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Nonlinear vibration of shallow cables with semiactive tuned mass damper

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Abstract

The nonlinear vibration of shallow cables, equipped with a semiactive control device is considered in this paper. The control device is represented by a tuned mass damper with a variable out-of-plane inclination. A suitable control algorithm is designed in order to regulate the inclination of the device and to dampen the spatial cable vibrations. Numerical simulations are conducted under free spatial oscillations through a nonlinear finite element model, solved in two different computational environments. A harmonic analysis, in the region of the primary resonance, is also performed through a control-oriented nonlinear Galerkin model, including detuning effects due to the cable slackening.

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Abbreviations

x, y, z, s:

Reference axes and curvilinear abscissa

t, τ, χ:

Time, normalized time and normalized abscissa

C0, C1:

Cable static and varied configurations

u, v, w:

Cable displacements functions

d, l:

Cable sag and cable span

E, S:

Elastic modulus and cross section

H :

Cable horizontal reaction

μ, c v , c w :

Mass and damping coefficients per unit length

p y , p z :

Distributed in-plane and out-of-plane loads

\(\bar{e}\) :

Constant Lagrangian measure of strain

ω iv , ω iw :

In-plane and out-of-plane natural circular frequencies

p iv , p iw , Ω:

Normalized modal loads and circular frequency

q v i , q w i :

In-plane and out-of-plane modal coordinates

n v , n w :

Number of in-plane and out-of-plane modes retained in the Galerkin models

a0ij , a1i , a2j :

Coefficients of the Galerkin models

a3i , b1j , b2ij , b3k :

Coefficients of the Galerkin models

ξ v i , ξ w i :

Damping coefficients in the Galerkin models

U, ΔU:

Vectors of nodal displacements in FEM models

\(\sigma,\ \tilde{\sigma}\) :

Error and error tolerance in the FEM procedure

n :

Number of unconstrained nodes in the FEM models

m, c, k:

Mass, damping coefficient and stiffness of the TMD

ξ d , ω d :

Damping ratio and circular frequency of the TMD

ι, ω:

Imaginary unit and complex circular frequency

β :

Fundamental complex eigenvalue

ω 0 :

First in-plane circular frequency of a cable without sag

α, ε:

In-plane and out-of-plane TMD inclinations

x0, r:

Position and local axis of the TMD

R :

Length of the TMD

η0, V:

Cable and TMD displacements

ξ, ζ:

Control forces

γ :

Mass ratio of the cable-TMD system

\(\tilde{\alpha}\) :

Scalar parameter of the time integration scheme

gε1, gε2:

Control gains

ψ, ν, λ2:

Nondimensional cable parameters

f v i , f w i :

Cable natural frequencies

v m , w m :

Cable mid-span displacements

v1q , w3q :

Cable observed displacements

α C , β C :

Rayleigh damping matrix parameters

\(\bar{q}_{1}^{w},\ \bar{q}_{2}^{w}\) :

Estimates of the first two out-of-plane modal amplitudes

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  1. Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Fabio Casciati & Filippo Ubertini

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  1. Fabio Casciati
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  2. Filippo Ubertini
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Correspondence to Filippo Ubertini.

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Casciati, F., Ubertini, F. Nonlinear vibration of shallow cables with semiactive tuned mass damper. Nonlinear Dyn 53, 89–106 (2008). https://doi.org/10.1007/s11071-007-9298-y

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