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3 Problem The rolling motion of a Mariner Class cargo ship can be approximated by the following equation: (144+A44) + B440+ C440 = Msin(wwavest) where is the roll angle, 144 is the roll inertia, A44 is the inertia of the added mass (i.e., the surrounding water has the effect of adding inertia to the vessel), B44 is the roll damping due to viscous shear forces, C44 is the hydrostatic restoring force. The right hand side represents sinusoidal forcing caused by waves at frequency waves, with M being the maximum applied moment to the ship. Assume 144 = 1.471 x 10¹0 kg-m², A44 = 2.1 x 10¹0 kg-m², C44 = 1.1852 x 1010 N-m/rad, B44 = 6.6018 × 10⁹ N-m/(rad/s). Determine the following parameters for the vessel by equation the coefficients of the above system to that of a damped harmonic oscillator: 1. undamped natural frequency (in Hz) 2. damping ratio 3. damped natural frequency (in Hz) 4. period of oscillation (in seconds, assuming the damped natural frequency) 5. If the waves suddenly stopped (M = 0) when the ship was at it's maximum roll angle, how long would it take the ship to settle down to 2% of this maximum roll angle? (Hint: Use your knowledge of the time constant for the decaying oscillations.)

3 Problem The rolling motion of a Mariner Class cargo ship can be approximated by the following equation: (144+A44) + B440+ C440 = Msin(wwavest) where is the roll angle, 144 is the roll inertia, A44 is the inertia of the added mass (i.e., the surrounding water has the effect of adding inertia to the vessel), B44 is the roll damping due to viscous shear forces, C44 is the hydrostatic restoring force. The right hand side represents sinusoidal forcing caused by waves at frequency waves, with M being the maximum applied moment to the ship. Assume 144 = 1.471 x 10¹0 kg-m², A44 = 2.1 x 10¹0 kg-m², C44 = 1.1852 x 1010 N-m/rad, B44 = 6.6018 × 10⁹ N-m/(rad/s). Determine the following parameters for the vessel by equation the coefficients of the above system to that of a damped harmonic oscillator: 1. undamped natural frequency (in Hz) 2. damping ratio 3. damped natural frequency (in Hz) 4. period of oscillation (in seconds, assuming the damped natural frequency) 5. If the waves suddenly stopped (M = 0) when the ship was at it's maximum roll angle, how long would it take the ship to settle down to 2% of this maximum roll angle? (Hint: Use your knowledge of the time constant for the decaying oscillations.)

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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3 Problem The rolling motion of a Mariner Class cargo ship can be approximated by the following equation: (I44 + A44) + B44 +C440 = M sin(wwavest) where is the roll angle, 144 is the roll inertia, A44 is the inertia of the added mass (i.e., the surrounding water has the effect of adding inertia to the vessel), B44 is the roll damping due to viscous shear forces, C44 is the hydrostatic restoring force. The right hand side represents sinusoidal forcing caused by waves at frequency waves, with M being the maximum applied moment to the ship. Assume 144 = 1.471 × 10¹0 kg-m², A44 = 2.1 × 10¹0 kg-m², C44 = 1.1852 × 1010 N-m/rad, B44 = 6.6018 × 10⁹ N-m/(rad/s). Determine the following parameters for the vessel by equation the coefficients of the above system to that of a damped harmonic oscillator: 1. undamped natural frequency (in Hz) 2. damping ratio 3. damped natural frequency (in Hz) 4. period of oscillation (in seconds, assuming the damped natural frequency) 5. If the waves suddenly stopped (M = 0) when the ship was at it's maximum roll angle, how long would it take the ship to settle down to 2 % of this maximum roll angle? (Hint: Use your knowledge of the time constant for the decaying oscillations.)
Transcribed Image Text:3 Problem The rolling motion of a Mariner Class cargo ship can be approximated by the following equation: (I44 + A44) + B44 +C440 = M sin(wwavest) where is the roll angle, 144 is the roll inertia, A44 is the inertia of the added mass (i.e., the surrounding water has the effect of adding inertia to the vessel), B44 is the roll damping due to viscous shear forces, C44 is the hydrostatic restoring force. The right hand side represents sinusoidal forcing caused by waves at frequency waves, with M being the maximum applied moment to the ship. Assume 144 = 1.471 × 10¹0 kg-m², A44 = 2.1 × 10¹0 kg-m², C44 = 1.1852 × 1010 N-m/rad, B44 = 6.6018 × 10⁹ N-m/(rad/s). Determine the following parameters for the vessel by equation the coefficients of the above system to that of a damped harmonic oscillator: 1. undamped natural frequency (in Hz) 2. damping ratio 3. damped natural frequency (in Hz) 4. period of oscillation (in seconds, assuming the damped natural frequency) 5. If the waves suddenly stopped (M = 0) when the ship was at it's maximum roll angle, how long would it take the ship to settle down to 2 % of this maximum roll angle? (Hint: Use your knowledge of the time constant for the decaying oscillations.)
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