Numerical investigation of the stability of forced nonlinear vibrations of a shallow cylindrical shell

Kiev Civil-Engineering Institute. Translated from Problemy Prochnosti, No. 4, pp. 74–78, April, 1989.     

A finite-element method for creep calculations on structures

A form of the initial-strain method is proposed that implements a general equation of state for a material during creep. A noniterative exact method has been devised for incorporating the effects of temperature on the elastic properties in a finite-element calculation.Translated from Problemy Prochnosti, No. 4, pp. 8–13, April, 1992.     

Quasiperiodic forced vibrations of a beam interacting with a nonlinear spring

Summary A simply supported beam with fixed ends and an attached strongly nonlinear spring is considered. As the forced bending vibrations of the beam have moderate amplitudes, the stretching force is a nonlinear function of the deflection. The vibrations are presented as a series with respect to the modes of the beam without attachment. Applying the Bubnov-Galerkin procedure, an infinite system of ordinary differential equations is derived. Using the multiple scales method, the quasiperiodic vibrations in the region of the combination resonance are analyzed in the paper. The system vibrations, when the nonlinear spring is attached in different points, are analyzed. Applicability of the nonlinear spring for mitigation of a combination resonance is discussed.     

Finite element method for non-linear forced vibrations of circular plates

Geometric non-linearities for large amplitude free and forced vibrations of circular plates are investigated. In-plane displacement and in-plane inertia are included in the formulation. The finite element method is used. An harmonic force matrix for non-linear forced vibration analysis is introduced and derived. Various out-of-plane and in-plane boundary conditions are considered. The relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented.     

Random excitation of nonlinear elastic structures with internal resonances

The influence of random vibration on the design of mechanical components has been restricted to the linear theory of small oscillations. However, this theory is inadequate and fails to predict the complex response characteristics which may be observed experimentally and can only be predicted by employing the nonlinear theory. This paper presents a brief overview of the basic nonlinear phenomena associated with nonlinear random vibration. An example of a clamped-clamped beam under filtered white noise excitation in the neighbourhood of 1:1 internal resonance condition is considered. Three approaches are employed to examine the response and stochastic bifurcation of the beam coupled modes. These are the Fokker-Planck equation together with closure schemes, Monte Carlo simulation, and experimental testing. The analytical results are compared with those determined by Monte Carlo simulation. It is found that above a critical static buckling load the analytical results fail to predict the snap-through phenomenon, while both Monte Carlo simulation and experimental results reveal the occurrence of snap-through. The bifurcation of second mode is studied in terms of excitation level, internal detuning and damping ratios. It is found that below the critical load parameter, the response statistics do not significantly deviate from normality. Above the critical value, where snap-through takes place, the response is strongly non-Gaussian. This research is supported by a grant from the National Science Foundation under grant number MSS-9203733 and by additional funds from the Institute for Manufacturing Research at Wayne State University.     

Stability of a compressed elastic plate with nonlinear supporting elements

An approach for studying the critical loading parameters of stability of an elastic plate with nonlinear supporting elements is developed. A general potential function of the system depending on state variables and control parameters is constructed. Classical methods of catastrophe theory including a determination of the different bifurcational sets and an analysis of the physical meaning of the corresponding stability criteria are used. The influence of the supporting elements on the behaviour of the plate under the loading is determined     

Geometrically nonlinear forced vibrations of the symmetric rectangular honeycomb sandwich panels with completed clamped supported boundaries

Geometrically nonlinear forced vibrations of symmetric rectangular honeycomb sandwich panels with clamped supported boundaries at the four edges are investigated using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose parameters of physical and mechanical properties are calculated by the corrected Gibson’s formula. The basic formulation of nonlinear forced vibrations has been developed based on the classical plate theory (CPT) and the nonlinear strain–displacement relation. The equilibrium equations have been obtained using Hamilton’s principle. Effects of axial half-waves, height and height ratio on the nonlinear free vibration response have been investigated for honeycomb sandwich panels.     

A BEM based domain decomposition method for nonlinear analysis of elastic space membranes

In this paper the Domain Decomposition Method (DDM) is developed for nonlinear analysis of both flat and space elastic membranes of complicated geometry which may have holes. The domain of the projection of the membrane on the xy plane is decomposed into non-overlapping subdomains and the membrane problem is solved sequentially in each subdomain starting from zero displacements on the virtual boundaries. The procedure is repeated until the traction continuity conditions are also satisfied on the virtual boundaries. The membrane problem in each subdomain is solved using the Analog Equation Method (AEM). According to this method the three coupled strongly nonlinear partial differential equations, governing the response of the membrane, are replaced by three uncoupled linear membrane equations (Poisson's equations) subjected to fictitious sources under the same boundary conditions. The fictitious sources are established using a meshless BEM procedure. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Geometrically nonlinear forced vibrations of the symmetric honeycomb sandwich panels affected by the water

Geometrically nonlinear forced vibrations of the symmetric rectangular honeycomb sandwich panels with the four edges simply supported and one surface affected by the water are investigated in this paper using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose physical and mechanical properties are determined using the Gibson correlations. The effect of water acting on honeycomb panels can be described as added mass, additional damping and additional stiffness coefficients which are obtained by the semi-analytical fluid pressures. The basic formulation of nonlinear forced vibrations has been developed base on the third-order shear deformation plate theory and Green Lagrange nonlinear strain–displacement relation. The equilibrium equations have been obtained using the Hamilton’s principle. Effects of water velocity, height and height ratio on the nonlinear forced vibration response have been studied for the honeycomb sandwich panels.     

Modeling of the three-dimensional thermal and stress-strain state of elastic bodies using mixed variational formulations of the finite-element method. Report 1

The basic solving relationships of the finite-element method for three-dimensional nonstationary problems of heat conduction using local mixed variational formulations for an octagonal isoparametric element are obtained. To reduce the order of the system of algebraic equations, the components of the thermalflux vector (CTFV) are expressed in terms of nodal temperature values from stationary conditions relative to the variation in the CTFV, which are written for a single element. The CTFV are approximated by linear Legendre polynomials. Consideration of the orthogonality of the approximating functions within the limits of an element makes it possible to avoid the formulation and inversion of the corresponding matrices relating the CTFV and temperature. The accuracy and stability of the various schemes of discretization are analyzed as an example of the determination of the thermal state in a cube with mixed boundary conditions.Translated from Problemy Prochnosti, No. 2, pp. 72–77, February, 1991.     

On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities

The work is devoted to calculation of effective elastic constants of homogeneous materials containing random or regular sets of isolated inclusions. Our approach combines the self-consistent effective field method with the numerical solution of the elasticity problem for a typical cell. The method also allows analysis of detailed elastic fields in the composites. By the numerical solution of the elasticity problem for a cell, integral equations for the stress field are used. Discretization of these equations is carried out by Gaussian approximating functions. For such functions, elements of the matrix of the discretized problem are calculated in explicit analytical forms. If the lattice of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure. The matrix-vector products with such matrices may be carried out by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Results are given for 2D-media with regular and random sets of circular inclusions, and compared with existing exact solutions.