2002 Sinusoidally Driven Systems Second Order LTI DE's

We start with the second order linear constant coefficient (CC) DE, which as we've seen can be interpreted as modeling a damped forced harmonic oscillator. If we further specify the oscillator to be a mechanical system with mass $m$, damping coefficient $b$, spring constant $k$, and with a sinusoidal driving force $B\cos \omega t$ (with $B$ constant), then the DE is $$mx''+bx'+kx=B\cos \omega t\tag{1}$$ For many applications it is of interest to be able to predict the periodic response of the system to various values of $\omega$.

In the sessions on Exponential Response and Gain & Phase Lag we worked out the general case of a sinusoidally driven LTI DE. Specializing these results to the second order case we have:
Characteristic polynomial: $$p(s)=ms^2+bs+k$$ Complex replacement: $$mz''+bz'+kz=Be^{i\omega t}, x=Re(z)$$ Exponential Response Formula: $$z_p=\frac{Be^{i\omega t}}{p(i\omega)}=\frac{Be^{i\omega t}}{k-m\omega^2+ib\omega}$$ $$\begin{aligned} x_p=Re(z_p)&=Re(\frac{k-m\omega^2-ib\omega}{(k-m\omega^2)^2+(b\omega)^2}(B(\cos \omega t+i\sin \omega t)))\\ &=\frac{B((k-m\omega^2)\cos \omega t + b\omega \sin \omega t)}{(k-m\omega^2)^2+(b\omega)^2}\\ &=\frac{B}{\sqrt{(k-m\omega^2)^2+(b\omega)^2}}\cos (\omega t - \phi) \end{aligned}$$ where $\phi=Arg(p(i\omega))=\tan^{-1}(\frac{b\omega}{k-m\omega^2})$. (In this case $\phi$ must be between 0 and $\pi$. We say $\phi$ is in the first or second quadrants.)
Letting $A=\frac{B}{\sqrt{(k-m\omega^2)^2+(b\omega)^2}}$, we can write the periodic response $x_p$ as $$x_p=A\cos (\omega t - \phi)$$

The complex gain, which is defined as the ratio of the amplitude of the output to the amplitude of the input in the complexified equation, is $$\widetilde g(\omega)=\frac{1}{p(i\omega)}=\frac{1}{k-m\omega^2+ib\omega}$$

The gain, which is defined as the ratio of the amplitude of the output to the amplitude of the input in the real equation, is $$g=g(\omega)=\frac{1}{\left\lvert p(i\omega)\right\rvert}=\frac{1}{\sqrt{(k-m\omega^2)^2+(b\omega)^2}}\tag{2}$$

The phase lag is $$\phi=\phi(\omega)=Arg(p(i\omega))=\tan^{-1}(\frac{b\omega}{k-m\omega^2})\tag{3}$$ and we also have the time lag = $\phi/\omega$.

Terminology of Frequency Response

We call the gain $g(\omega)$ the amplitude response of the system. The phase lag $\phi(\omega)$ is called the phase response of the system. We refer to them collectively as the frequency response of the system.