C.A. Tan

Free vibration analysis of planar curved beams by wave propagation

In this paper, a systematic approach for the free vibration analysis of a planar circular curved beam system is presented. The system considered includes multiple point discontinuities such as elastic supports, attached masses, and curvature changes. Neglecting transverse shear and rotary inertia, harmonic wave solutions are found for both extensional and inextensional curved beam models. Dispersion equations are obtained and cut-off frequencies are determined. Wave reflection and transmission matrices are formulated, accounting for general support conditions. These matrices are combined, with the aid of field transfer matrices, to provide a concise and efficient method for the free vibration problem of multi-span planar circular curved beams with general boundary conditions and supports. The solutions are exact since the effects of attenuating wave components are included in the formulation. Several examples are presented and compared with other methods.

On asymptotics of the solution of the moving oscillator problem

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.

Revisiting the moving force problem

Abstract The problem of the vibration of a beam subject to a travelling force is considered. The purpose of the study is to develop simple tools for finding the maximum deflection of a beam for any given velocity of the travelling force. It is shown that, for given boundary conditions, there exists a unique response–velocity dependence function. A technique to determine this function is suggested, which is based on the assumption that the maximum beam response can be adequately approximated by means of the first beam mode. To illustrate this, the maximum response function is calculated analytically for a simply supported (SS) beam and constructed numerically for a clamped–clamped beam. The effect of the higher modes on the maximum response is investigated, and the relative error of the one-mode approximation for a SS beam is constructed. The estimates obtained substantiate the assumption about adequacy of the one-mode approximation in a wide range of velocities; in particular, the relative error in the neighborhood of the velocity that results in the largest response is less than one percent.

A new method for calculating bending moment and shear force in moving load problems

In this paper, a new series expansion for calculating the bending moment and the shear force in a proportionally damped, one-dimensional distributed parameter system due to moving loads is suggested. The number of moving forces, which may be functions of time and spatial coordinate, and their velocities are arbitrary. The derivation of the series expansion is not limited to moving forces that are a priori known, making this method also applicable to problems in which the moving forces depend on the interactions between the continuous system and the subsystems it carries, e.g., the moving oscillator problem. A main advantage of the proposed method is in the accurate and efficient evaluation of the bending moment and shear force, and in particular, the shear jumps at the locations where the moving forces are applied. Numerical results are presented to demonstrate the rapid convergence of the new series representation.

Smart Suspension Systems for Bridge-Friendly Vehicles

In this paper, the effects of using semi-active control strategy (such as MR dampers) in vehicle suspensions on the coupled vibrations of a vehicle traversing a bridge are examined in order to develop various designs of smart suspension systems for bridge-friendly vehicles. The bridge-vehicle coupled system is modeled as a simply supported beam traversed by a two-degree-of-freedom quarter-car model. The surface unevenness on the bridge deck is modeled as a deterministic profile of a sinusoidal wave. As the vehicle travels along the bridge, the system is excited as a result of the surface unevenness and this excitation is characterized by a frequency defined by the speed of travel and the wavelength of the profile. The dynamic interactions between the bridge and the vehicle due to surface deck irregularities are obtained by solving the coupled equations of motion. Numerical results of a passive control strategy show that, when the lower natural frequency of the vehicle matches with a natural frequency (usually the first frequency) of the bridge and the excitation frequency, the maximum response of the bridge is large while the response of the vehicle is relatively smaller, meaning that the bridge behaves like a vibration absorber. This is undesirable from a bridge design viewpoint. Comparative studies of passive and semi-active controls for the vehicle suspension are performed. It is demonstrated that skyhook control can significantly mitigate the response of the bridge, while ground-hook control reduces the tire force impacted onto the bridge.

A novel approach to the calculation of pothole-induced contact forces in MDOF vehicle models

Abstract A technique is developed to predict the dynamic contact forces arising after passing road surface irregularities by a vehicle modelled as an undamped multiple-degrees-of-freedom (MDOF) system. An MDOF system moving along an uneven profile is decomposed into an aggregate of independent oscillators in the modal space, such that the response of each oscillator can be calculated independently. An equation relating the contact forces in the physical space to the modal forces is established. The technique developed is applied to the calculation of the coefficients of the harmonic components of the contact forces arising after the passage of a “cosine” pothole. The application of the technique to various problems, such as evaluation of the effect of parameter modifications on the vehicle dynamics and reduction of vehicle models in bridge-related problems, as well as its extension to the damped case, are also discussed. One interesting phenomenon reported in the DIVINE project [1], regarding the replacement of a steel suspension by an air suspension, resulting in an increase of the maximum response of short-span bridges, is explained by applying the technique suggested. The discussion is amply illustrated by examples of the application of the technique to the calculation of the tire forces due to a pothole for two simple—quarter-car and half-car—vehicle models.

A Technique for Assessing Tire Forces in a Vehicle Moving Along a Harmonic Profile and Its Application to Low-Speed Testing

The paper is concerned with the assessment of dynamic tire forces that arise when a vehicle traverses a road with a harmonic profile. In this case, solutions of all equations are found analytically. The technique developed is not specific to a particular linear vehicle model, which is represented by its mass, stiffness, and damping matrices. A factored representation of the vector of the tire forces as the product of a matrix and a vector, where both the matrix and the vector are functions of only one variable, is derived. The matrix is constructed either by the mathematical model of the vehicle or by results of special tests on the vehicle at several values of speed. The vector is given by an explicit formula and requires knowledge of only the wheelbase distances. An application of the technique discussed to the so-called low-speed testing is discussed. It is shown that, in spite of the wheelbase filtering phenomenon, it is possible to replace testing vehicles at highway speeds by tests at artificially low speeds. The results of the latter tests can then be recalculated into those corresponding to the highway conditions. The discussions are illustrated by numerical examples.Copyright

Dynamic tire forces in vehicles with semi-active suspensions

Assessment of vehicle tire forces is important in problems related to the structural health monitoring of highway bridges, damage to road pavements, design of suspensions, and road safety issues. In this paper, the effects of using semi-active control strategies, such as MR dampers, in vehicle suspensions on the dynamic tire forces are examined for the development of smart suspension systems for pavement- and bridge-friendly vehicles. The vehicle dynamics is described by a general linear MDOF model with multiple contacts (i.e., a multiple-axle vehicle) with the road. It is assumed that the tires are always in contact with the road surface. In particular, we are interested in the evaluation of the tire forces due to a harmonic excitation. A technique is developed to analytically assess the magnitude of the resulting tire force in the case of a passive suspension. Although the technique discussed cannot directly be applied to the calculation of tire forces in vehicles with controlled suspensions, it can efficiently be used for design purposes, which is demonstrated by an example of a semi-active suspension based on the sky-hook control. The discussion is amply illustrated by numerical examples.

Effect of a pothole on the dynamics of an MDOF vehicle

A technique is developed to assess the dynamic contact forces arising from passage of road surface irregularities by a vehicle modeled as a general MDOF system. The equations governing vibration of a vehicle moving along an uneven profile are, first, transformed to the state-space form and, then, to a system of uncoupled first-order complex differential equations. For a local roadway irregularity described functionally, all equations are solved analytically. The solutions obtained are combined to give dependencies of the harmonic components of the contact forces on the vehicle speed and irregularity dimensions. The technique developed is applied to the calculation of the coefficients of the harmonic components of the contact forces arising after the passage of a “cosine” pothole. One interesting phenomenon reported in the DIVINE project [1], regarding the replacement of a steel suspension by an air suspension resulting in an increase in the maximum response of short-span bridges, is explained by applying the technique suggested. The discussion is amply illustrated by examples of the application of the technique to the calculation of the tire forces due to a pothole for two simple vehicle models.Copyright

A New Approach to the Modeling of MDOF Vehicles Traversing Rough Surfaces

A technique is developed to predict the dynamic contact forces arising after passing road surface irregularities by a vehicle modeled as an undamped multiple-degrees-of-freedom (MDOF) system. An MDOF system moving along an uneven profile is decomposed into an aggregate of independent oscillators in the modal space, such that the response of each oscillator can be calculated independently. The technique developed is applied to the calculation of the coefficients of the harmonic components of the contact forces arising after the passage of a “cosine” pothole. The application of the technique to other problems is also discussed. One interesting phenomenon reported in the DIVINE project [1], regarding the replacement of a steel suspension by an air suspension resulted in increase of the maximum response of short-span bridges is explained by applying the technique suggested. The discussion is amply illustrated by examples of the application of the technique to the calculation of the tire forces due to a pothole for two simple—quarter-car and half-car—vehicle models.Copyright