Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.     

Free vibration of embedded double-walled carbon nanotubes considering nonlinear interlayer van der Waals forces

In this article, nonlinear free vibration of embedded double-walled carbon nanotubes (DWCNTs) duo to the nonlinear interlayer van der Waals (vdW) force is studied based on the nonlocal Euler-Bernoulli beam theory. The interlayer vdW force is modeled as a nonlinear function of inner and outer tubes deflections considering the variation of the interlayer distance along the circumference of DWCNTs. The harmonic balance method is applied to analyze the relationship between the deflection amplitudes and the frequencies of in-phase and out-of-phase free vibrations for DWCNTs. Finally, the influences of the nonlocal parameter, surrounding elastic medium, nanotube length, end condition and vibrational mode on the nonlinear free vibration properties of DWCNTs are discussed in detail.     

Forced oscillations of nonlinear damped equation of suspended string

We shall study the existence of time-periodic solutions of nonlinear damped equation of suspended string to which a periodic nonlinear force works. We shall be conterned with weak, strong and classical time-periodic solutions and also the regularity of the solutions. To formulate our results, we shall take suitable weighted Sobolev-type spaces introduced by [M. Yamaguchi, Almost periodic oscillations of suspended string under quasiperiodic linear force, J. Math. Anal. Appl. 303 (2) (2005) 643-660; M. Yamaguchi, Infinitely many time-periodic solutions of nonlinear equation of suspended string, Funkcial. Ekvac., in press]. We shall study properties of the function spaces and show inequalities on the function spaces. To show our results we shall apply the Schauder fixed point theorem and the fixed point continuation theorem in the function spaces.     

Periodic solution for nonlinear vibration of a fluid-conveying carbon nanotube,based on the nonlocal continuum theory by energy balance method

A nonlinear model is developed for the vibration of a single-walled carbon nanotube (SWCNT) based on Eringen’s nonlocal elasticity theory. The nanotube is assumed to be embedded in a Pasternak-type foundation with simply supported boundary conditions. The nonlinear equation of motion is solved by the energy balance method (EBM) to obtain a sufficiently accurate flow-induced frequency. It is demonstrated that the nonlinearity of the model makes a reasonable change to the frequency at high flow velocity and for the large deformations. Furthermore, the deviation of the frequency from the linear frequency will be exaggerated with an increase in the nonlocal parameter and a decrease of the Pasternak parameters. Ultimately, the results show that the nonlinearity of the model can be effectively tuned by applying axial tension to the nanotube.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Sound absorption of a quadratic and cubic nonlinearly vibrating curved panel absorber

This study investigated the sound absorption of a nonlinearly vibrating curved panel backed by a cavity. Very few studies on similar nonlinear structural-acoustic problems have been conducted to date, although there have been many on nonlinear plate or linear structural-acoustic problems. A curved panel is considered because the overall absorption bandwidth can be designed by appropriately adjusting the panel curvature, which is a key factor in controlling the structural resonant frequencies and absorption peaks. The theoretical formulation is developed based on the assumptions of quadratic and cubic nonlinear structural vibrations, and the linear acoustic pressure induced within the cavity. An approach based on the numerical integration method is developed to solve the nonlinear governing equation of the structural-acoustic problem. In the parametric study, the panel displacement amplitude converges with an increasing number of modes. The effects of excitation level, cavity depth, and damping factor are also examined. The quadratic and cubic nonlinearities and their effects on the sound absorption are also investigated. An experiment was conducted. The theoretical and experimental observations correspond reasonably with each other.     

Free and forced vibrations of nonlinear wave equations in ball

First, we shall deal with the free vibrations of a nonlinear radially symmetric wave equation (∂_t²−△)u=f(r,u) in n-dimensional ball B_a with center at the origin and radius a, where f is smooth, monotone decreasing in u, and satisfies f(r,0)=0. f(r,u) has asymptotic properties . For n=1,3 we shall show the existence of infinitely many radially symmetric time-periodic solutions with different periods of irrational multiple of a. Second, we shall deal with BVP for a forced nonlinear wave equation (∂_t²−△)u=εg(r,t,u), where g is T-periodic in t and ε is a small parameter. Under some Diophantine condition on a/T we shall show the existence of time-periodic solutions of the BVP. For 1?n?5 we shall construct infinitely many T satisfying the above Diophantine inequality, using asymptotic expansions of the zero points of the Bessel functions.     

Global smooth solutions of IBVP to nonlinear equation of suspended string

We shall consider IBVP to a nonlinear equation of suspended string with uniform density to which a nonlinear time-independent outer force works. The nonlinear term is smooth but not monotone. We shall show that IBVP has a unique time-global smooth solution. The regularity of the solutions shall be also studied.     

Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes

The aim of this paper is to show how the concept of nonlinear normal modes (NNMs) can be used to characterize the nonlinear dynamical behaviors of double-walled carbon nanotubes (DWCNTs). DWCNTs are modeled as double simply supported elastic beams with van der Waals (vdW) forces between the inner and outer walls. The multiscale method for deriving the approximate solutions of NNMs is applicable to the nonlinear systems. According to the procedure, the typical features – coaxial and noncoaxial vibrations of the system are exhibited in the literature. Moreover, the case of 1:3 internal resonance is discussed in detail, which can give rise to much more complex phenomenon for DWCNTs systems. Meanwhile, the amplitude-time curves of the nonlinear vibration with different initial conditions are presented, and the amplitude-frequency characteristic curves of the nonlinear vibration are also obtained.     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

On the Vykhandu-Levin iterative method for numerical solution of nonlinear systems

An iterative method for the solution of systems of nonlinear equations initiated by Vykhandu and investigated by Levin is discussed. It is shown that there is a flaw in the proof of Levin that the method is third-order convergent. Moreover, it is proved that the correct order of the method is only two.     

Variable structure control for a class of nonlinear systems with mismatched uncertainties

A scheme to stabilize nonlinear time-varying systems with both matched and mismatched uncertainties is proposed in this paper by switching between two control laws: a first-order sliding-mode control and a second-order sliding-mode control. Based on this idea, a variable structure control algorithm is designed for a class of second-order systems. The closed-loop system is globally or locally asymptotically stable. It has been proven that the stability region has relation with the order of the boundary function and the region can be obtained by solving an inequality. The uncertainty considered in this work is also more general than those in the existing works.     

Solvability and stability of a semilinear wave equation with nonlinear boundary conditions

We discuss solvability for the semilinear equation of the vibrating string x_tt(t,y)−Δx(t,y)+f(t,y,x(t,y))=0 in a bounded domain, and certain type of nonlinearity on the boundary. To this effect we derive a new dual variational method. Next we discuss stability of solutions with respect to initial conditions.     

利率期限结构的主成分分析

本文采用主成分分析的方法对我国的利率期限结构进行了研究。在采用这种方法的同时,结合非线性变换BOX—COX得出了我国的利率期限结构具有代表性的三个主成分:利率期限结构曲线的平移、斜率的变化以及曲率的变化。同时通过实证分析证实了这种方法的有效性。     

A modification of the Adomian decomposition method for a class of nonlinear singular boundary value problems

In this paper, the Adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems. The effectiveness of the modified method is verified by three examples.     

Numerical analysis of the atomic structure and shape of metal nanoparticles

Numerical simulation methods in the framework of molecular dynamics are used to study the emergence and development of amorphous, crystalline, and polycrystalline phases and their spread to the entire volume of a nanoparticle as it increases. Numerical results are presented for the parameters characterizing these processes in metal nanoparticles produced by top-down techniques. The basic features of nanoparticle formation in top-down processes and the properties of the nanoparticle structure are described.     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

Approximation procedures for the optimal control of bilinear and nonlinear systems

Optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems. For a given bilinear optimal control problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. Also, the direct solution to the Hamilton-Jacobi equation is analyzed. A power-series approach is presented which requires offline calculations as in the linear case (Riccati equation). The methods are compared and illustrated. Relations to classical linear systems theory are discussed.     

Analytical solitary wave solutions for the nonlinear analogues of the Boussinesq and sixth-order modified Boussinesq equations

Using tanh function and polynomial function methods, analytical solitary wave solutions have been found for the nonlinear analogues of Boussinesq and sixth-order modified Boussinesq equations where the nonlinearity is in the time-derivative term.