Steady-State Vibrations of a Beam on a Pasternak Foundation for Moving Loads

1980 ◽  
Vol 47 (4) ◽  
pp. 879-883 ◽  
Author(s):  
H. Saito ◽  
T. Terasawa
Keyword(s):  
Steady State ◽  
Constant Velocity ◽  
Moving Load ◽  
Equations Of Motion ◽  
Pasternak Foundation ◽  
Moving Loads ◽  
Euler Beam ◽  
Exponential Fourier Transform ◽  
Steady State Solutions ◽  
Beam Theories

The response of an infinite beam supported by a Pasternak-type foundation and subjected to a moving load is investigated. It is assumed that the load is uniformly distributed over the finite length on a beam and moves with constant velocity. The equations of motion based on the two-dimensional elastic theory are applied to a beam. Steady-state solutions are determined by applying the exponential Fourier transform with respect to the coordinate system attached to the moving load. The results are compared with those obtained from the Timoshenko and the Bernoulli-Euler beam theories, and the differences between the displacement and stress curves obtained from the three theories are clarified.

Linear Dynamics of an Elastic Beam Under Moving Loads

2000 ◽  
Vol 122 (3) ◽  
pp. 281-289 ◽  
Author(s):  
G. Visweswara Rao
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Moving Load ◽  
Mass Parameter ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Mass Effect ◽  
Moving Loads ◽  
Parameter Ratio ◽  
Load Mass

The dynamic response of an Euler-Bernoulli beam under moving loads is studied by mode superposition. The inertial effects of the moving load are included in the analysis. The time-dependent equations of motion in modal space are solved by the method of multiple scales. Instability regions of parametric resonance are identified and the moving mass effect is shown to significantly affect the transient response of the beam. Importance of modal interaction arising out of the possible internal resonance is highlighted. While the external resonance is due to the gravity effects of the moving load, the parametric and internal resonance solely depends on the load mass parameter—ratio of the moving load mass to the beam mass. Numerical results show the influence of the load inertia terms on the beam response under either a single moving load or a series of moving loads. [S0739-3717(00)01703-7]

Modal Analysis to Interpret Localization Phenomena in Two Nonlinear Tuned Mass Dampers

2021 ◽  
Author(s):  
Yuji Harata ◽  
Takashi Ikeda
Keyword(s):  
Steady State ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Mode Shapes ◽  
Response Curves ◽  
Tuned Mass Dampers ◽  
Second Mode ◽  
Steady State Solutions ◽  
Modal Coordinates

Abstract This study investigates localization phenomena in two identical nonlinear tuned mass dampers (TMDs) installed on an elastic structure, which is subjected to external, harmonic excitation. In the theoretical analysis, the mode shapes of the system are determined, and the modal equations of motion are derived using modal analysis. These equations are demonstrated as forming an autoparametric system in which external excitation directly acts on the first and third vibration modes, whereas the second vibration mode is indirectly excited due to the nonlinear coupling with the other modes. Van der Pol’s method is employed to obtain the frequency response curves for both physical and modal coordinates. The two TMDs vibrate in phase for the first and third modes, but vibrate out of phase for the second mode. Consequently, when all modes appear, the two TMDs may vibrate at different amplitudes, i.e., localization phenomena may occur because the TMD motions are expressed by the summation of motions for all modes. The numerical calculations clarify that the localization phenomena may occur in the two TMDs when all three modes appear simultaneously. Moreover, there are two steady-state solutions of the harmonic oscillations for the second mode with identical amplitudes; however, their phases differ by π. Hence, which TMD vibrates at higher amplitudes depends on which of these two steady-state solutions for the phase.

Steady-State Vibrations of Beam on Elastic Foundation for Moving Load

1954 ◽  
Vol 21 (4) ◽  
pp. 359-364
Author(s):  
J. T. Kenney
Keyword(s):  
Steady State ◽  
Elastic Foundation ◽  
Critical Velocity ◽  
Constant Velocity ◽  
Analytic Solution ◽  
Moving Load ◽  
Viscous Damping ◽  
Beam On Elastic Foundation ◽  
Bending Waves ◽  
Free Bending

Abstract This paper presents an analytic solution and resonance diagrams for a constant-velocity moving load on a beam on an elastic foundation including the effect of viscous damping. The limiting cases of no damping and critical damping are investigated. The possible velocities for the propagation of free bending waves are found and their relation to the critical velocity of the beam is studied.

Effects of Bearing and Shaft Asymmetries on the Resonant Oscillations of a Rotor-Dynamic System

1996 ◽  
Vol 118 (1) ◽  
pp. 107-114
Author(s):  
R. Ganesan
Keyword(s):  
Steady State ◽  
Dynamic System ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Mass Unbalance ◽  
Asymmetric Rotor ◽  
Amplitude And Frequency Modulation ◽  
Rotor Dynamic ◽  
Steady State Solutions

Parametric steady-state vibrations of an asymmetric rotor while passing through primary resonance and the associated stability behavior are analyzed. The undamped case is considered and the equations of motion are rewritten in a from suitable for applying the method of multiple scales. Sensitivity to the bearing as well as shaft asymmetries of the oscillations due to unbalance excitation is evaluated. Expressions for amplitude and frequency modulation functions are obtained and are specialized to yield the steady-state solutions near primary resonance. Frequency-amplitude relationships that result from combined parametric and mass unbalance excitations are derived. Stability regions in the parameter space are obtained based on the time evolution of the amplitude and phase of the steady-state motions. The effects of bearing asymmetry on the amplitude and phase of the resonant oscillations are brought out. The sensitivity of vibrational and stability characteristics to various rotor-dynamic system parameters is illustrated through a numerical investigation.

On the magnetic resonance of an alined spin-assembly

1972 ◽  
Vol 330 (1581) ◽  
pp. 265-270 ◽  
Keyword(s):  
Steady State ◽  
Magnetic Resonance ◽  
Equations Of Motion ◽  
Double Resonance ◽  
Relaxation Times ◽  
Spin Orientation ◽  
Optical Double Resonance ◽  
Set Up ◽  
Slow Passage ◽  
Steady State Solutions

The state of a spin-assembly of arbitrary J , undergoing magnetic resonance, is characterized by the multipole components p q k of the instantaneous spin-polarization which describe spin-orientation ( k = 1), spin-alinement ( k = 2), etc. Equations of motion analogous to Bloch’s equations ( k = 1) are set up for the multipole components of different k , introducing terms which describe phenomenologically ( a ) the pumping of the longitudinal multipole components ( q = 0), and ( b ) the independent but anisotropic relaxation of multipole components of different k . Steady-state solutions are obtained. In particular, the slow-passage magnetic resonance functions for the alinement components, which involve three relaxation times, are calculated explicitly. For the particular case of isotropic relaxation, these resonance functions reduce to the form originally derived for optical double resonance for a J = 1 assembly. It is emphasized that the damping constant which is involved is that for alinement.

Analytical and Numerical Validation of a Moving Modes Method for Traveling Interaction on Long Structures

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Antonio M. Recuero ◽  
José L. Escalona
Keyword(s):  
Coordinate System ◽  
Moving Load ◽  
Equations Of Motion ◽  
Winkler Foundation ◽  
Analytical Mechanics ◽  
Arbitrary Lagrangian Eulerian ◽  
Moving Loads ◽  
Computational Dynamics ◽  
Straight Beam ◽  
Selection Of

This work is devoted to the validation of a computational dynamics approach previously developed by the authors for the simulation of moving loads interacting with flexible bodies through arbitrary contact modeling. The method has been applied to the modeling and simulation of the coupled dynamics of railroad vehicles moving on deformable tracks with arbitrary undeformed geometry. The procedure presented makes use of a fully arbitrary Lagrangian–Eulerian (ALE) description of the long flexible solid (track) whose mechanical properties may be captured using a dynamics-preserving selection of modes, e.g., via a Padé approximation of a transfer function. The modes accompany the contact interaction rather than being referred to a fixed frame, as it occurs in the finite-element floating frame of reference formulation. In the method discussed in this paper, the mesh, which moves through the long flexible solid, is defined in the trajectory coordinate system (TCS) used to describe the dynamics of the set of bodies (vehicle) that interact with the long flexible structure. For this reason, the selection of modes can be focused on the preservation of the dynamics of the structure instead of having to ensure the structure's static displacement convergence due to the motion of the load. In this paper, the validation of the so-called trajectory coordinate system/moving modes (TCS/MM) method is performed in four different aspects: (a) the analytical mechanics approach is used to obtain the equations of motion in a nonmaterial volume, (b) the resulting equations of motion are compared to the classical discretization procedures of partial differential equations (PDE), (c) the suitability of the moving modes (MM) to describe deformation due to variable-velocity moving loads, and (d) the capability of the finite nonmaterial volume to describe the dynamics of an infinitely long flexible body. Validation (a) is completely general. However, the particular example of a moving load applied to a straight beam resting on a Winkler foundation, with known semi-analytical solution, is used to perform validations (b), (c), and (d).

Autoparametric interaction in a double pendulum system

2012 ◽  
Vol 226 (8) ◽  
pp. 1971-1986
Author(s):  
Matthew P Cartmell ◽  
Ivana Kovacic ◽  
Miodrag Zukovic
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Double Pendulum ◽  
Pendulum System ◽  
Right Hand ◽  
The Right ◽  
Steady State Solutions

This article investigates a four-degree-of-freedom mechanical model comprising a horizontal bar onto which two identical pendula are fitted. The bar is suspended from a pair of springs and the left-hand-side pendulum is excited by means of a harmonic torque. The article shows that autoparametric interaction is possible by means of typical external and internal resonance conditions involving the system natural frequencies and excitation frequency, yielding an interesting case when the right-hand-side pendulum does not oscillate, but stays at rest. It is demonstrated that applying the standard method of multiple scales to this system leads to slow-time and subsequently steady-state equations representative of periodic responses; however, in common with previous findings reported in the literature for systems of four or more interacting modes, global solutions are not obtainable. This article then concentrates on discussing a proposed new modification to the method of multiple scales in which the effect of detuning is accentuated within the zeroth-order perturbation equations and it is then demonstrated that the numerical solutions from this approach to multiple scales yield results that are virtually indistinguishable from those obtained from direct numerical integration of the equations of motion. It is also shown that the algebraic structure of the steady-state solutions for the modified multiple scales analysis is identical to that obtained from a harmonic balance analysis for the case when the right-hand-side pendulum is decoupled. This particular decoupling case is prominent from examination of both the original equations of motion and the steady-state solutions irrespective of the analysis undertaken. This article concludes by showing that the translation and rotation of the bar are, in this particular case, mutually coupled and opposite in sign.

scholarly journals Finite element modelling of moving loads on structures

2021 ◽  
Author(s):  
Gareth Forbes
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Moving Load ◽  
Equations Of Motion ◽  
Moving Loads ◽  
The Finite Element Method ◽  
Structure Development ◽  
Simply Supported ◽  
Ansys Apdl ◽  
Element Method

This paper provides a breif description of the moving load problem (force or mass) across a structure. Development of a matlab script to solve the analytical equations of motion is provided. The method of implementation to solve this type of structural dynamics, using the Finite Element Method is then described with a matlab script for a simply supported beam provided. Additionally, a script and method for implementing the Finite Element Method using ANSYS APDL is also given.

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