Question

Find, by the Fourier-series method, the steady-state solution for an undamped harmonic oscillator subject to a force having the form of a rectified sin-wave: F(t)=F₀|sinω₀t|, where ω₀ is the natural frequency of the oscillator.

Answer 1

Answer on Question55157 - Physics / Mechanics — Kinematics — Dynamics - for correction

Find, by the Fourier-series method, the steady-state solution for an undamped harmonic oscillator subject to a force having the form of a rectified sin-wave: $F(t)=F|sin(\omega_{0}t)|$, where is the natural frequency of the oscillator.

Solution

We want to obtain the steady-state solution to

$mx^{\prime\prime}+kx=F_{0}|sin(\omega_{0}t)|$

where $\omega_{0}$ is the natural frequency of the oscillator.

Let’s obtain the complex Fourier series representation for the driver

$F_{0}|sin(\omega_{0}t)|=\sum_{n=-\infty}^{+\infty}c_{n}e^{in\omega t}$

The period of $F_{0}|sin(\omega_{0}t)|$ is $\frac{\pi}{\omega_{0}}$. The frequency is $2\pi/T=2\omega_{0}$

Let’s find the coefficients of the series.

$c_{n}=\frac{\omega_{0}}{Pi}\int_{0}^{\pi/\omega_{0}}F_{0}sin(\omega_{0}t)e^{-2in\omega_{0}t}dt=$

$=\frac{\omega_{0}F_{0}}{2\pi i}\int_{0}^{\pi/\omega_{0}}(e^{i\omega_{0}t}-e^{-i\omega_{0}t})e^{-2in\omega_{0}t}=$

$=\frac{\omega_{0}F_{0}}{2\pi i}\int_{0}^{\pi/\omega_{0}}(\frac{e^{i\omega_{0}t(1-2n)}}{i\omega_{0}(1-2n)}+\frac{e^{-i\omega_{0}t(2n+1)}}{i\omega_{0}(2n+1)})=$

$=\frac{2F_{0}}{\pi(1-4n^{2})}$

The solution of the problem $x(t)$ can be represented as the following sum:

$x(t)=\sum_{n=-\infty}^{+\infty}x_{n}(t)$

where $x_{n}(t)$ is the solution of this equation:

$mx_{n}^{\prime\prime}+kx_{n}=c_{n}e^{in\omega t}$

General solution of this problem looks like $x_{n}=a_{n}e^{in\omega t}$. After substitution $x_{n}=a_{n}e^{in\omega t}$ into the left side we find:

$a_{n}=\frac{c_{n}}{k(1-2n^{2})}=\frac{2F_{0}}{\pi k(1-4n^{2})(1-2n^{2})}$

The final solution of the initial equation is:

$x(t)=\sum_{n=-\infty}^{+\infty}\frac{c_{n}}{k(1-2n^{2})}\frac{2F_{0}}{\pi k(1-4n^{2})(1-2n^{2})}e^{in\omega t}$