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]

Effects of Tuned Mass Damper on Fixed-Fixed 3D Nonlinear String Resting on Nonlinear Elastic Foundation

2017 ◽  
Vol 17 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Yi-Ren Wang ◽  
Li-Ping Wu
Keyword(s):  
Elastic Foundation ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Maximum Amplitude ◽  
Tuned Mass Damper ◽  
Mode Shapes ◽  
Nonlinear Elastic Foundation ◽  
Mass Damper ◽  
Nonlinear Elastic

This paper studies the vibration of a nonlinear 3D-string fixed at both ends and supported by a nonlinear elastic foundation. Newton’s second law is adopted to derive the equations of motion for the string resting on an elastic foundation. Then, the method of multiple scales (MOMS) is employed for the analysis of the nonlinear system. It was found that 1:3 internal resonance exists in the first and fourth modes of the string when the wave speed in the transverse direction is [Formula: see text] and the elasticity coefficient of the foundation is [Formula: see text]. Fixed point plots are used to obtain the frequency responses of the various modes and to identify internal resonance through observation of the amplitudes and mode shapes. To prevent internal resonance and reduce vibration, a tuned mass damper (TMD) is applied to the string. The effects of various TMD masses, locations, damper coefficients ([Formula: see text]), and spring constants ([Formula: see text]) on overall damping were analyzed. The 3D plots of the maximum amplitude (3D POMAs) and 3D maximum amplitude contour plots (3D MACPs) are generated for the various modes to illustrate the amplitudes of the string, while identifying the optimal TMD parameters for vibration reduction. The results were verified numerically. It was concluded that better damping effects can be achieved using a TMD mass ratio [Formula: see text]–0.5 located near the middle of the string. Furthermore, for damper coefficient [Formula: see text], the use of spring constant [Formula: see text]–13 can improve the overall damping.

Multimode Interactions in Suspended Cables

2002 ◽  
Vol 8 (3) ◽  
pp. 337-387 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat ◽  
Char-Ming Chin ◽  
Walter Lacarbonara
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Internal Resonances ◽  
Out Of Plane ◽  
Space Formulation ◽  
Suspended Cables ◽  
One To One ◽  
Plane Mode

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

Analysis of Elastic Beam Subjected to Moving Dynamic Loads

2011 ◽  
Vol 250-253 ◽  
pp. 1187-1191 ◽  
Author(s):  
Ren Zuo Wang ◽  
Shi Kai Chen ◽  
Chung Yue Wang ◽  
Bin Chin Lin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Large Deformation ◽  
Moving Load ◽  
Elastic Beam ◽  
Dynamic Loads ◽  
Frame Structures ◽  
Moving Loads ◽  
Vector Form ◽  
Rigid Frame

The main object of this paper is to apply the vector form intrinsic finite element (VFIFE, or V-5) techniques in nonlinear large deformation dynamic analysis for the responses of moving loads on rigid frame structures. In this study, the simulation of moving loading is brought into the vector form intrinsic finite element method. It can effectively simulate the moving load. Comparing the results of the numerical simulations by VFIFE with the results obtained from other literatures, they are very close. It proved that VFIFE can effectively simulate the nonlinear large deformation dynamic problem.

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).

Dynamic Model of a Discrete-Pontoon Floating Bridge Subjected by Moving Loads

2010 ◽  
Vol 29-32 ◽  
pp. 732-737 ◽  
Author(s):  
Zhang Jun ◽  
Jun Liu ◽  
Xiao Lu Ni ◽  
Wei Li ◽  
Rong Mu
Keyword(s):  
Moving Load ◽  
Elastic Beam ◽  
Beam Model ◽  
Moving Loads ◽  
Concentrated Force ◽  
Vehicle Load ◽  
Comparison Results ◽  
Floating Bridge ◽  
Mass Spring ◽  
Damping Systems

A discrete-pontoon floating bridge is studied based on the beam model with assumption of the bridge deck as a elastic beam with uniform section, live load such as vehicle as moving concentrate forces, and pontoons as independent mass-spring-damping systems with singular degree of freedom. The comparison results of between vehicles and moving concentrated force show that a vehicle load can be simplified as one moving concentrated force. The present model can study not only a single moving load but also multiple moving loads.

Internal Resonance of Finite Beams on Nonlinear Foundations Traversed by a Moving Load

2008 ◽  
Author(s):  
Masoud Ansari ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Nonlinear Vibration ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Moving Load ◽  
Closed Form Solution ◽  
Form Solution ◽  
Resonance Effects ◽  
Frequency Responses ◽  
Railway System ◽  
Multiple Scales Perturbation Method

Vibration analysis of beams traversed by moving load is an old and well known topic in structural mechanics, and has been of great interest for researchers of different fields, such as mechanical, railway and civil engineering. Many researchers have conducted different investigations in this field. In the present research, the nonlinear vibration of the system is studied and consequently the response of the system to a moving load is determined as a closed form solution. Furthermore, the effects of load amplitude on the response of the system are investigated. Galerkin’s method is first utilized to truncate the governing equation of motion and then MMS (Method of Multiple Scales) perturbation method is applied to study the nonlinear vibration of the system, in the presence of the internal resonance. Effects of damping of the foundation as well as magnitude of the moving load on the frequency responses are investigated. The proposed methodology and obtained results can be used to investigate the behavior several systems among which railway system shows a good compatibility.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

2009 ◽  
Vol 19 (01) ◽  
pp. 225-243 ◽  
Author(s):  
D. X. CAO ◽  
W. ZHANG
Keyword(s):  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Coupled System ◽  
Dynamic Responses ◽  
Governing Equations ◽  
Chaotic Motions

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

Periodic and Chaotic Oscillations of an Axially Moving Viscoelastic Belt in Three-Dimensional Space

2007 ◽  
Author(s):  
Wei Zhang ◽  
Yan-Qi Liu ◽  
Li-Hua Chen ◽  
Ming-Hui Yao
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Out Of Plane ◽  
Axially Moving

Periodic and chaotic space oscillations of an axially moving viscoelastic belt with one-to-one internal resonance are investigated for the first time. The Kelvin viscoelastic model is introduced to describe the viscoelastic property of the belt material. The external damping and internal damping of the material for the axially moving viscoelastic belt are considered simultaneously. The nonlinear governing equations of motion of the axially moving viscoelastic belt for the in-plane and out-of-plane are derived by the extended Hamilton’s principle. The method of multiple scales and Galerkin’s approach are applied directly to the partial differential governing equations of motion to obtain four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance of the first order modes for the in-plane and out-of-plane oscillations. Numerical method is used to investigate periodic and chaotic space motions of the axially moving viscoelastic belt. The results of numerical simulation demonstrate that there exist periodic, period-2, period-3, period-4, period-6, quasiperiodic and chaotic motions of the axially moving viscoelastic belt.

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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