Moving Timoshenko Beam Research Articles

Nonlinear dynamic modeling of planar moving Timoshenko beam considering non-rigid non-elastic axial effects

Nonlinear dynamic modeling of planar moving Timoshenko beam considering non-rigid non-elastic axial effects

Effect of Axial Load and Thermal Heating on Dynamic Characteristics of Axially Moving Timoshenko Beam

A thermoelastic coupling model of axially moving beam based on Timoshenko beam model is established, which comprehensively takes into account the length–diameter ratio, axial load and moving speed, and unify the axial tension and compression load into one governing equation, and study the influence of the length–diameter ratio, axial load, motion speed and thermal load of the beam model. The differential governing equations of transverse vibration of axially moving beam with considering the axial tensile and compressive loads are established based on the Timoshenko beam theory and Hamilton’s principle. The dynamic characteristics of different slender beams with axial load and pinned–pinned, clamped–clamped and clamped-free boundary conditions are investigated, respectively. The dimensionless frequencies of beam calculated numerically with the differential quadrature method (DQM) are compared with analytical solutions for some special cases. For axially moving beams working in the thermal environment, the temperature field inside the beam is simulated by the heat conduction equations, and the thermal effect on dynamic characteristics of beam is studied under different thermal heating cases.

Equilibrium solutions of axially moving Timoshenko beam with a supercritical speed

Analytical solution of the equilibrium configuration of an axially moving Timoshenko beam in supercritical regime is studied. Three kinds of classical boundary conditions are considered. An analytical solution of the equilibrium configuration in terms of the axial velocity in supercritical regime is determined. But most of all, for an axially moving Timoshenko beam with the fixed ends in supercritical regime, an anti-symmetric configuration is firstly detected. And, its analytical solution is solved. Besides, numerical example shows that the solution of the equilibrium configuration bifurcates with axially velocity. And the critical velocity derived from the Timoshenko beam theory is smaller than that derived from Euler-Bernoulli beam theory. The amplitude of the solution of equilibrium configuration derived from Timoshenko beam is larger.

A New Three-Dimensional Moving Timoshenko Beam Element for Moving Load Problem Analysis

Abstract A new three-dimensional moving Timoshenko beam element is developed for dynamic analysis of a moving load problem with a very long beam structure. The beam has small deformations and rotations, and bending, shear, and torsional deformations of the beam are considered. Since the dynamic responses of the beam are concentrated on a small region around the moving load and most of the long beam is at rest, owing to the damping effect, the beam is truncated with a finite length. A control volume that is attached to the moving load is introduced, which encloses the truncated beam, and a reference coordinate system is established on the left end of the truncated beam. The arbitrary Lagrangian–Euler method is used to describe the relationship of the position of a particle on the beam between the reference coordinate system and the global coordinate system. The truncated beam is spatially discretized using the current beam elements. Governing equations of a moving element are derived using Lagrange’s equations. While the whole beam needs to be discretized in the finite element method or modeled in the modal superposition method (MSM), only the truncated beam is discretized in the current formulation, which greatly reduces degrees-of-freedom and increases the efficiency. Furthermore, the efficiency of the present beam element is independent of the moving load speed, and the critical or supercritical speed range of the moving load can be analyzed through the present method. After the validation of the current formulation, a dynamic analysis of three-dimensional train–track interaction with a non-ballasted track is conducted. Results are in excellent agreement with those from the commercial software simpack where the MSM is used, and the calculation time of the current formulation is one-third of that of simpack. The current beam element is accurate and more efficient than the MSM for moving load problems of long three-dimensional beams. The derivation of the current beam element is straightforward, and the beam element can be easily extended for various other moving load problems.

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries. Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli (EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.

Complex modes and traveling waves in axially moving Timoshenko beams

Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice.

Wavelet-based spectral finite element dynamic analysis for an axially moving Timoshenko beam

In this article, wavelet-based spectral finite element (WSFE) model is formulated for time domain and wave domain dynamic analysis of an axially moving Timoshenko beam subjected to axial pretension. The formulation is similar to conventional FFT-based spectral finite element (SFE) model except that Daubechies wavelet basis functions are used for temporal discretization of the governing partial differential equations into a set of ordinary differential equations. The localized nature of Daubechies wavelet basis functions helps to rule out problems of SFE model due to periodicity assumption, especially during inverse Fourier transformation and back to time domain. The high accuracy of WSFE model is then evaluated by comparing its results with those of conventional finite element and SFE results. The effects of moving beam speed and axial tensile force on vibration and wave characteristics, and static and dynamic stabilities of moving beam are investigated.

Equilibrium bifurcation of high-speed axially moving Timoshenko beams

In this paper, equilibrium bifurcations of an axially moving Timoshenko beam are studied in the supercritical region. For the first time, Timoshenko beam theory is applied to investigate nonlinear dynamics of high-speed axially moving structures. The static equilibrium equation is deduced from the governing equation of transverse vibration of the axially moving Timoshenko beam. Two kinds of boundary conditions are considered. The non-trivial equilibrium solutions are analytically determined. Moreover, the equilibrium equations are discretized by using the finite difference method. Therefore, equilibrium configurations are numerically verified by proposing an iterative scheme. This investigation shows that non-trivial equilibrium solutions of Timoshenko beams bifurcate with axially moving speed. By comparing with Euler–Bernoulli beam theory, this study finds that the critical speed, determined by the Timoshenko beam, is remarkably smaller. Nevertheless, the equilibrium deformation of the moving Timoshenko beam is obviously larger. Furthermore, the present work derives the critical speed of the axially moving Timoshenko beam. At last, the effects of the system parameters on the equilibrium bifurcation and the critical speed are presented.

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique (GITT) is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations (ODEs) in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.

Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam

This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincare maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.

Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support

In this paper, the nonlinear forced vibrations and stability of an axially moving Timoshenko beam with an intra-span spring-support are investigated numerically. Taking into account the shear deformation and rotary inertia, three coupled nonlinear partial differential equations of motion are obtained using Hamilton's principle along with stress–strain relations. These equations are discretized into a set of coupled nonlinear ordinary differential equations via the Galerkin method. The pseudo-arclength continuation technique is used to solve the governing set of equations. The frequency–response curves of the system in the subcritical regime are obtained and examined. In addition, direct time integration is employed to investigate the global dynamics of the system through constructing the bifurcation diagrams of Poincaré maps.

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 Poincare 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.

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 Poincare 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.

Nonlinear free transverse vibrations of axially moving Timoshenko beams with two free ends

In this paper, nonlinear transverse vibrations of axially moving Timoshenko beams with two free ends are investigated. The governing equations and the associated boundary conditions are derived by the extended Hamilton principle. The method of multiple scales is applied to analyze the nonlinear partial differential equation. The natural frequencies and modes are investigated by performing the complex mode approach. The effect of natural frequencies with the stiffness and the axial speeds are numerically demonstrated. The solvability conditions are established for the cases of without and with 3:1 internal resonances. The relationships between the nonlinear frequencies and the initial amplitudes at different axial speeds and the nonlinear coefficients are showed for the case of without internal resonances. The effects of the related coefficients are demonstrated for the case of 3:1 internal resonances.

Dynamic modeling of flexible beam with considering shear deformation in non-inertial reference frame

In this paper, the effect of shear deformation is considered into flexural deflection of the geometrically nonlinear deformation of a flexible beam. Then, considering the coupling effect of deformation in to the extensional and flexural deflection, the second-order coupling terms of deformation in two displacement fields are developed and the axial inertial force and transverse distributed force are considered. The finite element method is used for the system discretization and the coupling dynamic equations of flexible beam are obtained by Lagrange’s equations. In this way, the new governing differential equations of the beam in the geometrically non-linear kinematics of deformation are derived. Numerical examples of a flexible beam are studied to analyze the effect of shear deformation on the dynamic character and to investigate the coupling effect. Furthermore, from this present method, a moving Timoshenko beam can also produce the dynamic stiffening phenomenon and some new properties can be shown.

Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams

Non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one, with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements. The set of non-linear partial-differential equations of both models are derived using an energy approach. The method of multiple scales is applied directly to both models, and using the equation order one, the mode shape equations and natural frequencies are obtained. Then, for the equation order epsilon, the solvability conditions are considered for the resonance case and the stability boundaries are formulated analytically via Routh–Hurwitz criterion. Eventually, some numerical examples are provided to show the differences in the behavior of the above-mentioned non-linear models.

Parametric resonance of axially moving Timoshenko beams with time-dependent speed

In this paper, parametric resonance of axially moving beams with time-dependent speed is analyzed, based on the Timoshenko model. The Hamilton principle is employed to obtain the governing equation, which is a nonlinear partial-differential equation due to the geometric nonlinearity caused by the finite stretch of the beam. The method of multiple scales is applied to predict the steady-state response. The expression of the amplitude of the steady-state response is derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by using the Lyapunov linearized stability theory. Some numerical examples are presented to demonstrate the effects of speed pulsation and the nonlinearity in the first two principal parametric resonances.

Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations

In this paper, nonlinear vibrations under weak and strong external excitations of axially moving beams are analyzed based on the Timoshenko model. The governing nonlinear partial-differential equation of motion is derived from Newton's second law, accounting for the geometric nonlinearity caused by finite stretching of the beams. The complex mode approach is applied to obtain the transverse vibration modes and the natural frequencies of the linear equation. The method of multiple scales is employed to investigate primary resonances, nonsyntonic excitations, superharmonic resonances, and subharmonic resonances. Some numerical examples are presented to demonstrate the effects of a varying parameter, such as axial speed, external excitation amplitudes, and nonlinearity, on the response amplitudes for the first and second modes, when other parameters are fixed. The stability of the response amplitudes is investigated and the boundary of instability is located.

Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.