Suspension Bridge Model Research Articles

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

Using a topological approach, we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ODE u ″ + b u + − a u − = p ( t ) . The choice of the equation is motivated by the studies about the Dancer–Fučik spectrum and the Lazer–McKenna suspension bridge model.

Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges

Cable-supported bridges typically exhibit minimal torsional motion under traffic and wind loads. If symmetry of the bridge about the deck's centerline is suddenly lost, such as by the failure of one or more cables or hangers (suspenders), torsional motion of the deck may grow and angles of twist may become large. The initiation of the disastrous torsional oscillations of the original Tacoma Narrows Bridge involved a sudden lateral asymmetry due to loosening of a cable band at midspan. The effects of these types of events on two-degree-of-freedom and four-degree-of-freedom section models of suspension bridges are analyzed. Vertical and rotational motions of the deck, along with vertical motions of the cables, are considered. A harmonic vertical force and an aerodynamic moment proportional to angular velocity are applied to the deck. Resistance is provided by translational and rotational springs and dashpots. Flutter instability and large oscillations occur under the aerodynamic moment, which provides “negative damping.” In order to model the occurrence of limit cycles, nonlinear damping of the van der Pol type is included in one case, and nonlinear stiffness of the hangers in others. The frequencies of the limit cycles are compared to the natural frequencies of the system.

Generalized nonlinear models of suspension bridges

This paper generalizes some results established in [J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., in press]. The geometric nonlinearity connected with the torsion of a road bed is included in the generalized model. The basic variational equations are derived from the principle of minimum energy. The existence of a solution to the generalized problem is proved. The existence is based on the Brouwer fixed-point theorem.

Nonlinear models of suspension bridges

Nonlinear variational equations describing one type of suspension bridges are proposed and studied. The variational equations describe the behaviour of road bed, main cables and cable stays. The road bed is described by two functions connected with vertical and horizontal deformation of any cross section. The main cable is considered to be perfectly flexible and inextensible. The cable stays only resist tensile forces. The variational equations are derived from the principle of minimum potential energy. The existence of solution is based on the Brouwer Fixed Point Theorem. The local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied. The local results are based on the Implicit Function Theorem for Banach spaces. A certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges.

Exact controllability of the suspension bridge model proposed by Lazer and McKenna

In this paper we give a sufficient condition for the exact controllability of the following model of the suspension bridge equation proposed by Lazer and McKenna in [A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537–578]: { w t t + c w t + d w x x x x + k w + = p ( t , x ) + u ( t , x ) + f ( t , w , u ( t , x ) ) , 0 < x < 1 , w ( t , 0 ) = w ( t , 1 ) = w x x ( t , 0 ) = w x x ( t , 1 ) = 0 , t ∈ R , where t ⩾ 0 , d > 0 , c > 0 , k > 0 , the distributed control u ∈ L 2 ( 0 , t 1 ; L 2 ( 0 , 1 ) ) , p : R × [ 0 , 1 ] → R is continuous and bounded, and the non-linear term f : [ 0 , t 1 ] × R × R → R is a continuous function on t and globally Lipschitz in the other variables, i.e., there exists a constant l > 0 such that for all x 1 , x 2 , u 1 , u 2 ∈ R we have ‖ f ( t , x 2 , u 2 ) − f ( t , x 1 , u 1 ) ‖ ⩽ l { ‖ x 2 − x 1 ‖ + ‖ u 2 − u 1 ‖ } , t ∈ [ 0 , t 1 ] . To this end, we prove that the linear part of the system is exactly controllable on [ 0 , t 1 ] . Then, we prove that the non-linear system is exactly controllable on [ 0 , t 1 ] for t 1 small enough. That is to say, the controllability of the linear system is preserved under the non-linear perturbation − k w + + p ( t , x ) + f ( t , w , u ( t , x ) ) .

Using a Gradient Vector to Find Multiple Periodic Oscillations in Suspension Bridge Models

(2005). Using a Gradient Vector to Find Multiple Periodic Oscillations in Suspension Bridge Models. The College Mathematics Journal: Vol. 36, No. 1, pp. 16-26.

Using a Gradient Vector to Find Multiple Periodic Oscillations in Suspension Bridge Models

Using a Gradient Vector to Find Multiple Periodic Oscillations in Suspension Bridge Models

A Mathematical Model of Suspension Bridges

We prove the existence of weak T-periodic solutions for a nonlinear mathematical model associated with suspension bridges. Under further assumptions a regularity result is also given.

Nonlinear Models of Suspension Bridges: Discussion of the Results

In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.

Non-stationary flow forces for the numerical simulation of aeroelastic instability of bridge decks

The aeroelastic wind forces acting on a streamlined bridge deck are analytically modeled and implemented in a finite element code for the investigation of aeroelastic instability and for structural analysis in non-linear dynamics. The present paper extends a previously introduced model, fully in the time domain, taking into account the “memory effect” of the aeroelastic processes, in order to reduce the computational effort. The implementation is tested on a two-DOF system and on a suspension bridge model.

Study of Passive Deck-Flaps Flutter Control System on Full Bridge Model. II: Results

A passive aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Control flap rotations are governed by prestressed springs and additional cables spanned between the control flaps and an auxiliary transverse beam supported by the main cables of the bridge. The rotational movement of the flaps is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time-domain formulation of self-excited forces for the whole three-dimensional suspension bridge model is obtained through a rational function approximation of the generalized Theodorsen function and implemented in the FEM formulation. This paper lays the theoretical groundwork for the one that follows.

Stability of Dynamic Models of Suspension Bridges

Stability of Dynamic Models of Suspension Bridges

Experimental investigation of the Saaremaa suspension bridge model

Experimental investigation of the Saaremaa suspension bridge model

Analytical study on flutter suppression by eccentric mass method on FEM model of long-span suspension bridge

A method of suppression of flutter in long-span bridges based on the concept of eccentric mass is presented in this paper. The auxiliary mass is placed on the windward side of a bridge deck to shift the center of gravity, and thus, the aerodynamic moment acting on the deck is reduced, resulting in an increase in the flutter wind speed. The state-space model of aerodynamic forces on the bridge deck section with eccentric mass is derived through a rational function approximation of unsteady aerodynamics. The parameters of the rational function model of a section of the bridge are obtained from the flutter derivatives experimentally determined. Then, the FEM model of full suspension bridge with main span of 2500m is derived. It is shown from numerical study that addition of an eccentric mass gives a significant improvement in the flutter wind speed. Distributing the eccentric mass in the center-span is found to be the most effective.

Adaptive neural networks for model updating of structures

A model updating methodology based on an adaptive neural network (NN) model is proposed in this study. The NN model has a feedforward architecture and is first trained off-line using some training data that are obtained from finite-element analyses and contain modal parameters as inputs and structural parameters as outputs. To reduce the number of training data while maintaining the data completeness, the variation of structural parameters is arranged using an orthogonal array. This NN model is then adaptively retrained on-line during the model updating process in order to eliminate the difference between the measured and the predicted modal parameters. A modified back-propagation algorithm is developed, in which the learning rate is dynamically adjusted once every few iterations. A jump factor is introduced to overcome the numerical difficulty caused by the saturation of the sigmoid function in order to improve the convergence performance of the NN model. The current adaptive NN updating procedure is applied to a suspension bridge model and verified both numerically and experimentally. The results indicate that by adaptively training the NN model and iteratively adjusting the structural parameters, it is possible to reduce the differences between the measured and the predicted frequencies from a maximum of 17% to 7% for the first eight vertical modes.

Finite element model updating for structures with parametric constraints

This paper presents a finite element (FE) model updating procedure applied to complex structures using an eigenvalue sensitivity-based updating approach. The objective of the model updating is to reduce the difference between the calculated and the measured frequencies. The method is based on the first-order Taylor-series expansion of the eigenvalues with respect to some structural parameters selected to be adjusted. These parameters are assumed to be bounded by some prescribed regions which are determined according to the degrees of uncertainty that exist in the parameters. The changes of these parameters are found iteratively by solving a constrained optimization problem. The improvement of the current study is in the use of an objective function that is the sum of a weighted frequency error norm and a weighted perturbation norm of the parameters. Two weighting matrices are introduced to provide flexibility for individual tuning of frequency errors and parameters' perturbations. The proposed method is applied to a 1/150 scaled suspension bridge model. Using 11 measured frequencies as reference, the FE model is updated by adjusting ten selected structural parameters. The final updated FE model for the suspension bridge model is able to produce natural frequencies in close agreement with the measured ones. Copyright © 2000 John Wiley & Sons, Ltd.

Bifurcation of periodic solutions in symmetric models of suspension bridges

We consider a nonlinear model for time-periodic oscillations of a suspension bridge. Under some additional restrictive assumptions we describe our model by a standard bifurcation scheme which allows us to use global bifurcation theorems and make some new conclusions.

Coupled String-Beam Equations as a Model of Suspension Bridges

We consider nonlinearly coupled string-beam equations modelling time-periodic oscillations in suspension bridges. We prove the existence of a unique solution under suitable assumptions on certain parameters of the bridge.

A Parametric Study on the Ultimate Strength of the Full Model of Two-Hinge Three-Span Suspension Bridges Due to Vertical Loads

A Parametric Study on the Ultimate Strength of the Full Model of Two-Hinge Three-Span Suspension Bridges Due to Vertical Loads

Mathematical Analysis of Dynamic Models of Suspension Bridges

In this paper we present some mathematical analysis of dynamic (PDE) models of suspension bridges as proposed by Lazer and McKenna. Our results are illustrated by numerical simulation with physical interpretation.