Q3 (a) (b) A new lunar lander is being designed for future missions to the moon,...

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Q3 (a) (b) A new lunar lander is being designed for future missions to the moon, illustrated schematically in Figure Q3a. The lander has four legs, each of which contains a linear spring of stiffness k 223 kNm and a damper with damping coefficient c = 15.8 kNsm, as illustrated in Figure Q3b Each spring and damper can be assumed to move vertically relative to the ground. The lander is designed to hover with zero vertical velocity at a distance 4 m above the ground, at which point the engine is switched off, and the lander drops to the ground. You may assume that the ground is level and that each leg impacts the ground surface at the same instant. Also, assume that the gravitational acceleration for the moon is 1.62 ms . i) ii) iii) The i) ii) If the lander has an overall mass of 7.6 tonnes when it is about to land, determine the velocity of the lander at the point of impact on the surface. [2] (HINT - Consider the kinetic energy of the lander to determine the velocity at impact) Derive the equation of motion of the lander after it has reached the surface. Hence determine the damping ratio of the system. [3] Assuming that the lander stays in contact with the surface and that each spring is displaced the same amount, using a Laplace Transform approach, show that the motion of the lander relative to the surface is given by: [10] x(t) = 0.0138[1 - e-4.158t (cos 10t 25.66 sin 10t)] $ k Figure Q3a Figure Q3b response of an underdamped system subject to a step input is given by x(t) = 1e-t [cos 4t + 0.75 sin 4t] What is the steady-state value of this system? [1] From first principles, derive an expression for the rise time of this system. [4] Q3 (a) (b) A new lunar lander is being designed for future missions to the moon, illustrated schematically in Figure Q3a. The lander has four legs, each of which contains a linear spring of stiffness k 223 kNm and a damper with damping coefficient c = 15.8 kNsm, as illustrated in Figure Q3b Each spring and damper can be assumed to move vertically relative to the ground. The lander is designed to hover with zero vertical velocity at a distance 4 m above the ground, at which point the engine is switched off, and the lander drops to the ground. You may assume that the ground is level and that each leg impacts the ground surface at the same instant. Also, assume that the gravitational acceleration for the moon is 1.62 ms . i) ii) iii) The i) ii) If the lander has an overall mass of 7.6 tonnes when it is about to land, determine the velocity of the lander at the point of impact on the surface. [2] (HINT - Consider the kinetic energy of the lander to determine the velocity at impact) Derive the equation of motion of the lander after it has reached the surface. Hence determine the damping ratio of the system. [3] Assuming that the lander stays in contact with the surface and that each spring is displaced the same amount, using a Laplace Transform approach, show that the motion of the lander relative to the surface is given by: [10] x(t) = 0.0138[1 - e-4.158t (cos 10t 25.66 sin 10t)] $ k Figure Q3a Figure Q3b response of an underdamped system subject to a step input is given by x(t) = 1e-t [cos 4t + 0.75 sin 4t] What is the steady-state value of this system? [1] From first principles, derive an expression for the rise time of this system. [4]