Question

Solve the problem of un-damped forced vibrations of a spring in the case where the forcing function is f(t)=A sin ωt. D.E associated with the problem is \(m \frac{d^2 y}{dt^2} + ky = f(t)\), with initial conditions as y(0)=y0 and y’(0)=y1 and assume λ^2 = k/m, μ=A/m.

(a) y = y0 cos⁡λt + y1  sin⁡λt + \(\frac{λ cos ωt}{-ω^2+λ^2}\)

(b) y = y0 cos⁡λt + (y1/μ)sin⁡λt + \(\frac{cos ωt}{ω^2+λ^2}\)

(c) y = y0 cos⁡λt + (y1 λ)sin⁡λt +\(\frac{sin ωt}{ω^2+λ^2}\)

(d) y = y0 cos⁡λt + (y1/λ)sin⁡λt + \(\frac{μ sin ωt}{-ω^2+λ^2}\)

This question was addressed to me in unit test.

This intriguing question originated from Harmonic Motion and Mass topic in section Linear Differential Equations – Second and Higher Order of Engineering Mathematics

Answer 1

Correct option is (d) y = y0 cos⁡λt + (y1/λ)sin⁡λt + \(\frac{μ sin ωt}{-ω^2+λ^2}\)

The best explanation: \(m \frac{d^2 y}{dt^2} + ky = A \,sin \,ωt\, \,or\, \frac{d^2 y}{dt^2} + \frac{k}{m} y = \frac{A}{m} sin ωt \rightarrow \frac{d^2 y}{dt^2} + λ^2 y = μ sin ωt\)

A.E is  m^2 + λ^2 = 0 –> m = ±λi –> y = c1 cos⁡λt + c2 sin⁡λt  at t=0 y=y0

–> c1 = y0 therefore yc = y0 cos⁡λt + c2 sin⁡λt

now differentiate and substituting y’(0)=y1

y’ = -y0  λsin⁡λt + λc2  cos⁡λt at t = 0 y1/λ = c2 and yc = y0 cos⁡λt + (y1/λ)sin⁡λt

to find particular solution \(y_p = \frac{μ sin ωt}{D^2+λ^2}\) assuming μ as constant \(y_p = \frac{μ sin ωt}{-ω^2+λ^2}\)

∴y=y0 cos⁡λt + (y1/λ)sin⁡λt + \(\frac{μ sin ωt}{-ω^2+λ^2}\).