Abstract
The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.
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Abbreviations
- c j (j=1,2,3,4):
-
The damping coefficient of the spring pendulum degree-of-freedom and the absorber (\(c_{j}=\varepsilon\hat{c}_{j})\)
- ω 1,ω 2 and ω 3 :
-
The natural frequency of the spring pendulum and absorber
- α,β :
-
The non-linear parameters (\(\beta =\varepsilon\hat{\beta})\)
- 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 :
-
Gravity acceleration
- M,m :
-
Masses of spring pendulum and absorber respectively
- l :
-
Statically stretched length of the pendulum
- l 1 :
-
Statically stretched length of the absorber
- \(x,\bar{x}\) :
-
Longitudinal response of the spring–pendulum (\(x=\bar{x}/l)\)
- \(u,\bar{u}\) :
-
Longitudinal response of the absorber (\(u=\bar{u}/l)\)
- φ :
-
Angular response of the pendulum
- k 1,k 2 :
-
Linear stiffness of the spring–pendulum and the absorber
- k i (i=3,4,5,6):
-
Spring stiffness of non-linear parameters
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Eissa, M., Kamel, M. & El-Sayed, A.T. Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber. Meccanica 46, 325–340 (2011). https://doi.org/10.1007/s11012-010-9311-2
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DOI: https://doi.org/10.1007/s11012-010-9311-2