Cruising and Hovering Response of a Tail-Stabilized Submersible 



I. INTRODUCTION 



In the present paper, motion equations are utilized to investigate some of 

 the stability and control problems that may be encountered in the operation of a 

 tail-stabilized submersible which must cruise and hover in an ocean current 

 environment. Most of the examples given to illustrate the behavior of the ves- 

 sel were obtained from a variation of parameter, digital computer study of the 

 vertical plane cruising and hovering limit maneuvers of a 43-ft-long, 8-ft- 

 diameter rescue submersible. The latter work was sponsored by the Special 

 Projects Office of the U.S. Navy, and the results were presented recently in a 

 Davidson Laboratory report (1). 



II. THE HYDRODYNAMIC SYSTEM 



The basic hull form is a streamlined body of revolution with a screw pro- 

 peller for forward thrust, bow and stern thrusters for hovering pitch, heave, 

 yaw, and sway control, a mercury flow system for roll control, and either a 

 movable ring tail or fixed tail fins with movable rudders and elevators for sta- 

 bilization and control in cruising operations. In the case of the rescue submers- 

 ible (2), a mating bell is near the center and on the bottom of the vessel. This 

 is to attach to the access hatch of a disabled submarine to permit the transfer 

 of twelve survivors at a time to the rescue submersible, and then to another 

 operational submarine. Thus, the need for both precise hovering control during 

 mating operations and the cruising maneuver capabilities. 



m. THE MATHEMATICAL MODEL 



Although a mathematical model representing the dynamic behavior of the 

 submersible in six-degrees-of -freedom was developed, this is simplified for 

 treatment of the three-degree-of-freedom motions in the vertical plane, and the 

 automatic control system is replaced by a series of simple control rules which 

 describe the limit overshoot maneuvers in cruising and hovering flight. This 

 simplified model is complete enough to show the underlying factors which can 

 cause difficulties with the motions of the submersible. 



During any given maneuver, it is assumed that the mass m, the pitch mo- 

 ment of inertia lyy, and the fluid current velocity Lf are constant. It also is 

 assumed that the submersible has geometric symmetry relative to its vertical 

 xz -plane and that the main hull is a body of revolution with fore -aft symmetry. 

 The latter assumption permits a relatively simple representation of the hydro- 

 dynamic forces and moments in hovering conditions where the angle of attack 

 can vary from to 2 tt. Extensive experimental data are lacking for these 

 conditions. 



The equations of motion in the vertical plane are written using the body 

 axes (x,y,z) and an inertial frame (xpy^zj) which is fixed in the fluid. The 

 fluid axes form an inertial frame because U= (Uf,0,wj)is constant relative to 

 an inertial frame (x^.y^,, z^) fixed in the earth. 



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