= Solution
Measure the car's vertical <displacement> $y$ from its static <equilibrium>, so gravity's constant contribution is already balanced. Both the spring and the <viscous damper> act on motion relative to the wheel. The <base-excited oscillator> therefore satisfies
$$
M\ddot y=-k(y-y_0)-\gamma(\dot y-\dot y_0).
$$
For dimensional clarity write the road profile as $B\sin(qx)$; the printed notation sets the implicit wave number $q=1$ in the chosen length units. Constant forward speed gives forcing <angular frequency> $\omega=qv$. With complex amplitudes $Y,Y_0$, the steady response is
$$
\frac Y{Y_0}=\frac{k+i\gamma\omega}{k-M\omega^2+i\gamma\omega},\qquad
\boxed{A(v)=|B|\sqrt{\frac{k^2+\gamma^2q^2v^2}{(k-Mq^2v^2)^2+\gamma^2q^2v^2}}.}
$$
Set $q=1$ for the expression in the question's notation. Assuming $M,k>0$ and $\gamma>0$, the <transient response> decays, leaving this <steady-state response>.
For a useful sketch define $\omega_n=\sqrt{k/M}$, $r=\omega/\omega_n$ and <damping ratio> $\zeta=\gamma/(2\sqrt{kM})$. The displacement <transmissibility> is
$$
\frac A{|B|}=\sqrt{\frac{1+4\zeta^2r^2}{(1-r^2)^2+4\zeta^2r^2}}.
$$
It starts at $1$, rises to a finite maximum for finite positive damping, crosses $1$ at $r=\sqrt2$, then tends to zero. Differentiation with respect to $r^2$ gives the maximum at $r^2=2/[1+\sqrt{1+8\zeta^2}]$.
\b[Very small damping permits large resonant oscillations] near $\omega_n$, producing discomfort and potentially large suspension travel. At exactly zero damping the assumed decay of transients fails, and resonant forcing produces secular growth. \b[Very large damping makes the car follow the bumps] over a wide range of speeds, sacrificing vibration isolation: at each fixed forcing frequency, $Y/Y_0\to1$ as $\gamma\to\infty$. For any fixed finite damping the response still eventually decreases at sufficiently high frequency. These two limits should not be interchanged.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-base-excited-car.png]
{title=Base-excited car displacement response for several damping ratios, including the weak-damping resonance and isolation threshold}
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