Question

A quarter-car suspension model consisting of a spring and a damper is shown in Figure 1. An active suspension element produces an input force F. Draw a free-body diagram for the sprung mass m, and hence derive a differential equation relating the input force F to the sprung mass displacement x. (a) (5 marks) (b) Assuming a mass m-250kg, spring coefficient k 100Nm-1 and damping coefficient of c-50Nsm1, show that the transfer function from the input force F to the sprung mass displacement from its static position x is: 0.004 s2 +0.2s +0.4 G(s) (5 marks) (c) Calculate the damning ratio and undamped natural frequency ??, and

Answer 1

F_d = force by damper = c(d/dt)x F_s = force by spring = kx By equilibrium state, + uparrow Sigma F_y = -F + F_d + F_s + M(d/dt)^2 x = 0 Therefore F = F_d + F_s + M(d/dt)^2 x -(1) F = c(d/dt)x + kx + M(d/dt)^2 x b) from equation (1) & taking all initial condition zero. Applying laplace transformation. F(s) = cs x(s) + k x(s) + Ms^2 x(s) Therefore G(s) = x(s)/F(s) = 1/Ms^2 + k + cs = Y_m/s^2 + c/m s + k/m = 1/250/s^2 + 50/250 s + 100/250 = 0.004/s^2 + 0.25 + 0.4 Proved c) Any second order transfer function is given as. c(s)/R(s) = k/s^2 + 2zw_ns + w^2_n Where, z = damping ratio w_n = natural frequency In the problem, transfer function given is G(s) = 0.004/s^2 + 0.2s + 0.4 Therefore w_n = squareroot 0.4 = 0.63 Ans