The cone, spider, and voice coil/motor structure of an electrodynamic loudspeaker all exhibit nonlinear, displacement-dependent behavior at low frequencies. Using a tensile/compression loadframe, the displacement-dependent dynamic lumped compliance and the normalized Bl-Product can be measured. The measured force versus displacement data can then be represented by truncated power series expansions. The cone and surround can be combined by the principle of superposition to achieve lumped parameter compliance. The power series coefficients can be inserted into the dynamic loudspeaker's differential equations of low-frequency motion in order to account for the components' nonlinearities. The average nonlinear, displacement-dependent behavior of a set of components is the term-by-term average of the components' power series coefficients. The extremes of the nonlinear behavior are found by multiplying the term-by-term standard deviation by plus and minus three (i.e., plus or minus three standard deviations is a widely used tolerance in automotive original equipment manufacturing). The nominal and extreme response of a loudspeaker in the constant displacement and resonance regions can be predicted by substituting the corresponding power series coefficients into the differential equations of motions and solved by numerical techniques.
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