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{"ops":[{"insert":"Question: - Derive 6DOF Equations of motion for the Vehicle. The equation of motion for the vehicle are mathematical models, which express the motion law of the vehicle. Based on the models, one may analyze and simulate the motion of a vehicle. In addition, based on small disturbance theory, one may derive linear longitudinal small disturbance motion equation and lateral small disturbance motion equations from the dynamic equation. Motion of the vehicle follows Newton's Laws. Newton's law formulates the relations between the summation of external forces, the acceleration, and the relations between the summation of external moments and the angular acceleration. Your work should follow the following assumptions 1. The earth is considered as an inertial reference, i.e. it is stationary. 2. Earth's curvature is neglected, and earth-surface is assumed to be flat. 3. The vehicle is assumed to be rigid body. Any two points on or within the airframe retain fixed with respect to each other. Ignore the aero-elastic effects of the vehicle. 4. The mass of the vehicle is assumed to retain constant. 5. The vehicle is considered as symmetry about Oxbym plane. The product of inertia ly and Izy vanish. Assume that the moving coordinate frame with an angular velocity o as shown in the figure 1. The vector w is resolved into three component p.q.r in this coordinate frame as follows. Where i, j, k are unit vectors respectively along Xb, yn and zo axes w = pi + aj +rk (1) i Figure 1. Component of angular velocity ( You have to derive relations for 1. Force Equations 2. Moment Equations 3. Kinematic Equations a. Equation for Center of Mass b. Angular Motion Equations Guess Consider changeable vector a(t). The a(t) is resolved into three component az, ay, a: in the coordinate frame thus a = axi + ayj + a_k (2) Taking derivative of a(t) with respect to time 1 yield da dk di dax day i + dt dt *j + daz k + ax + ay ana + az (3) dt dt dt Theoretical mechanics presents that if a rigid body rotates at an angular velocity w about fixed point, the velocity of arbitrary point P in rigid body is given by dr = wxr dt (4) Where r is a vector radius from the origin point to the point P. For vector radius one = w xi (5) =w xj (6) dt dk = w xk dt (7) Similarly da dag i + days + daz di k + W X (ax + dt dt ay dj + az dt aza) (8) dt dt i.e. da \u03b4\u03b1 +wxa St (9) dt\n"}]}

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