= Solution
Model the <AFM cantilever> as a driven <damped harmonic oscillator>. If the drive stays at the free-space <resonance> and the <spring constant> $k$ and effective <mass> $m$ are unchanged, its steady amplitude is
$$
A=\frac{F_d}{\sqrt{(k-m\omega^2)^2+(\gamma\omega)^2}},\qquad A(\omega_0)=\frac{F_d}{(\gamma_0+\gamma_{\rm ts})\omega_0}.
$$
Here $\gamma_0$ is intrinsic damping and $\gamma_{\rm ts}$ the added dissipative interaction with the surface. At large average separation there is no tapping, so $\gamma_{\rm ts}\simeq0$ and $A=A_{\rm free}$. As the lower turning point reaches the surface, intermittent contacts dissipate <energy> and the amplitude decreases. With further approach the oscillation becomes strongly constrained, eventually approaching a continuously contacting, nearly quenched response. \b[The damping-only sketch has a free-amplitude plateau followed by a descending amplitude on approach.] The rounded transition in the sketch is illustrative; <adhesion> and actual contact mechanics determine its shape. The deliberately omitted force-gradient-induced <resonance> shift can make real approach curves more complicated.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-1-tapping-control.png]
{title=Damping-only <tapping-mode atomic force microscopy> response and amplitude feedback}
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