Effect of a concentrated mass on large amplitude,free flexural vibrations of elastic plates and beams

This paper presents an approximate solution to the title problem for the following situations: (a) supported and clamped rectangular plates, (b) circular plates with edges elastically restrained against rotation, and (c) beams with ends flexibly attached.The approach is quite simple and it is shown that the engineering accuracy achieved is satisfactory from the point of view of the determination of fundamental frequencies in a wide variety of structural elements commonly used in transducer design and aerospace and naval engineering applications.     

The role of median surface curvature in large amplitude flexural vibrations of thin shells

General conclusions regarding the non-linear vibration of structural components like curved beams, rings and thin shells are derived from the study of two specific examples, the circular ring and shallow spherical shell. It is shown that whereas the non-linear behaviour of flat plates and straight bars is generally of a hardening type, the behaviour of thin structural elements that have a finite curvature of the undeformed median surface in one or both principal axis directions may be of the hardening or softening type, depending on the structural parameters as well as on whether the shell is open or closed. It is seen that with careful judgment in the use of mode shapes of one or more terms, the resulting modal equations help one to appreciate much better the physics of the problem, whereas sophisticated mathematical models tend to obscure this.     

Shear and rotatory inertia effects on large amplitude vibration of skew plates

The large amplitude free flexural vibration of elastic, isotropic skew plates is investigated, the effects of transverse shear and rotatory inertia being included. By use of Galerkin's method and the extended Berger approximation, solutions are obtained on the basis of an assumed vibration mode. The non-linear period vs. amplitude behavior is of the hardening type and the non-linear period is found to increase when the effects of transverse shear and rotatory inertia are considered in the analysis. The influence of these effects on aspect ratios and skew angles of thin and moderately thick skew plates is investigated both at small and large amplitudes.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

On the analysis of large amplitude vibrations of non-uniform rectangular plates

An analysis of the large amplitude vibrations of non-uniform rectangular plates is presented. The method is based on Berger's assumption of neglecting the second invariant of the middle surface strain in the expression corresponding to the total potential energy of the system, in conjunction with a Galerkin procedure.     

Effect of in-plane inertia on buckling of imperfect plates with large deformations

The influence of in-plane inertia on non-linear dynamic buckling of rectangular initially-imperfect plates is studied. A regular, consistent perturbation technique is used, and both the equations of motion and the boundary conditions are perturbed, yielding a relatively simple procedure for solving an otherwise very involved non-linear problem. Both pulse and vibration buckling during parametric resonance are analyzed. It is shown that, in the former case, the in-plane inertia can be disregarded whereas in the latter it plays an important role for non-slender plates.     

Large amplitude vibrations and damage detection of rectangular plates

In this work, geometrically nonlinear vibrations of fully clamped rectangular plates are used to study the sensitivity of some nonlinear vibration response parameters to the presence of damage. The geometrically nonlinear version of the Mindlin plate theory is used to model the plate behaviour. Damage is represented as a stiffness reduction in a small area of the plate. The plate is subjected to harmonic loading with a frequency of excitation close to the first natural frequency leading to large amplitude vibrations. The plate vibration response is obtained by a pseudo-load mode superposition method. The main results are focussed on establishing the influence of damage on the vibration response of the plate and the change in the time-history diagrams and the Poincaré maps caused by the damage. Finally, a criterion and a damage index for detecting the presence and the location of the damage is proposed. The criterion is based on analysing the points in the Poincaré sections of the damaged and healthy plate. Numerical results for large amplitude vibrations of damaged and healthy rectangular and square plates are presented and the proposed damage index for the considered cases is calculated. The criterion demonstrates quite good abilities to detect and localize damage.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

Negative longitudinal expansion and the amplitude of longitudinal vibrations in poly(ethylene) crystals

The experimental temperature dependences of the thermal expansion coefficient and the mean-square amplitude of atomic vibrations in the longitudinal direction (along the axes of chain molecules) in poly(ethylene) crystals of different sizes are measured using x-ray diffraction in the temperature range 5–350 K. Theoretical calculations of the temperature dependences of the thermal expansion coefficient and the mean-square amplitude of atomic vibrations in poly(ethylene) crystals are carried out. It is shown that the results of calculations are in good agreement with experimental data. The temperature ranges in which the quantum character of lattice dynamics affects the thermal expansion coefficient and the mean-square amplitude of vibrations are determined. It is revealed that the shear longitudinal modes play an important role in the lattice dynamics of crystals with a chain structure.     

Nonlinear capillary vibrations and stability of a highly charged drop under single-mode initial deformation of large amplitude

Nonlinear vibrations of a charged drop that are caused by a virtual initial perturbation of the equilibrium spherical shape of the drop were considered. The perturbation can be proportional to any of the free vibration modes. The spectrum and stability of the vibrations were studied correct to second order of smallness.     

Transverse vibrations of beams with variable moment of inertia and an intermediate support

The above problem is tackled using the Ritz method and approximating the deflection function by means of polynomial co-ordinate functions which satisfy the essential boundary conditions. The presence of an axial force and concentrated masses is considered. Fundamental frequency coefficients are determined in the case of a rather complex structure taking into account different types of boundary conditions.