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{"ops":[{"insert":"Question: - Derive 6DOF Equations of motion for the Vehicle.\nThe equation of motion for the vehicle are mathematical models, which express the motion law of\u00a0\nthe vehicle. Based on the models, one may analyze and simulate the motion of a vehicle. In\u00a0\naddition, based on small disturbance theory, one may derive linear longitudinal small disturbance\u00a0\nmotion equation and lateral small disturbance motion equations from the dynamic equation.\u00a0\nMotion of the vehicle follows Newton\u2019s Laws. Newton\u2019s law formulates the relations between the\u00a0\nsummation of external forces, the acceleration, and the relations between the summation of\u00a0\nexternal moments and the angular acceleration.\nYour work should follow the following assumptions\n1. The earth is considered as an inertial reference, i.e. it is stationary.\n2. Earth\u2019s curvature is neglected, and earth-surface is assumed to be flat.\n3. The vehicle is assumed to be rigid body. Any two points on or within the airframe retain\u00a0\nfixed with respect to each other. Ignore the aero-elastic effects of the vehicle.\n4. The mass of the vehicle is assumed to retain constant.\n5. The vehicle is considered as symmetry about Oxbyb plane. The product of inertia Ixy and Izy\nvanish.\nAssume that the moving coordinate frame with an angular velocity \u03c9 as shown in the figure 1. The\u00a0\nvector \u03c9 is resolved into three component p, q, r in this coordinate frame as follows. Where i, j, k\nare unit vectors respectively along xb, yb and zb axes\n\ud835\udf4e = \ud835\udc5d\ud835\udc8a + \ud835\udc5e\ud835\udc8b + \ud835\udc5fk\n\nYou have to derive relations for\n1. Force Equations\n2. Moment Equations\n3. Kinematic Equations\na. Equation for Center of Mass\nb. Angular Motion Equations\nGuess\nConsider changeable vector a(t). The a(t) is resolved into three component ax, ay, az in the\u00a0\ncoordinate frame thus\n\ud835\udc82 = \ud835\udc4e\ud835\udc65\ud835\udc8a + \ud835\udc4e\ud835\udc66\ud835\udc8b + \ud835\udc4e\ud835\udc67\ud835\udc8c (2)\nTaking derivative of a(t) with respect to time t yield\u00a0\n\ud835\udc51\ud835\udc82\n\ud835\udc51\ud835\udc61\n=\n\ud835\udc51\ud835\udc4e\ud835\udc65\n\ud835\udc51\ud835\udc61\n\ud835\udc8a +\n\ud835\udc51\ud835\udc4e\ud835\udc66\n\ud835\udc51\ud835\udc61\n\ud835\udc8b +\n\ud835\udc51\ud835\udc4e\ud835\udc67\n\ud835\udc51\ud835\udc61\n\ud835\udc8c + \ud835\udc4e\ud835\udc65\n\ud835\udc51\ud835\udc8a\n\ud835\udc51\ud835\udc61\n+ \ud835\udc4e\ud835\udc66\n\ud835\udc51\ud835\udc8b\n\ud835\udc51\ud835\udc61\n+ \ud835\udc4e\ud835\udc67\n\ud835\udc51\ud835\udc8c\n\ud835\udc51\ud835\udc61\n(3)\nTheoretical mechanics presents that if a rigid body rotates at an angular velocity \ud835\udf4e about fixed\u00a0\npoint, the velocity of arbitrary point P in rigid body is given by\n\ud835\udc51\ud835\udc93\n\ud835\udc51\ud835\udc61\n= \ud835\udf4e \u00d7 \ud835\udc93 (4)\nWhere r is a vector radius from the origin point O to the point P\n\nFor vector radius\n\ud835\udc51\ud835\udc8a\n\ud835\udc51\ud835\udc61\n= \ud835\udf4e \u00d7 \ud835\udc8a (5)\n\ud835\udc51\ud835\udc8b\n\ud835\udc51\ud835\udc61\n= \ud835\udf4e \u00d7 \ud835\udc8b (6)\n\ud835\udc51\ud835\udc8c\n\ud835\udc51\ud835\udc61\n= \ud835\udf4e \u00d7 \ud835\udc8c (7)\nSimilarly\n\ud835\udc51\ud835\udc82\n\ud835\udc51\ud835\udc61\n=\n\ud835\udc51\ud835\udc4e\ud835\udc65\n\ud835\udc51\ud835\udc61\n\ud835\udc8a +\n\ud835\udc51\ud835\udc4e\ud835\udc66\n\ud835\udc51\ud835\udc61\n\ud835\udc8b +\n\ud835\udc51\ud835\udc4e\ud835\udc67\n\ud835\udc51\ud835\udc61\n\ud835\udc8c + \ud835\udf4e \u00d7 (\ud835\udc4e\ud835\udc65\n\ud835\udc51\ud835\udc8a\n\ud835\udc51\ud835\udc61\n+ \ud835\udc4e\ud835\udc66\n\ud835\udc51\ud835\udc8b\n\ud835\udc51\ud835\udc61\n+ \ud835\udc4e\ud835\udc67\n\ud835\udc51\ud835\udc8c\n\ud835\udc51\ud835\udc61) (8)\ni.e.\n\ud835\udc51\ud835\udc82\n\ud835\udc51\ud835\udc61\n=\n\ud835\udeff\ud835\udc82\n\ud835\udeff\ud835\udc61 + \ud835\udf4e \u00d7 \ud835\udc82 (9)\nWhere\n\ud835\udeff\ud835\udc82\n\ud835\udeff\ud835\udc61 =\n\ud835\udc51\ud835\udc4e\ud835\udc65\n\ud835\udc51\ud835\udc61\n\ud835\udc8a +\n\ud835\udc51\ud835\udc4e\ud835\udc66\n\ud835\udc51\ud835\udc61\n\ud835\udc8b +\n\ud835\udc51\ud835\udc4e\ud835\udc67\n\ud835\udc51\ud835\udc61\n\ud835\udc8c\n\ud835\udeff\ud835\udc82\n\ud835\udeff\ud835\udc61 is called \u201crelative derivative\u201d\n\ud835\udc51\ud835\udc82\n\ud835\udc51\ud835\udc61\nis called \u201cabsolute derivative\n"}]}

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