The tapping-mode atomic force microscopy response has been analyzed by the Krylov–Bogolubov–Mitropolsky asymptotic method. Due to the presence of repulsive force, attractive force and damping in the tip-sample interaction, the response exhibits complicate nonlinear phenomena. The numerical and experimental results shows that the phenomena can be well described by this approximation solution when the driving frequency is close to the free resonance frequency and the setpoint amplitude ratio is larger than 0.4. ©1998 American Institute of Physics.