Stochastic response of an axially moving viscoelastic beam with fractional order constitutive relation and random excitations

A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relationship, where the random excitation can be decomposed as a nonstationary stochastic process, Mittag-Leffler internal noise, and external stationary noise excitation. Then, based on the Laplace transform approach, we derived the mean value function, variance function and covariance function through the Green's function technique and the residue calculus method, and obtained theoretical results. In some special case of fractional order derivative α , the Monte Carlo approach and error function results were applied to check the effectiveness of the analytical results, and good agreement was found. Finally in a general-purpose case, we also confirmed the analytical conclusion via the direct Monte Carlo simulation.     

Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance

This paper investigates the nonlinear forced dynamics of an axially moving Timoshenko beam. Taking into account rotary inertia and shear deformation, the equations of motion are obtained through use of constitutive relations and Hamilton’s principle. The two coupled nonlinear partial differential equations are discretized into a set of nonlinear ordinary differential equations via Galerkin’s scheme. The set is solved by means of the pseudo-arclength continuation technique and direct time integration. Specifically, the frequency-response curves of the system in the subcritical regime are obtained via the pseudo-arclength continuation technique; the bifurcation diagrams of Poincaré maps are obtained by means of direct time integration of the discretized equations. The resonant response is examined, for the cases when the system possesses a three-to-one internal resonance and when not. Results are shown through time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The results indicate that the system displays a wide variety of rich dynamics.     

Internal resonance of an axially transporting beam with a two-frequency parametric excitation

This paper investigates the transverse 3:1 internal resonance of an axially transporting nonlinear viscoelastic Euler-Bernoulli beam with a two-frequency parametric excitation caused by a speed perturbation. The Kelvin-Voigt model is introduced to describe the viscoelastic characteristics of the axially transporting beam. The governing equation and the associated boundary conditions are obtained by Newton’s second law. The method of multiple scales is utilized to obtain the steady-state responses. The Routh-Hurwitz criterion is used to determine the stabilities and bifurcations of the steady-state responses. The effects of the material viscoelastic coefficient on the dynamics of the transporting beam are studied in detail by a series of numerical demonstrations. Interesting phenomena of the steady-state responses are revealed in the 3:1 internal resonance and two-frequency parametric excitation. The approximate analytical method is validated via a differential quadrature method.     

Stability and bifurcations of an axially moving beam with an intermediate spring support

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton??s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.     

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is investigated.The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved.The nonlinear equations governing the transverse vibration are derived from the dynamical,constitutive,and geometrical relations.The method of multiple scales is used to determine the steady-state response.The modulation equation is derived from the solvability condition of eliminating secular terms.Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation.The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.     

Coupled global dynamics of an axially moving viscoelastic beam

The nonlinear global forced dynamics of an axially moving viscoelastic beam, while both longitudinal and transverse displacements are taken into account, is examined employing a numerical technique. The equations of motion are derived using Newton′s second law of motion, resulting in two partial differential equations for the longitudinal and transverse motions. A two-parameter rheological Kelvin–Voigt energy dissipation mechanism is employed for the viscoelastic structural model, in which the material, not partial, time derivative is used in the viscoelastic constitutive relations; this gives additional terms due to the simultaneous presence of the material damping and the axial speed. The equations of motion for both longitudinal and transverse motions are then discretized via Galerkin’s method, in which the eigenfunctions for the transverse motion of a hinged-hinged linear stationary beam are chosen as the basis functions. The subsequent set of nonlinear ordinary equations is solved numerically by means of the direct time integration via modified Rosenbrock method, resulting in the bifurcation diagrams of Poincaré maps. The results are also presented in the form of time histories, phase-plane portraits, and fast Fourier transform (FFTs) for specific sets of parameters.     

Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

Vibration and stability of an axially moving beam immersed in fluid

Vibration and stability are investigated for an axially moving beam in fluid and constrained by simple supports with torsion springs. The equations of motion of the beam with uniform circular cross-section, moving axially in a horizontal plane at a known rate while immersed in an incompressible fluid are derived first. An “axial added mass coefficient” and an initial tension are implemented in these equations. Based on the Differential Quadrature Method (DQM), a solution for natural frequency is obtained and numerical results are presented. The effects of axially moving speed, axial added mass coefficient, and several other system parameters on the dynamics and instability of the beam are discussed. Particularly, natural frequency in terms of the moving speed is presented for fixed–fixed, hinged–hinged and hybrid supports with torsion spring. It is shown that when the moving speed exceeds a certain value, the beam becomes subject to buckling-type instability. The variations of the lowest critical moving speed with several key parameters are also investigated.     

Vibration of axially moving beam supported by viscoelastic foundation

In this paper, transverse vibration of an axially moving beam supported by a viscoelastic foundation is analyzed by a complex modal analysis method. The equation of motion is developed based on the generalized Hamilton's principle. Eigenvalues and eigenfunctions are semi-analytically obtained. The governing equation is represented in a canonical state space form, which is defined by two matrix differential operators. The orthogonality of the eigenfunctions and the adjoint eigenfunctions is used to decouple the system in the state space. The responses of the system to arbitrary external excitation and initial conditions are expressed in the modal expansion. Numerical examples are presented to illustrate the proposed approach. The effects of the foundation parameters on free and forced vibration are examined.     

Principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string

To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string, the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string. To derive the governing equation, Newton‘s second law, Lagrangean strain, and Kelvin‘s model are respectively used to account the dynamical relation, geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms, closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance. The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance. Some numerical examples are presented to show the effects of the mean transport speed, the amplitude and the frequency of speed variation.     

Vibration and stability of an axially moving viscoelastic beam with hybrid supports

Vibration and stability are investigated for an axially moving beam constrained by simple supports with torsion springs. A scheme is proposed to derive natural frequencies and modal functions from given boundary conditions of an elastic beam moving at a constant speed. For a beam constituted by the Kelvin model, effects of viscoelasticity on the free vibration are analyzed via the method of multiple scales and demonstrated via numerical simulations. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams in parametric resonance. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.     

Forced vibration control of an axially moving beam with an attached nonlinear energy sink

This paper investigates a highly efficient and promising control method for forced vibration control of an axially moving beam with an attached nonlinear energy sink(NES).Because of the axial velocity,external force and external excitation frequency,the beam undergoes a high-amplitude vibration.The Galerkin method is applied to discretize the dynamic equations of the beam–NES system.The steady-state responses of the beams with an attached NES and with nothing attached are acquired by numerical simulation.Furthermore,the fast Fourier transform(FFT)is applied to get the amplitude–frequency responses.From the perspective of frequency domain analysis,it is explained that the NES has little effect on the natural frequency of the beam.Results confirm that NES has a great potential to control the excessive vibration.     

Multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam

The multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam are investigated in the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method is employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, leading to chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

Modeling and analysis of an axially acceleration beam based on a higher order beam theory

In this paper, a higher order model equation is presented for an axially accelerating beam. Based on a new kinematic frame of the beam and continuum mechanics theory, the coupled governing equations of nonlinear vibration for axially accelerating beam are obtained with the aid of the generalized Hamilton principle. The governing equations take into account the characteristic of the material, the shear strain, the rotation strain and the effect of longitudinally varying tension due to the axial acceleration. The equations are decoupled into a nonlinear partial-integro-differential equations when the transverse nonlinear vibration is small. For the principal parametric resonances, the steady-state frequency responses are obtained by the multiple scales method. The stable and unstable interval are analyzed for the trivial and nontrivial steady-state response. Effects of the system parameters on the amplitude have been investigated. The results show that the material parameter (i.e, in-plane Poisson ratio) has a significant effect on the amplitude and the nonlinear vibration behavior type. The amplitude decrease with the growth of the in-plane Poisson ratio. The total potential energy has play a very important role in determining the amplitude of frequency response according to model analysis. Lastly, comparisons among the analytical solutions and numerical solutions are made and good agreements for the amplitude are found.     

简谐激励作用下轴向运动梁横向振动特性分析

为分析简谐激励作用下轴向运动梁的横向振动问题,采用单元数目及长度固定不变、节点参数在不同时间步下无缝传递的节点生死方法,建立了时变系统的动力学有限元模型,通过已有实例验证了模型的准确性和有效性。在此基础上,分析了架设速度、激励力频率和幅值对某型平推式军用桥梁架设过程横向动力响应的影响规律。结果表明,在架设过程中,当桥梁的时变固有频率与激励力频率接近时,桥梁位移动态响应呈共振的特点,据此提出了减小某型平推式军用桥梁架设过程动力响应的措施。