A Bifurcation Analysis of the Quadratically Damped Mathieu Equation and Its Applications to the Dynamics of Submarine Towed-Array Lifting Devices

This work concerns the dynamics of towed array lifting devices (LFDs), which help to maintain the position of a tow line being dragged behind a vessel. The LFD satisfies the non-dimensional quadratically damped Mathieu equation ẍ+ (δ + 2 cos t)x+ ẋ |ẋ| = 0. Numerical study of this equation shows a wide array of dynamical features. The dynamical features of the system are exploited to obtain a control strategy for maintaining the LFD in the desired state. Introduction Submarine use of passive towed arrays affords increased sonar capability [1]. The objective here is to deploy a multiline array which can be remotely configured for optimum acoustic sensing capabilitiy. That is, a number of individual lines deploy through a single port and fan out to form a three-dimensional, volumetric array of individual sensors. By maintaining a fixed ship bearing and line configuration, composite sensor signals can be analyzed to determine the location and bearing of any acoustic emission source. Deploying and maintaining the position of individual lines comprising a volumetric array requires knowledge of the instantaneous position of each line relative to a fixed point on the ship or relative to the other lines. This must be done in a complex, unsteady ocean environment which is complicated by the turbulent flows associated with the towing vessel and the line themselves. Aperture generation is currently accomplished through the use of small lifting devices, called “lateral force devices” or LFDs. The dynamics of an LFD are complicated by changes in the tow line tension due to flow-induced vibration caused by coherent turbulent structures. These structures can result from the turbulent boundary layer on the tow line upstream of the LFD and from vortex shedding off of the tow line due to crossflow. Full scale experiments in a towing tank have shown that an LFD can exhibit unstable motions under particular conditions. Previous work on this problem presented in [2] and [3] derived the equation of motion and carried out both a linear stability analysis of the quadratic Mathieu equation and a nonlinear analysis for small values of 2. The goal of the current work is to extend the numerical treatment of the problem to better understand both the bifurcations in the system and their impact on the physical system dynamics. Simplified Model We investigate the motion of a simplified model of an LFD. The system along with the acting forces is shown in Figure 1. We assume that the towline connecting the LFD to the submarine is rigid, and can therefore withstand compression. The tension, T , is assumed to have a sinusoidal forcing function