NON-LINEAR VIBRATION AND THERMAL BUCKLING OF AN ORTHOTROPIC ANNULAR PLATE WITH A CENTRIC RIGID MASS

A computational analysis of the non-linear vibration and thermal post-buckling of a heated orthotropic annular plate with a central rigid mass is examined for the cases of immovably hinged as well as clamped constraint conditions of the outer edge. First, based on von Karman's plate theory and Hamilton's principles, the governing equations, in terms of the displacements of the middle plane, of the problem are derived. Then, upon assuming that harmonic responses of the system exist, the non-linear partial differential equations are converted into the corresponding non-linear ordinary differential equations through elimination of the time variable by using the Kantorovich time-averaging method. Finally, by applying a shooting method, the fundamental responses of the non-linear vibration and thermal post-buckling of the plate are numerically obtained. For some prescribed values of the parameters, such as the material rigidity ratio, temperature rise and so on, the curves of the fundamental frequency versus specified amplitude and the thermal post-buckled equilibrium paths of the plate are numerically presented.     

GEOMETRICALLY NON-LINEAR FREE VIBRATION OF FULLY CLAMPED SYMMETRICALLY LAMINATED RECTANGULAR COMPOSITE PLATES

The geometrically non-linear free vibration of thin composite laminated plates is investigated by using a theoretical model based on Hamilton's principle and spectral analysis previously applied to obtain the non-linear mode shapes and resonance frequencies of thin straight structures, such as beams, plates and shells (Benamar et al. 1991Journal of Sound and Vibration149 , 179-195; 1993, 164, 295-316; 1990 Proceedings of the Fourth International Conference on Recent Advances in Structural Dynamics, Southampton; Moussaoui et al. 2000 Journal of Sound and Vibration232, 917-943 [1-4]). The von Kármán non-linear strain-displacement relationships have been employed. In the formulation, the transverse displacement W of the plate mid-plane has been taken into account and the in-plane displacements U and V have been neglected in the non-linear strain energy expressions. This assumption, quite often made in the literature has been adopted in reference [2] and (El Kadiri et al. 1999 Journal of Sound and Vibration228, 333-358 [5]), in the isotropic case and has been mentioned here because the results obtained have been found to be in very good agreement with those based on the hierarchical finite element method (HFEM). In a previous study, it was assumed, based on the analogy with the isotropic case, that the fundamental carbon fibre reinforced plastic (CFRP) plate non-linear mode shape could be well estimated, by using nine plate functions, obtained as products of clamped-clamped beam functions in the x and y directions, symmetric in both the length U001and width directions [3]. In the present work, a convergence study has been performed and has shown that, although such an assumption may yield a good estimate for the non-linear resonance frequency, 18 plate functions should be taken into account instead of nine in the first non-linear mode shape and associated bending stress patterns calculations. This allows the anisotropy induced by the fibre orientations to be taken into account. Results are given for the fundamental mode of fully clamped CFRP rectangular plates, for various plate aspect ratios and vibration amplitudes. The non-linear mode shows a higher bending stress near the clamps at large deflection, compared with that predicted by linear theory. Some experimental measurements are presented which are in good qualitative agreement with the theory.     

VIBRATION AND STABILITY OF ROTATING POLAR ORTHOTROPIC ANNULAR DISKS SUBJECTED TO A STATIONARY CONCENTRATED TRANSVERSE LOAD

In the present paper, natural frequencies and stability of a spinning polar orthotropic disk subjected to a stationary concentrated transverse load are investigated. The analysis of the free vibration of a spinning disk is performed first to find natural frequencies and corresponding vibration modes. The resulting eigenfunctions obtained from the free vibration are used as deflection functions of the forced vibration of a disk where the load is modelled as a mass-spring-dashpot system fixed in space. By using the Galerkin approximation method, eigenvalues of the whole system are determined. Results show that disks with higher values of modulus ratios or the Poisson ratios have higher natural frequencies, and the stability of the whole system can be improved by raising the value of the modulus ratio or lowering that of the Poisson ratio.     

VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS

Free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation. The plate is assumed to be clamped on two opposite edges with the remaining two others either free, simply supported or clamped. One-dimensional differential quadrature technique, Galerkin approach, and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads. Comprehensive numerical results for silicon nitride/stainless-steel rectangular plates are presented in dimensionless tabular and graphical forms. The roles played by the constituent volume fraction index, temperature rise, shape and duration of dynamic loads, initial membrane stresses as well as the character of boundary conditions are studied. The results reveal that, when thermal effects are considered, functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses.     

电荷在一类点电荷系统中的非线性振动

本文用数值计算的方法讨论了电荷在一类点电荷系统中的非线性振动及其特点。     

ANALYSIS OF VIBRATIONS AND ENERGY FLOWS IN SANDWICH PLATES BEARING CONCENTRATED MASSES AND SPRING-LIKE INCLUSIONS IN HEAVY FLUID-LOADING CONDITIONS

Vibrations of and the energy propagation in an infinitely long fluid-loaded sandwich beam (a plate of the sandwich composition in one-dimensional cylindrical bending) bearing concentrated masses and supported by springs are described in the framework of the sixth order theory of multilayered plates coupled with the standard theory of linear acoustics. A sandwich plate is loaded by a layer of a compressible fluid which is bounded opposite to a plate side by a rigid baffle. The dispersion equation for a fluid-loaded sandwich plate is derived. The wave numbers (complex, pure real and pure imaginary) and relevant normal modes (both the travelling and the evanescent ones) are obtained. Their dependence on the parameter of a fluid's depth is studied. Then the Green matrix is constructed analytically as a linear combination of normal modes to describe the response of a plate and an acoustic medium to the point loading by a force or a moment. Continuity conditions at the loaded cross-section of a plate and in a fluid are formulated. Attention is focused at the selection of roots of the dispersion relation for the formulation of the continuity condition for a fluid at the loaded cross-section. The convergence rate of an approximate solution based on the modal composition of the Green matrix is estimated. The parametric study of the “structural” and the “fluid” energy flows in a fluid-loaded sandwich plate without inclusions is performed for various excitation conditions. Then the Green matrix method is applied to analyze the influence of a pair of identical inclusions on localization of vibrations (modal trapping) and energy flows. Conditions of localization of flexural waves at these inhomogeneities are explored.     

环形液层内热毛细对流的线性稳定性分析

为了了解液层深度对热毛细对流不稳定性的影响,利用线性稳定性理论分析了内径为20 mm、外径为40 mm、深度为1～20 mm的环形液层内硅油(Pr=6.7)的热毛细对流,重点考察了发生热流体波的临界温差及热流体波的临界周向波数,结果表明,临界温差随液层深度的增加而降低.讨论了浮力对热流体波及其临界条件的影响,并证实了常重力条件下当液层厚度大于5 mm时,热流体波呈静止状态这一现象.     

边界条件与一维高分子系统的稳定性

讨论了SSH模型在一维有限系统上的稳定性，提出了几种稳定边界的办法，并对边界条件的作用进行了分析讨论 关键词：     

NON-LINEAR FORCED VIBRATIONS OF PLATES BY AN ASYMPTOTIC-NUMERICAL METHOD

Non-linear forced vibrations of thin elastic plates have been investigated by an asymptotic-numerical method (ANM). Various types of harmonic excitation forces such as distributed and concentrated are considered. Using the harmonic balance method and Hamilton's principle, the equation of motion is converted into an operational formulation. Based on the finite element method a starting point corresponding to a non-linear solution associated to a given frequency and amplitude of excitation is computed. Applying perturbation techniques in the vicinity of this solution, the non-linear governing equation obtained is transformed into a sequence of linear problems having the same stiffness matrix. Employing one matrix inversion, a large number of terms of the perturbation series of the displacement and frequency can be easily computed with a small computation time. Iterations of this method lead to a powerful path-following technique. Comprehensive numerical tests for forced vibrations of plates subjected to time-harmonic lateral excitations are reported.     

复相系平衡条件及平衡稳定性条件的分析

复相系平衡条件及平衡稳定性条件的分析曾丹苓（重庆大学热力工程系重庆６３００４４）关键词复相系，平衡条件，平衡稳定性条件１引言汽液相变是工程上常见的现象，它是一个复杂的物理过程，涉及到传热传质、相转变、表面现象、流体流动及工质热物性等领域，但从热力学的...     

一类非线性弹簧振子的周期性振动

本文用数值计算的方法讨论了一类非线性弹簧振子的周期性振动及其特点．     

VIBRATION OF PLATES IN DIFFERENT SITUATIONS USING A HIGH-PRECISION SHEAR DEFORMABLE ELEMENT

A high-precision thick plate element proposed by the last author of this paper has been applied to free vibration analysis of plates to study its performance. The element has a triangular shape and it has three nodes at its corners, three mid-side nodes on each side and four nodes within the element. The transverse displacement and rotations of the normal have been taken as independent field variables and they have been approximated with polynomials of different orders. This has not only helped to include the effect of shear deformation but also made the element free from locking in shear. Initially, the number of degrees of freedom of the element is 35, which is reduced to 30 by eliminating the degrees of freedom of the internal nodes. This has been done through static condensation. To facilitate the condensation process, efficient mass lumping schemes have been recommended to form the mass matrix having zero mass for the internal nodes. Recommendation has also been made for the inclusion of mass for rotary inertia in a lumped mass matrix. Numerical examples of plates having different shapes and boundary conditions have been solved by this element. Examples of plates having internal cutout and concentrated mass have also been studied. The results obtained in all the cases have been compared with the published results to show the accuracy and range of applicability of the present element.     

PERFORMANCE AND DYNAMIC STABILITY OF GENERAL-PATH CENTRIFUGAL PENDULUM VIBRATION ABSORBERS

Centrifugal pendulum vibration absorbers are a type of tuned dynamic absorber used for the attenuation of torsional vibrations in rotating and reciprocating machines. They consist of masses that are constrained to move along specific paths relative to the rotational axis of the machine. Previous analytical studies have considered the performance of single absorber systems with general paths and of multi-absorber systems with a specific path type. In this paper, we investigate the performance and dynamic stability of systems comprised of multiple, identical centrifugal pendulum vibration absorbers riding on quite general paths. The study is carried out by considering a scaling of the system parameters, based on physically realistic ranges of dimensionless parameters, which permits application of the method of averaging. It is found that the performance of these systems is limited by two distinct types of instabilities. In one type, the system of absorbers lose their synchronous character, while in the other a classical non-linear jump affects all absorbers identically, leading to highly undesirable system behavior. These results are used to evaluate two common types of absorber paths, namely circles and cycloids, including intentional mistuning of the absorber frequencies. The results are used to make some recommendations about the selection of paths to achieve design goals in terms of absorber performance and operating range. The analytical predictions are confirmed by numerical simulations.     

VIBRATION PROTECTION OF CRITICAL COMPONENTS OF ELECTRONIC EQUIPMENT IN HARSH ENVIRONMENTAL CONDITIONS

Vibration protection of sensitive electronic equipment operating in harsh environments often relies on resilient mounts. The traditional optimal design for vibration isolation from random vibration is based on a trade-off choice of damping and stiffness properties of mounts, and is focussed primarily on optimizing the dynamic response of the electronic boxes, subject to limitations imposed on their rattle space. However, the reliability of the electronic equipment depends primarily on the vibration responses of the internal components that are often lightly damped and extremely responsive over a wide frequency range. The traditional approach, hence, completely ignores the presence of such components. Consequently, the traditionally designed vibration isolators are often insufficient for maintaining a fail-safe vibration environment for electronic equipment. The new design approach focuses basically on dynamic properties and responses of the critical internal components of an electronic device. The optimally chosen elastic and damping properties of the vibration isolators allow the vibration experienced by the above internal components to be minimised, subject to restraints imposed on the peak deflections of the electronic box.     

DYNAMIC STABILITY OF SKEW PLATES SUBJECTED TO AERODYNAMIC AND RANDOM IN-PLANE FORCES

This paper presents an investigation into the dynamic stability of skew plates acted upon simultaneously by an aerodynamic force in the chordwise direction and a random in-plane force in the spanwise direction. Due to this random in-plane force, the plate may become unstable before the aerodynamic force reaches its critical value. In this work, the finite element formulation is applied to obtain the discretized system equations. The system equations are then partially uncoupled and reduced in size by the modal truncation method. Finally, the unsmoothed and the smoothed versions of the stochastic averaging are used to calculate the system response, and the second-moment stability criterion is utilized to determine the stability boundary of the system. Numerical results show that the stability boundary obtained by the smoothed stochastic averaging is less conservative than that obtained by the unsmoothed version, and the former is the tangent of the latter at zero spectral density of the random in-plane force.     

FREE NON-LINEAR VIBRATION OF A ROTATING THIN RING WITH THE IN-PLANE AND OUT-OF-PLANE MOTIONS

Free non-linear vibration of a rotating thin ring with a constant speed is analyzed when the ring has both the in-plane and out-of-plane motions. The geometric non-linearity of displacements is considered by adopting the Lagrange strain theory for the circumferential strain instead of the infinitesimal strain theory. By using Hamilton's principle, the coupled non-linear partial differential equations are derived, which describe the out-of-plane bending and torsional motions as well as the in-plane bending and extensional motions. During deriving the equations of motion, we discuss how to model the circumferential stress and strain in order to consider the geometric non-linearity. Four models are established: three non-linear models and one linear model. For the four models, the linearized equations of motion are obtained in the neighbourhood of the steady state equilibrium position. Based on the linearized equations of the four cases, the natural frequencies are computed at various rotational speeds and then they are compared. Through the comparison, this study recommends which model is appropriate to describe the non-linear behaviour more precisely.     

IN-SERVICE IDENTIFICATION OF NON-LINEAR DAMPING FROM MEASURED RANDOM VIBRATION

A non-linearly damped single-degree-of-freedom (s.d.o.f.) system under broadband random excitation is considered. A procedure for in-service identification of the damping characteristic from measured stationary response is described. The procedure is based on the stochastic averaging method. The explicit analytical solution is obtained for the integral equation, which relates the desired damping characteristics to the apparent force in the shortened equation for the slowly varying response amplitude, and thus to the measured probability density of the amplitude. The approach is of a non-parametric nature, which makes it convenient for testing hypotheses of damping mechanisms from measured random vibration data. Extensive results of numerical tests for the procedure are presented.