\documentclass[a4paper,10pt,fleqn]{article}
\usepackage[margin=20mm]{geometry}
\usepackage{mathpazo,amsmath,amsfonts,amssymb}
\newcommand{\ddfrac}[2]{\dfrac{\mathrm{d}~#1}{\mathrm{d}~#2}}
\newcommand{\dt}{\Delta{}t}
\newcommand{\LM}{\varLambda_6W_1}
\newcommand{\LC}{\varLambda_5W_2}
\newcommand{\LK}{\varLambda_3W_3}
\begin{document}
\title{GSSSS}\date{}
\maketitle
\section{Integration}
The GSSSS method assumes that the displacement $d$ and the velocity $v$ are integrated as such,
\begin{gather}\label{EQ1}
d_{n+1}=d_n+\lambda_1\dt{}v_n+\lambda_2\dt^2a_n+\lambda_3\dt^2\left(a_{n+1}-a_n\right),\\
v_{n+1}=v_n+\lambda_4\dt{}a_n+\lambda_5\dt\left(a_{n+1}-a_n\right),
\end{gather}
where $\lambda_n$ are parameters associated with spectral radii.
\section{Updating}
The incremental form is then
\begin{gather}
\Delta{}a=\dfrac{1}{\lambda_3\dt^2}\Delta{}d-\dfrac{\lambda_1}{\lambda_3\dt}v_n-\dfrac{\lambda_2}{\lambda_3}a_n,\\
\Delta{}v=\lambda_4\dt{}a_n+\lambda_5\dt\Delta{}a,
\end{gather}
Thus, according to chain rule
\begin{gather}
\ddfrac{a_{n+1}}{d_{n+1}}=\dfrac{1}{\lambda_3\dt^2},\qquad{}\ddfrac{v_{n+1}}{d_{n+1}}=\dfrac{\lambda_5}{\lambda_3\dt}.
\end{gather}
\section{EOM}
The EOM is expressed at somewhere between $t_n$ and $t_{n+1}$.
\begin{multline}
M\left(\left(1-\LM\right)a_n+\LM{}a_{n+1}\right)+
C\left(\left(1-\dfrac{\LC}{\lambda_5}\right)v_n+\dfrac{\LC}{\lambda_5}v_{n+1}\right)+\\
K\left(\left(1-\dfrac{\LK}{\lambda_3}\right)d_n+\dfrac{\LK}{\lambda_3}d_{n+1}\right)=
\left(1-W_1\right)F_n+W_1F_{n+1}
\end{multline}
\section{Effective Stiffness}
The effective stiffness can be computed as
\begin{gather}
\bar{K}=\dfrac{\LM}{\lambda_3\dt^2}M+\dfrac{\LC}{\lambda_3\dt}C+\dfrac{\LK}{\lambda_3}K
\end{gather}
with $\Delta{}d=d_{n+1}-d_n$ be the unknown variable. Once converges,
\begin{gather}
\Delta{}a=\dfrac{1}{\lambda_3\dt^2}\Delta{}d-\dfrac{\lambda_1}{\lambda_3\dt}v_n-\dfrac{\lambda_2}{\lambda_3}a_n,\\
\Delta{}v=\lambda_4\dt{}a_n+\lambda_5\dt\Delta{}a,
\end{gather}
\end{document}