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Research Papers

# Steady-State Dynamic Response of a Bernoulli–Euler Beam on a Viscoelastic Foundation Subject to a Platoon of Moving Dynamic Loads

[+] Author and Article Information
Lu Sun

Transportation College, Southeast University, Nanjing, 210096, P.R.C.; Department of Civil Engineering, Catholic University of America, Washington, DC 20064sunl@cua.edu

Feiquan Luo

Department of Civil Engineering, Catholic University of America, Washington, DC 2006428luo@cua.edu

J. Vib. Acoust 130(5), 051002 (Aug 12, 2008) (19 pages) doi:10.1115/1.2948376 History: Received September 29, 2006; Revised February 01, 2008; Published August 12, 2008

## Abstract

A Bernoulli–Euler beam resting on a viscoelastic foundation subject to a platoon of moving dynamic loads can be used as a physical model to describe railways and highways under traffic loading. Vertical displacement, vertical velocity, and vertical acceleration responses of the beam are initially obtained in the frequency domain and then represented as integrations of complex function in the space-time domain. A bifurcation is found in critical speed against resonance frequency. When the dimensionless frequency is high, there is a single critical speed that increases as the dimensionless frequency increases. When the dimensionless frequency is low, there are two critical speeds. One speed increases as the dimensionless frequency increases, while the other speed decreases as the dimensionless frequency decreases. Based on the fast Fourier transform, numerical methods are developed for efficient computation of dynamic response of the beam.

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## Figures

Figure 1

A beam on a viscoelastic foundation subjected to a platoon of line loads

Figure 2

Bifurcation of critical speed and resonance frequency

Figure 3

Dynamic responses of beam to a single moving concentrated load with frequency Ω=0 (××, v=0m∕s; --, v=10m∕s; ○○, v=20m∕s; …, v=30m∕s; ●●, v=40m∕s; ̱, v=50m∕s)

Figure 12

3D space-time evolution of dynamic responses of the beam to a platoon of five moving line loads (left column: load frequency Ω=0Hz; right column: load frequency Ω=10Hz)

Figure 14

Dynamic responses of the beam at a fixed location to a platoon of five moving line loads with frequency Ω=10Hz at various spacings (●●, l=0.25m; --, l=0.5m; -⋅-, l=2m; ̱, l=5m; …, l=10m)

Figure 6

Dynamic responses of beam to a concentrated moving load at Ω=0Hz (- -, C=0Ns∕m2; --, C=104Ns∕m2; -⋅-, C=105Ns∕m2; —, C=106Ns∕m2; ….C=107Ns∕m2)

Figure 7

Dynamic responses of beam to a concentrated moving load at Ω=400Hz (- -, C=0Ns∕m2; --, C=104Ns∕m2; -⋅-, C=105Ns∕m2; —, C=106Ns∕m2; …. C=107Ns∕m2)

Figure 8

Dynamic responses of beam to a single point moving load (●●, Ω=0Hz, v=30m∕s; --, Ω=10Hz, v=30m∕s; —, Ω=0Hz, v=0m∕s; -⋅-, Ω=10Hz, v=0m∕s)

Figure 9

A platoon of five equally spaced moving line loads

Figure 10

Dynamic response of the beam to a platoon of five moving line loads (left column: load frequency Ω=0Hz; right column: load frequency Ω=10Hz)

Figure 11

Dynamic responses of the beam to a platoon of five moving line loads (+++, Ω=0Hz, v=30m∕s; -⋅-, Ω=10Hz, v=30m∕s; --, Ω=0Hz, v=0m∕s; —, Ω=10Hz, v=0m∕s)

Figure 13

Dynamic responses of the beam at a fixed location to a platoon of five moving line loads with frequency Ω=0Hz at various spacings (●●, l=0.25m; --, l=0.5m; -⋅-, l=2m; ̱, l=5m; …, l=10m)

Figure 4

Dynamic responses of beam to a concentrated moving load with v=10m∕s (- -, Ω=0Hz; --, Ω=50Hz; -⋅-, Ω=100Hz; —, Ω=800Hz)

Figure 5

Dynamic responses of beam to a concentrated moving load with v=30m∕s (- -, Ω=0Hz; --, Ω=100Hz; -⋅-, Ω=800Hz; —, Ω=3200Hz)

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