Forced Oscillations

Natural frequency is the frequency at which an oscillator vibrates freely without external forces or damping. Driving frequency is the frequency of the periodic external force applied to the system. When pendulum A oscillates freely, it establishes a natural frequency. The energy transferred to other pendulums forces them to oscillate at the driving frequency imposed by pendulum A's motion.

When driving frequency is much smaller or larger than natural frequency, the amplitude remains small. As the driving frequency approaches the natural frequency, amplitude increases significantly. Maximum amplitude occurs when driving frequency equals natural frequency. This relationship is described by the amplitude equation derived from the forced oscillator's equation of motion.

When a periodic driving force is first applied, the oscillator exhibits chaotic or irregular motion called transients. These transients gradually die out as damping dissipates energy. After transients fade, the oscillator reaches steady state, where motion becomes periodic and displacement is determined by the driving force frequency and amplitude.

Resonance occurs when the driving frequency equals the system's natural frequency. At resonance, the oscillator responds with maximum amplitude, and oscillations increase with each cycle as long as the driving force continues. This phenomenon is demonstrated by a paddle ball suspended from a finger: when the finger moves at the ball's natural frequency, the ball's oscillations grow dramatically larger.

A free oscillator's equation of motion includes only restoring and damping forces. The forced oscillator's equation incorporates an additional periodically varying driving force with amplitude F0 and driving angular frequency. This external force term is added to the damped harmonic oscillator equation to obtain the complete forced oscillator equation of motion.

When driving frequency is much higher than natural frequency, the difference between the two angular frequencies is large and positive. This large difference appears in the denominator of the amplitude equation, resulting in a small amplitude. At very high driving frequencies, the oscillator cannot respond quickly enough, so oscillations nearly disappear and the system barely moves.

When pendulum A undergoes free oscillation at its natural frequency, it generates a driving force that acts on the other pendulums through the connecting string. This external driving force causes pendulums B, C, and D to oscillate, even though they are not directly displaced. The energy from pendulum A's motion is transferred to the other pendulums, forcing them to oscillate at the driving frequency established by pendulum A.