Nonlinear Vibration Frequency Research Articles

Acoustic streaming with allowance for heat transfer

Acoustic streaming in a gas-filled cavity under vibration is numerically investigated. The case of thermally insulated cavity walls is compared with the case of walls maintained at a constant temperature. A strong influence of heat transfer on the acoustic streaming pattern is revealed for weakly nonlinear processes and vibration frequencies below resonance. Cavities of different diameters are considered.

Large amplitude free vibration of micro/nano beams based on nonlocal thermal elasticity theory

This paper is concerned with the nonlinear free vibration of a heated micro/nano beam modeled after the nonlocal continuum elasticity theory and Euler-Bernoulli beam theory. The governing partial differential equations are derived from the Hamilton variational principle and von Karman geometric nonlinearity, in which the effects of the nonlocality and ambient temperature are inclusive. These equations are converted into ordinary forms by employing the Kantorovich method. The solutions of nonlinear free vibration are then sought through the use of shooting method in spatial domain. Numerical results show that the proposed treatment provides excellent accuracy and convergence characteristics. The influences of the aspect ratio, nonlocal parameter and temperature rise parameter on the dimensionless radian frequency are carefully investigated. It is concluded that the nonlocal and temperature rise parameters lead to reductions of the nonlinear vibration frequency, while the influence of the nonlocal effect decreases with an increase in the aspect ratio.

Nonlinear Vibration of PZT4/PZT-5H Monomorph and Bimorph Beams with Graded Microstructures

Monomorph and bimorph actuators made of piezoelectric materials are widely used in smart materials and structural control. Recent studies show that their performances can be remarkably improved by the use of functionally graded piezoelectric materials (FGPMs) whose compositional profile and effective material properties are graded along the thickness direction. This paper investigates the linear and nonlinear vibration behaviors of monomorph and bimorph made from a mixture of PZT4 and PZT-5H with material composition following a power law distribution. Theoretical formulations are based on von Kármán kinematic relationships and Timoshenko beam theory to account for the effect of transverse shear deformation. The differential quadrature method (DQM) and the second-order backward difference scheme are employed to obtain the linear and nonlinear vibration frequencies. A parametric study is conducted to investigate the influences of volume fraction index, slenderness ratio, nonlinear deformation and different loading conditions.

Nonlinear vibration characteristics of graphene/piezoelectric sandwich films under electric loading based on nonlocal elastic theory

This paper reports the result of an investigation into the nonlinear vibration frequencies of graphene/piezoelectric sandwich films under electrical loading based on nonlocal elastic theory by utilizing a global residual harmonic balance method. Based on the Galerkin method and global residual harmonic balance method, the nonlinear resonant frequencies of graphene/piezoelectric sandwich films under electric exciting loads are obtained with a set of factors: the ratio of the oscillating amplitude to the thickness of sandwich films, small scale effect, electric loading exerted on piezoelectric layer, mode number and size length. Results indicate that the electric exciting load enhances the nonlinear resonant frequency of graphene/piezoelectric sandwich films, the nonlinear resonant frequency decreases as the scale effect increases, the scale effect has a more significant effect on higher mode resonant frequency and linear resonant frequency, and the effect of scale on the nonlinear resonant frequency is independent on the electric exciting load and the boundary conditions exerted on the sandwich films.

Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method

This study investigates the small scale effect on the nonlinear bending vibration of a rotating cantilever and propped cantilever nanobeam. The nanobeam is modeled as an Euler---Bernoulli beam theory with von Karman geometric nonlinearity. The axial forces are also included in the model as the true spatial variation due to the rotation. Hamilton's principle is used to derive the governing equation and boundary conditions for the Euler---Bernoulli beam based on Eringen's nonlocal elasticity theory. The differential quadrature method as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeam. The effect of nonlocal small---scale, angular speed, hub radius and nonlinear amplitude of rotary nanobeam is discussed.

Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory

In this paper, the nonlinear bending and free vibration responses of a simply supported functionally graded (FG) microplate lying on an elastic foundation are studied within the framework of the modified couple stress theory and the Kirchhoff/Mindlin plate theory together with the von Karman’s geometric nonlinearity. The equations of motion and boundary conditions for the FG microplate are derived from the Hamilton’s principle. Due to introducing the physical neutral surface, there is no stretching-bending coupling in the constitutive equations, and the equations of motion become simpler. By using the Galerkin method, the equations of motion are reduced to nonlinear algebraic equations and ordinary differential equations (ODEs) for the bending and vibration problems respectively. By solving the algebraic equations, closed-form solutions for the nonlinear bending deflection of the microplate are derived. Closed-form solutions for the nonlinear vibration frequency are also obtained by applying He’s variational method to the ODEs. Based on the obtained closed-form solutions, numerical examples are further presented to investigate the effects of the material length scale parameter to thickness ratio, the length to thickness ratio, the power law index and the elastic foundation on the nonlinear bending and free vibration responses of the microplate.

Nonlinear Vibration Analysis of Stochastic Thin Plates Based on Lindstedt-Poincar\'{e} Perturbation Method

Nonlinear vibration computational problem of stochastic thin plates in large amplitude was investigated here. Based on the nonlinear theory of large deflection of thin plates, we derived the governing equations of nonlinear free vibration of isotropic thin plates. And solved the governing equations by Lindstedt- Poincare Perturbation Method Presented herein are asymptotic analytical solutions for the frequency function of the thin plates with fixed parameters. Based on the theory of random field, after mathematical treatments of the probability density function of random vector expressions, such as determining the interval of integrating, substituting variable and transforming integral order, then the probability density function of nonlinear vibration frequency of stochastic thin plates can be obtained directly. And combined with Monte Carlo Simulation analysis and our previous work (1), the result of the method in this paper is validated, which testifies that the method in this paper is effective and of preferable precision.

Improved Shock Load Model of Stranded Wires Helical Springs Based on Perturbation Method

On the basis of original shock load model of stranded wires helical springs, the improved model by introducing nonlinear stiffness is established. An improved method of multiple scales is also presented based on the resolution principles of strongly nonlinear oscillation, and the perturbation solution of nonlinear response of improved model is derived correspondingly. Then, comparison between the two models had been made according to the experimental data. The results show that the improved model reflects the characteristics of nonlinear stiffness and nonlinear vibration frequency and improves the fitting precision of experimental data. It can not only describe the dynamic movement characteristics of spring more accurately, but also is especially significant to research the nonlinear characteristics of the spring under the condition of shock load and large amplitude. This study can also provide the theoretical foundation for the dynamic response characteristics of stranded wires helical springs which is being studied.

张拉平面膜结构非线性振动频率计算分析

利用冯∙卡门薄膜大挠度理论，结合达朗贝尔原理，建立正交异性张拉平面膜结构非线性自由振动的控制方程。然后利用伽辽金法对其进行简化，并采用同伦摄动法进行求解，得到振动频率的近似解析解。通过算例，计算了结构的非线性振动频率，并将本文结果与精确解进行比较分析。分析表明：本文所求得的近似解析解与精确解之间的最大误差小于4%。因此本文的近似解析解与精确解非常接近，且本文所得解形式更为简单，计算也更方便，有利于在工程中进行推广应用。 The nonlinear free vibration governing differential equations for the orthotropic tensioned plane membrane structure are established by Von Karman’s membrane large deflection theory and D’Alembert’s principle. Then the governing differential equations are simplified by Bubnov-Ga- lerkin method and solved by the homotopy perturbation method (HPM), and obtained the ap-proximate analytical solution of the vibration frequency. In the computational example, the non-linear vibration frequency of the structure is computed, and the results of this paper are analyzed and compared with the exact solution. The analysis shows that the approximate analytical solution obtained in this paper is very close to the exact solution (the maximum error is less than 4%), and the approximate analytical solution obtained in this paper is more simple and convenient. This is favorable for the popularization and application in engineering.

Finite Element Formulation for the Large-Amplitude Vibrations of FG Beams

Abstract On the basis of Euler-Bernoulli beam theory, the large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The von Karman type nonlinear strain-displacement relationship is employed where the ends of the beam are constrained to move axially. The material properties are assumed to be graded in the thickness direction according to the power-law and sigmoid distributions. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law index, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible.

Linear and Nonlinear Vibrations of a Column with an Internal Crack

In this paper, the results of numerical studies on linear and nonlinear transverse vibrations and instability of a geometrically nonlinear column with an internal crack subjected to Euler’s load are presented. The investigated column is composed of two members. The internal member consists of two rods connected by a pin and strengthened by a rotational spring of stiffness C. The stiffness C of the rotational spring reflects the size of the internal crack. The Hamilton principle was used to formulate the boundary problem. Because of the geometric nonlinearity, the solution to the problem was achieved by means of the perturbation method. The natural vibration frequencies (the linear problem) were computed after obtaining the equations from the first power of the small parameter ε. The orthogonality condition was applied to find and calculate the first correction of vibration frequency. The nonlinear vibration frequency of a column is derived as dependent on vibration amplitude. The results of the numerical calculation are based on vibration frequency, critical loading, and the amplitude–vibration frequency relation.

Nonlinear Free Vibration Analysis of Axisymmetric Polar Orthotropic Circular Membranes under the Fixed Boundary Condition

This paper presents the nonlinear free vibration analysis of axisymmetric polar orthotropic circular membrane, based on the large deflection theory of membrane and the principle of virtual displacement. We have derived the governing equations of nonlinear free vibration of circular membrane and solved them by the Galerkin method and the Bessel function to obtain the generally exact formula of nonlinear vibration frequency of circular membrane with outer edges fixed. The formula could be degraded into the solution from small deflection vibration; thus, its correctness has been verified. Finally, the paper gives the computational examples and comparative analysis with the other solution. The frequency is enlarged with the increase of the initial displacement, and the larger the initial displacement is, the larger the effect on the frequency is, and vice versa. When the initial displacement approaches zero, the result is consistent with that obtained on the basis of the small deflection theory. Results obtained from this paper provide the accurate theory for the measurement of the pretension of polar orthotropic composite materials by frequency method and some theoretical basis for the research of the dynamic response of membrane structure.

Study nonlinear vibration of cross‐ply laminated plates using scale models

This article describes the establishment of necessary similarity conditions for geometrically nonlinear vibrations of thin cross‐ply laminated plates. The Von‐karman's strain‐displacement relations have been employed to model structural nonlinearity of the system. The Galerkin procedure is used to reduce the nonlinear partial differential equations to a nonlinearly second‐order ordinary differential equation. An analytical investigation based on the direct use of the governing equations of the systems is undertaken to derive the necessary scaling laws and similarity conditions. In this study, a set of scaling laws are introduced that can predict the nonlinear free vibration frequency of the prototypes by projecting the frequency of the model, accurately. The effects of distorted models with relaxations in the number of plies, stacking sequence, and different vibration amplitudes are studied. The results presented herein indicate that models with different relaxations can predict the nonlinear vibration frequency of prototypes with good accuracy. POLYM. COMPOS., 35:752–758, 2014. © 2013 Society of Plastics Engineers

Nonlinear vibration of stringer shell by means of extended Hamiltonian approach

In this paper, the nonlinear free vibration of a stringer shell is studied. The mathematical model of the string shell, which is the most convenient for frequency analysis, is considered. Due to the geometrical properties of the vibrating shell, strong nonlinearities are evident. Approximate analytical expressions for the nonlinear vibration are provided by introducing the extended version of the Hamiltonian approach. The method suggested in the paper gives the approximate solution for the differential equation with dissipative term for which the Lagrangian exists. The aim of this study is to provide engineers and designers with an easy method for determining the shell nonlinear vibration frequency and nonlinear behavior. The effects of different parameters on the ratio of nonlinear to linear natural frequency of shells are studied. This analytical representation gives excellent approximations to the numerical solutions for the whole range of the oscillation amplitude, reducing the respective error of the angular frequency in comparison with the Hamiltonian approach. This study shows that a first-order approximation of the Hamiltonian approach leads to highly accurate solutions that are valid for a wide range of vibration amplitudes.

Nonlinear vibration and instability of fluid-conveying DWBNNT embedded in a visco-Pasternak medium using modified couple stress theory

Nonlinear free vibration and instability of fluid-conveying double-walled boron nitride nanotubes (DWBNNTs) embedded in viscoelastic medium are studied in this paper. The effects of the transverse shear deformation and rotary inertia are considered by utilizing the Timoshenko beam theory. The size effect is applied by the modified couple stress theory and considering a material length scale parameter for beam model. The nonlinear effect is considered by the Von Karman type geometric nonlinearity. The electromechanical coupling and charge equation are employed to consider the piezoelectric effect. The surrounding viscoelastic medium is described as the linear visco-Pasternak foundation model characterized by the spring and damper. Hamilton’s principle is used to derive the governing equations and boundary conditions. The differential quadrature method (DQM) is employed to discretize the nonlinear higher-order governing equations, which are then solved by a direct iterative method to obtain the nonlinear vibration frequency and critical fluid velocity of fluid-conveying DWBNNTs with clamped-clamped (C-C) boundary conditions. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, spring and damping constants of surrounding viscoelastic medium and fluid velocity on the nonlinear free vibration, instability and electric potential distribution of DWBNNTs. This study might be useful for the design and smart control of nano devices.

Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.

Dynamic Nonlinearity and Nonlinear Single-Degree-of-Freedom Model for Cable Net Glazing

AbstractThe nonlinear vibration differential equation and vibration frequency of cable net glazing subject to earthquake loading was determined, and a geometrically nonlinear single-degree-of-freedom model for cable net glazing was developed. The nonlinear response spectra were established, and nonlinear time history analysis with finite element (FE) models was conducted to verify them. The nonlinear vibration differential equation and frequency obtained as described in this paper provide a basis for the nonlinear single-degree-of-freedom model for cable net glazing. The analytical formula for the nonlinear frequency with a simplified expression is highly precise and convenient for use in engineering practice. For larger-amplitude seismic waves, the difference between the linear and nonlinear response spectra are more obvious. As the natural period of cable net glazing is always less than 2 s, the linear response spectra in the Chinese code for the seismic design of buildings can be used in the seismic de...

Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory

The present investigation proposes a nonclassical mathematical model and an algorithm for the axisymmetrically nonlinear free vibration analysis of a circular microplate. Based on the modified couple stress theory and von Kármán geometrically nonlinear theory, the governing equations for microplates are established in variational form from Hamilton principle containing only one material length scale parameter which can capture the size-dependent behavior. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed simple harmonic function in time. The resulting nonlinear spatial boundary value problem is then solved numerically by shooting method, and the size-dependent characteristic relationships of nonlinear vibration frequency versus central amplitude of the microplates are obtained. The parametric studies are conducted for immovable clamped and simply supported edge conditions, some of the results in special cases are verified by comparing with those in the literature. The numerical results indicate that the microplates modeled by the modified couple stress theory cause more stiffness than modeled by the classical continuum plate theory, such that the differences between the results of these two theories are large when the thickness to material length scale ratio is small, whereas they are decreasing with the increase of the ratio.