DYNAMIC RESPONSE OF LAMINATED ORTHOTROPIC SPHERICAL SHELLSINCLUDING TRANSVERSE SHEAR DEFORMATION AND ROTATORY INERTIA

（杨宜谦）（马和中）（王俊奎）ＤＹＮＡＭＩＣＲＥＳＰＯＮＳＥＯＦＬＡＭＩＮＡＴＥＤＯＲＴＨＯＴＲＯＰＩＣＳＰＨＥＲＩＣＡＬＳＨＥＬＬＳＩＮＣＬＵＤＩＮＧＴＲＡＮＳＶＥＲＳＥＳＨＥＡＲＤＥＦＯＲＭＡＴＩＯＮＡＮＤＲＯＴＡＴＯＲＹＩＮＥＲＴＩＡ￥Ｙａｎｇ...     

Vibration analysis of nonhomogeneous orthotropic cylindrical shells including combined effect of shear deformation and rotary inertia

This paper is the result of an investigation on the vibration of non-homogeneous orthotropic cylindrical shells, based on the shear deformation theory. Assume that the Young’s moduli, shear moduli and density of the orthotropic material are continuous functions of the coordinate in the thickness direction. The basic equations of non-homogeneous orthotropic cylindrical shells with the shear deformation and rotary inertia are derived in the framework of Donnell-type shell theory. The ends of a non-homogeneous orthotropic cylindrical shell are considered as simply supported. The basic equations are reduced to the sixth-order algebraic equation for the frequency using the Galerkin method. Solving this algebraic equation, the lowest values of non-dimensional frequency parameters for non-homogeneous orthotropic cylindrical shells with and without shear deformation and rotary inertia are obtained. Calculations, effects of shear stresses and rotary inertia, orthotropy, non-homogeneity and shell geometry parameters on the lowest values of non-dimensional frequency parameter are described. The results are verified by comparing the obtained values with those in the existing literature.     

Fluid inertia in large amplitude oscillatory shear

Homogeneous shearing is required in sliding plate flow experiments with one plate fixed and the other oscillating. However, when fluid inertia becomes significant, the velocity gradient and the stress will not be uniform. MacDonald et al. (1969) and Schrag (1977) investigated this effect for a linear viscoelastic fluid. However, linear viscoelasticity does not describe the behavior of melts in large amplitude oscillatory shear (LAOS). Jeyaseelan et al. (1993) have shown that the Berkeley kinetic network model does accurately describe the LAOS behavior of polymer melts. In this work, the Berkeley model is solved for LAOS in sliding plate flow with fluid inertia, by numerical integration of spatially discretized forms of the governing equations. Nonlinear viscoelasticity is predicted to aggravate the effects of fluid inertia in LAOS and experiments confirm this. Specifically, fluid inertia amplifies the first harmonic and produces no even harmonics. Operating limits are presented graphically for minimizing inertial effects in LAOS experiments. Received: 2 January 1998 Accepted: 27 April 1998     

Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia

The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov–Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov–Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.     

Influence of rotatory inertia on dynamic stability of the viscoelastic symmetric cross-ply laminated plates

The dynamic stability problem of the symmetrically laminated cross-ply plates made of the viscoelastic Voigt–Kelvin material, compressed by time-dependent stochastic membrane forces, is investigated. The effect of rotatory inertia is included in the present formulation. It is assumed that all elastic moduli have the same retardation times. By using the direct Liapunov method, bounds of the almost sure stability of cross-ply plates as a function of viscous damping coefficient, retardation time, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, cross-ply ratio and intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as an harmonic process with random phase.     

Influence of rotatory inertia and transverse shear on stochastic instability of the cross-ply laminated beam

The stochastic instability problem associated with an axially loaded cross-ply laminated beam is formulated. The effects of shear deformation and rotatory inertia are included in the present formulations. The beam is subjected to time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds for the almost sure instability of beams as a function of viscous damping coefficient, variance of the stochastic force, ratio of principal lamina stiffnesses, shear correction factor, number of layers, mode numbers and geometrical ratio, are obtained. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ² as well as for harmonic process with an amplitude A.     

Dynamic stability of viscoelastic circular cylindrical shells taking into account shear deformation and rotatory inertia

The present work discusses the problem of dynamic stability of a viscoelas- tic circular cylindrical shell,according to revised Timoshenko theory,with an account of shear deformation and rotatory inertia in the geometrically nonlinear statement.Pro- ceeding by Bubnov-Galerkin method in combination with a numerical method based on the quadrature formula the problem is reduced to a solution of a system of nonlinear integro-differential equations with singular kernel of relaxation.For a wide range of vari- ation of physical mechanical and geometrical parameters,the dynamic behavior of the shell is studied.The influence of viscoelastic properties of the material on the dynamical stability of the circular cylindrical shell is shown.Results obtained using different theories are compared.     

Modified Kirchhoff's theory of plates including transverse shear deformations

A primary flexure problem defined by Kirchhoff theory of plates in bending is considered. Significance of auxiliary function introduced earlier in the in-plane displacements in resolving Poisson-Kirchhoff's boundary conditions paradox is reexamined with reference to reported sixth order shear deformation theories, in particular, Reissner's theory and Hencky's theory. Sixth order modified Kirchhoff's theory is extended here to include shear deformations in the analysis.     

Shock and acceleration waves with large amplitudes in laminated composite plates

Propagation of shock and acceleration waves with large amplitudes is studied. The geometrical nonlinearity in the von Karman sense is included in deriving the plate equations. The dynamical conditions on the wave fronts are derived from the three-dimensional conditions in a way consistent with the derivation of the plate equations. General equations governing the propagation velocities are obtained. Solutions are presented for the case where the plates are initially at rest. It is found that, in this case, the large amplitude has a substantial effect only on the transverse shear shock wave. Finally, stability of the wave front is discussed.     

Stability of clamped skew plates

The stability problems of clamped skew plates are considered with the inplane stresses represented in terms of oblique components. Deflection is expressed in terms of a double series of beam characteristic functions of clamped-clamped beam. Energy method is used to obtain buckling coefficients under individual loadings and for a few cases of combined loading. Convergence is examined in a few representative cases. For buckling in shear, two critical values exist the magnitude of negative shear being much larger than that of positive shear.     

Large-deflection bending of symmetrically laminated rectilinearly orthotropic elliptical plates including transverse shear

Summary The nonlinear bending theory for symmetrically laminated elliptical plates exhibiting rectilinear orthotropy with transverse shear deformation is developed. Using Galerkin's method, the paper solves the problem of large deflections for plates under uniform lateral pressure. The special case of symmetrically laminated rectilinearly orthotropic circular plates is also discussed. Analytical solutions obtained may be applied directly to the design of engineering structures. Received 6 March 1995; accepted for publication 8 January 1997     

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.     

Complex potential formalisms for bending of inhomogeneous monoclinic plates including transverse shear deformation

This paper develops complex potential formalisms for the solution of the bending problem of inhomogenoeus anisotropic plates, on the basis of the most commonly used refined plate theories. Being an initial step in that direction, it works out such formalisms only in connection with the bending problem of shear deformable homogeneous plates as well as plates having a special type of inhomogeneity along their thickness direction. The adopted type of inhomogeneity is however still general enough to include certain classes of plates made of functionally graded material as well as the classes of cross- and angle-ply symmetric laminates as particular cases. The basic formalism, similar to that developed by Stroh in plane strain elasticity, is detailed in relation with the equilibrium equations of a generalized plate theory that accounts for the effects of transverse shear deformation and includes conventional, refined theories as particular cases. Some interesting specializations, related to the most important of those conventional plate theories, are then presented and discussed separately. Hence, the outlined formalisms provide, for the first time in analytical form, the general solution of the partial differential equations associated with the most commonly used refined, elastic plate theories.     

Natural frequencies of composite beams with a variable fiber volume fraction including rotary inertia and shear deformation

The present study is concerned with the vibration analysis of symmetric composite beams with a variable fiber volume fraction through thickness. First-order shear deformation and rotary inertia have been included in the analysis. The solution procedure is applicable to arbitrary boundary conditions. Continuous gradation of the fiber volume fraction is modeled in the form of an m-th power polynomial of the coordinate axis in the thickness direction of the beam. By varying the fiber volume fraction within the symmetric composite beam to create a functionally graded material （FGM）, certain vibration characteristics are affected. Results are presented to demonstrate the effects of shear deformation, fiber volume fraction, and boundary conditions on the natural frequencies and mode shapes of composite beams.