Driven undamped harmonic oscillator x x cos2t x0 1 x0 0

Question

Consider the following differential equation form:

x" + 2λx' + ω²x = F(t)

Use springmass.m to generate a picture of motion for each question below (1-3). Provide the final solution y(t) for each problem below, using the general solution statement for questions I, 2 and 3.

(I) Under-damped harmonic oscillator x" + 0.2x' + 2x = 0, x(0) = 1,x'(0) = 2, t_f = 12Π.

Hints: λ = 0.1 & ω = √2. For λ < ω,

x(t) = Ae^-λt cos( √(ω²-λ²t-Φ))

(2) Critically damped harmonic oscillator x" + 2x' + x = 0, x(0) = 1, x'(0) = 2, t_f = 12Π.

Hint: For λ = ω,

x(t) = Ae^-λt + Bte^-λt

(3) Over-damped harmonic oscillator x" + 4x' + x = 0,x(0) = 1,x'(0) = 2, tf = 12Π.

Hint: For λ> ω,

x(t) = Ae^r+t + Be^r-t , r = -λ±√(λ²- ω²)

For questions 4 , 5, and 6, use springmassdriven.m to generate a picture of motion. For each question below provide a picture as stated above, the final solution y(t) for each problem and provide a breakdown of the transient and steady-state solutions with a description of motion.

(4) Driven undamped harmonic oscillator x" + x = cos(2t), x(0) = 1, x'(0) = 0, t_f = 24Π.

(5) Resonantly driven undamped harmonic oscillator x" + x = cos(t), x(0) = 1,x:(0)= 0, tf = 24Π.

(6) Resonantly driven damped harmonic oscillator x" + 0.2x' + 2x = cos(t), x(0) = 1, x'(0) = 0, tf = 24Π.