A novel order reduction method for nonlinear dynamical system under external periodic excitations

作者:CAO DengQing;WANG JinLin & HUANG WenHu School of Astronautics;Harbin Institute of Technology;PO Box 137;Harbin 150001;China 刊名:中国科学:技术科学 上传者:周培祺

【摘要】The concept of approximate inertial manifold (AIM) is extended to develop a kind of nonlinear order reduction technique for non-autonomous nonlinear systems in second-order form in this paper. Using the modal transformation, a large nonlinear dynamical system is split into a 'master' subsystem, a 'slave' subsystem, and a 'negligible' subsystem. Accordingly, a novel order reduction method (Method I) is developed to construct a low order subsystem by neglecting the 'negligible' subsystem and slaving the 'slave' subsystem into the 'master' subsystem using the extended AIM. As a comparison, Method II accounting for the effects of both 'slave' subsystem and the 'negligible' subsystem is also applied to obtain the reduced order subsystem. Then, a typical 5-degree-of-freedom nonlinear dynamical system is given to compare the accuracy and efficiency of the traditional Galerkin truncation (ignoring the contributions of the slave and negligible subsystems), Method I and Method II. It is shown that Method I gives a considerable increase in accuracy for little computational cost in comparison with the standard Galerkin method, and produces almost the same accuracy as Method II. Finally, a 3-degree-of- freedom nonlinear dynamical system is analyzed by using the analytic method for showing predominance and convenience of Method I to obtain the analytically reduced order system.

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IntroductionThestructureofmodernmachineandtheconnectionofcomponentassemblyarehighlycom-plicatedatpresent.Thefiniteelementmethodhasmanyadvantagesintheanalysisofcomplexmachinesystemsanditisprobablythebestchoiceforpracticalstructuraldesign.Butinthemachinestructureanalysisarefineddiscretizationisusuallynecessarytoobtainareliableresponse,especiallyifthenonlineareffectsaretakenintoaccount.Inthisprocess,alargesetofsecondordernonlinearordinarydifferentialequationisestablished.Thelargeordernonlinearsystemsofequationareusuallycostlytosolveintermsofcomputertimeandstorage,especiallyoverlong-timeintervals.Hence,itisessentialtoreducetheorderofthesystem.Theobjectiveofthemodelreductionconsideredhereistoobtainasmallsubsystem,representativeoftheoriginalsystem,ofvari-ables.Thelinearorderreductionmethods,suchastheGuyanorderreductionmethod[1],internalbalancingtechnique[2]andstandardGalerkinmethod(SGM)mayalsobeappliedtononlineardynamicsystems.Althoughthelinearorderreductionmethodshavebeenprovedtobeefficientinconstructingapproximatesolutionsfornonlineardynamicsystemsitself,ithasownlimitationthatsmallersubsystemoftensacrificesaccuracy.AnovelorderreductionmethodfornonlineardynamicalsystemunderexternalperiodicexcitationsTheconceptofapproximateinertialmanifold(AIM)isextendedtodevelopakindofnonlinearorderreductiontechniquefornon-autonomousnonlinearsystemsinsecond-orderforminthispaper.Usingthemodaltransformation,alargenonlineardynamicalsystemissplitintoa'master'subsystem,a'slave'subsystem,anda'negligible

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