Bifurcation Analysis On Vibration Of Shape Memory Alloy Beam

In this paper, the bifurcation characteristics of single mode vibration of the SMA laminated beam are analyzed with average method and constrained bifurcation theory. The main work in this article is divided into the following five aspects:Part 3 A brief introduction of SMA, the characters of SMA and one-dimensional constitutive of SMA is given in this part. Then the research background, engineering significance and research prospects are briefly described.Part II SMA laminated beam model is established firstly. Then SMA laminated beam can be seen as a continuous one-dimensional elastic body. And through the stress analysis of micro-unit of the beam, the differential equation of motion of the beam can be got. The following, after taking the first mode of the beam, using Galerkin method, the ordinary differential equation of vibration can be got. In the case of 1 / 2 sub-harmonic resonance, using SMA double-flag style hysteretic character and average method, the amplitude-frequency response equation of the beam and the solutions of the equation can be obtained.Part III Using bifurcation theory and Maple software, the transition boundary and the regional bifurcation diagram of the whole system can be got. In the bifurcation diagram, obvious hysteretic phenomenon can be observed, and isolated solutions can also be found. The delay like this is significant for the vibration control of dynamic systems. Some special isolated solutions are also of great significance to the practical problems. By observing the hysteretic phenomenon, the conclusion that SMA can play the most important role to vibration control, when SMA is in incomplete state of phase transition.Part IV The theoretical analysis is verified by numerical simulation with Matlab software. Firstly, SMA double-flag style hysteretic model is expressed by absolute value functions; then the vibration equation got in partâ…¡is solved by numerical method; finally,the theoretical and numerical results are compared. At last, the conclusion that the theoretical results and numerical results are basically identical can be got.Part V Summary of the thesis.