Abstract
Cables’ in-plane nonlinear vibrations under multiple support motions with a phase lag are investigated in this paper, which is a continuation of our previous work (Guo et al. in Arch Appl Mech 1647–1663, 2016). The main results are twofold. Firstly, asymptotically reduced models for the cable’s in-plane nonlinear vibrations, with or without internal resonance, excited by multiple support motions, are established using a boundary modulation approach. Two in-plane boundary dynamic coefficients are derived for characterizing moving supports’ effects, which depend closely on the cable’s initial sag, distinct from the out-of-plane ones Guo et al. (2016), and are also found to be equal for symmetric in-plane dynamics while opposite for antisymmetric dynamics. Secondly, cable’s in-plane nonlinear responses due to multiple support motions are calculated and phase lag’s dynamic effects are fully investigated. Two important factors associated with phase lags, i.e., the excitation-reduced and excitation-amplified factors, are both derived analytically, indicating theoretically that the phase lags would weaken the cable’s single-mode symmetric dynamics but amplify the antisymmetric dynamics. Furthermore, through constructing frequency response diagrams, the phase lag is found to change the characteristics of cables’ two-to-one modal resonant dynamics, both qualitatively and quantitatively. All these semi-analytical results obtained from the reduced models are also verified by applying the finite difference method to the cable’s full model, i.e., the continuous partial differential equation.
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Acknowledgements
This study is funded by National Science Foundation of China under Grant No. 11502076 and No.11572117, and also by Program for Supporting Young Investigators, Hunan University.
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Appendices
Appendix 1 [29]
Using Hamiltonian principle or Newton’s law, the cable’s in-plane dynamic equations are derived as
where the nonlinear Lagrangian strain \(\varepsilon _\mathrm{d}\) is used
And v(x, t) is the cable’s longitudinal displacement, \(H={mgl}^{2}/8b\) is the initial tension’s horizontal component. By neglecting the longitudinal inertia, we can solve \(\varepsilon _\mathrm{d} \left( {x,t} \right) \) from Eq. (66), i.e.,
Using Eq. (68), we can transform Eq. (65) into
After further non-dimensionalizations using the following scales
We can obtain the non-dimensional cable model denoted by Eq. (1). Note that in Eq. (70) the quantities with and without ‘\(\sim \)’ are dimensional and non-dimensional ones, respectively.
Linear modal results for suspended cables are presented below. The in-plane symmetric modes are given by
where \(c_\mathrm{i}\) is the normalization constants. And the associated eigenfrequencies are determined by
where \({\lambda }^{2}={EA}/{mgl}(8b/l)^{3}\) is the elasto-geometric parameter. The above nonlinear transcendental equations can be solved by the Newton–Raphson method.
And the in-plane antisymmetric modes are
with the associated eigenfrequencies
Appendix 2
Appendix 3
To verify the semi-analytical results produced by our reduced models, based on the \(2\mathrm{nd}\)-order center-finite-difference scheme [36], we simulate directly the cable’s dynamic responses by using a time-marching/stepping program (coded by C++). Briefly, using the following approximations
where the big O(...) above means the order of (...). Explicitly, the above finite difference schemes have errors of first order and of second order, respectively (i.e., \(O\left( {\Delta x} \right) ,O\left( {\Delta ^{2}x} \right) \) for space discretization, and \(O\left( {\Delta t} \right) ,O\left( {\Delta ^{2}t} \right) \) for time discretization).
we derive the discretized version for cable dynamics in Eq. (1)
where the integral term S is computed using the Simpson’s integral rule
And the boundary conditions in Eq. (2) are discretized as
where the phase lag between supports is denoted by \(\theta _0\).
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Guo, T., Kang, H., Wang, L. et al. An investigation into cables’ in-plane dynamics under multiple support motions using a boundary modulation approach. Arch Appl Mech 87, 989–1006 (2017). https://doi.org/10.1007/s00419-017-1226-0
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DOI: https://doi.org/10.1007/s00419-017-1226-0