Vibration of a beam due to a random stream of moving forces with random velocity

doi:10.1016/0022-460X(84)90464-4

Journal of Sound and Vibration

Volume 97, Issue 1, 8 November 1984, Pages 23-33

https://doi.org/10.1016/0022-460X(84)90464-4 Get rights and content

Abstract

The problem of dynamic response of a beam to the passage of a train of concentrated forces with random amplitudes and velocities is considered. Force arrivals at the beam are assumed to constitute the point stochastic process of events. Thus, the excitation process is an idealization of vehicular traffic loads on a bridge. An analytical technique is developed to determine the response of the beam. Explicit expressions for the expected value and the variance of the beam deflection are provided. As an example, the response of a beam to a stationary stream of forces is determined for some practical situations, and discussed.

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Most of the existing work, follows a similar approach of computing the statistics of the structural responses due to moving loads using modal decomposition, where the loading is represented as functions of the structure’s mode shapes [9–11]. The modal approach has been expanded to include different arrival processes and loading pulses [12], as well as to the case of random velocities [13,14] and random load durations using an Erlang renewal process rather than Poisson process [15,16]. Others have also used modal decomposition to compute the power spectral densities (PSD) of the response [14,17].
While the problem of random loads moving across a beam has been studied extensively, the existing methods are limited in terms of their applicability. The most common approach uses modal superposition, which requires that the load be represented in terms of the structure’s mode shapes, making the representation of the load intrinsically a function of the structure to which it is applied. To address this problem, this paper approximates the discrete stochastic moving load as a white noise passed through filters constructed with Padé approximants. The resulting model of the load is independent of the structure and enables efficient random vibration methods to be applied for solution of the problem. By representing the structure and the loading together in an augmented state space system, the variance of responses can be found directly and accurately, enabling the traffic-induced responses of a broader class of bridge structures to be analyzed. The effectiveness and usefulness of the proposed approach is demonstrated through two examples of bridge structures.
Dynamic response of a beam to the train of moving forces driven by an Erlang renewal process
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A simplified model of moving loads on beams in the form of a stream of point-wise random forces was dealt with already in the sixties of the last century [12–14]. Vibration and reliability problem of beams and bridges subjected to a stream of random forces was analyzed and discussed in many papers [15–21], also for suspension bridges [22,23]. It should be noted that renewal processes are more adequate models of traffic loads than the Poisson process.
Dynamic response of a beam to a random train of moving forces moving with the same velocity is considered. Unlike a widely used Poisson process model, a more adequate Erlang renewal process is used as a process driving the train of forces. Normal-mode approach is used to convert the problem into that of a renewal driven train of general pulses. Consequently the modal responses are the filtered renewal processes and are expressed as integrals with respect to the response to a single pulse (passage of a force) and to the increments of the counting renewal process. The expressions for the mean values and cross-correlations of modal responses are obtained as single and double integrals, respectively. The results are presented in terms of the renewal density of the underlying Erlang counting process. Mean value and variance of the mid-span deflection of the beam are determined by numerical evaluation of the pertinent integrals. Numerical analysis is carried out for the Erlang processes with integer parameter $k = 2$ , $k = 3$ and 4 and, for comparison, also for a Poisson process. Different traffic conditions such as the velocity and the mean arrival rate of vehicles are taken into account.
Identification of horseshoes chaos in a cable-stayed bridge subjected to randomly moving loads
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Chang et al. [7] investigated the dynamic response of a fixed–fixed beam with an internal hinge on an elastic foundation, which is subjected to a moving mass oscillator with uncertain parameters such as random mass, stiffness, damping, velocity and acceleration. In the same impetus, Śniady et al. [8,9] and Rystwej et al. [10] investigated on the problem of a dynamic response of a beam and a plate to the passage of a train of random forces. In this study they assumed that the random train of forces idealizes the flow of vehicles having random weights and travelling at the stochastic velocity.
In this paper, the dynamic response of cable-stayed bridge loaded by a train of moving forces with stochastic velocity is investigated. The cable-stayed bridge is modelled by Rayleigh beam with linear elastic supports. The stochastic Melnikov method is derived and the mean-square criterion is used to determine the effects of stochastic velocity and cables number on the threshold condition for the inhibition of smale horseshoes chaos in the system. The results indicate that the intensity of the random component of the loads velocity can be contributed to the enlargement of the possible chaotic domain of the system, and/or increases the chances to have a regular behavior of the system. On the other hand, the presence of cables in cable-stayed bridges system increases it degree of safety and paradoxically can be contributed to its destabilization. Numerical simulations of the governing equations are carried out to confirm the analytical prediction. The effect of loads number on the system response is also investigated.
Structural modification formula and iterative design method using multiple tuned mass dampers for structures subjected to moving loads
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The response of linear systems to different types of Poisson excitations was studied and different methods were reported in many studies [29,30,31]. The probabilistic problem of moving loads where the excitation is described by a random process was studied by many authors [32–34]. In this numerical example the second order statistics of the response are determined by simulation and used as a comparison criterion in order to assess the effect of two absorber configurations on the beam response.
Passive vibration control based on multiple tuned mass dampers is widely used in structural dynamics. They consist of a series of single tuned mass dampers with closely spaced frequencies attached to a primary structure and have the advantage of a wider effective frequency band. In this paper an iterative design method based on updating the receptances of a primary structure by controlled modifications introduced at every step by an individual tuned mass damper is proposed. There are some similarities between this method and the system control theory which allow the modifications to be represented as feedback elements. The implementation of the method is based on an approximate modification formula which is proposed and applied to continuous structures modelled as Euler–Bernoulli beams. The method is exemplified and used to improve the vibration response of a bridge-like structure induced by a train of masses moving independently at random speeds in the same direction.
Dynamic response of micro-periodic composite rods with uncertain parameters under moving random load
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In this paper, axial vibrations of a finite micro-periodic composite rod with uncertain parameters under a moving random load are investigated. The solution of the problem was found by using the random dynamic influence function and applying the perturbation method. The average tolerance approach was also used to pass from differential equations with periodic coefficients to differential equations with constant coefficients. Two types of the moving random load are considered: a normal stationary or non-stationary process, the continuous load model (CLM), and a random train of moving forces, the discrete load model (DLM). Finally, a numerical example is provided to demonstrate the algorithm in the context of a computer implementation. With slight modification, the algorithm for longitudinal vibrations of a rod could also be applied to the dynamics of periodic beams, plates and shells with random parameters subjected to stochastic moving loads.
Spectral density of the bridge beam response with uncertain parameters under a random train of moving forces
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Citation Excerpt :
Tung [2–4] was probably the first author to publish papers on the stochastic vibrations and reliability of a bridge beam subjected to a random train of moving point forces. In the papers by Śniady and co-authors [5–9] the analysis of the beam's vibrations, the estimation of the beam's reliability and fatigue modelled as the first crossing problem have been presented. The vibrations of a beam with various boundary conditions due to a train of random forces moving along the beam with a constant speed and in the same direction have been analysed by Zibdeh and Rackwitz [10, 11].
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Vibration of a beam due to a random stream of moving forces with random velocity

Abstract

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

Vibration of Solids and Structures under Moving Loads

On the dynamic response of a beam to randomly moving load

Transactions of the American Society of Mechanical Engineers Journal of Applied Mechanics

An Introduction to Random Vibration

Statistical Methods in Structural Mechanics

Random response of highway bridges to vehicle loads