The equation of motion for a damped simple harmonic oscillator is ?where k and ? are constants, m is the mass and x is the displacement of the system. Describe the conditions for lightly damped, critically damped and over-damped oscillations. Draw diagrams to show how the displacement varies with time in the three cases, given that in each case the system is initially displaced and then released from rest. A system whose natural frequency in the absence of damping is 4 rad s^?1 is subject to a damping force such that k/m = 10 s^?1. Show that the system is over damped and that the general solution for the displacement is x = A exp (? 2t) + B exp (?8t). The mass is initially at x = + 0.5m and given an initial velocity V towards the equilibrium position. Find the smallest value of V that will produce a negative displacement.

Transcribed Image Text:

dx + ix = 0, d?x dt dr?