Tutorial Week 10
From UBCMATH WIKI
Worksheet Questions
- A undamped mass-spring system with mass 1 kg and spring constant 16 kg/s$^2$, is initially at rest. At $t=3$, a linearly increasing force is applied until the force reaches $F_0 = 10$ N at $t=8$. After that moment, the force remains constant at that level ($F_0$).
- Write down the forcing function for this scenario in terms of Heaviside functions.
- Write down the ODE for this mass-spring system subject to the given forcing function.
- What is the transfer function ($H(s)$) and the impulse response ($h(t)$) for this ODE?
- Use the impulse response and convolution to solve the ODE from part (b).
- Consider the following differential equation:$$ y''+\omega^2 y=\frac{2}{\pi}\cos(t)$$ where $\omega=2.01$.
- Find a particular solution $y_p(t)$ using the method of undetermined coefficients.
- Now consider the equation $$ y''+\omega^2 y=\sum_{n=1}^{10}\frac{2}{n\pi}\cos(nt).$$ Use your solution in (a), and again the method of undetermined coefficients, to determine a particular solution $y_p(t)$.
- Describe how the magnitude of the coefficients in your particular solution varies with n.
