Solve Example 3.20 where the structure has viscous damping, that is, the governing equation of motion is \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2}=F(t)\), the forcing \(F(t)\) is the same square wave, and \(\omega_{n}=2 \omega_{T} \mathrm{rad} / \mathrm{s}\).

Example 3.20 Response to a Square Wave Using Fourier Series 47 Consider the simple undamped oscillator + wx F(t)/m, where F(t) is the force represented by the square wave function in Figure 3.49. The square wave is the sim- plest of all non-harmonic, periodic functions since the amplitude over each half-period is constant. Solve for the response using the Fourier series representation for F(t), assuming zero initial conditions. Fo F(t) T T -Fo 2 Figure 3.49: A square wave forcing function F (t).