Analysis for buckling and vibrations of composite stiffened shells and plates

This paper deals with development of triangular finite element for buckling and vibration analysis of laminated composite stiffened shells. For the laminated shell, an equivalent layer shell theory is employed. The first-order shear deformation theory including extension of the normal line is used. In order to take into account a non-homogeneous distribution of the transverse shear stresses a correction of transverse shear stiffness is employed. Based on the equivalent layer theory with six degrees of freedom (three displacements and three rotations), a finite element that ensures C⁰ continuity of the displacement and rotation fields across inter-element boundaries has been developed. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Results of vibration and buckling analysis of stiffened plates and shells are discussed.     

Stability of steady-state vibrations of a nonlinear,imperfectly elastic two-mass system subjected to harmonic excitation

A study is made of the stability of steady-state vibrations of harmonically excited two-mass systems with allowance for the imperfect elasticity of the materials and cubic nonlinearities. It is shown that, in the general case, when imperfect elasticity is accounted for by the model constructed by G. S. Pisarenko, an increase in the inflection of the skeleton curves with an increase in the amplitude of the vibrations may lead to the appearance of regions of instability and amplitude jumps. The averaging method is used to determine the conditions under which the amplitudes will be stable. Resonance curves of amplitude are constructed for a two-mass system modeling an imperfectly elastic dynamic vibration damper. The range of stable amplitudes is also determined for this system. It is shown that a cubic nonlinearity can either reinforce or offset the inflections caused by inelasticity.Translated from Problemy Prochnosti, No. 8, pp. 69–76, August, 1994.     

Nonlinear vibrations of shells and plates with account for energy dissipation in the layers

We solve the problem of the vibrations of shells and plates with large deflections with account for energy dissipation in the layers and analyze the influence of the structure of the packet of layers on the vibration decrement.Translated from Problemy Prochnosti, No. 3, pp. 61–66, March, 1993.     

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.     

Nonlinear vibrations of angle-ply laminated circular cylindrical shells: Skewed modes

Skewed modes in geometrically nonlinear forced vibrations of angle-ply laminated circular cylindrical shells are investigated in the present study by using the Amabili–Reddy higher-order shear deformation theory. An harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retains dissipation. Numerical results are obtained by using the pseudo-arclength continuation method and bifurcation analysis.     

Steady-state longitudinal vibrations of sectional,physically nonlinear plates

The artile deals with steady-state longitudinal vibrations of plates consisting of curvilinear rings placed one inside the other without interference, under the effect of a harmonic distorting load acting along the outer periphery. The rings are made from different isotropic materials that are physically nonlinear. The problem is solved by the method of successive approximations. In each approximation the sought functions are represented in the form of expansions with respect to a small parameter. Numerical investigations were carried out for plates consiting of two concentrically arranged circular rings.Translated from Problemy Prochnosti, No. 3, pp. 24–27, March, 1991.     

On the validity of nonlinear shear deformation theories for laminated plates and shells

This paper addresses the validity of the recently introduced so-called nonlinear shear deformation theories for laminated composite plates and shells. The finite element method is used to determine the maximum stresses for a wide range of statically loaded plate and shell panels. Various thickness ratios are included. This paper concludes that for the vast majority of composite materials and for moderately thick plates and shells. stresses normally reach the maximum allowable stress before nonlinear terms can become important. This has been demonstrated by showing that for the limiting case of shear deformation theories (in which the minimum span length (or radius) to thickness ratio is 20), the material usually fails before the maximum deflection reaches the magnitude of the thickness (where nonlinear terms start to become significant). Therefore, the nonlinear shear deformation theories, which are considerably more complicated than linear ones, have limited applications.     

Active damping of geometrically nonlinear vibrations of doubly curved laminated composite shells

This paper deals with the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of doubly curved laminated composite shells. Vertically/obliquely reinforced 1–3 piezoelectric composite (PZC) and active fiber composite (AFC) materials are used as the materials of the constraining layer of theACLD treatment. The Golla–Hughes–McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. The first-order shear deformation theory (FSDT) and the Von Kármán type non-linear strain displacement relations are used for analyzing this coupled electro-elastic problem. A three dimensional finite element (FE) model of doubly curved laminated smart composite shells integrated with ACLD patches has been developed to investigate the performance of these patches for controlling the geometrically nonlinear transient vibrations of the shells. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the doubly curved laminated cross-ply and angle-ply shells for suppressing their geometrically nonlinear transient vibrations. It is found that the performance of the ACLD patch with its constraining layer being made of the AFC is significantly higher than that of the ACLD patch with vertically/obliquely reinforced 1–3 PZC constraining layer. The effects of variation of piezoelectric fiber orientation in both the obliquely reinforced 1–3 PZC and the AFC constraining layers on the control authority of the ACLD patches have also been investigated.     

Parametric vibrations and stability of viscoelastic shells

The problem of dynamic stability of viscoelastic extremely shallow and circular cylindrical shells with any hereditary properties, including time-dependence of Poisson’s ratio, are reduced to the investigation of stability of the zero solution of an ordinary integro-differential equation with variable coefficients. Using the Laplace integral transform, an integro-differential equation is reduced to the new integro-differential one of which the main part coincides with the damped Hill equation and the integral part is proportional to the product of two small parameters. Changing this equation for the system of two linear equations of the first order and using the averaging method, the monodromy matrix of the obtained system is constructed. Considering the absolute value of the eigen-values of monodromy matrix is greater than unit, the condition for instability of zero solution is obtained in the three-dimensional space of parameters corresponding to the frequency, viscosity and amplitude of external action. Analysis of form and size of instability domains is carried out.     

An analysis of nonlinear thickness-shear vibrations of quartz crystal plates using the two-dimensional finite element method

A nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut quartz crystal plates is presented with the two-dimensional finite element method. The Mindlin plate equations are truncated to the first-order ones as an approximation, and then they are used for the formulation of nonlinear finite element analysis with all zero- and first-order displacements. The matrix equation of motion is established with the first-order harmonic approximation, and the generalized nonlinear eigensystem is solved by a direct iterative procedure. A displacement amplitude versus frequency curve and corresponding mode shapes are obtained and examined. The nonlinear finite element program is developed based on the earlier linear edition and can be utilized to predict nonlinear characteristics of miniaturized quartz crystal resonators in the design process.     

Flexural vibrations of poroelastic plates

Summary The governing equations of flexural vibrations of thin, fluid-saturated poroelastic plates are derived in detail. The plate material obeys Biot's theory of poroelasticity with one degree of porosity, while the plate theory employed is the one due to Kirchhoff. These governing equations are compared with the corresponding ones for thermoelastic plates and a poroelastic-thermoelastic analogy for flexural plate dynamics is established in the frequency domain. The dynamic response of a rectangular, simply supported, poroelastic plate to harmonic load is obtained analytically-numerically and the effects of inertia as well as of porosity and permeability on the response is assessed.     

Extensional vibrations of piezoelectric plates

Summary We consider piezoelectric plates with a thickness small with respect to the lateral dimensions. The surfaces of these plates are partly coated with electrodes. Equations are derived which describe the lateral vibrations of these plates approximately. The first resonance-frequency of a circular plate as a function of the radius of the electrodes is computed and compared with measured values.