Nonlinear Dynamics

, Volume 61, Issue 1–2, pp 109–121

# Vibration reduction of multi-parametric excited spring pendulum via a transversally tuned absorber

Original Paper

## 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.

Nonlinearity Passive control Stability Pitch and roll motion

## Nomenclature

cj (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 ( $$\beta_{1}=\varepsilon\hat{\beta}_{1})$$

fj

the forcing amplitude of the main system ( $$f_{j}=\varepsilon^{2}\hat{f}_{j})$$

Ωj

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

l1

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

ki (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

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