Question

anding-gear unit of a small spacecraft is tested by performing drop tests from height, h. The the below shows a schematic of the landing gear unit at the instant when it touches the rtr) Initially i.e. at the point of being in contact with the ground, the spring is fully extended and the velocity is given as /2gh. Given that the mass of the vehicle is 50kg, the damping constant, B = 180 N s/m and the spring constant k = 474.5 N/m. Answer the following questions: a) Obtain a differential equation that represents the motion of the landing gear b) Using Laplace transform, show that the displacement in the s - domain may be represented as: 2gh Where B -the damping coefficient, k-the spring constant and M-the mass. This section analyses the stability of the system. 0 Using the values of B, k and m, determine the transfer function poles and zeros. (i) Illustrate the system poles and zeros in the s-plane. (ili) Hence, determine the nature of the system response. By choosing a suitable value for the height h, determine an equation representing the displacement, x(t), of the vehicle for t2 0. c) d)

Answer 1

Given that The landing gear unit of small spacecraft is tests form height h. The velocity is given as = squareroot 2 gh the mass of the vehicle is 50 kg the damping constant B = 180 N s/m the sparing constant = 474.5 N/m by Newton second law tells us sigma F = m_M_a = F_damp + F_spring M_x = -Bx - Kr Mx + Bx - Kr = 0 \r\n Mx + B squareroot 2 gh + Kr = 0 x = e^mt x = me^mt x = m^2 e^mt M m^2 e^mt + Bme^mt + Ke^ml = 0 (m^2 + Bm/M + K/M) e^mt = 0 m^2 + Bm/M + K/M = 0 Roots are b plusminus squareroot b^2 - 4ac/2a -B/M plusminus squareroot B^2 /M^2 - 4K/M /2 -B/M plusminus squareroot B^2 - 4KM/M^2 /2 m = -B plusminus squareroot B^2 - 4KM/2M -B plusminus squareroot B^2 - 4 KM/2M t - B - squareroot B^2 - 4 KM/2M + e x - c \r\n Laplace