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\begin{document}
\title{Generalised-$\alpha$}\date{}
\maketitle
\section{Integration}
The generalized alpha method assumes that the displacement $d$ and the velocity $v$ are integrated as such,
\begin{gather}\label{EQ1}
d_{n+1}=d_n+\Delta{}tv_n+\Delta{}t^2\left(\left(\dfrac{1}{2}-\beta\right)a_n+\beta{}a_{n+1}\right),\\
v_{n+1}=v_n+\Delta{}t\left(1-\gamma\right)a_n+\Delta{}t\gamma{}a_{n+1},
\end{gather}
where $\beta$ and $\gamma$ are two parameters. The integration relationship is identical to that of Newmark emthod.
\section{Updating}
From \eqref{EQ1}, the acceleration $a_{n+1}$ can be expressed as
\begin{gather}
a_{n+1}=\dfrac{1}{\beta\Delta{}t^2}d_{n+1}-\dfrac{1}{\beta\Delta{}t^2}d_n-\dfrac{1}{\beta\Delta{}t}v_n+\left(1-\dfrac{1}{2\beta}\right)a_n.
\end{gather}
Thus, according to chain rule
\begin{gather}
\ddfrac{a_{n+1}}{d_{n+1}}=\dfrac{1}{\beta\Delta{}t^2},\qquad{}\ddfrac{v_{n+1}}{d_{n+1}}=\dfrac{\gamma}{\beta\Delta{}t}.
\end{gather}
\section{EOM}
The EOM is expressed at somewhere between $t_n$ and $t_{n+1}$.
\begin{gather}
Ma_{n+1-\alpha_m}+Cv_{n+1-\alpha_f}+Kd_{n+1-\alpha_f}=F_{n+1-\alpha_f},
\end{gather}
which can also be explicitly shown as
\begin{gather}
\begin{split}
M\left(\left(1-\alpha_m\right)a_{n+1}+\alpha_ma_n\right)+C\left(\left(1-\alpha_f\right)v_{n+1}+\alpha_fv_n\right)+K\left(\left(1-\alpha_f\right)d_{n+1}+\alpha_fd_n\right)\\=\left(1-\alpha_f\right)F_{n+1}+\alpha_fF_n,
\end{split}
\end{gather}
where $\alpha_m$ and $\alpha_f$ are two additional parameters.
\section{Effective Stiffness}
The effective stiffness can be computed as
\begin{gather}
\bar{K}=\left(1-\alpha_m\right)\dfrac{1}{\beta\Delta{}t^2}M+\left(1-\alpha_f\right)\dfrac{\gamma}{\beta\Delta{}t}C+\left(1-\alpha_f\right)K,
\end{gather}
with $\Delta{}d=d_{n+1}-d_n$ be the unknown variable. Once converges,
\begin{gather}
a_{n+1}=\dfrac{1}{\beta\Delta{}t^2}\Delta{}d-\dfrac{1}{\beta\Delta{}t}v_n+\left(1-\dfrac{1}{2\beta}\right)a_n,\\
v_{n+1}=v_n+\Delta{}t\left(1-\gamma\right)a_n+\Delta{}t\gamma{}a_{n+1}.
\end{gather}
Or,
\begin{gather}
\Delta{}a=\dfrac{1}{\beta\Delta{}t^2}\Delta{}d-\dfrac{1}{\beta\Delta{}t}v_n-\dfrac{1}{2\beta}a_n,\\
\Delta{}v=\Delta{}ta_n+\Delta{}t\gamma{}\Delta{}a.
\end{gather}
\end{document}