• Korean Journal of Mathematics
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• v.16 no.3
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• pp.389-402
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• 2008

We show the existence of at least two solutions for a class of systems of the critical growth nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition. We first show that the system has a positive solution under suitable conditions, and next show that the system has another solution under the same conditions by the linking arguments.

• Interaction and multiscale mechanics
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• v.2 no.3
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• pp.263-276
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• 2009

This paper is intended to investigate interaction response of a train running over a suspension bridge undergoing support settlements. The suspension bridge is modeled as a single-span suspended beam with hinged ends and the train as successive moving oscillators with identical properties. To conduct this dynamic problem with non-homogeneous boundary conditions, this study first divides the total response of the suspended beam into two parts: the static and dynamic responses. Then, the coupled equations of motion for the suspended beam carrying multiple moving oscillators are transformed into a set of nonlinearly coupled generalized equations by Galerkin's method, and solved using the Newmark method with an incremental-iterative procedure including the three phases: predictor, corrector, and equilibrium-checking. Numerical investigations demonstrate that the present iterative technique is available in dealing with the dynamic interaction problem of vehicle/bridge coupling system and that the differential movements of bridge supports will significantly affect the dynamic response of the running vehicles but insignificant influence on the bridge response.

• The Pure and Applied Mathematics
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• v.16 no.2
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• pp.199-212
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• 2009

We investigate the multiplicity of the nontrivial periodic solutions for a class of the system of the nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition. We show that the system has at least two nontrivial periodic solutions by the abstract version of the critical point theory on the manifold with boundary. We investigate the geometry of the sublevel sets of the corresponding functional of the system and the topology of the sublevel sets. Since the functional is strongly indefinite, we use the notion of the suitable version of the Palais-Smale condition.

• Honam Mathematical Journal
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• v.31 no.1
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• pp.75-85
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• 2009

We show the existence of the nontrivial periodic solution for a class of the system of the nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition by critical point theory and linking arguments. We investigate the geometry of the sublevel sets of the corresponding functional of the system, the topology of the sublevel sets and linking construction between two sublevel sets. Since the functional is strongly indefinite, we use the linking theorem for the strongly indefinite functional and the notion of the suitable version of the Palais-Smale condition.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.339-345
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• 2008

We prove the existence of the positive solution for the nonlinear system of suspension bridge equations with Dirichlet boundary condition and periodic condition $$\{u_{tt}+u_{xxxx}+av^+=1+{\epsilon}_1h_1(x,t)\text{ in }(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\v_{tt}+v_{xxxx}+bu^+=1+{\epsilon}_2h_2(x,t)\text{ in }(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ where $u^+={\max}\{u,0\},\;{\epsilon}_1,\;{\epsilon}_2$ are small numbers and $h_1(x,t)$, $h_2(x,t)$ are bounded, ${\pi}$-periodic in t and even in x and t and ${\parallel}h_1{\parallel}={\parallel}h_2{\parallel}=1$.

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.355-362
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• 2008

We prove the existence of a unique positive solution for a class of systems of the following nonlinear suspension bridge equation with Dirichlet boundary conditions and periodic conditions $$\{{u_{tt}+u_{xxxx}+\frac{1}{4}u_{ttxx}+av^+={\phi}_{00}+{\epsilon}_1h_1(x,t)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\{v_{tt}+v_{xxxx}+\frac{1}{4}u_{ttxx}+bu^+={\phi}_{00}+{\epsilon}_2h_2(x,t)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ where $u^+={\max}\{u,0\},\;{\epsilon}_1,\;{\epsilon}_2$ are small number and $h_1(x,t)$, $h_2(x,t)$ are bounded, ${\pi}$-periodic in t and even in x and t and ${\parallel} h_1{\parallel}={\parallel} h_2{\parallel}=1$. We first show that the system has a positive solution, and then prove the uniqueness by the contraction mapping principle on a Banach space

• Interaction and multiscale mechanics
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• v.3 no.4
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• pp.333-342
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• 2010

The Pseudo-Excitation Method (PEM) is applied to study the stochastic space vibration responses of train-bridge coupling system. Each vehicle is modeled as a four-wheel mass-spring-damper system with two layers of suspension system possessing 15 degrees-of- freedom. The bridge is modeled as a spatial beam element, and the track irregularity is assumed to be a uniform random process. The motion equations of the vehicle system are established based on the d'Alembertian principle, and the motion equations of the bridge system are established based on the Hamilton variational principle. Separate iteration is applied in the solution of equations. Comparisons with the Monte Carlo simulations show the effectiveness and satisfactory accuracy of the proposed method. The PSD of the 3-span simply-supported girder bridge responses, vehicle responses and wheel/rail forces are obtained. Based on the $3{\sigma}$ rule for Gaussian stochastic processes, the maximum responses of the coupling system are suggested.

• Interaction and multiscale mechanics
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• v.3 no.4
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• pp.309-320
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• 2010

With taking the geometric nonlinearity of bridge structure into account, a framework is presented for predicting the dynamic responses of a long-span suspension bridge subjected to running train and turbulent wind. The nonlinear dynamic equations of the coupled train-bridge-wind system are established, and solved with the Newmark numerical integration and direct interactive method. The corresponding linear and nonlinear processes for solving the system equation are described, and the corresponding computer codes are written. The proposed framework is then applied to a schemed long-span suspension bridge with the main span of 1120 m. The whole histories of the train passing through the bridge under turbulent wind are simulated, and the dynamic responses of the bridge are obtained. The results demonstrate that the geometric nonlinearity does not influence the variation tendency of the bridge displacement histories, but the maximum responses will be changed obviously; the lateral displacement of bridge are more sensitive to the wind than the vertical ones; compared with wind velocity, train speed affects the vertical maximum responses a little more clearly.

• Journal of the Korean Society for Railway
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• v.9 no.5 s.36
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• pp.561-568
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• 2006

Dynamic equations of motion for the interaction system of bridge and vehicle are derived to investigate the dynamic responses of bridge and vehicles induced by moving automated guide-way transit(AGT) vehicle and surface roughness of bridge. The vehicle model for ACT vehicle is idealized as 11 DOF including yawing, lateral translation and steering of wheels, and the bridges are modeled with finite element method. The AGT vehicle model was verified by experimental study. Parametric studies are carried out to investigate the effect of vehicle speed, surface roughness, stiffness and damping of the suspension system, AGT vehicles and dynamic wheel loads of the AGT vehicles. From the parametric study it can be seen that the dynamic incremental factor of the bridge and dynamic responses of vehicles have a tendency to increase with vehicle speeds, surface roughness and the stiffness of AGT vehicle suspension system. On the other hand those dynamic wheel loads have tendencies to decrease in according to increase of damping of the suspension system.

• Wind and Structures
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• v.23 no.1
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• pp.19-43
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• 2016

In the present study, a new methodology is presented to study the ride comfort and bridge responses of a long-span bridge-traffic-wind coupled vibration system considering stochastic characteristics of traffic flow and bridge surface progressive deterioration. A three-dimensional vehicle model with 24 degrees-of-freedoms (DOFs) including a three-dimensional non-linear suspension seat model and the longitudinal vibration of the vehicle is firstly presented to study the ride comfort. An improved cellular automaton (CA) model considering the influence of the next-nearest neighbor vehicles and a progressive deterioration model for bridge surface roughness are firstly introduced. Based on the equivalent dynamic vehicle model approach, the bridge-traffic-wind coupled equations are established by combining the equations of motion of both the bridge and vehicles in traffic using the displacement relationship and interaction force relationship at the patch contact. The numerical simulations show that the proposed method can simulate rationally the ride comfort and bridge responses of the bridge-traffic-wind coupled system; and the vertical, lateral, and longitudinal vibrations of the driver seat model can affect significantly the driver's comfort, as expected.