GENERALIZED MULTI-SYMPLECTIC METHOD FOR DYNAMIC RESPONSES OF CONTINUOUS BEAM UNDER MOVING LOAD

2013 ◽  
Vol 05 (03) ◽  
pp. 1350033 ◽  
Author(s):  
WEIPENG HU ◽  
ZICHEN DENG ◽  
HUAJIANG OUYANG
Keyword(s):  
Moving Load ◽  
Dynamic Responses ◽  
Viscous Damping ◽  
Symplectic Method ◽  
Continuous Beam ◽  
Local Conservation ◽  
Structure Preserving ◽  
Symplectic Scheme ◽  
Continuous Beams ◽  
Conservation Property

Based on the multi-symplectic idea, a generalized multi-symplectic integrator method is presented to analyze the dynamic response of multi-span continuous beams with small damping coefficient. Focusing on the local conservation properties, the generalized multi-symplectic formulations are introduced and a fifteen-point implicit structure-preserving scheme is constructed to solve the first-order partial differential equations derived from the dynamic equation governing the dynamic behavior of continuous beams under moving load. From the results of the numerical experiments, it can be concluded that, for the cases considered in this paper, the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3751 when the continuous beam is under a constant-speed moving load and the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3095 when the continuous beam is under a variable-speed moving load with fixed step lengths Δt = 0.05 and Δx = 0.025. Similar to a multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property.

Singular Function Models of a Simply Continuous Beam Bridge under a Moving Load

2012 ◽  
Vol 204-208 ◽  
pp. 2240-2243
Author(s):  
Jun Zhang ◽  
Ming Kang Gou ◽  
Chuan Liang ◽  
Xiao Lu Ni ◽  
Zi Wen Zhou
Keyword(s):  
Present Model ◽  
Dynamic Models ◽  
Moving Load ◽  
Dynamic Responses ◽  
Singular Function ◽  
Moving Mass ◽  
Continuous Beam ◽  
Concentrated Force ◽  
Continuous Beam Bridge ◽  
Beam Bridge

The system of a simply continuous beam was looked on as one span beam with several internal elastic supports of inexhaustible stiffness. There were two types of models such as the dynamic models by a moving concentrated force and by a moving mass. A three-span beam was introduced as example solved with the present model by a moving concentrated force and FEM, which verified that the present model was correct. Two cases of the example bridge by a moving concentrated force and by a moving mass were considered. The results indicate that mass of the moving load has little influence over the dynamic responses of the simply continuous beam bridge.

scholarly journals Application of Volterra Integral Equations in Dynamics of Multispan Uniform Continuous Beams Subjected to a Moving Load

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Filip Zakęś ◽  
Paweł Śniady
Keyword(s):  
Integral Equations ◽  
Dynamic Behavior ◽  
Moving Load ◽  
Numerical Procedure ◽  
Volterra Integral Equations ◽  
Continuous Beam ◽  
Concentrated Force ◽  
Numerical Examples ◽  
Continuous Beams ◽  
Distributed Load

The dynamic behavior of multispan uniform continuous beam arbitrarily supported on its edges subjected to various types of moving noninertial loads is studied. Problem is solved by replacing a multispan structure with a single-span beam loaded with a given moving load and redundant forces situated in the positions of the intermediate supports. Redundant forces are obtained by solving Volterra integral equations of the first or the second order (depending on the stiffness of the intermediate supports) which are consistent deformation equations corresponding to each redundant. Solutions for the beam arbitrarily supported on its edges (pinned or fixed) due to a moving concentrated force and moving distributed load are given. The difficulty of solving Volterra integral equations analytically is bypassed by proposing a simple numerical procedure. Numerical examples of two- and three-span beam have been included in order to show the efficiency of the presented method.

Multi-Symplectic Method for the Zakharov-Kuznetsov Equation

2015 ◽  
Vol 7 (1) ◽  
pp. 58-73 ◽  
Author(s):  
Haochen Li ◽  
Jianqiang Sun ◽  
Mengzhao Qin
Keyword(s):  
Solitary Wave ◽  
Error Analysis ◽  
Symplectic Method ◽  
Backward Error Analysis ◽  
Backward Error ◽  
Wave Evolution ◽  
Zk Equation ◽  
Symplectic Scheme ◽  
Conservation Property ◽  
Accuracy Order

AbstractA new scheme for the Zakharov-Kuznetsov (ZK) equation with the accuracy order of is proposed. The multi-symplectic conservation property of the new scheme is proved. The backward error analysis of the new multi-symplectic scheme is also implemented. The solitary wave evolution behaviors of the Zakharov-Kunetsov equation is investigated by the new multi-symplectic scheme. The accuracy of the scheme is analyzed.

scholarly journals Application of Laplace Transform to the Free Vibration of Continuous Beams

2017 ◽  
Vol 63 (1) ◽  
pp. 163-180 ◽  
Author(s):  
H.B. Wen ◽  
T. Zeng ◽  
G.Z. Hu
Keyword(s):  
Free Vibration ◽  
Laplace Transform ◽  
Reaction Force ◽  
Bending Moment ◽  
Frequency Equation ◽  
Simplified Model ◽  
Continuous Beam ◽  
Transform Method ◽  
Continuous Beams ◽  
Vibration Problems

AbstractLaplace Transform is often used in solving the free vibration problems of structural beams. In existing research, there are two types of simplified models of continuous beam placement. The first is to regard the continuous beam as a single-span beam, the middle bearing of which is replaced by the bearing reaction force; the second is to divide the continuous beam into several simply supported beams, with the bending moment of the continuous beam at the middle bearing considered as the external force. Research shows that the second simplified model is incorrect, and the frequency equation derived from the first simplified model contains multiple expressions which might not be equivalent to each other. This paper specifies the application method of Laplace Transform in solving the free vibration problems of continuous beams, having great significance in the proper use of the transform method.

scholarly journals Effect of intermediate supports location on natural frequencies of multiple cracked continuous beams

2018 ◽  
Author(s):  
Do Nam ◽  
Nguyen Tien Khiem ◽  
Le Khanh Toan ◽  
Nguyen Thi Thao ◽  
Pham Thi Ba Lien
Keyword(s):  
General Solution ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Continuous Beam ◽  
Cracked Beam ◽  
Continuous Beams ◽  
Simplified Method ◽  
Rigid Supports ◽  
Natural Frequency Analysis ◽  
Eigenfrequency Spectrum

The present paper deals with free vibration of multiple cracked continuous beams with intermediate rigid supports. A simplified method is proposed to obtain general solution of free vibration in cracked beam with intermediate supports that is then used for natural frequency analysis of the beam in dependence upon cracks and support locations. Numerical results show that the support location or ratio of span lengths in combination with cracks makes a significant effect on eigenfrequency spectrum of beam. The discovered effects of support locations on eigenfrequency spectrum of cracked continuous beam are useful for detecting not only cracks but also positions of vanishing deflection on the beam.

scholarly journals FORCED RESPONSE VIBRATION OF SIMPLY SUPPORTED BEAMS WITH AN ELASTIC PASTERNAK FOUNDATION UNDER A DISTRIBUTED MOVING LOAD

2020 ◽  
Vol 4 (2) ◽  
pp. 1-7
Author(s):  
Fatai Hammed ◽  
M. A. Usman ◽  
S. A. Onitilo ◽  
F. A. Alade ◽  
K. A. Omoteso
Keyword(s):  
Shear Modulus ◽  
Moving Load ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Forced Response ◽  
Dynamic Responses ◽  
Moving Force ◽  
Pasternak Elastic Foundation

In this study, the response of two homogeneous parallel beams with two-parameter Pasternak elastic foundation subjected to a constant uniform partially distributed moving force is considered. On the basis of Euler-Bernoulli beam theory, the fourth order partial differential equations of motion describing the behavior of the beams when subjected to a moving force were formulated. In order to solve the resulting initial-boundary value problem, finite Fourier sine integral technique and differential transform scheme were employed to obtain the analytical solution. The dynamic responses of the two beams obtained was investigated under moving force conditions using MATLAB. The effects of speed of the moving force, layer parameters such as stiffness (K_0) and shear modulus (G_0 ) have been conducted for the moving force. Various values of speed of the moving load, stiffness parameters and shear modulus were considered. The results obtained indicates that response amplitudes of both the upper and lower beams increases with increase in the speed of the moving load. Increasing the stiffness parameter is observed to cause a decrease in the response amplitudes of the beams. The response amplitudes decreases with increase in the shear modulus of the linear elastic layer.

Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

2018 ◽  
Vol 25 (3) ◽  
pp. 593-611
Author(s):  
Xiayang Zhang ◽  
Haoquan Liang ◽  
Meijuan Zhao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Impact Load ◽  
Moving Load ◽  
Numerical Results ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Load Case ◽  
Inverse Laplace Transformation ◽  
Damping Effect

This paper, taking the clamped boundary condition as an example, develops Su and Ma's fundamental solutions of the dynamic responses of a Timoshenko beam subjected to impact load. Based on that, a further extension regarding the general moving load case is also established. Kelvin–Voigt damping, whether proportionally or nonproportionally damped, is incorporated into the model, making it more comprehensive than the model of Su and Ma. Numerical inverse Laplace transformation is introduced to obtain the time-domain solution, where Durbin's formula and the corresponding convergence criteria are utilized in numerical experiments. Further, the real modal superposition method is applied at an analytical level to validate the numerical results by applying a proportionally damped condition. Total comparisons are made between the methods by sufficient case studies. The dynamic responses with and without damping effect are computed with wider slenderness to verify the correctness and effectiveness of the numerical results. Furthermore, parametric studies regarding the damping coefficients are performed to explore the nonproportional damping effect. The results show that the structural damping has significant influences on the dynamic behaviors and is especially stronger at small slender ratios. As the damping decreases the inherent frequencies and excites the low-frequency modal components more actively, a resonant phenomenon appears in high slenderness case when the beam experiences a low-speed moving load. Additionally, the computations in the moving load case indicate that the algorithm convergence is preferable when the number of grids exceeds 1000.

scholarly journals Appropriate Matching Locations of Rail Expansion Regulator and Fixed Bearing of Continuous Beam Considering the Temperature Change of Bridge

2020 ◽  
Vol 10 (17) ◽  
pp. 6046
Author(s):  
Ping Lou ◽  
Te Li ◽  
Xinde Huang ◽  
Ganggui Huang ◽  
Bin Yan
Keyword(s):  
Temperature Change ◽  
Element Model ◽  
Continuous Beam ◽  
Fixed Bearing ◽  
Effective Measure ◽  
Long Span ◽  
Continuous Beams ◽  
Additional Force ◽  
Continuously Welded Rail ◽  
Nonlinear Finite Element Model

Due to the temperature change of bridges, there is a great additional force in continuously welded rails on continuous bridges. Laying rail expansion regulators is an effective measure to reduce the additional force. The nonlinear finite element model is presented for a continuously welded rail track with a rail expansion regulator resting on the embankment and simple and continuous beams, considering the temperature change of the bridge. Then, a method is proposed to determine the locations of the rail expansion regulator and the fixed bearing of the continuous beam, corresponding to the maximum additional forces of rail reaching minimum values. Their appropriate matching locations are recommended based on the obtained influence laws of any locations of the rail expansion regulator and the fixed bearing of the continuous beam on the maximum additional forces of rail. The results can provide the theoretical basis for the design of the rail expansion regulator and the fixed bearing of long-span continuous bridges.

Temperature Effect on Dynamic Behaviors of Cis-Polyisoprene Chain

2016 ◽  
Vol 08 (01) ◽  
pp. 1650012
Author(s):  
Weipeng Hu ◽  
Zichen Deng ◽  
Hailin Zou ◽  
Tingting Yin
Keyword(s):  
Temperature Effect ◽  
Dynamic Characteristics ◽  
Symplectic Form ◽  
Volume Effect ◽  
Symplectic Method ◽  
Linear Polymers ◽  
Dynamic Behaviors ◽  
Structure Preserving ◽  
Langevin Model ◽  
Novel Structure

A novel structure-preserving method, named as stochastic generalized multi-symplectic method, is proposed to analyze the temperature effect on the dynamic characteristics hided in the motion of the cis-polyisoprene chain in this paper. Ignoring the dynamic backflow and the exhaust volume effect, the motion of the Gaussian chain in linear polymers can be described as the Langevin model, which can be written into the stochastic generalized multi-symplectic form. For this stochastic generalized multi-symplectic form, a box structure-preserving scheme is constructed to simulate the motion of the cis-polyisoprene chain. From the simulation results, the temperature effects on the dynamic behaviors around the glass transition temperature and the viscous flow temperature of cis-polyisoprene are investigated. The structure-preserving method for analyzing the temperature effect on the dynamic characteristics of the cis-polyisoprene chain presented in this paper proposes a new way to study some dynamic characteristics of complex fluid systems.

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