Free vibrations

A system is in equilibrium when it is at its undisturbed position. If it is displaced from that position and released, a restoring influence acts towards equilibrium. The system moves back towards equilibrium, but its motion carries it past that position. The restoring influence then acts in the opposite direction, causing the system to move back again.

This repeated motion is a free vibration: the system oscillates after release without a continuing periodic driving force.

Natural frequency of free vibrations

The frequency of a free vibration is the number of complete oscillations per second. When damping is weak, this frequency is approximately the system’s natural frequency. The natural frequency depends on the system’s physical properties; for example, the mass and stiffness of a mass–spring system affect its natural frequency.

Damping of free vibrations

Damping transfers energy away from the oscillation. As a result, the amplitude—the maximum displacement from equilibrium—decreases with time. With no continuing driving force to replace the transferred energy, the vibration gradually dies away.

In an ideal system with no damping, no energy is transferred away from the oscillation. The free vibration therefore continues at constant amplitude.

Forced vibrations

A vibrating system can be made to move by a force that repeats regularly. For example, a swing can be driven by repeated pushes. The resulting oscillations are called forced vibrations, and the repeating external force is the driving force.

The driver transfers energy to the system. This can maintain the vibration even when damping dissipates energy, for example through friction or air resistance. In a steady vibration, the energy supplied by the driver balances the energy dissipated over time.

Frequency and amplitude of forced vibrations

When the driving force first acts, the motion may include a temporary response called a transient. In a damped system, this transient dies away. The remaining steady response has the same frequency as the driving force, which need not be the system’s natural frequency.

The amplitude of the steady response is not fixed by the driving frequency alone. It also depends on the strength of the driving force and on properties of the system, including its natural frequency and damping. For the same system and driving frequency, a stronger driving force produces a larger steady amplitude.

Resonance and damping

A system can oscillate at a natural frequency when it is disturbed and then left to oscillate freely. If a repeated external force drives the system at a frequency close to a natural frequency, the oscillations can build to a large amplitude. This response is called resonance.

The driving force transfers energy to the system. Near resonance, it transfers energy effectively over successive oscillations, so the amplitude grows. Damping removes energy from the oscillation, limiting the amplitude that can be reached.

Resonance peak and damping

The amplitude of a driven system depends on the driving frequency. A graph of response amplitude against driving frequency shows a resonance peak near a natural frequency: the response is largest in this region and smaller at frequencies farther from it.

For the same driving force, the amount of damping changes the shape of the peak:

• Weaker damping gives a higher, sharper peak. Large amplitudes occur over a narrower range of driving frequencies.
• Greater damping gives a lower, broader peak. The maximum amplitude is smaller and the response changes less sharply with frequency.

Resonance in a mass–spring system and a string

A mass attached to a spring can oscillate at its natural frequency. If a periodic force drives the mass close to that frequency, the system can resonate and the mass can oscillate with a large amplitude. Damping limits that amplitude.

A driven string can also resonate, but it supports stationary waves at more than one natural frequency. When the driving frequency matches one of these frequencies, a large-amplitude stationary-wave pattern can form. The pattern has nodes, which do not move, and antinodes, where the oscillation amplitude is greatest. Damping limits the amplitude of the pattern.