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Mechanical Impedance

Let us rewrite the amplitude of a driven oscillator as following:

Z_m is called the mechanical impedance of the oscillator. (We used above = k/m, 2 = a/m). The impedance consists of two parts, a resistive part a, and a reactive part X_m = ( m ω – k/ω). The resistive part arises from damping force and causes loss of energy during oscillations.

The reactive part is dependent on the frequency of the applied force. It contains two terms: a term m ω which is controlled by mass (or inertia) of the oscillator, and another term k/ω controlled by elasticity of the spring (or restoring force.) At high frequencies the inertia term dominates the impedance.

Z_m = m ω        (ω large)

At low frequencies, impedance is controlled by elasticity or spring constant:

Z_m = k/ω          (ω small)

Impedance is minimum when,

k/ω = m ω,     or ω = ω₀

so that     (Zm)_min = a

The frequency ω = ω₀ is called frequency of velocity resonance. At this frequency, amplitude of velocity of the oscillator attains its maximum value. To show that, we have

Hence, velocity amplitude becomes maximum when Z_m is minimum.

Further, we note that phase difference arises because of reactive part of the impedance:

or the velocity remains in phase with force at ω = ω₀.