Abstract
The chaotification method can be used to weaken the line spectra component of the ship's radiated noise. The main obstacle that plagues the engineering realization of this innovative concept is how to realize the small-amplitude chaotic motion of the nonlinear vibration isolation system (NVIS) in a large parameter range and how to maintain the chaotic state under variable operating conditions. In this paper, the dimensionless dynamic equation of the NVIS with a flexible base is derived. A control method similar to generalized synchronization is used to realize the continuous chaos of the system. The maximum Lyapunov exponent, the conditional Lyapunov exponent, and disturbances are introduced to verify the effectiveness of the control method and its stability and robustness in a large range of parameters. Aiming at the coexistence of multiple attractors, a new global analysis method is proposed, which is used to study the distribution of basin of attraction (BA). According to the changes of BA before and after the system is controlled, the BA is partitioned. Initial conditions from different partitions are selected to study the attenuation effect of line spectra and vibration, and the favorable initial conditions for realizing the small-amplitude chaotic motion of the system are given. The method may be extended to the global analysis of other controlled systems.
Highlights
Rough global analysis and drawing the diagram of the basin of attraction (BA), the initial conditions for generating various response behaviors can be visually displayed. e most intuitive method to determine the BA is the Poincaremapping method [12, 13], which discretizes the time domain of the system for sampling and preserves the continuity of the phase space
E advantage of this method is the high accuracy in determining the distribution of the BA, but it has the limitation of large computational effort. e simple cell mapping method proposed by Hsu et al [14] discretizes the time domain and phase space to ensure a certain accuracy and at the same time improve the computational efficiency
Compared with the point mapping method and the cell mapping method, this method splits the process of determining attractor and BA, reduces the cycle operation, and greatly improves the computational efficiency. e types of coexisting attractors and their distribution in phase space are visualized by path tracking and phase trajectory reconstruction
Summary
It is necessary to consider the flexibility of the base of the marine mechanical NVIS. Schematic diagram of the controlled NVIS is shown in Figure 1; M1 and M2 represent the device and the base, respectively. M1 is supported by a vibration isolator that combines a linear damper and a nonlinear spring with cubic nonlinearity. Where M1 is the mass of the device; M2 is the mass of the base; K1, K3, and C1 are linear stiffness, cubic nonlinear stiffness, and damping coefficient of nonlinear isolator, respectively. K2 and C2 are stiffness coefficients of linear spring and damping coefficients of damper between base and fixed plane, respectively. Y2 − h2, where Y1 and Y2 are the new displacement variables of the device and the base; h1 and h2 are the compressions of nonlinear spring and linear spring. Substituting (5) into (4), the first-order form of the dimensionless dynamic equation can be obtained y_1 y2, y_2 −ξ1 y2 − y4 − k1 y1 − y3
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