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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
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
References
Rega, G.: Nonlinear vibrations of suspended cables, part I: modeling and analysis. Appl. Mech. Rev. 57, 443–478 (2004)
Rega, G.: Nonlinear vibrations of suspended cables, part II: deterministic phenomena. Appl. Mech. Rev. 57, 479–514 (2004)
Larsen, J.W., Nielsen, S.R.K.: Non-linear stochastic response of a shallow cable. Int. J. Non-Linear Mech. 41, 327–344 (2004)
Perkins, N.C.: Modal interactions in the non-linear response of elastic cables under parametric/external excitation. Int. J. Non-Linear Mech. 27, 233–250 (1992)
Nayfeh, A.H., Arafat, H.N., Chin, C.M., Lacarbonara, W.: Multimode interactions in suspended cables. J. Vib. Control 8, 337–387 (2002)
Benedettini, F., Rega, G., Alaggio, R.: Nonlinear oscillations of a four-degree of freedom model of a suspended cable under multiple internal resonance conditions. J. Sound Vib. 182, 775–798 (1995)
Luongo, A., Rega, G., Vestroni, F.: Parametric analysis of large amplitude free vibrations of a suspended cable. Int. J. Solids Struct. 20, 95–105 (1984)
Irvine, H.M., Caughey, T.K.: The linear theory of free vibrations of suspended cables. Proc. R. Soc. Lond. 341, 299–315 (1974)
Arafat, H.N., Nayfeh, A.H.: Nonlinear responses of suspended cables to primary resonance excitations. J. Sound Vib. 266, 325–354 (2003)
Desai, Y.M., Popplewell, N., Shah, A.H., Buragohain, D.N.: Geometric nonlinear analysis of cable supported structures. Comput. Struct. 29(6), 1001–1006 (1988)
Desai, Y.M., Popplewell, N., Shah, A.H.: Finite element modeling of transmission line galloping. Comput. Struct. 57, 407–420 (1995)
Gattulli, V., Martinelli, L., Perotti, F., Vestroni, F.: Nonlinear oscillations of cables under harmonic loading using analytical and finite element models. Comput. Methods Appl. Mech. Eng. 193, 69–85 (2004)
Cluni, F.: Studio del comportamento dinamico dei cavi strutturali: modelli numerici e prove sperimentali. Dissertation, University of Perugia (2004) (in Italian)
Cluni, F., Gusella, V., Ubertini, F.: A parametric investigation of wind-induced cable fatigue. Eng. Struct. (2007). doi: 10.1016/j.engstruct.2007.02.010
Canbolat, H., Dawson, D., Rahn, C.D., Nagarkatti, S.: Adaptive boundary control of out-of-plane cable vibration. ASME J. Appl. Mech. 65, 963–969 (1998)
Susumpow, T., Fujino, Y.: Active control of multimodal cable vibrations by axial support motion. Earthq. Eng. Struct. Dyn. 5, 283–292 (1995)
Gattulli, V., Pasca, M., Vestroni, F.: Nonlinear oscillations of a nonresonant cable under in-plane excitation with a longitudinal control. Nonlinear Dyn. 14(2), 139–156 (1997)
Alaggio, R., Gattulli, V., Potenza, F.: Experimental validation of longitudinal active control strategy for cable oscillations. In: Proceedings of the 9th Italian Conference on Wind Engineering (INVENTO), Pescara (2006)
Gattuli, V., Vestroni, F.: Nonlinear strategies for longitudinal control in the stabilization of an oscillating suspended cable. Dyn. Control 10(4), 359–374 (2000)
Xu, Y.L., Yu, Z.: Non-linear vibration of cable-damper system, part I: formulation. J. Sound Vib. 225, 447–463 (1999)
Xu, Y.L., Yu, Z.: Non-linear vibration of cable-damper system, part II: application and verification. J. Sound Vib. 225, 465–481 (1999)
Zhou, Q., Nielsen, S.R.K., Qu, W.L.: Semi-active control of three-dimensional vibrations of an inclined sag cable with magnetorheological dampers. J. Sound Vib. 296, 1–22 (2006)
Claren, R., Diana, G.: Vibrazioni dei conduttori. Energ. Elettr. 11, 677–688 (1966) (in Italian)
Gattulli, V., Lepidi, M., Luongo, A.: Controllo con una massa accordata dell’instabilità aeroelastica di un cavo sospeso. In: Proceedings of the 16th Italian Conference on Theoretic and Applied Mechanics (AIMETA) (2003) (in Italian)
Pacheko, B.M., Fujino, Y., Sulekh, A.: Estimation curve for modal damping in stay cables with viscous dampers. ASCE J. Struct. Eng. 119(6), 1961–1979 (1990)
Wu, W.J., Cai, C.S.: Experimental study of magnetorheological dampers and application to cable vibration control. J. Vib. Control 12(1), 67–82 (2006)
Abdel-Rohman, M., Spencer, B.F.: Control of wind-induced nonlinear oscillations in suspended cables. Nonlinear Dyn. 37, 341–355 (2004)
Markiewikz, M.: Optimum dynamic characteristics of stockbridge dampers for dead-end spans. J. Sound Vib. 188, 243–256 (1995)
Cai, C.S., Wu, W.J., Shi, X.M.: Cable vibration reduction with a hung-on TMD system, part I: theoretical study. J. Vib. Control 12(7), 801–814 (2006)
Cai, C.S., Wu, W.J., Shi, X.M.: Cable vibration reduction with a hung-on TMD system, part II: parametric study. J. Vib. Control 12(8), 881–899 (2006)
Wu, W.J., Cai, C.S.: Cable vibration control with a magnetorheological fluid based tuned mass damper. In: Proceedings of the 10th Biennial ASCE Aerospace Division International Conference on Engineering, Construction, and Operations in Challenging Environments, League City/Houston, TX, USA, 5–8 March 2006
Casciati, F., Magonette, G., Marazzi, F.: Technology of Semiactive Devices and Applications in Vibration Mitigation. Wiley, Chichester (2006)
Casciati, F., Ubertini, F.: Control of cables nonlinear vibrations under turbulent wind action. In: Deodatis G., Spanos P. (eds.) Computational stochastic mechanics, 5th International Conference on Computational Stochastic Mechanics, Rodos, June 2006, pp. 169–178. Millpress, Rotterdam (2007)
ANSYS Inc.: ANSYS and CivilFEM 9.0 User Manual. Madrid (2005)
The Mathworks Inc.: Matlab and Simulink. Natick (2002)
Hilber, H.M., Hughes, T.J.R., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5, 283–292 (1977)
Doedel, E.J., Paffenroth, R.C., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B.E., Sandstede, B., Wang, X.: AUTO2000: continuation and bifurcation software for ordinary differential equations. Available online from http://indy.cs.concordia.ca/auto/
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-007-9298-y