Study On Dynamics Of The Non-conservative Viscoelastic And Piezoelectric Composite Plate With Cracks

The dynamic analysis of the non-conservative systems are widely applied in the practical engineering, such as moving systems in the damping medium, pipe conveying fluid, bridge, aerofoil and brake drum. Therefore, analyses of the dynamic stability of the non-conservative structure have important theoretical significance and widespread application value. The research objects of this paper are the viscoelastic and piezoelectric composite plate, the dynamic characteristics and dynamic stability of the viscoelastic and piezoelectric composite plate containing cracks subjected to non-conservative forces are analyzed. The main research work is as follows.(1) The transverse vibration characteristics of a viscoelastic rectangular thin plate and a viscoelastic plate with linearly varying thickness and cracks are investigated. The presence of cracks can result in discontinuity of the slope compatibility condition at the two sides of the crack location. Based on fracture mechanics theory, the additional rotation induced by the crack is given.Based on the thin plate theory and the two-dimensional viscoelastic differential type constitutive relation, the differential equation of motion of the viscoelastic plate and a plate with linearly varying thickness containing the all-over part-through crack are established by continuity conditions at the crack location. The generalized complex eigenvalue of the cracked viscoelastic plate are calculated, and the curves of complex frequencies versus are obtained.The transverse vibration characteristics of the viscoelastic plate containing single and multiple all-over part-through cracks are analyzed.(2) The vibration characteristics and dynamic stability of the viscoelastic rectangular plate and varying thickness viscoelastic plate with crack and subjected to uniformly distributed tangential follower forces are analyzed. The differential equations of the cracked viscoelastic rectangular, linearly varying thickness viscoelastic plate and parabolicly varying thickness viscoelastic plate subjected to follower force are established. The complex eigenvalue equations of the cracked viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt laws for distortion subjected to follower force are obtained by the differential quadrature method, and the 5 method is adopted at the crack continuity conditions.The general eigenvalue equations of cracked viscoelastic plate subjected to follower force are calculated. The effects of the geometric parameter, the dimensionless delay time and the crack parameters on the transverse vibration characteristics, type of instability and the corresponding critical loads of the non-conservative viscoelastic plates are analyzed.(3) The free vibration problems of a rectangular thin plate with finite elastic point supports and the edge attached to distributed elastic restraint are studied. The element-free Galerkin method is proposed to solve the transverse vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges. Based on the extended Hamilton's principle for the elastic dynamics system, the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established by the element-free Galerkin method. The eigenvalue equations are presented. Via numerical calculation, the curves of the natural frequency of thin plates versus the spring constant, locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained. As a conclusion, the effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.(4) The element free Galerkin method is proposed to solve the transverse free vibration of the composite cracked plate with surface bonded piezoelectric sheet. First, the energy function of the system is established, utilizing the extended Hamilton's principle for the elastic dynamics system and introduced the dimensionless variables and parameters, the dimensionless variational expression of the piezoelectric composite plate with cracks is deduced. Then, the dimensionless equations of motion and general eigenvalue equations of transverse vibration of the piezoelectric composite plate with multiple cracks are obtained. The vibration model functions are given. By number calculating the curves of the dimensionless natural frequency of thin plate versus the geometry parameters of cracks and piezoelectric materials are plotted. The effects of piezoelectric sheet on the transverse vibration characteristics of the thin plate and the changes of the natural characteristics of a rectangular plate due to the presence of cracks are analyzed.(5) The dynamic stability of the piezoelectric composite viscoelastic plate with crack and subjected to uniformly distributed tangential follower forces are analyzed. The active force of surface bonded piezoelectric materials to the plate is the typical non-conservative force.The research object is the composite plate of the viscoelastic plate with surface bonded piezoelectric sheet, the differential equation of motion of the piezoelectric composite viscoelastic plate subjected to follower force are established. The weak integral form of the differential equations and the boundary conditions are obtained, and the equations of equivalent weak integral form are discreted and calculated. The complex eigenvalue equations of the cracked piezoelectric composite viscoelastic plate subjected to follower force are obtained by the element-free Galerkin method. The effects of the piezoelectric sheet with voltage on the dynamic stability of the non-conservative viscoelastic plates are analyzed.