MECH2300 S2 2014 Problem Solving Exercise #8

Consider a harmonically forced mass-spring system with m = 5 kg, k = 20,000 N/m, and f(t) = 20 sin(t) N.

(a) Derive the transfer function that relates input force to output displacement as a function of frequency ratio .

(b) Find the natural frequency n of the system. (c) For what range of will the amplitude of the response be less than 20 mm? (d) For what range of forcing frequency will the amplitude of the response be less than 20 mm?

Consider a harmonically forced mass-spring-damper system with m = 1.5 kg, k = 300 N/m, c = 5 Ns/m.

(e) Derive the transfer function that relates input force to output displacement as a function of frequency ratio .

(f) Find the magnitude and phase of the transfer function from (e) and sketch plots vs . (g) Find the steady state response x1(t) to the force f1(t) = 0.2 cos(10 t) N. (h) Find the steady state response x2(t) to the force f2(t) = 0.3 cos(15 t) N. (i) Find the steady state response x3(t) to the force f3(t) = 0.2 cos(10 t) + 0.3 cos(15 t) N. (j) Find the magnitude of the acceleration response to f1(t).