A Hybrid Pseudo-Force/Laplace Transform Method for Non-Linear Transient Response of a Suspended Cable

Ni, Yiqing; Lou, Wenjuan; Ko, J. M.

doi:10.1006/jsvi.2000.3082

Cited by 20 publications

(6 citation statements)

References 34 publications

(28 reference statements)

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“…To avoid some or nearly all of the aforesaid hypotheses, direct computational treatments of the approximate [7,8] or exact [10,11] PDEs of cable motion have recently been accomplished based on a space-time finite difference (FD) procedure confronting the finite-amplitude free vibration problems of sagged and arbitrarily inclined cables with/without internal resonances. In the meantime, several FD-based implementations have been used successfully to deal with a range of problems in nonlinear forced vibrations, including cables subject to random excitation [12], highly-extensible cable mechanics [13], low-tension cables with large displacements [14] or semi-active vibration control strategies [15]. Overall, the robustness, utility and versatility of FD algorithms have been evidenced.…”

Section: Introductionmentioning

confidence: 99%

Space–time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

Srinil

Rega

2008

Journal of Sound and Vibration

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This paper presents direct numerical simulation and validation of analytical prediction of the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By counting on an approximate, kinematically non-condensed, planar modeling, a simply-supported horizontal cable subject to a primary external resonance and a 1:1 (or 1:1 vs. 2:1) internal resonance is analyzed. To obtain analytical solutions, a secondorder multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of damped forced cable motion. Accounting for weakly quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, direct numerical simulations of associated nonlinear partial-differential equations describing the high-dimensional, multi-degree-offreedom, system dynamics are carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable space-time varying nonlinear responses of amplitudes, displacements and tensions are numerically assessed, thoroughly validating the analytically predicted solutions as regards actual existence, meaningful role and predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with an insightful discussion on kinematically approximate/exact and planar/non-planar cable modeling. Keywords suspended cable, direct numerical simulation, analytical prediction, reduced-order model, internal resonance, nonlinear forced vibration 1 1 , , J J J J m m m m m m , 2 cos 2 cos J J J r r r s s s J J J J s s ss ss r r rr rr J J (5) Here, γ r = (σ f +σ)t -β r , γ s = σ f t -β s in Eq.(4), whereas γ r = σt -2β r +β s , γ s = σ f t -β s in Eq.(5), with β r (β s ) being the phase of associated a r (a s ) amplitude. In addition to the first-order superimposition of resonant ( , ) J J r s ζ ζ modal functions with their correlated phases (e.g., Figs.1b, c), the spatial displacement distributions in both Eqs.(4) and (5) further depend on second-order shape functions assembling quadratic nonlinear effects of every retained resonant/non-resonant mode via , J J ij ij

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Section: Introductionmentioning

confidence: 99%

Space–time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

Srinil

Rega

2008

Journal of Sound and Vibration

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“…e governing differential equations of the tether motion can be derived using the Hamilton principle [24]:…”

Section: Vibration Equationmentioning

confidence: 99%

“…in which δ is the symbol for variation; t 1 and t 2 are the limits of the time interval; L is the unstretched length of tether; and dt is the differential of time. Q is the kinetic energy density [24]:…”

Section: Vibration Equationmentioning

confidence: 99%

Nonlinear Vibration Analysis of Submerged Floating Tunnel Tether

Sun

2020

Advances in Civil Engineering

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To study the nonlinear vibration characteristics of submerged floating tunnel tether, the nonlinear vibration partial differential equation of tether is set up, in which the inclination, sag, initial tension force, and length of tether are taken into account. According to the linear vibration mode of tether, the partial differential equation is converted into ordinary differential equations by Galerkin method and the nonlinear vibration equation of tether for the in-plane first four modes and the out-of-plane first four modes are obtained. The results show that the change of inclination results is in the change of sag. The initial configuration of sag is symmetric. Thus, the sag only affects the damping ratio of the in-plane symmetric mode, namely, that of the first and the third mode. The sag has no effect on the second and forth order modes which are antisymmetric modes. Therefore, the sag is only existed in plane and has no effect on the out-of-plane mode. The first and the third in-plane modal damping ratios of tether are in direct proportion to their inclination, whereas in inverse proportion to their sag. The first modal damping ratio of tether (both in-plane and out-of-plane) is in direct proportion to its length, whereas in inverse proportion to its initial tension.

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“…Since the initiating work of Skutch in 1897, much research has been done to study transverse vibrations of axially moving strings [1] . Recently, due to the increasing of the complexity of the studied problems and nonlinear characteristic of the models, semi-analytic methods and numerical methods have become important research tools, such as multiscale method [2,3] , difference method [4,5] , finite element method [6] , Laplace transformation method [7] , Galerkin method [8][9][10][11][12][13] and so on. For the real-mode Galerkin method, the basis function is essentially required to meet the boundary conditions, and the modal function of linear static strings is usually taken as the basis function in order to improve the convergence of series solution.…”

Section: Introductionmentioning

confidence: 99%

Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

2007

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Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.

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A Hybrid Pseudo-Force/Laplace Transform Method for Non-Linear Transient Response of a Suspended Cable

Cited by 20 publications

References 34 publications

Space–time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

Space–time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

Nonlinear Vibration Analysis of Submerged Floating Tunnel Tether

Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

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