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A forced harmonic oscillator is a linear restoring system acted on by an external force:Its general response is the sum of a particular forced response and a homogeneous transient fixed by the initial conditions. It need not always contain both a natural-frequency oscillation and a forcing-frequency oscillation: the transient can vanish, and the force can be nonperiodic.

For $F(t)=F_0\cos \omega t$ with $b>0$, a steady response is $x=\mathrm{Re}(\chi (\omega )F_0{e}^{i\omega t})$, whereSubstitution verifies this mechanical susceptibility. The natural motion decays, leaving the forcing frequency in the long-time response. With no damping and forcing exactly at ${\omega }_0=\sqrt{k/m}$, zero initial data instead give $x=F_{0}t\sin ({\omega }_{0}t)/(2m{\omega }_0)$. Its linearly growing envelope is resonance, not a bounded steady oscillation. The model separates the drive, the initial transient and the role of energy loss.