The driven damped oscillator can be represented as a linear input-output system. For

$$m\ddot x+c\dot x+kx=F(t),$$

take the applied force as the input and displacement as the output. Under zero initial conditions, the force-to-displacement transfer function is

$$\boxed{H(s)=\frac{X(s)}{F(s)}=\frac{1}{ms^2+cs+k}}.$$

This representation does not introduce new oscillator physics. It packages the same mass, damping, stiffness, transient modes, and steady forced response into a form that can be analyzed with general LTI-system tools.

Frequency response

For sinusoidal forcing at angular frequency $\omega$, evaluate the transfer function on the imaginary axis:

$$H(i\omega)=\frac{1}{k-m\omega^2+i c\omega}.$$

Its magnitude is

$$|H(i\omega)| =\frac{1}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}},$$

so if the force amplitude is $F_0$, the displacement amplitude is

$$A=F_0|H(i\omega)|.$$

The phase of $H(i\omega)$ is the displacement phase relative to the applied force. Thus the amplitude and phase relations of the driven oscillator are exactly the magnitude and phase of its frequency response.

Poles and transients

The poles satisfy

$$ms^2+cs+k=0.$$

They are the same characteristic roots that determine free damped motion. Consequently:

• complex-conjugate poles correspond to underdamped transients;
• a repeated real pole corresponds to critical damping;
• two real negative poles correspond to overdamped transients.

The transfer-function description therefore connects the oscillator's transient behavior and its forced steady-state response in one model.

Why the frequency-response viewpoint is useful

Once vibration is expressed through $H(s)$, the same analysis tools used for filters, actuators, sensors, and feedback systems can be applied to mechanical structures. Resonant peaks can be identified on magnitude plots, phase changes can be tracked with frequency, and modifications to $m$, $c$, or $k$ can be interpreted as movements of poles and changes in the frequency response.

For example, a rotating imbalance or periodic machine load may excite a structure weakly at one operating speed and strongly near a resonant region. The mechanical model determines the physical parameters; the transfer function makes that response compatible with the broader language of linear systems and control.