A spring/mass/dashpot system with oscillating forcing function is modeled by the ODE:

$\displaystyle {x}{''}+{4}{x}'+{225}{x}={\cos{{\left(\omega{t}\right)}}}$

where $\displaystyle \omega$ represents the frequency of the forcing term. We will determine what value of $\displaystyle \omega$ corresponds to the maximal amplitude for the response oscillation.

Using the real part of the solution to the complexified ODE we have:

$\displaystyle {x}_{{p}}=\text{Re}{\left(\frac{{e}^{{{i}{15}{t}}}}{{{225}-\omega^{{2}}+{4}{i}\omega}}\right)}$

This can be expressed in form of a pure oscillation:

$\displaystyle {A}{\cos{{\left({15}{t}-\phi\right)}}}$

where $\displaystyle {A}$ is the amplitude and $\displaystyle \phi$ is the phase lag for the response oscillation. Now the amplitude is:

$\displaystyle {A}=\frac{{1}}{\sqrt{{{\left({225}-\omega^{{2}}\right)}^{{2}}+{\left({4}\omega\right)}^{{2}}}}}$

This is maximal when the radicand in the denominator is minimal. Using calculus, determine what positive value of $\displaystyle \omega$ makes the expression $\displaystyle {\left({225}-\omega^{{2}}\right)}^{{2}}+{\left({4}\omega\right)}^{{2}}$ minimal.

$\displaystyle \omega$ = Preview Question 6