Equations For Free Vibration Research Articles

The Analyses for Natural Frequency of Laminated Plate Using an Improved Higher-Order Theory

An improved higher order method was used in calculating the natural frequency of laminated composite plate. The function of equation of free vibration of laminated plate was established using Hamilton's principle and calculated the natural frequencies of plates. The influence of the laying-angle and thickness ratio and cross-elastic modulus ratio to the frequency of the plate were analyzed. It is shown that the natural frequency of antisymmetric angle-ply laminated plates and antisymmetric cross-ply laminated plate can be calculated by using this paper theory; in addition, it is feasible to calculate the natural frequency of the random form of laying laminate.

Free Vibration of Elastic Timoshenko Beam on Fractional Derivative Winkler Viscoelastic Foundation

The fractional derivative Winkler viscoelastic foundation model is established by introducing the concept of fractional derivative. The control equations of free vibration of elastic Timoshenko beam on fractional derivative Winkler viscoelastic foundation are also built by considering the shear deformation and rotary inertia, and the control equations of elastic Timoshenko beam are decoupled by using the deformation function and considering the properties of fractional derivative, and the expressions of deflection and section corner of elastic Timoshenko beam on fractional derivative Winkler viscoelastic foundation are obtained. The influences of fractional derivative order and shear shape factor on the free vibration of elastic Timoshenko beam are discussed by numerical example.

Solution of Free Vibration Equations of Euler-Bernoulli Cracked Beams by Using Differential Transform Method

In this paper, free vibration differential equations of cracked beam are solved by using differential transform method (DTM) that is one of the numerical methods for ordinary and partial differential equations. The Euler–Bernoulli beam model is proposed to study the frequency factors for bending vibration of cracked beam with ant symmetric boundary conditions (as one end is clamped and the other is simply supported). The beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in both vertical displacement and rotational due to bending. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section by using DTM and analytical solution. The results show that DTM provides simple method for solving equations and the results obtained by DTM converge to the analytical solution with much more accurate for both shallow and deep cracks. This study demonstrates that the differential transform is a feasible tool for obtaining the analytical form solution of free vibration differential equation of cracked beam with simple expression.

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.

Exact Analysis of Axisymmetric Dynamic Response of Functionally Graded Cylinders (or Disks) and Spheres

Free and forced vibration analyses of functionally graded hollow cylinders and spheres are performed and analytical benchmark solutions are presented. The material is assumed to be graded in the radial direction according to a simple power law. The Laplace transform method is used, and the inversion into the time domain is performed exactly using calculus of residues. The Complex Laplace parameter in the free vibration equation has directly given natural frequencies, and the results are given in tabular form. On the inner surface, various axisymmetric dynamic pressures are applied, and radial displacement and hoop stress are presented in the form of graphs. The exponent in the power law, called the inhomogeneity parameter, essentially refers to the degree of inhomogeneity. Increasing the inhomogeneity parameter provides a stress-shielding effect. Closed-form solutions obtained in the present paper are tractable, and they allow for further parametric studies. The inhomogeneity constant is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution.

Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells

This paper presents an analytical procedure and closed-form vibration solutions with analytically determined coefficients for orthotropic circular cylindrical shells having classical boundary conditions. This analysis is based upon the Donnell–Mushtari shell theory. This is the simplest thin shell theory and its results for the lowest frequencies of a closed cylinder may not be as accurate. It is known that the exact procedure is complicated for orthotropic shells and this complexity has apparently prevented most researchers from getting results. Using the separation of variables method, the closed-form natural frequencies are successfully obtained in this work. They are found in a compact form. Moreover, the characteristics of the eigenvalues are examined. The exact solutions are validated through numerical comparisons with available solutions in literatures and the semi-analytical differential quadrature finite element method (S-DQFEM) solutions calculated by the authors.

Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method

► We investigate analytical and DTM solutions of free vibration of a Timoshenko column. ► We examine the variations in free vibration natural frequencies of the column. ► We observe good agreement in the results of DTM with the results of analytical method. In this study free vibration analysis of a Timoshenko column with a tip mass having rotary inertia is carried out by both exact solution and differential transform method (DTM). The support of the system is modeled by an elastic spring against rotation. Fixity factor is used to define the relative stiffness of the elastic connection and the column. The governing equation of the column is solved by the separation of variables method including the rotatory inertia to get the exact solution. The same problem is also solved by DTM algorithm and the results are compared with the ones of exact solution. The comparison tables are presented in numerical analysis to show that a very good agreement is observed for DTM.

Power Series Solution of Nonlinear Free Vibration Frequency of Isotropic Rectangular Thin Plates in Large Amplitude

Nonlinear vibration computational problem of isotropic thin plates in large amplitude was investigated here. We applied the Von Kármán’s theory of thin plates to derive the governing equations of nonlinear free vibration of isotropic thin plates, and solved the governing equations by direct integration method combined with power series expansion method. We obtained the power series solution of the nonlinear vibration frequency of the rectangular thin plates with four edges simply supported. Finally, the paper gave the computational example and compared the two results from the large amplitude theory and the small one, respectively. Results obtained from this paper provide a new analytical computational approach for calculating the frequency of nonlinear free vibration of isotropic thin plates in large amplitude, and provide more accurate theoretical basis for the vibration control and dynamic design of plate structures.

Study on Free Vibrations of Self-Anchored Suspension Bridges

Study on characters of suspension bridges free vibration is basic to analyze wind resistance, shock resistance, and other dynamic behaviors. Current studies are usually focused on earth-anchored suspension bridges. Compared with earth-anchored suspension bridges, self-anchored suspension bridges are anchored on the girder end which makes girder under the compression condition, cable longitudinal displacement can not be neglected, in addition, large ratio of rise to span leads to cable large tilt angle. Based on functional analysis method, it uses variation calculus to derive differential equation of two-tower and three-span continuous self-anchored suspension bridges free vibrations, taking into account cable tilt angle and cable longitudinal displacement. According to the simplified differential equation, it draws to a formula that can calculate self-anchored longitudinal and vertical vibration frequencies. Finally, two examples are carried out to check the formula’s reasonableness.

Rayleigh's quotient for multiple cracked beam and application

Rayleigh's quotient for Euler-Bernoulli multiple cracked beam with different boundary conditions has been derived from the governed equation of free vibration. An appropriate choosing of approximate shape function in terms of mode shape of uncracked beam and specific functions satisfying conditions at cracks and boundaries leads to an explicit expression of natural frequencies through crack parameters that can simplify not only the analysis of natural frequencies of cracked beam but also the crack detection problem. Numerical analysis of natural frequencies of the cracked beam by using the obtained expression in comparison with the well-known methods such as the characteristic equation and finite element method shows their good agreement. The analytical expression of natural frequencies applied to the crack detection problem allows the result of detection to be improved.

Integral Equation for Symmetrical Free Vibration of Levy – Plate Having Discontinuous Simple Supports

he principal concerned in this paper is the finite Hankel integral transform method for solving the symmetrical free vibration problem of rectangular thin plate. The considered plate has two opposite simply supported edges and the others are partially simply supported mixed with free. Based on the Levy – plate solution, the mixed boundary conditions between the simple and free supports on the same side of the plate can be written as dual – series equations. However, with the advantages of the present method, these mentioned equations are analytically reduced to homogeneous integral equation of Fredholm – type. Significantly, the nature of moment singularities where the simple support changes to a free edge has been brought into consideration. They can be isolated and treated analytically with a weight function. From this aspect, the attention is placed specifically on deriving an integral equation of the problem considered here instead of the partial differential equation. As a result, those with mixed boundary conditions and the moment singularities in the order of an inverse – square – root type have been successfully obtained.

Influence of the Tension Variety on Non-Linear Natural Frequency of an Inclined Cable

According to the vibration characteristics of small sag cable, an equation of non-linear free transverse in-plane vibration of an inclined cable is presented considering the effects of the flexural rigidity. In the equation the influence of the tension variety along the inclined cable was taken into account by using the precise parabola equation of the taut cable. Taking single mode of a cable in consideration, the governing equation was simplified as a traditional Duffing equation with a quadratic term. It was solved by using the method of Fourier series. Combined with engineering practice, an example was provided to study nonlinear natural vibration characteristic of an inclined cable in plane when the length, the inclination angle, the tension were changed separately, whether considering the influence of tension variety along the inclined cable or not.

Curvature Mode Characteristics of the Simple Beam with a Local Damaged Crack

General expressions of crack perturbations are given with perturbation method. With the delta function to express the crack position at the simple beam, the expressions of cross-section rotary inertia and mass per unit length of the whole beam are founded, and then the transverse free vibration equation of the simple beam with a damaged crack is established. The first and second order perturbations are considered as the linear combinations of corresponding mode shapes of intact state, according to the boundary conditions, and theoretical solutions about eigenvalues and mode shapes of the damage state are finally obtained. One simple beam is taken as a numerical example to calculate the modal parameters; the effects of first and second order perturbations on the results are compared. It is believed that the approach proposed in this paper could provide the necessary theoretical background for damage identification in simple beam.

Approximate solutions for transient response of constrained damping laminated cantilever plate

The series composed by beam mode function is used to approximate the displacement function of constrained damping of laminated cantilever plates, and the transverse deformation of the plate on which a concentrated force is acted is calculated using the principle of virtual work. By solving Lagrange's equation, the frequencies and model loss factors of free vibration of the plate are obtained, then the transient response of constrained damping of laminated cantilever plate is obtained, when the concentrated force is withdrawn suddenly. The theoretical calculations are compared with the experimental data, the results show: both the frequencies and the response time of theoretical calculation and its variational law with the parameters of the damping layer are identical with experimental results. Also, the response time of steel cantilever plate, unconstrained damping cantilever plate and constrained damping cantilever plate are brought into comparison, which shows that the constrained damping structure can effectively suppress the vibration.

Analysis of thermoelastic damping in microresonators by considering the stretching effect

In this paper, the previous theory for obtaining the quality factor of thermoelastic damping in microresonators has been modified. Here, the stretching term for the neutral axis of bending is considered, which has been neglected in previous work. The Taylor expansion of electric actuation has described the static position of microbeam, which is more accurate than previous work that extends it about the straight position of a microbeam. Also, the theory of this work may be used for two layered microresonators without this assumption that the length of two layered electrically actuated microresonators may be obtained, for the first time. To obtain this complete theory firstly the equations of motion coupled with thermal effects are obtained in a two layered microbeam where the second layer is deposited on a part of the first layer length. Then, the quality factor of thermoelastic damping of the system, i.e. the half of the ratio of the real part of the free vibration resonance frequency to its imaginary part, has been obtained by solving the free vibration equation using the strained parameter perturbation method. It has been shown that neglecting the stretching of neutral axis of bending and expanding the electrostatic actuation about the straight position of the microbeam considered in previous researches cause an error that is not negligible for the voltage larger than 30% of the pull-in voltage. Also, it is shown that the quality factor of thermoelastic damping in two layered microresonators depends on the value of length, thickness and the kind of material of the deposited layer.

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.

L‐P Perturbation Solution of Nonlinear Free Vibration of Prestressed Orthotropic Membrane in Large Amplitude

This paper reviewed the research on the nonlinear free vibration of pre‐stressed orthotropic membrane, which is commonly applied in building membrane structures. We applied the L‐P perturbation method to solve the governing equations of large amplitude nonlinear free vibration of rectangular orthotropic membranes and obtained a simple approximate analytical solution of the frequency and displacement function of large amplitude nonlinear free vibration of rectangular membrane with four edges simply supported. By giving computational examples, we compared and analyzed the frequency results. In addition, vibration mode of the membrane and displacement and time curve of each feature point on the membrane surface were analyzed in the computational example. Results obtained from this paper provide a simple and convenient method to calculate the frequency and lateral displacement of nonlinear free vibration of rectangular orthotropic membranes in large amplitude. Meanwhile, the results provide some theoretical basis for solving the response of membrane structures under dynamic loads and provide some computational basis for the vibration control and dynamic design of building membrane structures.

Theoretical study regarding the modes shapes of wood beam having different boundary conditions using the Eigenvalues problems.

In engineering practice, however, beams problems often involve consideration of dynamic disturbances, produced by time-dependent external forces or displacements. Structural dynamics deals with time-dependent motions of structures and analyzes the internal forces associated with them. Thus, its objective is to determine the effect of vibrations on the performance of the structure. It can be started from the case of a homogeneous beam with different boundary conditions at the ends. By applying the principle of D’Alembert for an infinitesimal beam element, the equation of free transverse vibrations of the beam. The equation of the normal vibration modes, along with the homogeneous boundary conditions describe a type Sturm-Liouville problem. Solving type Sturm –Liouville problems allows the determination of inherent values , and the functions of the vibration shapes of beams. The normal vibration modes for several/more combinations of conditions at the ends of the beam are determined.

Element Level Identification of a Viscously Damped System

The modal data-based system identification (SI) algorithms normally consider the free vibration equation of motion of the dynamic system without damping for the identification of damage of existing structures. The present study intends to assess the damage of existing structures, including the effect of damping. The primary objective is to develop the equation error approach-based SI algorithm to identify both the stiffness properties and the damping parameters. The viscous damping model, which is computationally affordable and widely applicable to non-conservative systems, is considered for this objective. The formulation is hinged on the use of free vibration response in which the error norm of the eigen equation with damping is minimized. The SI algorithm is demonstrated through numerically-simulated modal data, which are obtained by the finite element analysis. The Monte Carlo simulation-based error sensitivity study confirms the consistency and robustness of the proposed algorithm.

Three-Step Method for Stiffness Identification of Inter-Story Shearing Structures Under Ambient Excitation

This paper presents a three-step method for the stiffness identification of inter-story shearing structures under ambient excitation. The transition from structural random response to deterministic response is realized by the natural excitation technique (NExT). The parameter-decoupled free vibration equation of inter-story shearing structures is deduced, which can simplify the procedure of identification and reduce the amount of calculation. Through the effective combination of the NExT, the parameter-decoupled method, and the extended Kalman filter (EKF) algorithm, a good scheme is presented for stiffness identification of inter-story shearing structures under ambient excitation. Using inter-story shearing structures with six-degree-of-freedom and ten-degree-of-freedom as examples, the scheme is tested by numerical simulation and experiment. The result shows that the maximum of identification error is 7.82%, and it can reach the requirement of engineering application basically.