Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

An analytical method for the three-dimensional vibration analysis of a functionally graded cylindrical shell integrated by two thin functionally graded piezoelectric （FGP） layers is presented. The first-order shear deformation theory is used to model the electromechanical system. Nonlinear equations of motion are derived by considering the von Karman nonlinear strain-displacement relations using Hamilton＇s principle. The piezoelectric layers on the inner and outer surfaces of the core can be considered as a sensor and an actuator for controlling characteristic vibration of the system. The equations of motion are derived as partial differential equations and then discretized by the Navier method. Numerical simulation is performed to investigate the effect of different para- meters of material and geometry on characteristic vibration of the cylinder. The results of this study show that the natural frequency of the system decreases by increasing the non-homogeneous index of FGP layers and decreases by increasing the non-homogeneous index of the functionally graded core. Furthermore, it is concluded that by increasing the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increase considerably.

Analytical and Experimental Study of Free Vibration of Beams Carrying Multiple Masses and Springs

The structures in engineering can be simplified into elastic beams with concentrated masses and elastic spring supports. Studying the law of vibration of the beams can be a help in guiding its design and avoiding resonance. Based on the Laplace transform method, the mode shape functions and the frequency equations of the beams in the typical boundary conditions are derived. A cantilever beam with a lumped mass and a spring is selected to obtain its natural frequencies and mode shape functions. An experiment was conducted in order to get the modal parameters of the beam based on the NExT-ERA method. By comparing the analytical and experimental results, the effects of the locations of the mass and spring on the modal parameter are discussed. The variation of the natural frequencies was obtained with the changing stiffness coefficient and mass coefficient, respectively. The findings provide a reference for the vibration analysis methods and the lumped parameters layout design of elastic beams used in engineering.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

Free vibration analysis of a spinning piezoelectric beam with geometric nonlinearities

The linear and non-linear free vibrations of a spinning piezoelectric beam are studied by considering geometric nonlinearities and electromechanical coupling effect. The non-linear differential equations of the spinning piezoelectric beam governing two transverse vibrations are derived by using two Euler angles transformation and extended Hamilton principle, wherein an additional piezoelectric coupling term and different linear terms are present in contrast to the traditional shaft model. Linear frequencies are obtained by solving the standard eigenvalues of the linearized system directly, and the non-linear frequencies and non-linear complex modes are achieved by using the method of multiple scales. For free vibrations analysis of a spinning piezoelectric beam, the non-linear modal motions are investigated as forward and backward precession with different spinning speeds. The responses to the initial conditions for such a gyroscopic system are studied,and a beat phenomenon is found, which are then validated by numerical simulation. The influences of some parameters such as electrical resistance, rotary inertia and spinning speeds to the non-linear dynamics of a spinning piezoelectric beam are investigated.

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.

Experimental and numerical study of effecting core configurations on the static and dynamic behavior of honeycomb plate with aluminum material

The sandwich panel incorporated a honeycomb core,a widely utilized composite structure recognized as a fundamental classification of composite materials.Comprised a core resembling a honeycomb,possessing thickness and softness,and is flank by rigid face sheets that sandwich various shapes and materials.This paper presents an examination of the static and dynamic analysis of lightweight plates made of aluminum honeycomb sandwich composites.Honeycomb sandwich plate samples are 300 mm long,and 300 mm wide,the heights of the core have been varied at four values ranging from 10 to 25 mm.The honeycomb core is manufactured from Aluminum material by using a novel technique namely resistance spot welding(RSW)instead of using adhesive material,which is often used when an industrial flaw is detected.Numerical optimization based on response surface methodology(RSM)and design of experiment software(DOE)was used to verify the current work.A theoretical examination of the crashworthiness behavior(maximum bending load,maximum deflection)and vibration attributes(natural frequency,damping ratio,transient temporal response)of honeycomb sandwich panels with different design parameters was also carried out.In addition,the finite element method-based ANSYS software was used to confirm the theoretical conclusions.The findings of the present work showed that the relationship between the natural frequency,core height,and cell size is direct.In contrast,the relationship between the natural frequency and the thickness of the cell wall is inverse.Conversely,the damping ratio is inversely proportional to the core height and cell size but directly proportional to the thickness of the cell wall.The study indicates that altering the core height within 10-25 mm leads to a significant increase of 82%in the natural frequency and a notable decrease of 49%in the damping ratio.These findings are based on a specific cell size value of 0.01 m and a cell wall thickness of 0.001 m.Also,the results indicate that for a given set of cell wall thickness and size values,an increase in core height from(0.01-0.025)m,leads to a reduction of the percentage of maximum response approX imately 76%.Conversely,the increasing thickness of the wall of cell wall,ranging 0.3-0.7 mm with a constant core height equal to 0.015 m,resulted in a de crease of maximum transient response by 7.8%.

Natural frequency analysis of laminated piezoelectric beams with arbitrary polarization directions

Piezoelectric devices exhibit unique properties that vary with different vibration modes,closely influenced by their polarization direction.This paper presents an analysis on the free vibration of laminated piezoelectric beams with varying polarization directions,using a state-space-based differential quadrature method.First,based on the electro-elasticity theory,the state-space method is extended to anisotropic piezoelectric materials,establishing state-space equations for arbitrary polarized piezoelectric beams.A semi-analytical solution for the natural frequency is then obtained via the differential quadrature method.The study commences by examining the impact of a uniform polarization direction,and then proceeds to analyze six polarization schemes relevant to the current research and applications.Additionally,the effects of geometric dimensions and gradient index on the natural frequencies are explored.The numerical results demonstrate that varying the polarization direction can significantly influence the natural frequencies,offering distinct advantages for piezoelectric elements with different polarizations.This research provides both theoretical insights and numerical methods for the structural design of piezoelectric devices.

Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM

The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.

Nonlinear vibration and buckling of circular sandwich plate under complex load

The nonlinear vibration fundamental equation of circular sandwich plate under uniformed load and circumjacent load and the loosely clamped boundary condi- tion were established by von Karman plate theory, and then accordingly exact solution of static load and its numerical results were given. Based on time mode hypothesis and the variational method, the control equation of the space mode was derived, and then the amplitude frequency-load character relation of circular sandwich plate was obtained by the modified iteration method. Consequently the rule of the effect of the two kinds of load on the vibration character of the circular sandwich plate was investigated. When circumjacent load makes the lowest natural frequency zero, critical load is obtained.

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.

Free Vibration of Sagging Pipelines in Water

Studied in this paper is free vibration of a long span pipeline with nonlinearities taken into account. The pipeline sags under gravity and takes the shape of a plane curve. Vibration in the plane and out of the plane is regarded as small motions about the large static deflection. Manifestations of nonlinearities such as amplitude-dependent frequencies and internal resonance are investigated.

An Effective Method for Free Vibration of Plate on Elastic Half Space

The vibration analysis of a plate on an elastic foundation is an important problem in engineering. It is the interaction of a plate with the three-dimensional half space and the plate is usually loaded from both the upper and lower surfaces. The contact pressure from the soil can not be predefined. According to Lamb's solution for a single oscillating force acting on a point on the surface of an elastic half space, and the relevant approximation formulae, a relation between the local pressure and the deflection of the plate has been proposed. Based on this analysis, the reaction of the soil can be represented as the deformation of the plate. Therefore, the plate can be separated from the soil and only needs to be divided by a number of elements in the analysis. The following procedure is the same as the standard finite element method. This is a semi-analytical and semi-numerical method. It has been applied to the dynamic analysis of circular or rectangular plates on the elastic half space, at low or high frequency vibration, and on rigid, soft or flexible foundations. The results show that this method is versatile and highly accurate.

Analysis of the Behavior of a Square Plate in Free Vibration by FEM in Ansys

In the realization of mechanical structures, achieving stability and balance is a problem commonly encountered by engineers in the field of civil engineering, mechanics, aeronautics, biomechanics and many others. The study of plate behavior is a very sensitive subject because it is part of the structural elements. The study of the dynamic behavior of free vibration structures is done by modal analysis in order to calculate natural frequencies and modal deformations. In this paper, we present the modal analysis of a thin rectangular plate simply supported. The analytical solution of the differential equation is obtained by applying the method of separating the variables. We are talking about the exact solution of the problem to the limit values. However, numerical methods such as the finite element method allow us to approximate these functions with greater accuracy. It is one of the most powerful computational methods for predicting dynamic response in a complex structure subject to arbitrary boundary conditions. The results obtained by MEF through Ansys 15.0 are then compared with those obtained by the analytical method.

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitudefrequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.

NONLINEAR VIBRATION OF CIRCULAR SANDWICH PLATES UNDER CIRCUMJACENT LOAD

Based on von Karman plate theory, the issue about nonlinear vibration for circular sandwich plates under circumjacent load with the loosely clamped boundary condition was researched. Nonlinear differential eigenvalue equations and boundary conditions of the problem were formulated by variational method and then their exact static solution can be got. The solution was derived by modified iteration method, so the analytic relations between amplitude and nonlinear oscillating frequency for circular sandwich plates were obtained. When circumjacent load makes the lowest natural frequency zero, critical load is obtained.

APPLICATION OF WEIGHTED NONOSCILLATORY AND NON-FREE-PARAMETER DISSIPATION DIFFERENCE SCHEME IN CALCULATING THE FLOW OF VIBRATING FLAT CASCADE

A dual-time method is introduced to calculate the unsteady flow in a certain vibrating flat cascade. An implicit lower-upper symmetric-gauss-seidel scheme（LU-SGS） is applied for time stepping in pseudo time domains, and the convection items are discretized with the spatial three-order weighted non-oscillatory and non-free-parameter dissipation difference （WNND） scheme. The turbulence model adopts q-co low-Reynolds-number model. The frequency specmuns of lift coefficients and the unsteady pressure-difference coefficients at different spanwise heights as well as the entropy contours at blade tips on different vibrating instants, are obtained. By the analysis of frequency specmuns of lift coefficients at three spanwise heights, it is considered that there exist obvious non-linear perturbations in the flow induced by the vibrating, and the perturbation frequencies are higher than the basic frequency. The entropy contours at blade tips at different times display an intensively unsteady attribute of the flow under large amplitudes.

FORMULATION AND EVALUATION OF AN ANALYTICAL STUDY FOR CYLINDRICAL HELICAL SPRINGS

The free vibration analysis of cylindrical helical springs is carried out by means of an analytical study. In the governing equations of the motion of the springs, all displacement functions are defined at the centroid axis and also the effects of the rotational inertia, axial and shear deformations are included in the proposed model. Explicit analytical expressions which give the vibrating mode shapes are derived by rigorous application of the symbolic computing package MATHEMATICA and a process of searching is used to determine the exact natural frequencies. Numerical examples are provided for fixed-fixed boundary conditions. The free vibrational pa- rameters are chosen as the number of coils （n = 4- 14）, the helix pitch angle （a = 5 - 30°） and as the ratio of the diameters of the cylinder and the wire （D/d = 4 - 18） in a wide range. Validation of the proposed model has been achieved through comparison with a finite element model using two-node standard beam elements and the results available in published literature, which in these cases indicates a very good correlation.

Various performance-enhancing effects from the same intensity of whole-body vibration training

Purpose:The purpose of this study was to compare the effects of an 8-week whole-body vibration training program in various frequency and amplitude settings under the same acceleration on the strength and power of the knee extensors.Methods:Sixty-four young participants were randomly assigned to 1 of 4 groups with the same acceleration(4 g):high frequency and low amplitude(n = 16,32 Hz,1 mm) group,medium frequency and medium amplitude(n = 16,18 Hz,3 mm) group,low frequency and high amplitude(n = 16,3 Hz,114 mm) group,and control(n = 16,no vibration) group.All participants underwent 8 weeks of training with body mass dynamic squats,3 sessions a week.Results:The results showed that the high frequency and low amplitude group increased significantly in isometric contraction strength and 120°/s isokinetic concentric contraction strength;the medium frequency and medium amplitude group increased significantly in 60°/s and 120°/s isokinetic strength of both concentric and eccentric contraction;and the low frequency and high amplitude group increased significantly in 60°/s and 120°/s isokinetic eccentric contraction strength.Conclusion:All frequency and amplitude settings in the 8-week whole-body vibration training increased muscle strength,but different settings resulted in various neuromuscular adaptations despite the same intensity.

Non-Linear Vibration of Rectangular Reticulated Shallow Shell Structures

This paper deals with non-linear vibration of rectangular reticulated shallow shells by applying non-linear elastic theory of such structures established by the author .Us-ing the assumed (generalized)Fourier series solutions for transverse deflection (latticejoint transverse displacement )and force function,weighted means of the trial functions lead to the relations among the coefficients related to the solutions and vibration equ-ation which determines the unknown time function,and then the amplitude -frequeney relations for free vibration and forced vibration due to harmonic force are derived withthe aid of the regular perturbation method and Galerkin procedure,respectively.Nu-merical examples are given as well.