algorithm or large vibration o lexible ompoite beam truture. The reult are reeived by analytial and numerial tete. The method o multiple ale perturbation tehnique i applied to determine the olution up to eond order approximation. The tability o the ytem i invetigated applying both requeny repone equation and phae-plane method. Alo we an determine on the requeny repone urve where the ytem beome table and where beome untable or any parameter. The eet o all dierent parameter on ytem behavior are tudied numerially. Variation o ome equation parameter lead to the bending o the requeny repone urve and hene to the jump phenomenon. II. MATHEMATICAL MODELING Let u onider a ompoite lexible beam with a retangular ro-etion preented in Fig.. Equation o motion o the preent beam model i derived by the Hamilton priniple o lat ation. The ame proedure but or more ompliated beam ytem i hown in Warminke et al. [9]. Two ubytem mehanial and eletrial are oupled by a en or and an atuator. Thu idea i inorporated in NSC algorithm. We preent dynami o the nonlinear beam together with NSC ontroller: U U U U ( UU U U ) o t U o t V (a) V V V UV (b) Where U and V denote repone o the plant and ontroller and are natural requenie o the ytem (Plant) and the ontroller repetively and uppoe that where i mall parameter. U ( t ; ) u ( T T ) u ( T T )... V ( t ; ) v ( T T ) v ( T T )... and the time derivative beame d dt d dt D D D D D D Subtituting equation () and () into equation () and equating the oeiient o ame power o in both ide we obtain: O( ) : () () ( D ) u () ( D ) v () O( ) : ( D ) u o( T ) u o( T ) v D D u D u u ( u ( D u ) u ( D u )) (6) ( D ) v D Dv D v u v (7) The general olution o equation () and () i given by i T i T u ( T T) A e A e i T i T v ( T T) A e A e (8) where A A are untion in T and an be determined by elimination the eular term rom the next order o perturbation. Subtituting equation (8) into equation (6) (7) yield i T i T i T i T ( D ) u ( e e ) ( Ae Ae ) i T i T i T i T ( e e ) ( A e A e ) i ( Ae Ae ) [( A e A e ) i T i T i T i T ( A e A e ) ( A e A e ) i T i T i T i T ( A e A e )] i ( A e A e ) i T i T i T i T ( A e A e ) (9) i T i T Fig.. Model o the beam A. Perturbation analyi Equation () an be olved analytially uing multiple time ale perturbation tehnique [] a: ( D ) v i ( Ae Ae ) i T i T [( A e A e )( A e A e )] i T i T i T i T ( i T i T ) i A e A e () Where the prime indiate the derivative with repet to T. We analyze the ae February IJENS

4 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: () and where are three detuning parameter. Subtituting equation () into equation (9) and () and eliminating the term that produe eular term we get: i A i A A A A A i T i T i T e Ae Ae () i T i A i A A A e () We expre the omplex untion A A in the polar orm a A A a ( T ) a ( T ) e e i ( T ) i ( T ) () Where a a and are real untion in T. Subtituting equation () into equation () and () and eparating real and imaginary part yield a a a in in a in a 8 a o a o (6) a a aa in (7) a ( ) o (8) T T ( ) ( ) a o where T and () a a then it For the teady tate olution ollow rom equation () - (8) that the teady tate olution are given by a a in in a in (9) a a ( ) a o o 8 a o () in () a ( ) o a a () From equation (9)-() we have the ollowing ae: Cae: a and From equation () and () we get a pratial ae (ytem with ontrol) in () a o a () Inerting equation () and () into equation (9) and () we get a a a in in a a () a ( ) a 8 a a o o (6) Squaring and adding both ide o equation () and (6) we obtain a a a a ( ) a a 8 a a a 6 (7) Equation (7) i alled the requeny repone equation that deribed the ytem teady tate olution behavior. B. Stability analyi The tability o a partiular equilibrium olution wa determined by examining the eigenvalue o the jaobian matrix o the right-hand ide o equation ()-(8). In thi etion we analyze the tability analyi and evaluate the ontrol law. To derive the tability riteria we need to examine the behavior o a a mall deviation rom the teady tate olution and. Thu we aume that a a a a a a (8) orrepond to a linear olution where a a and and a a and are perturbation whih are aumed to be mall ompared with a a and.subtituting rom equation (8) into equation ()-(8) February IJENS

5 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: expanding or mall a a and term in a a and we obtain keeping linear a a a (9) a a () a a a () a a () Where ( i j...) are given in the appendix. ij The tability o the teady-tate motion depend on the eigenvalue o the above ytem o equation (9)-(). The eiginvalue are obtain orm the determined at primary and ub-harmoni reonane ae where ( ). It i oberved rom thi Fig. that the teady tate amplitude o the beam i about 9 time o the exitation ore amplitude. The olution i table and the phae plane how multi limit yle. () The neeary and uiient ondition or the teady tate olution to be table i that the real part o all eigenvalue o the Jaobian () are le than zero. Expanding the determinant () yield L L L L () Where L L L and L are oeiient o equation (). Routh-Hurwitz riterion i ued to etablih the tability o the equilibrium olution. Aordingly the neeary and uiient ondition or table ytem i L > L L L > L (L L L ) L L > L > () In the requeny repone urve the table (untable) teady-tate olution have been repreented by olid (dahed) line. III. NUMERICAL RESULTS The repone o the two-degree-o-reedom non-linear ytem under external and parametri exitation i tudied. The olution o thi ytem i determined up to the eond order approximation applying the multiple time ale perturbation. To tudy the behavior o the ytem () the Runge-Kutta o ourth order method wa applied to determine the numerial olution o the given ytem. Reult are preented in graphial orm a teady tate amplitude againt detuning parameter and time hitory or the repone or both beam and ontroller under the ollowing value o the ytem parameter =. =.98 β =. =. =. = = =. =.. Fig. illutrate the repone and the phae-plane or the beam without ontroller Fig.. Sytem behavior without ontroller at ub-harmoni reonane ( =. β =. =. =. =.) A. Eet o the NSC ontroller Fig. illutrate the repone or the beam with ontroller at imultaneou primary ub-harmoni and internal reonane ( = ). The energy input orward to the truture U (beam) by the exitation i largely tranerred to the ontroller V. The eetivene o the ontroller E a (E a = teady tate amplitude o the beam without ontroller / teady tate amplitude o the beam with ontroller) i about whih mean that the teady tate amplitude i redued to le than o the exitation amplitude. The aturation our or the beam and the ontroller at (t = e). Alo the oillation are inreaing dynami hao and beome table or the beam and the ontroller. February IJENS

6 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: the amplitude i dereaed and the urve are bent to the right a hown in Fig. (g). a a Fig. a. Eet o the detuning parameter Fig.. Sytem behavior with NSC ontroller at ( and ) B. Repone urve and eet o dierent parameter In thi etion the tability zone and eet o the dierent parameter are diued uing requeny repone equation. Alo the tability o the numerial olution i tudied uing the phae-plane method. The teady tate repone o the given ytem at variou parameter near the imultaneou primary ub-harmoni and internal reonane ae i invetigated and tudied. The requeny repone equation given by (7) i olved numerially at the ame value o the parameter o Fig.. In all Fig. the olid line tand or the table olution and the dotted line or the untable olution. Fig. (a) how the eet o the detuning parameter on the teady tate amplitude o the beam or the pratial ae where ( a and a ). In thi ig. the repone amplitude onit o a ontinuou urve whih i bent to the right and ha hardening eet and there exit jump phenomena. Thi ontinuou urve ha table and untable olution. Fig. (b) how that the teady tate amplitude o the beam i a monotoni dereaing untion in the linear damping oeiient μ. Fig. ( d) how that or poitive and negative value o the non-linear parameter and produe either hard or ot pring repetively a the urve are either bent to the right or to the let leading to the appearane o the jump phenomenon multi-valued amplitude and the untable region i inreaed. Fig. (e ) how that the teady tate amplitude o the beam i a monotoni inreaing untion in the exitation amplitude and the gain alo the untable region i inreaed. For inreaing value o the natural requeny a a Fig. b. Eet o the damping oeiient Fig.. Eet o the non-linear parameter Fig. d. Eet o the non-linear parameter February IJENS

7 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: a = =. = =. = =. untable i dereaed a hown in Fig. ( d e). Fig ( g h) how that the teady tate amplitude are monotoni inreaing untion in the exitation amplitude and the gain and the region o untable are inreaed. 8 6 a Fig. e. Eet o the exitation amplitude a Fig. a. Eet o the detuning parameter a Fig.. Eet o the gain w =. w = Fig. b. Eet o the damping oeiient a w = Fig. g. Eet o the natural requeny Fig.. Eet o ytem parameter on the requeny repone urve o the plant at ( a and a ) a Fig. how the eet o the detuning parameter on the teady tate amplitude o the ontroller or the pratial ae where ( a and a ) at the ame value o the parameter o Fig.. In Fig. (a) we note that the repone urve o the ontroller a have two branhe ymmetri about and have ingle-valued. Fig (b) how that at the interval [-.8.8] the teady tate amplitude i a monotoni dereaing untion in the damping oeiient otherwie the teady tate amplitude i a monotoni inreaing. Alo or inreaing value o the damping oeiient the natural requenie and the gain the teady tate amplitude are dereaed the region o a 8 6 Fig.. Eet o the damping oeiient w =w =. w =w =. w =w = Fig. d. Eet o the natural requenie and February IJENS

8 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: 6 a Fig. 6 how the omparion between the repone o the beam without ontroller at the ame parameter magnitude ued or tability and the requeny repone equation given by equation (7) at the primary and ub-harmoni reonane ae where. In the Fig. 6(a) the teady tate amplitude o the beam i about. whih i in a good agreement with the amplitude o the beam at a hown in Fig. 6(b). Fig. e. Eet o the gain a = =. = (a) Fig.. Eet o the exitation amplitude. a = = a (b) Fig. 6. The repone o the plant at the ame tability parameter magnitude a 8 6 Fig. g. Eet o the exitation amplitude Fig. h. Eet o the gain Fig.. Eet o ytem parameter on the requeny repone urve o the ontroller at ( a and a ) IV. CONCLUSIONS Ue An ative vibration ontroller or uppreing the vibration o the non-linear beam ubjeted to external and parametri exitation ore ha been tudied in the preene o : internal reonane. The olution o thi ytem i determined up to the eond order approximation applying the multiple time ale perturbation tehnique. The aturation phenomenon and internal reonane are ued to ontrol the teady tate and tranient vibration. The numerial reult how that the aturation ontrol o teady tate vibration i eiient. Both the requeny repone equation and the phae-plane tehnique are applied to tudy the tability o the ytem. The eet o the dierent parameter o the ytem and ontroller are tudied numerially. From the above tudy the ollowing may be onluded: February IJENS

9 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: 7. The amplitude o the beam i inreaed to about 9 time o the exitation ore amplitude at primary and ub-harmoni reonane.. For the eet o NSC ontroller at the imultaneou primary ub-harmoni and internal reonane ) The eetivene o the ontroller E a i about whih mean that the teady tate amplitude i redued to le than o the exitation amplitude. The aturation our or the beam and the ontroller at (t = e). For the main ytem:. For the requeny repone urve o the beam the teady tate amplitude o the beam i a monotoni dereaing untion in the linear damping oeiient μ the natural requenie and monotoni inreaing untion in the exitation amplitude and the gain.. For poitive and negative value o the non-linear parameter and lead to the appearane o the jump phenomenon and multi-valued amplitude.. The region o intability are inreaed or inreaing the linear damping oeiient μ and the gain. For the ontroller: 6. At the interval [-.8.8] the teady tate amplitude i a monotoni dereaing untion in the damping oeiient otherwie the teady tate amplitude i a monotoni inreaing. 7. The teady tate amplitude i a monotoni dereaing untion in the linear damping oeiient the natural requenie and the gain but the teady tate amplitude i a monotoni inreaing untion in the exitation amplitude and the gain. 8. The region o intability are dereaed or inreaing value o in the linear damping oeiient the Future work: natural requenie and the gain. Alo or inreaing value o the exitation amplitude and the gain the region o intability are inreaed. - In the uture work we an deal with the ame ytem ubjeted to imultaneou tuned and external or tuned and external and parametri exitation. - The ytem an be tudied at another dierent reonane ae. - The ytem an be tudied at paive ontroller. in Appendix a o a o a a 9 ( ) a o a a in in in o a a o in a a a in a aa in o o a o a a a a in REFERENCES [] A. H. Nayeh Perturbation method. New York: Wiley; 97. Chap. 6 pp. 8. [] S. S. Oueini A. H. Nayeh and J. R. Pratt A nonlinear vibration aborber or lexible truture Nonlinear Dynami 998 vol. pp.9 8. [] H. A. Sodano G. Park and D. j. Inman An invetigation into the perormane o maro-iber ompoite or ening and trutural vibration appliation Mehanial Sytem Signal Proe vol. 8 pp [] O. N. Ahour A. H. Nayeh Adaptive ontrol o lexible truture uing a nonlinear vibration aborber Nonlinear Dynami vol. 8 pp. 9. [] J. Shan H. Liu and D. Sun Slewing and vibration ontrol o a ingle-link lexible manipulator by poitive poition eedbak (PPF) Mehatroni vol. pp. 87. [6] M. K. Kwak and S. Heo Ative vibration ontrol o mart grid truture by multi-input and multi-output poitive poition eedbak ontroller Journal o Sound Vibration 7 vol. pp.. [7] L. Jun H. Hongxing and S. Rongying Sat uration-baed ative aborber or a non-linear plant to a prinipal external exitation Mehanial Sytem Signal Proe 7 vol. pp February IJENS

10 International Journal o Bai & Applied Siene IJBAS-IJENS Vol: No: 8 [8] J. Warminki M. P. Cartmell M. Bohenki and I. Ivanov Analytial and experimental invetigation o an autoparametri beam truture Journal o Sound Vibration 8 vol. no. pp [9] J. Warminki M. Bohenki W. Jarzyna P. Filipek and M. Augutyniak Ative uppreion o nonlinear ompoite beam vibration by eleted ontrol algorithm Communiation in Nonlinear Siene and Numerial Simulation vol. 6 no. pp [] M. M. Kamel and Y. A. Amer Repone o parametrially exited one-degree-o-reedom ytem with non-linear damping and tine Phyia Sripta vol. 66 pp. 6. [] M. Eia and Y. A. Amer Vibration ontrol o a antilever beam ubjet to both external and parametri exitation Applied Mathemati and Computation vol. pp [] M. Eia and M. Sayed "A omparion between paive and ative ontrol o non-linear imple pendulum Part-I" Mathematial and Computational Appliation 6 vol. pp [] M. Eia and M. Sayed "A omparion between paive and ative ontrol o non-linear imple pendulum Part-II" Mathematial and Computational Appliation 6 vol. pp. 6. [] M. Eia and M. Sayed "Vibration redution o a three DOF nonlinear pring pendulum" Communiation in Nonlinear Siene and Numerial Simulation 8 vol. pp [] Y. A. Amer "Vibration ontrol o ultraoni utting via dynami aborber" Chao Solution & Fratal 7 vol. pp [6] X. Y. Guo W. Zhang and M. Yao "Nonlinear dynami o angleply ompoite laminated thin plate with third-order hear deormation" Siene China Tehnologial Siene vol. pp [7] W. Zhang Z. Yao and M. Yao "Periodi and haoti dynami o ompoite laminated piezoeletri retangular plate with one-totwo internal reonane" Siene in China Serie E: Tehnologial Siene 9 vol. pp [8] W. Zhang M. Gao M. Yao and Z. Yao "Higher-dimenional haoti dynami o a ompoite laminated piezoeletri retangular plate" Siene in China Serie G: Phyi Mehani and Atronomy 9 vol. pp [9] W. Zhang J. Yang and Y. Hao "Chaoti vibration o an orthotropi FGM retangular plate baed on third-order hear deormation theory" Nonlinear Dynami vol. 9 pp [] W. Zhang and S. B. Li "Reonant haoti motion o a bukled retangular thin plate with paramet rially and externally exitation" Nonlinear Dynami vol. 6 pp [] Y. S. Hamed W. A. EL-Ganaini and M. M. Kamel Vibration uppreion in ultraoni mahining deribed by non-linear dierential equation Journal o Mehanial Siene and Tehnology 8 vol. no. 8 pp. 8-. [] Y. S. Hamed W. A. EL-Ganaini and M. M. Kamel Vibration uppreion in multi-tool ultraoni mahining to multi-external and parametri exitation Ata Mehania Sinia 9 vol. pp.. [] Y. S. Hamed W. A. EL-Ganaini and M. M. Kamel Vibration redution in ultraoni mahine to external and tuned exitation ore Applied Mathematial Modeling 9 vol. pp [] M. Sayed and Y. S. Hamed "Stability and repone o a nonlinear oupled pith-roll hip model under parametri and harmoni exitation" Nonlinear Dynami vol. 6 pp. 7. [] M. Sayed and M. Kamel "Stability tudy and ontrol o heliopter blade lapping vibration" Applied Mathematial Modelling vol. pp [6] M. Sayed and M. Kamel ": and : internal reonane ative aborber or non-linear vibrating ytem" Applied Mathematial Modelling vol. 6 pp.. February IJENS