Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).     

Nonlinear dynamics of high-dimensional models of in-plane and out-of-plane vibration in an axially moving viscoelastic beam

Nonlinear dynamics of high-dimensional models of an axially moving viscoelastic beam with in-plane and out-of-plane vibration with combined parametric and forcing excitations are investigated by the incremental harmonic balance (IHB) method in this paper. Governing equations of transverse in-plane and out-of-plane and longitudinal vibration are obtained basing on the Hamilton's principle. The Galerkin method is used to separate time variable and spatial variable to obtain a set of multi-order differential equations. The IHB method with the fast Fourier transform (FFT) is used to solve periodic response of high-dimensional models of the beam for which convergent mode is reached. Stability of the steady-state periodic solutions is analyzed using the multivariable Floquet theory. Particular attention is paid to in-plane and out-of-plane vibration on convergent mode of the beam with combined parametric and forcing excitations. Multiple solutions are observed, and jump phenomena between in-plane and out-of-plane vibration with different transverse cross sections are discovered.     

On the dynamics of axially moving strings

We investigate an axially moving cable with large sag modelled as string and calculate the steady state solution between two rolls. Alternatively we model the cable as beam with very small bending stiffness and compare this solution with the string solution. The validity of the analytically found boundary layer solution is verified by numerical methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration and stability of an axially moving Rayleigh beam

In this paper, the vibration and stability of an axially moving beam is investigated. The finite element method with variable-domain elements is used to derive the equations of motion of an axially moving beam based on Rayleigh beam theory. Two kinds of axial motions including constant-speed extension deployment and back-and-forth periodical motion are considered. The vibration and stability of beams with these motions are investigated. For vibration analysis, direct time numerical integration, based on a Runge–Kutta algorithm, is used. For stability analysis of a beam with constant-speed axial extension deployment, eigenvalues of equations of motion are obtained to determine its stability, while Floquet theory is employed to investigate the stability of the beam with back-and-forth periodical axial motion. The effects of oscillation amplitude and frequency of periodical axial movement on the stability of the beam are discussed from the stability chart. Time histories are established to confirm the results from Floquet theory.     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

Vibration of axially moving hyperelastic beam with finite deformation

In this paper, we study the vibration of an axially moving hyperelastic beam under simply supported condition. The kinematic of the axially moving beam have been described by Eulerian-Lagrangian formulation. In continuum mechanics frame, the finite deformation formula and a higher order shear deformation beam theory are applied to describe the deformation of the axially moving hyperelastic beam. In these formulas the material parameter, shear deformation and the geometric non-linearity have been taken into account. Through the Hamilton principle, the governing equations of nonlinear vibration are obtained, where the transverse vibration is coupled with the longitudinal vibration. When the velocity is a constant, the critical speed and natural frequencies are determined by solving the corresponding linear equations. Meantime, effects of the geometrical and material parameters on the critical speed and natural frequencies have been investigated. Comparisons among the critical velocities of the hyperelastic and Euler linear beam are also made. The results show that the critical velocity of hyperelastic beam is larger than that of linear Euler–Bernoulli beam. For the natural frequencies, we have the same conclusions. Lastly, by the multiple scales method, the leading order analytical solutions of the equilibrium state of axially moving hyperelastic beam in the supercritical regime are obtained. Furthermore the amplitudes of analytical solutions of the hyperelastic beam have been compared with that of linear Euler–Bernoulli beam. The effects of the material and geometrical parameters on the asymptotic solutions and the amplitude has been analyzed.     

Modeling problems of steeply inclined cableway subjected to moving load

Problems of structural modeling of multi-span bicable ropeway systems due to significant inclination of a track are discussed in this paper with reference to the vibration analysis. The analytical continuous model of a cableway is formulated in a form which enables to describe in-plane vibrations of steeply inclined carrying cable subjected to traveling passenger carriers modeled by moving pendulums. The nonlinear interaction between carrying cable and moving carriers is taken into account in the moving load definition. Effects of a steep track slope are discussed on the basis of equations of motion of the system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.     

On the vibrations of axially graded Rayleigh beams under a moving load

In the present investigation, the forced and free vibrations of axially functionally graded (AFG) Rayleigh and Euler-Bernoulli (EB) beams subjected to a moving load are studied and compared, aiming at performance enhancement of transportation systems. Also, for the first time, a precise mathematical modeling is obtained to analyze the influence of various key factors such as axial material gradation and rotary inertia factor on the critical speed, dynamic magnification factor, mechanisms of cancellation, and maximum free vibration of the system. Model verification is performed with the available results in the literature, and a good agreement is observed. Furthermore, the dynamical responses of the system acquired from the analytical and numerical approaches are in good agreement. It is demonstrated that for the gradient parameter, which is lower and higher than the critical value, the material properties variation has a reverse effect on the forced-free vibration amplitudes. Besides, it is concluded that, compared with the conventional isotropic EB beams, by selecting appropriate values of rotary inertia factor and gradient parameter in the AFG Rayleigh beams, the cancellation and maximum free vibration phenomena can be controlled. The results of this study can serve as a comprehensive benchmark to optimally design inhomogeneous structures under moving loads.     

Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate are studied by the extended Melnikov method.The rectangular thin plate is subject to transversal and i...     

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

Analytical study on dynamic responses of a curved beam subjected to three-directional moving loads

Considering the warping resistance, inertia force and moving three-directional loads, a more comprehensive set of governing equations for vertical, torsional, radial and axial motions of the curved beam are derived. The analytical solutions for vertical, torsional, radial and axial responses of the curved beam subjected to three-directional moving loads are obtained, using the Galerkin method to discretize the partial differential equations and the modal superposition method to decouple the ordinary differential equations. The analytical results are compared with the numerical integration and a published work to verify the validity of the proposed solutions. Effects of Galerkin truncation terms and damping ratio on solution convergence are also discussed. Considering first-mode and higher-mode truncation respectively, the conditions of resonance and cancellation are analyzed for vertical, torsional, radial and axial motions of the curved beam. Taking a curved bridge under passage of a vehicle as an example, the influences of system parameters, such as vehicle speed, braking acceleration, bridge curve radius, bridge span and bridge deck elastic modulus, on bridge midpoint vibration are explored. The proposed approach and results may be beneficial to enhance understanding the three-directional vehicle-induced dynamic responses of curved bridges. It is shown that when the axial motion, or the multiple moving loads are involved, the first-order truncation are not accurate enough and one should use higher-mode truncation to study the responses of curved beams. In addition, it is necessary to consider damping in the vibration study of curved beams.     

Using spectral element method for analyzing continuous beams and bridges subjected to a moving load

Spectral element method in frequency domain is employed to analyze continuous beams and bridges subjected to a moving load. The formulation is developed for an Euler beam under a moving load with an arbitrary amplitude and velocity. It is shown that the procedure is simplified for a moving load with a constant amplitude and velocity. Static Green’s function is used as a modifying function to improve the moment and shear force results. It is further shown that while modifying function is used in conjunction with spectral element method, fewer elements will be required to achieve proper results. The numerical examples show the accuracy of the method.     

Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity

Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.     

Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.     

Transient magneto-thermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source

The aim of this paper is to study the transient wave propagations in a non-homogeneous anisotropic thermoviscoelastic thick strip placed in a constant primary magnetic field and subjected to a moving heat source. The governing equations for temperature and displacement fields are solved by means of a dual reciprocity boundary element method (DRBEM). In the case of plane deformation, a numerical scheme for the implementation of the method is presented and the numerical computations are carried out for the temperature, displacement components and thermal stress components. The validity of DRBEM is examined by considering a magneto-thermo-visco-elastic thick strip occupies a rectangular region and good agreement is obtained with existent results. The results obtained are presented graphically to show the effect of inhomogeneity on the displacement components and thermal stress components. Relevant results of previous investigations are deduced as special cases from this study.