3 - Perturbation Moment

Given:
\begin{align*}
C_{M_\beta} &= C_{M_\beta}\left(\alpha, C_\mu, M\right) \\
C_\mu &= \frac{\dot{m}V_j}{\frac{1}{2}\rho V^2 c}
\end{align*}
The moment is given by
\begin{equation}
M = \frac{1}{2}\rho V^2c^2C_{m_\beta}-Lx_a
\end{equation}
where \(x_a\) is the distance between the aerodynamic center (location of lift) and the elastic axis. Using trim values for the coefficients,
\begin{equation}
M_{trim} = \frac{1}{2}\rho V^2c^2C_{M_\beta}-\frac{1}{2}\rho V^2 c C_l x_a
\end{equation}
where \(C_l = C_l(\alpha,C_\mu)\).
Also,
\begin{align}
V &= \sqrt{U_T^2+U_P^2} \\
\alpha &= \theta - \tan^{-1}\frac{U_P}{U_T} \approx \theta - \frac{U_P}{U_T} \\
M &= M_{tip}\frac{V}{\Omega R}
\end{align}
The circulatory perturbation equations are
\begin{align}
\label{eq:deltaM}
\delta M_C &= \rho V\delta V c^2C_{M_\beta} + \frac{1}{2}\rho V^2c^2\delta C_{M_\beta} - \rho V \delta V c C_l x_a - \frac{1}{2}\rho V^2 c \delta C_l x_a \\
\label{eq:deltaV}
\delta V &= \frac{U_P\delta U_P + U_T\delta U_T}{V} \\
\delta C_{M_\beta} &= C_{M_{\beta_\alpha}} \delta \alpha + C_{M_{\beta_\mu}} \delta C_\mu + C_{M_{\beta_M}} \delta M \\
\delta \alpha &= \delta\theta - \frac{U_T\delta U_P - U_P\delta U_T}{U_T^2} \\
\delta C_\mu &= \frac{-2C_\mu \delta V}{V} \\
\delta M &= \frac{M_{tip}}{\Omega R}\delta V \\
\label{eq:deltaCl}
\delta C_l &= C_{l_\alpha}\delta \alpha + C_{l_\mu}\delta C_\mu
\end{align}
The non-circulatory moment is
\begin{equation}
M_{NC} = \frac{1}{4}\pi\rho\Omega^2c^3\left[r\left(\frac{1}{4}+\frac{x_a}{c}\right)\ddot{\beta}-r\left(\frac{1}{2}+\frac{x_a}{c}\right)\dot{\theta}-c\left(\frac{3}{32}+\frac{1}{2}\frac{x_a}{c}\right)\ddot{\theta}\right]
\end{equation}
Substituting equations \eqref{eq:deltaV} through \eqref{eq:deltaCl} into \eqref{eq:deltaM}
\begin{align}
\delta M_C &= \rho c^2C_{M_\beta} \left(U_P\delta U_P + U_T\delta U_T\right) \\
&+ \frac{1}{2}\rho \left(U_T^2+U_P^2\right)c^2C_{M_{\beta_\alpha}}\left(\delta \theta - \frac{U_T\delta U_P - U_P\delta U_T}{U_T^2}\right) \nonumber \\
&-\rho c^2C_{M_{\beta_\mu}}C_\mu \left(U_P\delta U_P + U_T\delta U_T\right) \nonumber \\
&+\frac{1}{2}\rho c^2C_{M_{\beta_M}} \sqrt{U_T^2+U_P^2}\left(\frac{M_{tip}}{\Omega R}\right)\left(U_P\delta U_P + U_T\delta U_T\right) \nonumber \\
&-\rho cC_lx_a\left(U_P\delta U_P + U_T\delta U_T\right) \nonumber \\
&-\frac{1}{2}\rho c x_a\left(U_T^2+U_P^2\right)C_{l_\alpha}\left(\delta\theta - \frac{U_T\delta U_P - U_P\delta U_T}{U_T^2}\right) \nonumber \nonumber \\
&+\rho c x_a C_{l_\mu}C_\mu\left(U_P\delta U_P + U_T\delta U_T\right) \nonumber
\end{align}
The terms could be rearranged to group each perturbation (i.e. \(\delta M = \delta U_P(\ldots) + \delta U_T(\ldots) + \delta \theta(\ldots)\)).

The final result is
\begin{equation}
\delta M = \delta M_C + M_{NC}
\end{equation}