$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

In the generalized-$\alpha$ method, we evaluate forces and moments at time $t_{n+\alpha_{f}}=\left(1-\alpha_{f}\right)t_{n}+\alpha_{f}t_{n+1}$ and the time rate of change of linear and angular momenta at time $t_{n+\alpha_{m}}=\left(1-\alpha_{m}\right)t_{n}+\alpha_{m}t_{n+1}$ , where $\alpha_{f}$ and $\alpha_{m}$ may be evaluated from the spectral radius for an infinite time step, $\rho_{\infty}$ (Section 3.9↑). For second-order systems these parameters may be evaluated from [151] $$\begin{equation} \alpha_{f}=\frac{1}{1+\rho_{\infty}}\,,\quad\alpha_{m}=\frac{2-\rho_{\infty}}{1+\rho_{\infty}}\,,\label{eq:ga-alphas-2nd} \end{equation}$$ Then, the Newmark parameters are given by $$\begin{equation} \begin{aligned}\beta & =\frac{1}{4}\left(1+\alpha_{m}-\alpha_{f}\right)^{2}\,,\\ \gamma & =\frac{1}{2}+\alpha_{m}-\alpha_{f}\,. \end{aligned} \label{eq:ga-Newmark-params} \end{equation}$$ Accordingly, to solve numerically for $\delta W=0$ over the time domain, we express (6.3.3-4) in the discretized time domain as $$\begin{equation} \delta\mathbf{r}\cdot\left(\mathbf{f}_{n+\alpha_{f}}^{ext}-\dot{\mathbf{p}}_{n+\alpha_{m}}\right)+\delta\boldsymbol{\theta}\cdot\left(\mathbf{m}_{n+\alpha_{f}}^{ext}-\dot{\mathbf{h}}_{n+\alpha_{m}}\right)=0\,,\label{eq:ga-virtual-work} \end{equation}$$ or equivalently, $$\begin{equation} \left[\begin{array}{cc} \delta\mathbf{r} & \delta\boldsymbol{\theta}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{n+\alpha_{f}}^{ext}-\dot{\mathbf{p}}_{n+\alpha_{m}}\\ \mathbf{m}_{n+\alpha_{f}}^{ext}-\dot{\mathbf{h}}_{n+\alpha_{m}} \end{array}\right]=0\,,\label{eq:ga-virtual-work-alt} \end{equation}$$ Thus, the residual vector is given by $$\begin{equation} \left[\mathbf{R}\right]=\left[\begin{array}{c} \mathbf{f}_{n+\alpha_{f}}^{ext}\\ \mathbf{m}_{n+\alpha_{f}}^{ext} \end{array}\right]-\left[\begin{array}{c} \dot{\mathbf{p}}_{n+\alpha_{m}}\\ \dot{\mathbf{h}}_{n+\alpha_{m}} \end{array}\right]\label{eq:ga-residual-vector} \end{equation}$$ where $$\begin{equation} \begin{aligned}\dot{\mathbf{p}}_{n+\alpha_{m}} & =\left(1-\alpha_{m}\right)\dot{\mathbf{p}}_{n}+\alpha_{m}\dot{\mathbf{p}}_{n+1}\\ \dot{\mathbf{h}}_{n+\alpha_{m}} & =\left(1-\alpha_{m}\right)\dot{\mathbf{h}}_{n}+\alpha_{m}\dot{\mathbf{h}}_{n+1} \end{aligned} \,.\label{eq:ga-momenta-interpolation} \end{equation}$$ According to the Newmark integration scheme, $$\begin{equation} \begin{aligned}\dot{\mathbf{p}}_{n+1} & =\frac{\mathbf{p}_{n+1}-\mathbf{p}_{n}}{\gamma\Delta t}+\left(1-\frac{1}{\gamma}\right)\dot{\mathbf{p}}_{n}\\ \dot{\mathbf{h}}_{n+1} & =\frac{\mathbf{h}_{n+1}-\mathbf{h}_{n}}{\gamma\Delta t}+\left(1-\frac{1}{\gamma}\right)\dot{\mathbf{h}}_{n} \end{aligned} \,,\label{eq:ga-momenta-Newmark} \end{equation}$$ where $\mathbf{p}_{n+1}=m\dot{\mathbf{r}}_{n+1}$ and $\mathbf{h}_{n+1}=\mathbf{J}_{n+1}\cdot\boldsymbol{\omega}_{n+1}$ .