Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading (original) (raw)
Related papers
Non-linear dynamic analysis of beams with variable stiffness
Journal of Sound and Vibration, 2004
In this paper the Analog Equation method (AEM), a BEM-based method, is employed to the nonlinear dynamic analysis of an initially straight Bernoulli-Euler beam with variable stiffness undergoing large deflections. In this case the cross-sectional properties of the beam vary along its axis and consequently the coefficients of the differential equations governing the dynamic equilibrium of the beam are variable. The formulation is in terms of the displacement components. Using the concept of the analog equation, the two coupled nonlinear hyperbolic differential equations are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious time dependent load distributions. A significant advantage of this method is that the time history of the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Beams with constant and varying stiffness are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.
The purpose of this paper is to develop an analysis method to solve the free vibration response for a continuous system subjected to an initial velocity profile using an initial velocity approximation based on an equivalent impulse load. It has been shown that for a single degree of freedom system, the initial velocity can be applied as an impulsive loading with a very short duration. The proposed analysis method in this paper is done for a continuous system to show that this approximation works not only for a single degree of freedom system, but for a continuous system as well. The assumed initial velocity profile is from a case of interest to the authors. The available analytical solution for a continuous system such as a simply supported beam subjected to an initial velocity is compared with the finite element solution determined from SAP 2000 using the initial velocity approximation. The SAP2000 solution using the proposed approximation showed an excellent agreement to the analytical solution. Finally, this method can be used to find the dynamic response of complex frames subjected to an initial velocity profile, where the analytical solution for such cases is difficult to find.
There are many engineering structures that can be modeled as beams, carrying one, two or multi degree-of-freedom spring-mass systems. Examples of such practical applications include components of buildings, machine tools, vehicle suspensions and rotating machinery accessories of machine structures. Because of these wide ranging applications, the vibration behavior of beams carrying discrete structural elements, such as beams carrying a two degree-of-freedom spring-mass system have received considerable attention for many years.
Acta Mechanica, 2011
A semi-analytical analysis for the transient elastodynamic response of an arbitrarily thick simply supported beam due to the action of an arbitrary moving transverse load is presented, based on the linear theory of elasticity. The solution of the problem is derived by means of the powerful state space technique in conjunction with the Laplace transformation with respect to the time coordinate. The inversion of Laplace transform has been carried out numerically using Durbin’s approach based on Fourier series expansion. Special convergence enhancement techniques are invoked to completely eradicate spurious oscillations and obtain uniformly convergent solutions. Detailed numerical results for the transient vibratory responses of concrete beams of selected thickness parameters are obtained and compared for three types of harmonic moving concentrated loads: accelerated, decelerated and uniform. The effects of the load velocity, pulsation frequency and beam aspect ratio on the dynamic response are examined. Also, comparisons are made against solutions based on Euler–Bernoulli and Timoshenko beam models. Limiting cases are considered, and the validity of the model is established by comparison with the solutions available in the existing literature as well as with the aid of a commercial finite element package.
Large deflection analysis of beams with variable stiffness
Acta Mechanica, 2003
In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.
PRECISE TIME-STEP INTEGRATION FOR THE DYNAMIC RESPONSE OF A CONTINUOUS BEAM UNDER MOVING LOADS
Journal of Sound and Vibration, 2001
The dynamic response of a non-uniform continuous Euler}Bernoulli beam is analyzed with Hamilton's principle and the eigenpairs are obtained by the Ritz method. A high-precision integration method is used to calculate the dynamic responses of this beam. Numerical results show that the method is more accurate in the prediction of the vibration responses under the moving loads than the Newmark method.
International Journal of Advances in Scientific Research and Engineering (ijasre), 2019
The finite difference equations of the successive approximation method (SAM) which substitute the differential equations of bending beam of a variable stiffness are obtained. Difference equations of SAM, which approximate the limit conditions of the hand ends of the beam, are also obtained: simply supported hand end; rigidly fixed hand end and free hand end. On the basis of the obtained equations, a numerical algorithm was developed for calculating beams of constant and variable thickness under the action of various static loads. According to this algorithm, a program for calculating beams on a computer was performed. Variable stiffness simple supported hand ends beams, rigidly fixed hand ends beams with uniformly distributed loads along their lengths, with concentrated force, were calculated. A cantilever beam of variable thickness was also calculated under the action of the uniformly distributed load over its entire length. The examples presented here show the accuracy of the results and the simplicity of the algorithm. Checks for integral equilibrium conditions of beams were performed to validate the newly obtained results.
A numerical–analytical combined method for vibration of a beam excited by a moving flexible body
International Journal for Numerical Methods in Engineering, 2007
The vibration of a beam excited by a moving simple oscillator has been extensively studied. However, the vibration of a beam excited by an elastic body with conformal contact has attracted much less attention. This is the subject of the present paper. The established model is more complicated but has a much wider range of applications than the moving-oscillator model.
Dynamics Analysis of a Damped Non uniform Beam subjected to Loads moving with Variable Velocity
Aims/ objectives : To obtain the analytical solutions of the governing fourth order partial differential equations with variable and singular coefficients of non-uniform elastic beams under constant and harmonic variable loads travelling at varying velocity. Methodology: The governing equation of the problem is a fourth order partial differential equation. In order to solve this problem, elegant technique called Galerkin's Method is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. Results: The results show that response amplitudes of the non uniform beam decrease as the value of the axial force N increases. Furthermore, for fixed value of axial force N, the displacements of the simply supported non uniform beam resting on elastic foundations decrease as the foundation modulus K increases. The results further show that, for fixed N and K, it is observed that higher values of the load longitudinal frequency produce more stabilizing effects on the elastic beam. Conclusion: Higher values of axial force N and foundation moduli K reduce the risk factor of resonance in a vibrating system. Also higher load longitudinal frequency produce more stabilizing effects on the elastic beam thereby reduce resonance in a vibrating system.
Dynamic Analysis of Nonuniform Beams With Time-Dependent Elastic Boundary Conditions
Journal of Applied Mechanics, 1996
The dynamic response of a nonuniform beam with time-dependent elastic boundary conditions is studied by generalizing the method of Mindlin-Goodman and utilizing the exact solutions of general elastically restrained nonuniform beams given by Lee and Kuo. The time-dependent elastic boundary conditions for the beam are formulated. A general form of change of dependent variable is introduced and the shifting polynomials of the third-order degree, instead of the fifth-order degree polynomials taken by Mindlin-Goodman, are selected. The physical meaning of these shifting polynomial functions are explored. Finally, the limiting cases are discussed and several examples are given to illustrate the analysis.