= Solution
The printed diagram and the word “parallel” describe a <Kelvin-Voigt model>, not a series <Maxwell fluid>. In the depicted model both elements have displacement $x$ and their <forces> add:
$$
\boxed{F=kx+\mu\dot x.}
$$
For $x=ae^{i\omega t}$ and the convention $G(\omega)=F_{\rm amplitude}/x_{\rm amplitude}$,
$$
\boxed{G(\omega)=k+i\omega\mu,\qquad G'=k,\quad G''=\omega\mu.}
$$
The <storage modulus> and <loss modulus> here are element stiffnesses with units $\mathrm{N\,m^{-1}}$; conversion to stress/<strain> moduli requires a geometric factor. If “response” instead denotes displacement per <force>, it is the compliance $1/(k+i\omega\mu)$. Specifying which ratio is used removes that convention ambiguity. The mean dissipated power is $\mu\omega^2|a|^2/2$, confirming the positive loss sign for $e^{i\omega t}$.
For completeness, the actual series <Maxwell fluid> analogue has a common <force> and additive <spring>/<dashpot> extensions. Thus $\dot x=\dot F/k+F/\mu$. With $\tau=\mu/k$,
$$
G_M(\omega)=\frac{k i\omega\tau}{1+i\omega\tau},\qquad G_M'=\frac{k\omega^2\tau^2}{1+\omega^2\tau^2},\qquad G_M''=\frac{k\omega\tau}{1+\omega^2\tau^2}.
$$
These different functions cannot both describe the printed parallel arrangement. The distinction also changes the <elastic beam>'s response to a constant load.
Back to article page
