A damped harmonic oscillator of weight mg 49 N spring const

A damped harmonic oscillator of weight mg = 4.9 N, spring constant k = 1000. N/m and damping constant ? = 7.0 N*sec/m, is driven by a periodic square-wave force,

(a) Find the complex-exponential Fourier series for this force.

(b) Determine the steady-state forced vibration of the oscillator, i.e., the particular solution to the inhomogeneous differential equation, as a Fourier series.

(c) Give the numerical values of the real parts of the Fourier amplitudes for the first eight terms in the particular solution series. Which mode (frequency) has the greatest response? Is it the driving frequency? If not, what multiple of the driving frequency is it?

Solution

² = k/m

= 26.46 rad/s

x(t) = A cos(t)

0.01 = 0.04cos(26.46t)

t = 0.050 sec

0.03 = 0.04cos(26.46t)

t = 0.0273 sec

(a)

E = ½kA²

E = ½35(0.04)²

E = 0.028 Joules = 28 mJ

(b)

v = -A sin(t)

v = -.04sin(26.46*.05)

|v| = 1.02 m/s

(c)

K = ½kA²sin²(t)

K = ½35(.04)²sin²(26.46*.0273)

K = .01225 Joules = 12.3 mJoules

(d)

U = ½kA²cos²(t)

U = ½35(.04)²cos²(26.46*.0273)

U = 15.8 mJ