Research On The Technique Of Restraining The Propagation Of Vibration In Beams Using The Theory Of Band Gap

The composite material with artificial periodicity, which has elastic band gaps, is called phononic crystal. The elastic wave with frequency following in the gaps could not propagate in phononic crystals. The locally resonant phononic crystal can restrain the propagation of the low frequency vibration with structures far smaller than the wave length. Phononic crystals with gap property offer new methods for restraining the propagation of the structural vibration. In this paper, the propagation of flexual wave in Timoshenko beam with local resonators is studied. We attempt to optimize the parameters of the local resonators of the Timoshenko beam with the genetic algorithm (GA) to obtain the largest attenuation of the propogating low frequency flexural waves in the beam. The main contents and results are as follows:1. Basing on the theory of transfer matrix (TM), we build the algorithm to calculate the vibration of Timoshenko beam with local resonators with one or two degrees of freedom. The algorithm is validated with finite element analysis of commercial software. It is illustrated that the transfer matrix method is accurate for calculating the vibration of Timoshenko beam attaching local resonators. Moreover, The TM method has advantages of precision and speed in calculation. In addition, It is very convenient to change the parameters of the attached local resonators and easy to be programmed. Therefore, The TM algorithm is integrated in the GA to optimize the structural parameters.2. We have discussed the influences of the parameters of the local resonators on the gaps of the Timoshenko beam. The conclusions are as follows: (1) Increasing the mass of resonators will decrease the beginning frequency of the gap and increase its attenuation, while increasing the mass too much will cause saturation. (2) Accretion of the springs'stiffness can widen the gap and enlarge its attenuation, but the beginning frequency of the gap will increase, which indicates that the stiffness of resonators must be chosen carefully to depress the flexural wave porpogation in beams. (3) Increasing the damp coefficients is an efficient method to weaken the formant but it also weakens the attenuation in the gap. From these analysises we conclude that these parameters are necessary to be optimized.3. Basing on the genetic algorithm, we build the algorithm for optimizing of the vibration characteristics of Timoshenko beam attaching local resonators. We employ insertion loss to evaluate the restration of the vibration propagation in the beam. After that, we develop an penalty strategy to take the constraints of attached mass and responsing displacement into account. Strategy of the optimization is worked out. The optimization method is implemented on Matlab based on its GA toolbox.4. Different constraints are considered when optimizing the insertion loss of periodically and nonperiodically attached local resonators on the Timshenko beam. The results are as follows: (1) Increasing the mass of resonators and damp coefficients can both increase the insertion loss, while too much increasing will cause saturation. (2) The maximal insertion loss won't be obtained whether the spring stiffness is too big or too small, thus there exists the most suitable spring stiffness. (3) When the boundaries are free, the periodic structures bring bigger insertion loss. (4) Limiting the parameters in a certain range makes the structures realizable but brings worse result of the optimization, what is worse, it may happen that the parameters can not satisfy the restraining.In this paper, we build the optimization algorithm for the effect of the restraining the propagating of low frequency vibration in Timoshenko beam attaching local resonators, when constraints are considered. It is important for the application of the theory of Locally Resonant band gaps to the subject of restraining the propagation of the low frequency structural vibration.