Abstract
The use of passive control strategy is a common way to stabilize and control dangerous vibrations in a nonlinear spring pendulum which is describing the ship’s roll motion. In this paper, a tuned absorber in the transversal direction is connected to a spring pendulum with multi-parametric excitation forces to control the vibration due to some resonance cases on the system. The method of multiple scale perturbation technique (MSPT) is applied to study the periodic solution of the given system near simultaneous sub-harmonic and internal resonance case. The stability of the steady-state solution near the resonance case is investigated and studied using frequency response equations. The effects of the absorber and some system parameters on the vibrating system are studied numerically. Optimal working conditions of the system are extracted when applying passive control methods. Comparison with the available published work is reported.
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Abbreviations
- c j (j=1,2,3,4):
-
the damping coefficient of the spring pendulum modes and the absorber ( \(c_{j}=\varepsilon\hat{c}_{j}\) )
- ω1,ω2 and ω3:
-
the natural frequency of the spring pendulum modes and absorber
- α,β:
-
the nonlinear parameters ( \(\beta_{1}=\varepsilon\hat{\beta}_{1})\)
- f j :
-
the forcing amplitude of the main system ( \(f_{j}=\varepsilon^{2}\hat{f}_{j})\)
- Ω j :
-
the frequencies of the main system
- ε :
-
a small perturbation parameter
- g :
-
the gravity acceleration
- M,m:
-
the masses of the spring pendulum and absorber, respectively
- l :
-
statically stretched length of the pendulum
- l 1 :
-
statically stretched length of the absorber
- \(x,\bar{x}\) :
-
the longitudinal response of the spring pendulum ( \(x=\bar{x}/l\) )
- \(u,\bar{u}\) :
-
the longitudinal response of the absorber ( \(u=\bar{u}/l\) )
- φ :
-
the angular response of the pendulum
- k1,k2:
-
the linear stiffness of the spring pendulum and the absorber
- k i (i=3,4,5,6):
-
the spring stiffness of nonlinear parameters
- M(t):
-
a moment acts at the point O
- F(t):
-
a force acts on mass M in the x direction
References
Mwad, D.J.: Passive Vibration Control. Wiley, Chichester (1988)
Meirovitch, L.: Fundamental of Vibrations. McGraw-Hill, New York (2001)
Nayfeh, A.H., Mook, D.T., Marshell, A.R.: Nonlinear coupled of pitch and roll modes in ship motions. J. Hydronaut. 7(4), 145–152 (1973)
Tondl, A., Nabergoj, R.: Dynamic absorbers for an externally excited pendulum. J. Sound Vib. 234(4), 611–624 (2000)
Lee, W.K.: A global analysis of a forced spring–pendulum system. Ph.D. Dissertation, University of California, Berkeley (1988)
Lee, W.K., Hsu, C.S.: A global analysis of a harmonically excited spring–pendulum system with internal resonance. J. Sound Vib. 171(3), 335–359 (1994)
Lee, W.K., Park, H.D.: Chaotic dynamics of a harmonically excited spring pendulum system with internal resonance. J. Non-linear Dyn. 14, 211–229 (1997)
Lee, W.K., Park, H.D.: Second order approximation for chaotic responses of a harmonically excited spring–pendulum system. Int. J. Non-Linear Mech. 34, 749–757 (1999)
Eissa, M.: Vibration control of non-linear mechanical system via a neutralizer. Electronic Bulletin No 16, Faculty of Electronic Engineering Menouf, Egypt, July (1999)
Eissa, M., EL-Serafi, S., EL-Sheikh, M., Sayed, M.: Stability and primary simultaneous resonance of harmonically excited non-linear spring–pendulum system. Appl. Math. Comput. 145, 421–442 (2003)
Eissa, M., Sayed, M.: A comparison between active and passive vibration control of non-linear simple pendulum, Part I: Transversally tuned absorber and negative \(G\dot{\varphi}^{n}\) feedback. Math. Comput. Appl. 11(2), 137–149 (2006)
Eissa, M., Sayed, M.: A comparison between active and passive vibration control of non-linear simple pendulum, Part II: Longitudinal tuned absorber and negative \(G\ddot{\varphi}\) and G φ n feedback. Math. Comput. Appl. 11(2), 151–162 (2006)
Sayed, M.: Improving the mathematical solutions of non-linear differential equations using different control methods. Ph.D. Thesis, Department of Mathematics, Faculty of Science, Menoufia, Egypt (2006)
Eissa, M., Sayed, M.: Vibration reduction of a three-DOF non-linear spring pendulum. Commun. Nonlinear Sci. Numer. Simul. 13, 465–488 (2008)
Ayaz, Z., Vassalos, D., Turan, O.: Parametrical studies of a new numerical model for controlled ship motions in extreme astern seas. J. Marine Sci. Technol. 11, 19–38 (2006)
Lee, D., Hong, S.Y., Lee, G.J.: Theoretical and experimental study on dynamic behavior of a damaged ship in waves. Ocean Eng. 34, 21–31 (2007)
Bayly, P.V., Virgin, L.N.: An empirical study of the stability of periodic motion in the forced spring–pendulum. Proc. R. Soc. Lond. A 443, 391–408 (1993)
Kamel, M.M.: Bifurcation analysis of a nonlinear coupled pitch–roll ship. Math. Comput. Simul. 73, 300–308 (2007)
Zhou, L., Chen, F.: Stability and bifurcation analysis for a model of a nonlinear coupled pitch–roll ship. Math. Comput. Simul. 79, 149–166 (2008)
Song, Y., Sato, H., Iwata, Y., Komatsuzaki, T.: The response of a dynamic vibration absorber system with a parametrically excited pendulum. J. Sound Vib. 259(4), 747–759 (2003)
Amer, T.S., Bek, M.A.: Chaotic responses of a harmonically excited spring pendulum moving in circular path. J. Nonlinear Anal. 10, 3196–3202 (2009)
Alasty, A., Shabani, R.: Chaotic motions and fractal basin boundaries in spring–pendulum system. J. Nonlinear Anal. 7, 81–95 (2006)
Vyas, A., Bajaj, K.: Dynamics of auto-parametric vibration absorbers using multiple pendulums. J. Sound Vib. 246(1), 115–135 (2001)
Osama, A.M., Nayfeh, A.H.: Control of ship roll using passive and active anti-roll tanks. Ocean Eng. 36, 661–671 (2009)
Kamel, M., Eissa, M., EL-Sayed, A.T.: Vibration reduction of a non-linear spring pendulum under multi-parametric excitations via a longitudinal absorber. Phys. Scr. 80, 025005 (2009) (12 pp.)
Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)
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Eissa, M., Kamel, M. & El-Sayed, A.T. Vibration reduction of multi-parametric excited spring pendulum via a transversally tuned absorber. Nonlinear Dyn 61, 109–121 (2010). https://doi.org/10.1007/s11071-009-9635-4
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DOI: https://doi.org/10.1007/s11071-009-9635-4