Dynamic Bifurcation And Control Of The Structures With Shape Memory Alloy(SMA)

The structures with shape memory alloys (SMA) may produce complex dynamicbifurcation with the variation of their parameters. Therefore, it is necessary to studythe parameters’ influences on the dynamic bifurcation in order to guide the structuredesign. Based on the basic mechanics principles, the paper establishes the models forseveral types of the SMA composite structures and investigates their bifurcations andHopf bifurcation control. The main achievements are listed below.1．An SMA oscillator involving the double-loop hysteresis is studied to revealtheir bifurcations under primary resonance excitation. Firstly the piecewiseamplitude-frequency equation is acquired by applying the averaging method. Then alltypes of the qualitatively different amplitude-frequency curves and theircorresponding parameter conditions are obtained by the constrained bifurcation theory.Finally, the effects of SMA on vibration reduction are analyzed. The results show thatthe SMA element works most effectively if its strain is less than the critical value forfinishing the martensite phase transformation. In addition, SMA’s effects areweakened as the temperature elevation.2．The SMA laminated beam is analyzed for its dynamical bifurcations underprimary resonance excitation. The partial differential dynamic equation of SMAlaminated beam is established. The model firstly accounts for the effect of thediscontinuity in the strain gradient on the axial space variable in the SMA layers,which is ignored in the existed literature. By using the Galerkin method, thesingle-mode dynamical equation is obtained in which the restoring force caused bythe SMA layers is expressed by the piecewisely nonlinear double-loop hysteresisrelationship with the deformation. The average method is used to obtain theamplitude-frequency response equation, and then the constrained bifurcation theory isemployed to study the influences of the elasticity modulus and the excitationamplitude on bifurcation modes. In addition, the results show that SMA layers canreduce the beam vibration significantly in the progress of SMA martensite phasetransformation through the comparison between the primary resonance responses ofthe system and its linearization system.3．The dynamics stability and robust control of SMA hybrid composite (SMAHC)panel under the aerodynamic load and thermal load are studied. A modified Brinson model is proposed to calculate the restoring force of SMA as the strain is constrained.By applying the Hamilton principle, the SMAHC panel model is established in whichthe aerodynamic load is described by the first-order piston theory and the deformationrelation by von Kármán nonlinear theory. Then the multi-mode version of this modelis deduced by the Galerkin method. Subsequently, the dynamic stability of SMAHCpanel is studied and the flutter critical conditions and the buckling critical conditionsare determined in the parameter plane composed with the surrounding temperatureand the aerodynamic pressure. The results show that the increasement of SMAcomponent can enlarge significantly the regions in the parameter plane correspondingto the panel in flat and stable state. The results also show that SMA’s control on thelimit-cycle flutter of the panel is robust to the parameters’ variation.