Tutorial Week 4

Worksheet Questions

1. 1. Please rewrite the function $y(t)=c_1\cos(\omega t)+c_2\sin(\omega t),$ in the form $y(t)=D\cos(\omega t-\gamma).$ (Find expressions of $D$ and $\gamma$ in terms of $c_1$ and $c_2$.)
   2. Find the steady state periodic solution to the mass spring system described by $x''+cx'+x=10\cos(\omega t),$ where $\omega$ and $c$ are both positive constants. Your answer should look like $x(t)=A(\omega)\cos(\omega t)+B(\omega)\sin(\omega t).$ (Find expressions of $A(\omega)$ and $B(\omega)$.)
   3. For what angular frequency $\omega$ is the amplitude $D(\omega)$ of the steady state periodic solution you obtained in part b maximized? Hint: use your result from part a. Also, note that $C/g(w)$ has a maximum value when $g(w)$ has a minimum value and $\sqrt{f(w)}$ has a minimum when $f(w)$ has a minimum. These last two facts will simplify the calculations required. Be sure to determine conditions under which your critical points are actually maxima.

Solutions

1. 1. $D=\sqrt{c_1^2+c_2^2}$, $\gamma=\arctan(\frac{c_2}{c_1})$.
   2. $A(\omega)=\frac{10(1-\omega^2)}{(1-\omega^2)^2+c^2\omega^2}$, $B(\omega)=\frac{10c\omega}{(1-\omega^2)^2+c^2\omega^2}$.
   3. $D(\omega)=\frac{10}{\sqrt{(1-\omega^2)^2+c^2\omega^2}}$. When $D(\omega)$ is maximized the denominator of it($\sqrt{g(\omega)}$) is minimized. As we know $\sqrt{g(\omega)}$ has a minimum when $g(\omega)$ has a minimum, and $g'(\omega)=2\omega(-2+2\omega^2+c^2).$ When $\omega=\sqrt{1-\frac{c^2}{2}}$, $g(\omega)$ obtain its minimum value. So if $1-\frac{c^2}{2}$ is positive, i.e. $0<c<\sqrt{2}$, then $\sqrt{1-\frac{c^2}{2}}$ is the practical resonance frequency. Otherwise, no maximum for $\omega>0$.