Analytical Solution For Vibration Research Articles

Analytical solution for vibration and buckling of cylindrical sandwich panels with improved FG metal foam core

In this research paper, a novel higher-order porosity distribution is proposed, and free vibration and buckling behavior of cylindrical sandwich panels with functionally graded metal foam core are investigated. It is assumed that the sandwich panel is made of two thin face metal sheets and a moderately thick functionally graded metal foam core. The dynamic governing equations are obtained based on the third-order shear deformation theory and utilizing the Hamilton principle. It is assumed that the voids are distributed non-uniformly along the thickness of the functionally graded metal foam core. The elastic modulus of the core depends on the square of the relative density, thus is changing nonlinearly through the thickness of the core. The Navier technique is implemented to derive closed form solutions for the natural frequencies and the buckling load of the simply supported cylindrical sandwich panels. The accuracy and effectiveness of the proposed model are verified by comparison with previous researches. Results revealed that the presented higher-order porosity distribution can increase the buckling load and fundamental natural frequency by 10% and 5% in comparison to the conventional porosity distributions, respectively. In addition, it is shown that if the flexural vibration is of interest, ignoring the in-plane and rotary inertias, which can significantly reduce the computational complexity, does not have any considerable effect on the natural frequencies.

An analytical solution for vibration in a functionally graded sandwich beam by using the refined zigzag theory

An analytical solution for vibration in a functionally graded sandwich beam by using the refined zigzag theory

Analytical solutions for vibrations and buckling analysis of laminated composite nanoplates based on third-order theory and strain gradient approach

A nonlocal model based on the strain gradient approach is developed within the framework of the Third-order Shear Deformation Theory (TSDT) for the investigation of the free vibrations and the critical buckling loads of laminated composite nanoplates. The theory is suitable to deal with thick and thin plates since it includes also the First-order Shear Deformation Theory (FSDT) and the Classical Laminated Plate Theory (CLPT). An analytical procedure based on the Navier approach is employed to obtain the solutions, which are discussed highlighting the effects of the strain gradient, as well as the influence of the geometric ratios and mechanical properties, on the results. The paper aims at providing reliable benchmarks for further developments of the topic to be used as references in future comparison tests.

Sound radiation of a beam with a wedge-shaped edge embedding acoustic black hole feature

As a potential intelligent structure which is capable of concentrating energy into local area, the Acoustic Black Hole (ABH) structure has high potential of application for sound radiation control. The ABH effect is caused by the infinitely small material attenuation in an inhomogeneous medium, such as the wedge-shaped beam with a diminishing thickness, giving rise to the reduction in the bending wave speed and no wave reflections in the tip. Therefore, the waves will be trapped in the edge portion. This paper investigates sound radiation problems of a beam embedding an ABH feature, focusing on characterizing and improving the low frequency performance of the beam on sound radiation reduction. The ABH beam is modeled and solved by transfer matrix method (TMM). Analytical solutions of vibration, wavenumber domain analysis and acoustic radiation phenomena are presented. A critical mode with very high order of the ABH beam is found, resulting in that its radiation efficiency is lower than that of the uniform beam in a specified range of frequencies. Generally, the low frequency performance will be improved effectively by enhancing the interaction between the bending waves and ABH element.

Vibro-acoustic response of a clamped rectangular sandwich panel in thermal environment

Composite sandwich structures are extensively applied in the automotive, marine and aircraft industries owing to their superior stiffness-to-weight ratio. This paper is concentrated on the vibro-acoustic characteristics of a clamped rectangular sandwich panel in high temperature environments. The analytical solution of vibration and acoustic responses for the fully clamped boundary condition is derived. Firstly, natural frequencies and corresponding modes in thermal environment are derived using the piecewise shear deformation theory. Thermal buckling temperatures and corresponding modes are obtained to investigate the buckling characteristics of the sandwich panel. The vibration response is acquired by applying the mode superposition method. The sound pressure distribution is derived by applying the Rayleigh integral, and the sound radiation efficiency is obtained in sequence. Sound transmission loss (STL) of the sandwich panel is also obtained in theory. Next, numerical simulations and experimental result are carried out to verify the accuracy of the analytical solutions. Finally, focus is placed on the influence of several key parameters including temperature, boundary condition, thickness of the core and dimension of the sandwich on vibro-acoustic characteristics of sandwich panel.

Analytical solution for vibration and buckling of annular sectorial porous plates under in-plane uniform compressive loading

In this article, an analytical solution for free vibration of moderately thick annular sectorial porous plates in the presence of in-plane loading is presented. Because of the in-plane loading, before the vibrational analysis, a buckling analysis is performed. To this end, equations of motion together with the stability equations are derived using Hamilton principle. Both the governing equations of motion and stability are highly coupled differential equations, which are difficult to solve analytically. So, they are decoupled through performing some mathematical operations. The decoupled equations are then solved analytically for annular plates with simply supported boundary conditions on the radial edges and different boundary conditions on the circumferential edges. Natural frequencies and also critical buckling load are obtained and the effects of thickness ratio, radii ratio, porosity, and boundary conditions are studied in detail. Finally, the effect of in-plane loading on the natural frequency of the plate is studied comprehensively. Numerical results show that the natural frequency decreases as the load ratio approaches one and vanishes as it reaches one.

Buckling and vibro-acoustic response of the clamped composite laminated plate in thermal environment

Composite structures are extensively applied in aircraft and marine industries because of their high stiffness-to-weight ratios and other advantages. The composite structures in the hypersonic aircraft are usually exposed to the thermal and noise environment. The paper is focused on the buckling and vibro-acoustic response of the clamped composite laminated plate excited by a concentrated harmonic force in the thermal environment. The analytical solution of vibration and acoustic response for fully clamped boundary condition is derived in the paper. Meanwhile, the natural frequencies and buckling temperatures of the plate in uniform temperature environment are also derived by applying the classical laminate theory (CLT) and first order shear deformation theory (FOSDT). The vibration response is obtained by applying the mode superposition method, while the sound pressure and radiation efficiency are calculated by Rayleigh integral in sequence. The accuracy of the presented method is verified by the numerical simulations. The structural and acoustic response affected by the thermal load is demonstrated through a numerical example.

Analytical Solution for Vibration of a Rotating Delaminated Composite Beam with End Mass

In this paper, the free vibration of rotating laminated composite beams (LCBs) with general lay-ups and single through-the-width delamination is analytically investigated. The Hamilton’s principle is used to derive the coupled governing differential equations and boundary conditions for the rotating delaminated beam, considering the effects of shear deformation, rotary inertia, material couplings (bending–tension, bending–twist and tension–twist couplings), and Poisson’s effect. Both the free mode and constrained mode assumptions are adopted. Analytical solution for the natural frequencies and mode shapes are presented by incorporating the constraint conditions using the Lagrange multipliers method. The accuracy is assured by the convergence of the natural frequencies, as well as by comparison with published results. The effects of various factors such as delamination parameter, fiber angle, hub radius, material anisotropy, end mass and rotating speed are studied in detail. The difference between the results based on the free mode and constrained mode assumptions is also investigated.

On the Flexural Vibration of Pre-Stressed Nanobeams Based on a Nonlocal Theory

In this study, a model based on a nonlocal theory is proposed to study the e ect of external loads on the free vibration of nanobeams. The governing equations for the initially pre-stressed beam are obtained by using the Hamilton principle. Analytical solution of vibration is presented using the Eringen nonlocal theory to bring out the e ect of the nonlocal behavior on natural frequencies. The performance of the present model is compared with that of others by the presentation of comparative results. It is suggested that the present model can be used as a benchmark in future studies.

On Approximate Analytical Solutions for Vibrations in Cracked Plates

Recent NATO funded research on methods for detection and interpretation methodologies for damage detection in aircraft panel structures has motivated work on low-order nonlinear analytical modelling of vibrations in cracked isotropic plates, typically in the form of aluminium aircraft panels. The work applies fundamental aspects of fracture mechanics to define an elliptical crack, and the local stress field and loading conditions, arbitrarily located at some point in the plate, and then derives an analytical expression for this that can be incorporated into the PDE for an edge loaded plate with various possible boundary conditions. The plate PDE is converted into a nonlinear Duffing-type ODE in the time domain by means of a Galerkin procedure and then an arbitrarily small perturbation parameter is introduced into the equation in order to apply an appropriate solution method, in this case the method of multiple scales. This is used to solve the equation for the vibration in the cracked plate for the chosen boundary conditions, which, in turn, leads to an approximate analytical solution. The solution is discussed in terms of the perturbation approximations that have been applied and highlights the phenomenology inherent within the problem via the specific structures of the analytical solution.

運転性能の予測できる歯形誤差の評価方法

The Linear Approximated Equation (LAE) of rotational vibration of a spur gear pair and its analytical solution of vibration have been developed by this research. In this report, by considering that the analytical solution gives the common relation between the vibration and gear dimensions (inertia, contact ratio, variable stiffness), running condition (rotational speed, load), tooth profile error of gear pair, the evaluation method of tooth profile error correspond to rotational vibration is developed. The average vibration of low and high speed running gear are easily estimated from the product of average dynamic factor by the amplitude of equivalent error. According to the magnitude of average vibration, the accuracy grade of profile error is decided. The validity of this new accuracy grade is confirmed both by the comparison with usual gear accuracy standard and the result of running experiment.