Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlin- ear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of mul- tiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequen- cies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.     

An investigation into non-linear vibrations of thin plates

The variational and modified forms of the von Kármán-type non-linear plate equations are considered in the context of the Rayleigh-Ritz and Galerkin methods. An approximate analysis of the non-linear vibrations of thin elastic plates including inplane inertia is presented. The quantitative study confirms that the inplane inertia effects are negligible for thin plates provided the non-linearity is not too large. It is observed that the non-linear inertia terms in the transverse equation of motion should be retained in any such study. The analysis is simplified by neglecting the inplane inertia and applied to constrained and unconstrained plates. A different type of inplane boundary condition termed ‘the partially constrained’ is studied, and the inadequacy of replacing the unconstrained condition by means of an average-zero stress condition is clearly demonstrated. It is observed that in most of the cases considered the Galerkin method yields lower bounds for the non-linear coefficient of the modal equation. In all cases the Galerkin results yield less stiff models than the Rayleigh-Ritz method. The general significance of the convergence of the two methods beyond the scope of the title problem is highlighted.     

Crack detection of a double-beam carrying a concentrated mass

This paper presents the influence of a concentrated mass location on the natural frequencies of a cracked double-beam. The double-beam consists of two different beams connected by an elastic medium. The concentrated mass is located on the main beam. The relationship between the natural frequency and the location of concentrated mass is established and called “Frequency–Mass Location” (FML). The numerical simulations show that when there is a crack, the frequency of the double-beam changes irregularly when the concentrated mass is attacked at the crack position. This irregular change can be amplified by the wavelet transform and this is useful for crack detection: the crack location can be detected by the location of peaks in the wavelet transform of the FML. Finite element model for the cracked double-beam carrying a concentrated mass is presented and numerical simulations are also provided.     

求解矩形夹层板在集中载荷下的稳定问题

在考虑了横向切应力和横向正应力对夹层板稳定影响的情况下,给出了矩形夹层板结构屈曲失稳的控制方程、基本解以及边界条件。应用功的互等定理求解了在均布载荷作用下的矩形夹层板的屈曲失稳问题。     

A numerical method for clamped thin rectangular plates carrying a uniformly distributed load

A numerical method for clamped thin rectangular plates carrying a uniformly distributed load and the exact solution of the governing equations are given. The solution is presented in a simple form suitable for direct practical use. The method is a very simple and practical approach. The results are compared with those reported in the previous papers Published in Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–143, June 2007.     

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.     

Generalized non-linear free vibrations of prestressed plates and shells

This paper investigates the non-linear free vibration of prestressed plates and shells in a general form. The analysis includes the effects of in-plane inertia. The analysis is based on the non-linear equations of motion and uses a perturbation procedure. No assumption is made a priori for the form of the time or space mode. The boundary conditions are treated in a general manner including boundary conditions where non-linear stress resultants are specified. The method is illustrated by three examples.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

Free vibrations of circular cylindrical shells with a small added concentrated mass

The effect of a small added mass on the frequency and shape of free vibrations of a thin shell is studied using shallow shell theory. The proposed mathematical model assumes that mass asymmetry even in a linear formulation leads to coupled radial flexural vibrations. The interaction of shape-generating waves is studied using modal equations obtained by the Bubnov–Galerkin method. Splitting of the flexural frequency spectrum is found, which is caused not only by the added mass but also by the wave-formation parameters of the shell. The ranges of the relative lengths and shell thicknesses are determined in which the interaction of flexural and radial vibrations can be neglected.     

Resonant vibrations of a clamped viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a viscoelastic rectangular plate undergoing resonant vibrations. The thermoviscoelastic behavior of materials is described using the concept of complex characteristics. The coupling of the electromechanical and thermal fields is taken into account. The nonlinear problem is solved by the Bubnov–Galerkin method. The effect of the mechanical boundary conditions and dissipative-heating temperature on the performance of the sensors is analyzed     

Free vibrations of a partially clamped trapezoidal plate

The natural frequencies and mode shapes of partially clamped trapezoidal cantilever plates having various swept-back angles are examined numerically and experimentally by a finite-element computer program, SAP6, and holographic interferometry, respectively. A good agreement is found between the numerical and the experimental results. The influence of the clamped position, the free-width ratio, and the swept-back angle on the vibratory behavior of the trapezoidal plates is investigated.Paper was presented at the 1990 SEM Spring Conference on Experimental Mechanics held in Albuquerque, NM on June 3–6.     

On the flexural vibration of an elastic plate carrying a concentrated mass

Summary The dynamic response of an elastic plate carrying a concentrated mass is analysed. Despite the presence of a singular mass distribution function, a rigorous analysis leading to a closed-form solution in the form of an infinite series has been made. By developing Green's function for the associated partial differential equation, any form of dynamic excitation is easily considered.

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Ein beitrag zu den biegeschwingungen einer elastischen platte mit einer punktförmigen zusatzmasse

Übersicht Untersucht wird das dynamische Verhalten einer elastischen Platte mit punktförmiger Zusatzmasse. Ungeachtet der Masseverteilung mit Singularität wird eine mathematisch strenge Behandlung, die zu einer geschlossenen Lösung in der Form unendlicher Reihen führt, unternommen. Durch Entwicklung der Greenschen Funktion für die assoziierte partielle Differentialgleichung lassen sich beliebige dynamische Erregungen behandeln.

     

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.     

Viscoelastic vibrations of a triangular plates

The problem of stationary vibrations of a viscoelastic equilateral triangular plate is treated. The vibrations are caused by the action of a uniformly distributed load which varies harmonically and by the motion of the simply supported boundary of the plate as a rigid body which vibrates at the same frequency. The level lines of the vibration amplitude are studied and the graphs of the amplitude distribution along the height of the triangle are given.