 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 isHere ${\gamma }_0$ is intrinsic damping and ${\gamma }_{\mathrm{ts}}$ the added dissipative interaction with the surface. At large average separation there is no tapping, so ${\gamma }_{\mathrm{ts}}\simeq 0$ and $A=A_{\mathrm{free}}$. 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. 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.