# Screw Dynamics of a Multibody System by a
## 2 Momentum Screw  
Screw theory is a useful mathematical tool used to represent motion and force of rigid bodies. In this paper, the concept of a momentum screw to connect the force screw (wrench) and velocity screw (twist) provides a new theoretical basis for the dynamics analysis of a multi-rigid-body system. Suppose that the velocity of a differential mass \(dm\) on the rigid body is \(\mathbf{v}\), then its momentum is  
\[d\mathbf{p}=vdm \tag{1}\]  
where \(d\mathbf{p}\) represents the differential of momentum. In the coordinate frame shown in Figure 1, the differential moment of the momentum of the differential mass is indicated by \(d(_{o}\mathbf{L})\) where \({}_{o}\mathbf{L}\) represents the moment of momentum of the particle about the origin of the coordinate frame:  
\[d(_{o}\mathbf{L})=\mathbf{r}\times\mathbf{v}dm \tag{2}\]  
where \(\mathbf{r}\) represents the position vector of the differential mass \(dm\). Define the differential momentum screw as  
\[d\mathbf{\Gamma}= \begin{bmatrix}d\mathbf{p}\\ d_{\left(o,\mathbf{L}\right)}\end{bmatrix} = \begin{bmatrix}\mathbf{v}dm\\ \mathbf{r}\times\mathbf{v}dm\end{bmatrix}  \tag{3}\]  
where \(\mathbf{\Gamma}\) represents the momentum screw. Therefore, the momentum screw of a rigid body is  
\[\mathbf{\Gamma}= \begin{bmatrix}\int\limits_{\Omega}\mathbf{v}dm\\ \int\limits_{\Omega}\mathbf{r}\times\mathbf{v}dm\end{bmatrix}  \tag{4}\]  
where \(\Omega\) represents the whole space of the rigid body. Because  
\[\mathbf{p}=\int\limits_{\Omega}\mathbf{v}dm=m\mathbf{v}_{C} \tag{5}\]  
where \(C\) represents the center of mass, \(\mathbf{v}_{C}\) is the linear velocity of the mass center. At the same time,  
\[ \begin{array}{ll}{}_{o}\mathbf{L}&=\rho\int\limits_{\Omega}\left(\mathbf{r}_{C}+\mathbf{ r}_{CA}\right)\times\left(\mathbf{v}_{C}+\mathbf{\omega}\times\mathbf{r}_{CA}\right)d \Omega\\ &=\rho\mathbf{r}_{C}\times\mathbf{v}_{C}\int\limits_{\Omega}d\Omega+\rho\int\limits_{ \Omega}\mathbf{r}_{CA}\times\left(\mathbf{\omega}\times\mathbf{r}_{CA}\right)d\Omega\\ &=\mathbf{r}_{C}\times m\mathbf{v}_{C}+\left\{\int\limits_{\Omega}\left[\left(\mathbf{r}_ {CA}^{T}\mathbf{r}_{CA}\right)\mathbf{I}-\left(\mathbf{r}_{CA}\mathbf{r}_{CA}^{T}\right)\right] \rho d\Omega\right\}\mathbf{\omega}\end{array}  \tag{6}\]  
where \(\mathbf{\omega}\) is the angular velocity of the rigid body, \(\mathbf{r}_{CA}\) is the vector from point \(C\) to point \(A\), and \(\mathbf{J}_{C}=\int\limits_{\Omega}\left[\left(\mathbf{r}_{CA}^{T}\mathbf{r}_{CA}\right) \mathbf{I}-\mathbf{r}_{CA}\mathbf{r}_{CA}^{T}\right]\rho d\Omega\). So, we obtain that  
\[{}_{o}\mathbf{L}=\mathbf{r}_{C}\times\mathbf{p}+\mathbf{I}_{C}\mathbf{\omega} \tag{7}\]  
As a result, the momentum screw can be defined as  
\[\mathbf{\Gamma}= \begin{bmatrix}\mathbf{p}\\ {}_{o}\mathbf{L}\end{bmatrix}  \tag{8}\]  
where \(\mathbf{p}=m\mathbf{v}\), which is the momentum of a rigid body; \({}_{o}\mathbf{L}=\mathbf{r}_{C}\times\mathbf{p}+\mathbf{J}_{C}\mathbf{\omega}\), which represents the moment of momentum of the rigid body; and the left subscript \(o\) represents the quantity in the coordinate frame \(oxyz\). Figure 1: Momentum and moment of momentum of a rigid body. Let the resultant force screw acting on the rigid body be  
\[\mathbf{\mathfrak{S}}= \begin{bmatrix}\mathbf{F}\\ \mathbf{r}_{A}\times\mathbf{F}\end{bmatrix}  \tag{9}\]  
where \(\mathbf{r}_{A}\) represents the position vector of an arbitrary point on the action line of the force in the coordinate frame.