Wave mode characteristics on piezo-electric stepped beam

In this paper, a systematic approach for the freevibration analysis and forced-response of the beam bonded with PZT patchesis presented employing the travelling wave method.The wave propagation characteristic of the stepped beam bonded with PZTpatches is studied based on distributed parameters theories. Neglecting theeffect of transverse shear and rotary inertia, harmonic wave solutions arefound for both flexible and axial vibration of beam models. Then, the systemis simplified into a node model considering multiple point discontinuitiesdue to attached masses, and actuated moment of PZT patches. And wavescattering matrices including wave reflection and transmission matrices innodes are formulated by applying the compatibility of displacements andequilibrium of forces at the junctions. Based on the above work, the conceptof the wave loop, which is the process when the vibration wave comes througha periods along the wave propagation paths, is introduced, and wave loopsand transmission matrices are derived accounting for general boundaryconditions. Therefore, the wave loops matrices combined with the aid offield transfer matrices provides a concise and efficient method to solve thefree vibration problem of beam bonded with PZT patches. The frequencies andresponse solutions are exact since the effects of attenuating wavecomponents are included in the formulation. Furthermore, the generalrelations between the flexural wave transmission factor and the position ofthe PZT actuator in structures is discussed too. The numerical results givetwo major conclusions: 1) the PZT patch bonded position near by thefixed-end in beam has the powerful actuated capability, because theattenuating wave components created by the active wave incident upon thediscontinuities boundary enhance the transmission effectiveness of theactive traveling wave propagation; 2) the modulus of the mode transmissionfactor has a close relation with the sensitivity of the nature frequencies.The bigger modulus of the mode transmission factor, the bigger sensitivitiesfactor of the nature frequencies is.In addition, a comparison of eigenvalues and frequency response functionobtained by finite element method (FEM) and the wave method respectively isalso presented. It is indicated that the result by the wave method is moreexact than one by FEM.