Reference no: EM131095268
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, tf = 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, tf = 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) = Aer+t + Ber-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, tf = 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.
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