Nonlinear Dynamics

, Volume 53, Issue 1–2, pp 89–106 | Cite as

Nonlinear vibration of shallow cables with semiactive tuned mass damper

Original Paper


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.


Bifurcations Galerkin approach Numerical simulations Structural cables Tuned mass damper 


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


Cable horizontal reaction

μ, cv, cw

Mass and damping coefficients per unit length

py, pz

Distributed in-plane and out-of-plane loads


Constant Lagrangian measure of strain

ωiv, ωiw

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

piv, piw, Ω

Normalized modal loads and circular frequency

qiv, qiw

In-plane and out-of-plane modal coordinates

nv, nw

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

ξiv, ξiw

Damping coefficients in the Galerkin models


Vectors of nodal displacements in FEM models

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

Error and error tolerance in the FEM procedure


Number of unconstrained nodes in the FEM models

m, c, k

Mass, damping coefficient and stiffness of the TMD

ξd, ωd

Damping ratio and circular frequency of the TMD

ι, ω

Imaginary unit and complex circular frequency


Fundamental complex eigenvalue


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


Length of the TMD

η0, V

Cable and TMD displacements

ξ, ζ

Control forces


Mass ratio of the cable-TMD system


Scalar parameter of the time integration scheme

gε1, gε2

Control gains

ψ, ν, λ2

Nondimensional cable parameters

fiv, fiw

Cable natural frequencies

vm, wm

Cable mid-span displacements

v1q, w3q

Cable observed displacements

αC, βC

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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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Structural MechanicsUniversity of PaviaPaviaItaly

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