The Mathematical Model for Naval Ship Types
- Last UpdatedAug 18, 2023
- 4 minute read
A naval ship has been considered as a rigid body with four degrees of freedom, in the X, Y and Z axes:
Co-ordinate System (Plan View)
Co-ordinate System (Transverse View)
The ship motions are translation and rotation about the X axis, translation along the Y axis and rotation about the Z axis. The ship motions in the other two degrees of freedom are considered small and have not been included in the mathematical model.
The equations of motion can be written in the following form, References 33:
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X-Equation |
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(53) |
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Y-Equation |
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(54) |
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Nz- Equation (turning about z-axis) |
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(55) |
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Nx- Equation (rolling about z-axis) |
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(56) |
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The left-hand side of the above equations represents the inertia terms and the right-hand side represents the hydrodynamic, rudder, propeller and external forces and moments acting on the ship, which are defined as follows:
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External Forces |
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(57) |
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(58) |
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(59) |
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(60) |
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Hull Hydrodynamic Side Forces and Moments |
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(61) |
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(62) |
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(63) |
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Rudder Forces and Moments |
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(64) |
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(65) |
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(66) |
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Propeller Forces |
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(67) |
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where the following definitions apply:
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mass moment of inertia, |
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added mass moment of inertia, |
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longitudinal and vertical positions of the center of gravity from the origin of the co-ordinate axes, see the Co-ordinate System (Plan View) image and the Co-ordinate System (Transverse View), |
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yaw rate, |
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ship velocity components along the x and y axes respectively, |
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represent the propeller, rudder and hull respectively, |
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roll angle. |
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interaction coefficients between hull, propeller and rudder in X, Y and Nx directions. |
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thrust deduction factor |
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propeller thrust |
See Estimation of Ship Resistance in Deep and Shallow Water for estimation of the hydrodynamic derivatives.