Vibration of Viscoelastic Beams Subjected to Moving Harmonic Loads (original) (raw)
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Journal of Sound and Vibration, 2006
The problem of lateral vibration of a beam subjected to an eccentric compressive force and a harmonically varying transverse concentrated moving force is analyzed within the framework of the Bernoulli-Euler beam theory. The Lagrange equations are used to examine the free vibration characteristics of an axially loaded beam and the dynamic response of a beam subjected to an eccentric compressive force and a moving harmonic concentrated force. The constraint conditions of supports are taken into account by using the Lagrange multipliers. In the study, trial function denoting the deflection of the beam is expressed in a polynomial form. By using the Lagrange equations, the problem is reduced to the solution of a system of algebraic equations. Results of numerical simulations are presented for various combinations of the value of the eccentricity, the eccentric compressive force, excitation frequency and the constant velocity of the transverse moving harmonic force. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with exact solutions based on the Bernoulli-Euler beam theory obtained for the special cases of the investigated problem.
Vibration of Beams with General Boundary Conditions Due to a Moving Harmonic Load
Journal of Sound and Vibration, 2000
Vibrational behavior of elastic homogeneous isotropic beams with general boundary conditions due to a moving harmonic force is analyzed. The analysis duly considers beams with four di!erent boundary conditions; these include pinned}pinned, "xed}"xed, pinned}"xed, and "xed}free. The response of beams are obtained in closed forms and compared for three types of the force motion: accelerated, decelerated, and uniform motion. The e!ects of the moving speed and the frequency of the moving force on the dynamic behavior of beams are studied in detail.
Dynamic response of Euler-Bernoulli beams to resonant harmonic moving loads
Structural Engineering and Mechanics, 2012
The dynamic response of Euler-Bernoulli beams to resonant harmonic moving loads is analysed. The non-dimensional form of the motion equation of a beam crossed by a moving harmonic load is solved through a perturbation technique based on a two-scale temporal expansion, which permits a straightforward interpretation of the analytical solution. The dynamic response is expressed through a harmonic function slowly modulated in time, and the maximum dynamic response is identified with the maximum of the slow-varying amplitude. In case of ideal Euler-Bernoulli beams with elastic rotational springs at the support points, starting from analytical expressions for eigenfunctions, closed form solutions for the time-history of the dynamic response and for its maximum value are provided. Two dynamic factors are discussed: the Dynamic Amplification Factor, function of the non-dimensional speed parameter and of the structural damping ratio, and the Transition Deamplification Factor, function of the sole ratio between the two non-dimensional parameters. The influence of the involved parameters on the dynamic amplification is discussed within a general framework. The proposed procedure appears effective also in assessing the maximum response of real bridges characterized by numerically-estimated mode shapes, without requiring burdensome step-by-step dynamic analyses.
Geometrically nonlinear free and forced vibrations of Euler-Bernoulli multi-span beams
MATEC Web of Conferences
The objective of this paper is to establish the formulation of the problem of nonlinear transverse forced vibrations of uniform multi-span beams, with several intermediate simple supports and general end conditions, including use of translational and rotational springs at the ends. The beam bending vibration equation is first written at each span and then the continuity requirements at each simple support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. The formulation is based on the application of Hamilton's principle and spectral analysis to the problem of nonlinear forced vibrations occurring at large displacement amplitudes, leading to the solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved here in the case of a four span beam in the free regime using an approximate method developed previously (second formulation) leading to the amplitude dependent fundamental nonlinear mode of the multi-span beam and to the corresponding backbone curves. Considering the nonlinear regime, under a uniformly distributed excitation harmonic force, the calculation of the corresponding generalised forces has led to the conclusion that the nonlinear response involves predominately the fourth mode. Consequently, an analysis has been performed in the neighbourhood of this mode, based on the single mode approach, to obtain the multi-span beam nonlinear frequency response functions for various excitation levels.
Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load
Iraqi journal of science, 2020
This paper investigates the modal analysis of vibration of Euler-Bernoulli beam subjected to concentrated load. The governing partial differential equation was analysed to determine the behaviour of the system under consideration. The series solution and numerical methods were used to solve the governing partial differential equation. The results revealed that the amplitude increases as the length of the beam increases. It was also found that the response amplitude increases as the foundation increases at fixed length of the beam.
Physical Nonlinear Analysis of a Beam Under Moving Harmonic Load
Abstract A prismatic beam made of a behaviorally nonlinear material is analyzed under a harmonic load moving with a known velocity. The vibration equation of motion is derived using Hamilton principle and Euler-Lagrange Equation. The amplitude of vibration, circular frequency, bending moment, stress and deflection of the beam can be calculated by the presented solution. Considering the response of the beam, in the sense of its resonance, it is found that there is no critical velocity when the behavior of the beam material is assumed to be physically nonlinear.
Journal of Sound and Vibration, 2007
The situation of structural elements supporting motors or engines attached to them is usual in technological applications. The operation of the machine may introduce severe dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. An exact solution for the title problem is obtained in closed-form fashion, considering general boundary conditions by means of translational and rotatory springs at both ends. The model allows to analyze the influence of the masses and their rotatory inertia on the dynamic behavior of beams with all the classic boundary conditions, and also, as particular cases, to determine the frequencies of continuous beams. r
Non-linear vibration of variable speed rotating viscoelastic beams
Nonlinear Dynamics, 2010
Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton's principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.
World journal of engineering and technology, 2024
The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with con
Analysis of Shearing Viscoelastic Beam under Moving Load
Shock and Vibration, 2012
In this paper the dynamic behavior of a viscoelastic beam subjected to a moving distributed load has been studied analytically. The viscoelastic properties of the beam have been considered as the linear standard model in shear and incompressible in bulk. The stress components have been separated to the shear and dilatation components then, the governing equation in viscoelastic form has been obtained with direct method and it has been solved with the eigenfunction expansion method. Using the obtained dimensionless coefficients from the governing equation, an analytical procedure has been presented and by parametric studies the effects of the load properties and viscoelastic materials on the amplitude and frequency of the response have been investigated. Such results can present an idea for selecting some parameters in engineering design.