What is the angular frequency of the damped oscillations

Question

Assignment:

Problem:

1. A simple harmonic oscillator consists of a 0.2-kg mass attached to a spring with force constant 1 N/m.

The mass is displaced by 5 cm and released from rest. Calculate

(a) the natural frequency ν₀ and the period τ₀,

(b) the total energy, and

(c) the maximum speed of the oscillator.

2. Now allow the motion of the previous problem to occur is a resisting medium. After oscillating for 10_s, the amplitude decreases to half the initial value. Calculate

(a) the damping parameter β, and

(b) the frequency ν₁ and compare it to the undamped frequency ν₀

3. Mathematica problem: Consider underdamped motion with amplitude A = 1 m. Use Mathematica to plot x(t) and its two components (e^-βt and cos(ω₁t - δ)) on the same plot as the solution for the undamped oscillator (β = 0). Take ω₀ = 1 rad/s. Make separate plots for β²/ω²₀ = 0.1, 0.5, and 0.9 and for the phase δ = 0, π/2 and π. Plot nine separate plots for each set of these β and δ values.

Discuss the results.

4. Mathematica problem: Now consider a driven oscillator with β = 0.2 s^-1. Plot x_p(t), x_c(t), and the sum x(t) on the same plot. Let m = 1 kg and k = 1 kg/s². Do this for ω/ω₁ = 1/9, 1/3, 1.1, 3, and 6. For the x_c(t) solution let the phase angle be 0 and the amplitude A = -1 m. For x_p(t) let A = 1 m/s2 but calculate δ. What do you observe about the relative magnitudes of the two solutions as ω increases? Why does this occur?

5. 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.

Questions:

(a) For this complete system, including both the spring and the damper, write the differential equation governing horizontal oscillations of the mass in terms of m, g, h, and u.

Now consider the special case where u = 3√ gh

(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 of 1/e?

(d) What is the Q-value (= ω₀/β) of this oscillator?

(e) This oscillator, initially in its rest position, is suddenly set in motion at t = 0 by an impulse which imparts a non-zero momentum in the x-direction. Find the value of the phase angle δ in the equation x(t) = Ae^-βt cos(ωt - δ) that describes the subsequent motion, and sketch x(t) vs t for the first few cycles.

(f) If the oscillator is driven with a force mg cos ωt, where ω =√2g/h, what is the amplitude of the steady-state response?