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RESEARCH PAPERS

# Forced Nonlinear Oscillations of a Semi-Infinite Beam Resting on a Unilateral Elastic Soil: Analytical and Numerical Solutions

[+] Author and Article Information
Giovanni Lancioni

Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italyg.lancioni@univpm.it

Stefano Lenci

Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italylenci@univpm.it

J. Comput. Nonlinear Dynam 2(2), 155-166 (Nov 30, 2006) (12 pages) doi:10.1115/1.2447406 History: Received August 18, 2006; Revised November 30, 2006

## Abstract

The dynamics of a semi-infinite Euler–Bernoulli beam on unilateral elastic springs is investigated. The mechanical model is governed by a moving-boundary hyperbolic problem, which cannot be solved in closed form. Therefore, we look for approximated solutions following two different approaches. From the one side, approximate analytical solutions are obtained by means of perturbation techniques. On the other side, numerical solutions are determined by a self-made finite element algorithm. The analytical and numerical solutions are compared with each other, and the effects of the problem nonlinearity on the beam motion are analyzed. In particular, the superharmonics oscillations and the resonances are investigated in depth.

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## Figures

Figure 1

The mechanical model

Figure 2

The function y0(V0)

Figure 3

Physical interpretation of the quantities involved in Eq. 20

Figure 5

The resonance frequencies as a function of the static TDP position

Figure 6

The functions ν2stat(y) for y0=1 (lower) and y0=2 (upper)

Figure 7

Simulation 1 (ω=0.4ωc): (a) Fourier transform Xc(ω) of the function xc(t); (b) oscillation of xc(t) in the time interval 3–3.5s

Figure 8

Simulation 1 (ω=0.4ωc): Fourier transform W(ω) of the displacement w(t) at: (a) x=3m and (c) x=10m; time history of the displacement at (b) x=3m and (d) x=10m (dotted line ≡ bilateral springs)

Figure 9

Simulation 2 (ω=0.8ωc): (a) Fourier transform Xc(ω) of the function xc(t); and (b) oscillation of xc(t) in the time interval 3–3.5s

Figure 10

Simulation 2 (ω=0.8ωc): time history of the displacement at: (a) x=4m and (b) x=20m (dotted line ≡ bilateral springs); Fourier transform W(ω) of the transverse displacement w(t) at: (a) x=4, 10, 20, 30m 10101010

Figure 11

Simulation 3 (ω=1.6ωc): (a) Fourier transform Xc(ω) of the function xc(t); (b) oscillation of xc(t) in the time interval 3–3.5s

Figure 12

Simulation 3 (ω=1.6ωc): Fourier transform W(ω) of the transverse displacement w(t) at (a) x=4m and (c) x=20m; time history of the displacement at (b) x=4m and (d) x=20m (dotted line ≡ bilateral springs)

Figure 13

The differences Δw=wl=30−wl=400 for: (a) ω=0.4ωc; (b) ω=0.8ωc; and (c) ω=1.6ωc

Figure 14

Function D evaluated for different values of y0 from the first-order asymptotic theory (dotted line) and from the numerical (solid line) model. V=0.01

Figure 15

High-order resonant frequencies

Figure 4

The functions D(0.5Ω) (dash-dot line), D(2Ω) (dash line), and D(6Ω) (solid line)

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