Response of a harmonic oscillator to a resonance frequency step

Abstract

Following eq. (14.4) we assume that the oscillator is at all times driven by the force 2α _p/mU ₀ cos (ω _e t + ø). Tip-sample interaction forces are neglected. For t < 0 we excite on resonance, i.e., ω _e = ω ₀ and assume that the oscillator performs its steadystate oscillations. At t = 0 we instantaneously change the resonance frequency of the oscillator to the new frequency Ω ₀ = ω ₀ + Δω and seek solutions of eq. (10.2), neglecting F _ts, of the form

$$\xi (t) = Y(t)\cos ({\omega _{e}}t + \phi ) + X(t)\sin ({\omega _{e}}t + \phi )$$

.