Driven systems oscillate at the drive frequency, not their natural frequency

Evidence

Off resonance, the drive frequency is “unnatural” for the system, but a strong drive (a high impedance drive that will exert its prescribed force no matter what a lower impedance system does) is the only frequency in town, so to speak. When the driven system settles down under continued driving, it will be oscillating at the drive frequency f, not the natural vibrational frequency f0 of the system. This is seen in figure 10.3, since the drive (red) and the velocity (green) are in lockstep. Colloquially, we may say that the oscillator registers its protest for being driven off resonance by responding weakly, not by oscillating at its natural frequency. It is sometimes difficult to remember this. Even Galileo got it wrong: First of all one must observe that each pendulum has its own time of vibration so definite and determinate it is not possible to make it move with any other period than that which nature has given it. — Eric J. Heller, Why You Hear What You Hear- An…, 975

Figure 10.3 A damped oscillator with a quality factor Q = 3 and resonance frequency f0 = 1 is kicked (left) and driven (right). The velocity (green) and drive force (red) are shown at preresonant, resonant, and postresonant frequencies. In preresonance, the force leads the velocity. At resonance, force and velocity are in phase, and the velocities reached are high. Above resonance or postresonance, the velocity leads the force. The amplitude of the velocity and the frequency are shown. Notice that the driven system in its steady state is oscillating at the drive frequency, not its natural frequency. — Eric J. Heller, Why You Hear What You Hear- An…, 966

Categorization

• XREF: Connects to driven-resonance mechanics used everywhere in acoustics and instrument design, and to the reader’s broader goal of why we hear what we hear.
• DEFINE: Defines the distinction between drive frequency f and natural frequency f0 for a forced oscillator.
• corroborated: 2 independent passages support this claim — Eric J. Heller, Why You Hear What You Hear- An…, 975; Eric J. Heller, Why You Hear What You Hear- An…, 966