= Solution
Let the Maxwell branch contain <spring> modulus $G$ and series viscosity $\eta_M$, in parallel with a <dashpot> of viscosity $\eta_s$. The branch <strains> are equal and stresses add. Writing $\sigma_M$ for the Maxwell <stress>,
$$
\dot\gamma=\frac{\dot\sigma_M}{G}+\frac{\sigma_M}{\eta_M},
\qquad \sigma=\sigma_M+\eta_s\dot\gamma.
$$
For $\gamma=\operatorname{Re}(\gamma_0e^{i\omega t})$ and $\tau=\eta_M/G$,
$$
\boxed{G^*(\omega)=i\omega\eta_s+
G\frac{i\omega\tau}{1+i\omega\tau}}.
$$
Multiplying the Maxwell fraction by $1-i\omega\tau$ gives
$$
\boxed{G'=\frac{G(\omega\tau)^2}{1+(\omega\tau)^2},\qquad
G''=\omega\eta_s+\frac{G\omega\tau}{1+(\omega\tau)^2}}.
$$
The sign of the imaginary coefficient depends on the phasor convention; these expressions use $e^{i\omega t}$, so $G''>0$ represents dissipation. In the time domain, the equivalent constitutive relation is
$$
\sigma+\tau\dot\sigma=(\eta_M+\eta_s)\dot\gamma+\tau\eta_s\ddot\gamma.
$$
The original mechanical schematic shows the series <spring>/<dashpot> branch separately from the parallel <dashpot>.
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