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Suppose an autonomous surface vessel (ASV) traveling with velocity TvG/O= vi₁ begins to make a turn by adjusting the thrust of its left and right thrusters, TA and TB, respectively. The center of mass of the ASV is located at G and the ASV is symmetric about its vertical axis. The ASV also experiences a drag force that is proportional to its speed and opposes its velocity. At the instant shown, the drag force is D = -kvi₁ where k is a drag coefficient. 1. To model the mass moment of inertia, approximate the ASV as consisting of three rigid bodies: a flat plate as a center body of mass 6m and two slender rods housing the propulsion assemblies, each of mass m, at the outboard sides of the vehicle. Determine the mass moment of inertia, IG, about the vertical axis passing through the center of mass G. (Hint: Use the parallel axis theorem.) 2. At the instant shown, determine the inertial acceleration vector Iac/o = axî₁ + ayi2 of the center of mass and the angular acceleration a of the ASV. (Hint: Solve the system of equations that result from applying Euler's 1st and 2nd Law.)

Suppose an autonomous surface vessel (ASV) traveling with velocity TvG/O= vi₁ begins to make a turn by adjusting the thrust of its left and right thrusters, TA and TB, respectively. The center of mass of the ASV is located at G and the ASV is symmetric about its vertical axis. The ASV also experiences a drag force that is proportional to its speed and opposes its velocity. At the instant shown, the drag force is D = -kvi₁ where k is a drag coefficient. 1. To model the mass moment of inertia, approximate the ASV as consisting of three rigid bodies: a flat plate as a center body of mass 6m and two slender rods housing the propulsion assemblies, each of mass m, at the outboard sides of the vehicle. Determine the mass moment of inertia, IG, about the vertical axis passing through the center of mass G. (Hint: Use the parallel axis theorem.) 2. At the instant shown, determine the inertial acceleration vector Iac/o = axî₁ + ayi2 of the center of mass and the angular acceleration a of the ASV. (Hint: Solve the system of equations that result from applying Euler's 1st and 2nd Law.)

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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### Educational Content: Dynamics of Autonomous Surface Vessels **Problem Statement:** Suppose an autonomous surface vessel (ASV) traveling with velocity \( ^I \mathbf{v}_{G/O} = \mathbf{v} \hat{i}_1 \) begins to make a turn by adjusting the thrust of its left and right thrusters, \( T_A \) and \( T_B \), respectively. The center of mass of the ASV is located at \( G \) and the ASV is symmetric about its vertical axis. The ASV also experiences a drag force that is proportional to its speed and opposes its velocity. At the instant shown, the drag force is \( \mathbf{D} = -k v^2 \hat{i}_1 \) where \( k \) is a drag coefficient. **Tasks:** 1. **Modeling the Mass Moment of Inertia:** - Approximate the ASV as consisting of three rigid bodies: a flat plate as a center body of mass \( 6m \) and two slender rods housing the propulsion assemblies, each of mass \( m \), at the outboard sides of the vehicle. - Determine the mass moment of inertia, \( I_G \), about the vertical axis passing through the center of mass \( G \). *(Hint: Use the parallel axis theorem.)* 2. **Calculating Inertial and Angular Accelerations:** - At the instant shown, determine the inertial acceleration vector \( ^I \mathbf{a}_{G/O} = a_x \hat{i}_1 + a_y \hat{i}_2 \) of the center of mass and the angular acceleration \( \alpha \) of the ASV. - *(Hint: Solve the system of equations that result from applying Euler’s 1st and 2nd Law.)* **Illustration Explanation:** - **Diagram:** - The diagram depicts the ASV viewed from above. It shows forces and dimensions relevant to the problem. - The center of mass \( G \) is located at the geometrical center. - Thrust forces \( T_A \) and \( T_B \) are shown acting on the ASV. - A drag force \( D \) is acting in the opposite direction of \( \hat{i}_1 \). - The distance from the center to the thrusters is denoted as \( L \). - The axes
Transcribed Image Text:### Educational Content: Dynamics of Autonomous Surface Vessels **Problem Statement:** Suppose an autonomous surface vessel (ASV) traveling with velocity \( ^I \mathbf{v}_{G/O} = \mathbf{v} \hat{i}_1 \) begins to make a turn by adjusting the thrust of its left and right thrusters, \( T_A \) and \( T_B \), respectively. The center of mass of the ASV is located at \( G \) and the ASV is symmetric about its vertical axis. The ASV also experiences a drag force that is proportional to its speed and opposes its velocity. At the instant shown, the drag force is \( \mathbf{D} = -k v^2 \hat{i}_1 \) where \( k \) is a drag coefficient. **Tasks:** 1. **Modeling the Mass Moment of Inertia:** - Approximate the ASV as consisting of three rigid bodies: a flat plate as a center body of mass \( 6m \) and two slender rods housing the propulsion assemblies, each of mass \( m \), at the outboard sides of the vehicle. - Determine the mass moment of inertia, \( I_G \), about the vertical axis passing through the center of mass \( G \). *(Hint: Use the parallel axis theorem.)* 2. **Calculating Inertial and Angular Accelerations:** - At the instant shown, determine the inertial acceleration vector \( ^I \mathbf{a}_{G/O} = a_x \hat{i}_1 + a_y \hat{i}_2 \) of the center of mass and the angular acceleration \( \alpha \) of the ASV. - *(Hint: Solve the system of equations that result from applying Euler’s 1st and 2nd Law.)* **Illustration Explanation:** - **Diagram:** - The diagram depicts the ASV viewed from above. It shows forces and dimensions relevant to the problem. - The center of mass \( G \) is located at the geometrical center. - Thrust forces \( T_A \) and \( T_B \) are shown acting on the ASV. - A drag force \( D \) is acting in the opposite direction of \( \hat{i}_1 \). - The distance from the center to the thrusters is denoted as \( L \). - The axes
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