= Solution
Measure the mass's displacement $x$ from its static equilibrium height. This removes the constant spring extension supporting its <weight>. If the spring's top moves with the box, its incremental extension is $y_2-x$, so its incremental upward <force> is $k(y_2-x)$. The specified damping force is against the mass's velocity in the static frame. <Newton's second law> therefore gives
$$
\boxed{m\ddot x+b\dot x+kx=ky_2=ka_s\cos\omega t.}
$$
In absolute height coordinates an additional constant on the right expresses the spring's rest length, its fixed attachment height and <gravity>; translating to static equilibrium removes it. Replacing the given damping by $-b(\dot x-\dot y_2)$ would describe a different <base-excited oscillator>.
After the decaying transient, write the <steady-state response> as $x=A(\omega)\cos(\omega t-\Phi)$. Using a complex sinusoid gives
$$
\widetilde x=\frac{ka_s}{k-m\omega^2+ib\omega},
$$
so
$$
\boxed{A(\omega)=\frac{ka_s}{\sqrt{(k-m\omega^2)^2+b^2\omega^2}},\qquad \Phi(\omega)=\operatorname{atan2}(b\omega,k-m\omega^2).}
$$
For positive <frequency>, the phase lag lies between zero and $\pi$, passing through $\pi/2$ at $\omega_0=\sqrt{k/m}$. For small $b$, the <frequency response> begins at $a_s$, develops a narrow <resonance> near $\omega_0$, and falls as
$$
A(\omega)\sim\frac{k a_s}{m\omega^2}\qquad(\omega\gg\omega_0).
$$
The precise amplitude-peak <frequency> is $\sqrt{k/m-b^2/(2m^2)}$ when that expression is positive.
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-seismograph-response.png]
{title=Absolute and relative motion of the stated seismograph model, and the phase lag of its inertial mass}
The pen records the relative displacement $z=x-y_2$. Its complex <transfer function> is
$$
\frac{\widetilde z}{a_s}=\frac{m\omega^2-ib\omega}{k-m\omega^2+ib\omega}.
$$
At high <frequency>, $x$ is small while $z\simeq-y_2$. \b[The inertial mass then stays nearly stationary and the recorded relative motion reproduces the ground displacement, with reversed sign.] At low <frequency> the mass follows the box and their relative motion is small; near <resonance> the response is strongly frequency-dependent. Here “high” means compared with the suspension's natural <frequency>, rather than an absolute seismic-frequency threshold.
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