The name mechanical impedance refers to various response functions of a mechanical system.

If we imagine the system as a "black box" that turns an oscillating quantity received as input into a different quantity as output, the response function gives, as the input frequency varies, the amplitude and phase shift of the output.

Depending on how input and output are defined, we will of course obtain different response functions. For example, in a gear we may be interested in turning one velocity into another velocity, or one displacement into another displacement, or a force into a displacement, etc.

In the canonical case the mechanical impedance of a system is defined as the ratio between the force applied at a point and the velocity of that point. In the mks system it is measured in mechanical ohms (kg/s).

We are usually interested in the linear frequency response of the system, which means that we study the motion of the system when we apply to it, as "input", a force of small amplitude oscillating periodically in time (in particular sinusoidally).

Impedance of simple elements

For a sinusoidal force of angular frequency ${\displaystyle \omega }$ the impedance has a magnitude, the ratio between the amplitudes of force and velocity, and a phase, the phase shift between the two. For the three simplest elements:

• mass m: ${\displaystyle |Z|=m\omega }$. Moving a mass takes more force the faster the oscillation; the velocity lags the force by a quarter of a period.
• spring with spring constant k: ${\displaystyle |Z|=k/\omega }$. The spring is "hard" at low frequencies and yields easily at high ones; the velocity leads the force by a quarter of a period.
• viscous friction ${\displaystyle F=\gamma v}$: ${\displaystyle Z=\gamma }$, independent of frequency; force and velocity are in phase. It is the only one of the three elements that dissipates energy.

The phase shifts of the mass and the spring are opposite: when the two elements are connected, as in the harmonic oscillator, their contributions subtract and the impedance is minimum at the frequency at which ${\displaystyle m\omega =k/\omega }$, that is at the resonance frequency.

Examples

• Calculation of the impedance of the harmonic oscillator
• Calculation of the impedance of a vibrating string

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