Skip to main content
Log in

An investigation into cables’ in-plane dynamics under multiple support motions using a boundary modulation approach

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Guo, T., Kang, H., Wang, L., Zhao, Y.: Cable’s non-planar coupled vibrations under asynchronous out-of-plane support motions: travelling wave effect. Arch. Appl. Mech. 86, 1647–1663 (2016)

  2. Rega, G.: Nonlinear vibrations of suspended cables–Part I: Modeling and analysis. Appl. Mech. Rev. 57, 443–478 (2004)

    Article  Google Scholar 

  3. Ibrahim, R.A.: Nonlinear vibrations of suspended cables–Part III: Random excitation and interaction with fluid flow. Appl. Mech. Rev. 57, 515–549 (2004)

    Article  Google Scholar 

  4. Irvine, H.M.: Cable Structures. Dover Publications, New York (1992)

    Google Scholar 

  5. Triantafyllou, M.: Dynamics of cables, towing cables and mooring systems. Shock Vib. Dig. 23, 3–8 (1991)

    Article  Google Scholar 

  6. Jin, D., Wen, H., Hu, H.: Modeling, dynamics and control of cable systems. Adv. Mech. 34, 304–313 (2004)

    Google Scholar 

  7. Hagedorn, P., Schäfer, B.: On non-linear free vibrations of an elastic cable. Int. J. Non Linear Mech. 15, 333–340 (1980)

    Article  MATH  Google Scholar 

  8. Luongo, A., Rega, G., Vestroni, F.: Planar non-linear free vibrations of an elastic cable. Int. J. Non Linear Mech. 19, 39–52 (1984)

    Article  MATH  Google Scholar 

  9. Benedettini, F., Rega, G.: Non-linear dynamics of an elastic cable under planar excitation. Int. J. Non Linear Mech. 22, 497–509 (1987)

    Article  MATH  Google Scholar 

  10. 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)

    Article  MATH  Google Scholar 

  11. Benedettini, F., Rega, G., Alaggio, R.: Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions. J. Sound Vib. 182, 775–798 (1995)

    Article  Google Scholar 

  12. Cai, Y., Chen, S.: dynamics of elastic cable under parametric and external resonances. J. Eng. Mech. ASCE 120, 1786–1802 (1994)

    Article  Google Scholar 

  13. Lilien, J.-L., Da Costa, A.P.: Vibration amplitudes caused by parametric excitation of cable stayed structures. J. Sound Vib. 174, 69–90 (1994)

    Article  MATH  Google Scholar 

  14. Costa, A.P.D., Martins, J., Branco, F., Lilien, J.-L.: Oscillations of bridge stay cables induced by periodic motions of deck and/or towers. J. Eng. Mech. ASCE 122, 613–622 (1996)

    Article  Google Scholar 

  15. El-Attar, M., Ghobarah, A., Aziz, T.: Non-linear cable response to multiple support periodic excitation. Eng. Struct. 22, 1301–1312 (2000)

    Article  Google Scholar 

  16. Berlioz, A., Lamarque, C.-H.: A non-linear model for the dynamics of an inclined cable. J. Sound Vib. 279, 619–639 (2005)

    Article  Google Scholar 

  17. Georgakis, C.T., Taylor, C.A.: Nonlinear dynamics of cable stays. Part 1: sinusoidal cable support excitation. J. Sound Vib. 281, 537–564 (2005)

    Article  Google Scholar 

  18. Wang, L., Zhao, Y.: Large amplitude motion mechanism and non-planar vibration character of stay cables subject to the support motions. J. Sound Vib. 327, 121–133 (2009)

    Article  Google Scholar 

  19. Gonzalez-Buelga, A., Neild, S., Wagg, D., Macdonald, J.: Modal stability of inclined cables subjected to vertical support excitation. J. Sound Vib. 318, 565–579 (2008)

    Article  Google Scholar 

  20. Luongo, A., Zulli, D.: Dynamic instability of inclined cables under combined wind flow and support motion. Nonlinear Dyn. 67, 71–87 (2012)

    Article  MathSciNet  Google Scholar 

  21. Guo, T., Kang, H., Wang, L., Zhao, Y.: Cable’s mode interactions under vertical support motions: boundary resonant modulation. Nonlinear Dyn. 84, 1259–1279 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo, T., Kang, H., Wang, L., Zhao, Y.: A boundary modulation formulation for cable’s non-planar coupled dynamics under out-of-plane support motion. Appl. Mech. Arch. (2015). doi:10.1007/s00419-00015-01058-00418

    Google Scholar 

  23. Shaw, S.W., Pierre, C.: Normal modes of vibration for non-linear continuous systems. J. Sound Vib. 169, 319–347 (1994)

    Article  MATH  Google Scholar 

  24. Nayfeh, A.H.: Nonlinear Interact. Wiley, New York (2000)

    Google Scholar 

  25. Ghobarah, A., Aziz, T., El-Attar, M.: Response of transmission lines to multiple support excitation. Eng. Struct. 18, 936–946 (1996)

    Article  Google Scholar 

  26. Rega, G., Lacarbonara, W., Nayfeh, A., Chin, C.: Multiple resonances in suspended cables: direct versus reduced-order models. Int. J. Non Linear Mech. 34, 901–924 (1999)

    Article  MATH  Google Scholar 

  27. Nayfeh, A.H., Arafat, H.N., Chin, C.-M., Lacarbonara, W.: Multimode interactions in suspended cables. J. Vib. Control 8, 337–387 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lacarbonara, W.: Direct treatment and discretizations of non-linear spatially continuous systems. J. Sound Vib. 221, 849–866 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, Hoboken (2008)

    MATH  Google Scholar 

  30. Zhao, Y., Wang, L.: On the symmetric modal interaction of the suspended cable: three-to-one internal resonance. J. Sound Vib. 294, 1073–1093 (2006)

    Article  Google Scholar 

  31. Srinil, N., Rega, G.: The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables. Int. J. Non Linear Mech. 42, 180–195 (2007)

    Article  MATH  Google Scholar 

  32. Lacarbonara, W., Rega, G., Nayfeh, A.: Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems. Int. J. Non Linear Mech. 38, 851–872 (2003)

    Article  MATH  Google Scholar 

  33. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  34. Ermentrout, B., Simulating, A.: Animating Dynamical Systems: A Guide To Xppaut for Researchers and Students. Society for Industrial and Applied Mathematics, Philadelphia (2002)

  35. Chen, L.Q., Zhang, Y.L., Zhang, G.C., Ding, H.: Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed. Int. J. Non Linear Mech. 58, 11–21 (2014)

    Article  Google Scholar 

  36. Abe, A.: Validity and accuracy of solutions for nonlinear vibration analyses of suspended cables with one-to-one internal resonance. Nonlinear Anal. Real World Appl. 11, 2594–2602 (2010)

    Article  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tieding Guo.

Appendices

Appendix 1 [29]

Using Hamiltonian principle or Newton’s law, the cable’s in-plane dynamic equations are derived as

$$\begin{aligned} m{\ddot{w}}+2c_w {\dot{w}}-\left\{ {H{w}'+EA\left( {{y}'+{w}'} \right) \varepsilon _\mathrm{d} } \right\} ^{\prime }= & {} 0 \end{aligned}$$
(65)
$$\begin{aligned} m{\ddot{v}}+2c_v {\dot{v}}-\left( {EA\varepsilon _\mathrm{d} } \right) ^{\prime }= & {} 0 \end{aligned}$$
(66)

where the nonlinear Lagrangian strain \(\varepsilon _\mathrm{d}\) is used

$$\begin{aligned} \varepsilon _\mathrm{d} \left( {x,t} \right) ={v}'+{y}'{w}'+\frac{{w}'^{2}+{v}'^{2}}{2} \end{aligned}$$
(67)

And v(xt) 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.,

$$\begin{aligned} \varepsilon _\mathrm{d} \left( {x,t} \right) =g\left( t \right) =\frac{1}{L}\int _0^L {\left( {y^{\prime }{w}'+\frac{1}{2}{w}'^{2}+\frac{1}{2}{v}'^{2}} \right) \hbox {d}x} \end{aligned}$$
(68)

Using Eq. (68), we can transform Eq. (65) into

$$\begin{aligned} m{\ddot{w}}+2c{\dot{w}}-H{w}''=\frac{EA}{L}\left( {y^{\prime \prime }+{w}''} \right) \int _0^L {\left( {y^{\prime }{w}'+\frac{1}{2}{v}'^{2}+\frac{1}{2}{w}'^{2}} \right) \hbox {d}x} =0 \end{aligned}$$
(69)

After further non-dimensionalizations using the following scales

$$\begin{aligned} w={{\tilde{w}}}/l,\;x={{\tilde{x}}}/l,\;t={\tilde{t}}\sqrt{g/{8b}},\;c={{\tilde{c}}\sqrt{{8b}/g}}/{\left( {ml} \right) },\;\alpha ={EA}/H \end{aligned}$$
(70)

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

$$\begin{aligned} \phi _{\mathrm{i}} \left( x \right) =c_{{i}} \left[ {1-\tan \left( {\frac{\omega _i }{2}} \right) \sin \omega _i x-\cos \omega _i x} \right] ,\;i=1,3,5\ldots \end{aligned}$$
(71)

where \(c_\mathrm{i}\) is the normalization constants. And the associated eigenfrequencies are determined by

$$\begin{aligned} \frac{1}{2}\omega _i -\tan \left( {\frac{1}{2}\omega _i } \right) -\frac{1}{2\lambda ^{2}}\omega _i^3 =0,\;i=1,3,5\ldots \end{aligned}$$
(72)

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

$$\begin{aligned} \phi _{\mathrm{i}} \left( x \right) =\sqrt{2}\sin \left( {i\pi x} \right) ,\;i=2,4,6\ldots \end{aligned}$$
(73)

with the associated eigenfrequencies

$$\begin{aligned} \omega _i =i\pi , i=2,4,6\ldots \end{aligned}$$
(74)

Appendix 2

$$\begin{aligned} {\varPi }_1 \left( x \right)= & {} \alpha /2\left\langle {{\phi }'_1, {\phi }'_1 \;} \right\rangle {y}''+\alpha \left\langle {{y}',{\phi }'_1 \;} \right\rangle {\phi }''_1 \end{aligned}$$
(75)
$$\begin{aligned} {\varPi }_2 \left( x \right)= & {} \alpha /2\left\langle {{\phi }'_1, {\phi }'_1 \;} \right\rangle {y}''+\alpha \left\langle {{y}',{\phi }'_1 \;} \right\rangle {\phi }''_1 \end{aligned}$$
(76)
$$\begin{aligned} \chi _{n} \left( x \right)= & {} \alpha \left[ {\frac{3}{2}\phi _n^{\prime \prime }\left\langle {\phi _n^{\prime },\phi _n^{\prime }} \right\rangle } \right. +\phi _n^{\prime \prime }\left\langle {{y}',{\varPsi }_1^{\prime }} \right\rangle +{y}''\left\langle {\phi _n^{\prime },{\varPsi }_1^{\prime }} \right\rangle +{\varPsi }_1^{\prime \prime }\left\langle {{y}',\phi _n^{\prime }} \right\rangle \nonumber \\&\left. +\,{2\phi _n^{\prime \prime }\left\langle {{y}',{\varPsi }_2^{\prime }} \right\rangle +2{y}''\left\langle {\phi _n^{\prime },{\varPsi }_2^{\prime }} \right\rangle +2{\varPsi }_2^{\prime \prime }\left\langle {{y}',\phi _n^{\prime }} \right\rangle } \right] \end{aligned}$$
(77)

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

$$\begin{aligned} \frac{\partial \left( \;\right) }{\partial x}\approx & {} \frac{\left( \;\right) _{i+1} -\left( \;\right) _{i-1} }{2\Delta x}+O\left( {\Delta x} \right) ,\quad \frac{\partial ^{2}\left( \;\right) }{\partial x^{2}}\approx \frac{\left( \;\right) _{i+1} -2\left( \;\right) _i +\left( \;\right) _{i-1} }{\Delta x^{2}}+O\left( {\Delta ^{2}x} \right) , \nonumber \\ \frac{\partial \left( \;\right) }{\partial t}\approx & {} \frac{\left( \;\right) _{j+1} -\left( \;\right) _{j-1} }{2\Delta t}+O\left( {\Delta t} \right) ,\quad \frac{\partial ^{2}\left( \;\right) }{\partial t^{2}}\approx \frac{\left( \;\right) _{j+1} -2\left( \;\right) _j +\left( \;\right) _{j-1}}{\Delta t^{2}}+O\left( {\Delta ^{2}t} \right) , \end{aligned}$$
(78)

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)

$$\begin{aligned} w_{i,j+1}= & {} \frac{1}{\left( {1+c\Delta t} \right) \Delta x^{2}}\left\{ {\left( {1+\alpha S\Delta t^{2}} \right) w_{i-1,j} -\left( {1-c\Delta t} \right) \Delta x^{2}w_{i,j-1} } \right. \nonumber \\&\left. {+\left( {2\Delta x^{2}-2\Delta t^{2}-2\alpha S\Delta t^{2}} \right) w_{i,j} +\left( {\Delta t^{2}+\alpha S\Delta t^{2}} \right) w_{i+1,j} -8\alpha fS\Delta t^{2}\Delta x^{2}} \right\} \end{aligned}$$
(79)

where the integral term S is computed using the Simpson’s integral rule

$$\begin{aligned} S= & {} \int _0^1 {\left( {{y}'{w}'+\frac{1}{2}{w}'^{2}} \right) \hbox {d}x} \approx \frac{\Delta x}{3}\left\{ {s_0 +2\sum _{i=1}^{n/2-1} {s_{2i} } +4\sum _{i=1}^{n/2} {s_{2i-1} } +s_n } \right\} \nonumber \\ s_{\mathrm{i}}\triangleq & {} 4f\left( {1-2\Delta x i} \right) \frac{w_{i+1,j} -w_{i-1,j} }{2\Delta x}+\frac{1}{2}\left( {\frac{w_{i+1,j} -w_{i-1,j} }{2\Delta x}} \right) ,i=0,1,\cdots n \end{aligned}$$
(80)

And the boundary conditions in Eq. (2) are discretized as

$$\begin{aligned} w_{0,j} =Z_0 \cos \left( {\Omega \;t_j -\theta _0 } \right) ,\quad w_{n,j} =Z_0 \cos \left( {\Omega \;t_j } \right) ,\quad t_j =j \Delta t \end{aligned}$$
(81)

where the phase lag between supports is denoted by \(\theta _0\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-017-1226-0

Keywords

Navigation