Coupled oscillators and normal modes — A.P. French
Physics — Waves · damped-forced-oscillations
Problem
A block of mass m is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made on this system:
(1) If the block is pushed horizontally with a force equal to mg, the static compression of the spring is equal to h.
(2) The viscous resistive force is equal to mg if the block moves with a certain known speed u.
(a) For this complete system (including both spring and damper) write the differential equation governing horizontal oscillations of the mass in terms of m, g, h, and u.
Answer the following for the case that u=3gh:
(b) What is the angular frequency of the damped oscillations?
(c) After what time, expressed as a multiple of h/g, is the energy down by a factor 1/e?
(d) What is the Q of this oscillator?
(e) This oscillator, initially in its rest position, is suddenly set into motion at t=0 by a bullet of negligible mass but nonnegligible momentum traveling in the positive x direction. Find the value of the phase angle δ in the equation x=Ae−γt/2cos(ωt−δ) that describes the subsequent motion, and sketch x versus t for the first few cycles.
(f) If the oscillator is driven with a force mgcosωt, where ω=2g/h, what is the amplitude of the steady-state response?