# Trajectory Tracking of a Near-Surface Torpedo using Numerical Methods

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2 ISSN (Print) : An ISO 3297: 27 Certified Organization Vol.4, Special Issue 12, September 215 surfaces on the vehicle due to hydrodynamic effects. Thus the vehicle acquires components of translational velocity u, v and w in x, y and z axes respectively. Also it acquires angular velocities p, q and r about x, y and z axes respectively. The linear and angular velocities are obtained by differentiating the corresponding linear and angular displacements. Differentiating the corresponding linear and angular velocities we get the respective linear and angular accelerations. In the real scenario the hydrodynamic forces and moments acting on all six degrees of freedom are important, but here we are considering the surfacing motion of an underwater body which involves only horizontal forces and moments. All vertical forces and moments are balanced by buoyancy. Therefore only surge, sway and yaw motion need to be considered. The equations of motion are hence given by Eq. 1 through 3. X = X u u + X u 2u 2 (1) Y = Y v v + Y v v + Y r r + Y r r (2) N = N v v + N v v + N r r + N r r (3) Where X u, X u 2 are the hydrodynamic coefficients or hydrodynamic derivatives for surge motion, Y v, Y v, Y r, Y r are the hydrodynamic coefficients for sway motion and N v, N v, N r, N r are the hydrodynamic coefficients for yaw motion. The equation for pure sway and yaw is comparable with Fourier series expansion up to 1 st harmonic; and the Fourier constants are determined by numerically integrating the results of computer simulations. Forward velocity test is used to find damping coefficient X u 2 acting in x direction. The added mass coefficient X u acting in x direction is obtained from some empirical relations. Thus we obtain all the hydrodynamic coefficients for surfacing motion of the given underwater vehicle. III. DESCRIPTION OF MODEL AND SPECIFICATIONS Here the Japanese type 91 torpedo is chosen for analysis with slight modification in the length and tail geometry to make it simplified with Length of model (L) = 5.94 m and mass of material m = kg Fig. 1: Actual and Modified Dimensions (in mm) Fig. 2: Co-ordinate system for simulations [5] IV. SIMULATIONS RANSE based CFD simulations have been carried out in pure sway (translation about y-axis) and pure yaw (rotation about z-axis) by inducing rigid body motions on the torpedo in its sea environment. The meshing strategy adopted here is in accordance with [9]. The boundary conditions are as in [1]. Various motion parameters used to accomplish the simulations are given in Table 1. Copyright to IJIRSET 16

3 ISSN (Print) : An ISO 3297: 27 Certified Organization Vol.4, Special Issue 12, September 215 Table 1 Motion Parameters Motion Parameters Values Forward Velocity, U 2 m/s Angular Velocity of Oscillation, ω.5 rad/s Amplitude of Pure Sway Motion,y.3 m Amplitude of Pure Yaw Motion, Ψ.873 rad Time histories of sway force and yaw moment in pure sway and yaw simulations have been shown in Fig. 3 and 4. (a) Sway Force (b) Yaw Moment Fig. 3: Time histories of forces and moments in pure sway simulation (a) Sway Force (b) Yaw Moment Fig. 4: Time histories of forces and moments in pure yaw simulation Resistance versus forward velocity has also been plotted to get the derivatives dependent on forward velocity. Fig. 5: Variation of resistance with forward velocity Copyright to IJIRSET 17

4 y position,m heading,deg heading rate,deg/s x position,m y position,m ISSN (Online) : ISSN (Print) : An ISO 3297: 27 Certified Organization Vol.4, Special Issue 12, September 215 V. RESULTS AND DISCUSSIONS Results of CFD simulations have been subjected to harmonic analysis and hydrodynamic derivatives have been determined. A Program has been written in MATLAB to plot the trajectory of torpedo which takes the hydrodynamic derivatives as well as the specifications of torpedo as input. The forward velocity of torpedo is 12 m/s. Rudder angle is incremented uniformly from to 5 through 1 seconds after which a constant rudder angle of 5 is maintained and obtained the turning circle manoeuvre plot of torpedo. Using the MATLAB results we study the behaviour of torpedo Fig. 6: Plots showing variations of x position (m) & y position (m) with time(s) Fig. 7: Plots showing variations of Heading Angle (deg) & Heading Rate (deg/s) respectively with time(s) x position,m Fig. 8: Plot showing y position (m) v/s x position (m) (turning circle manoeuvre). VI. CONCLUSION The results obtained after simulations are found to be satisfactory and it gives the standard manoeuvre of a particular torpedo. Hence it is seen that using this numerical technique we can predict the path of torpedo in standard and desired manoeuvres. The method can be also be used to predict the trajectory of an enemy submarine if its dimensions are known. Now that we have assessed the maneuverability of the torpedo and also the done the prediction of its path, guidance of the torpedo to hit its target can be achieved through an autopilot. Copyright to IJIRSET 18

5 ISSN (Print) : An ISO 3297: 27 Certified Organization Vol.4, Special Issue 12, September 215 REFERENCES [1] Bertram, V., Practical ship hydrodynamics, Butterworth Publications, 2. [2] Crane, C.L., Eda, H. and Landsburg, A. Controllability, Principles of Naval Architecture, vol.3, pp , [3] Cura-Hochbaum, A., Prediction of hydrodynamic coefficients for a passenger ship model, International Conference in Marine Hydrodynamics, pp , 26. [4] Cura-Hochbaum, A., Vogt, M. and Gatchell, S., Maneuvering prediction for two tankers based on RANS simulations, SIMMAN Workshop on Verification and Validation of Ship Maneuvering Simulation Methods, pp. F 23-28, 28. [5] Fossen, T.I., Guidance and Control of Ocean Vehicles, Published by John Wiley & Sons Ltd, pp no: , [6] Nonaka, K., MiyazakI, H., Nimura, T., Ueno, M., Hino, T. and Kodama, Y., Calculation of hydrodynamic forces acting on a ship in maneuvering motion, Research Paper, Ship Research Institute, Japan, pp , 27. [7] Ohmori, T., Finite-volume simulation of flows about a ship in maneuvering motion, Journal of Marine Science and Technology, vol.3, pp.82-93, [8] Sheeja, J. and Krishnankutty P., Prediction of ship maneuvering hydrodynamic coefficients using numerical towing tank model tests, 12 th Numerical Towing Tank Symposium, Tuscany, Italy, 4-7 th October, 29. [9] Sheeja, J, Rudra UARVR, Elangovan, M and Ram Kumar J, (211) Maneuverability Assessment of a Container Ship Using CFD, International Conference on Developments in Marine CFD, Chennai. [1] Sulficker, N., RANSE based estimation of hydrodynamic forces acting on Ships in dynamic conditions, M.S. Thesis, Department of Ocean Engineering, IIT Madras, pp.5-82, 27. [11] Toxopeus, S.L., Deriving mathematical maneuvering models for bare ship hulls using viscous flow calculations, Journal of Marine Science and Technology, vol. 14, pp. 3-38, 29. Copyright to IJIRSET 19

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