## 4 Spatial Discretization  
Spatial discretization is more adaptable than modal discretization when treating structures with complicated shapes, whose explicit mode functions cannot be obtained in a straightforward manner. In this study, we took the beam model with distributed gyroscopes to show the technique of spatial discretization. The segment of beam and segment of gyroscopes were considered as presented in Figure 3. This spatial discretization technique can also be expanded to other irregular structures. Every node of the beam element has six DOFs, three translational displacements (\(u\), \(v\), \(w\)), and three rotational displacements (\(\theta_{x}\), \(\theta_{y}\), \(\theta_{z}\)) along the three coordinates \(x\), \(y\), and \(z\), respectively. The transversal rotational angles are  
\[\theta_{y}=\frac{\partial v}{\partial x}\,,\,\theta_{z}=-\frac{\partial w}{ \partial x} \tag{25}\]  
The displacement vector of an arbitrary position in element \(e\) with length \(l_{e}\) is  
\[[\Delta^{e}(x)]=\left[u,v,w,\theta_{x},\theta_{y},\theta_{z}\right]^{T} \tag{26}\]  
The displacement vector can be expressed using the classical finite element cubic interpolating equation for bending deflections and linear interpolating equation for axial and torsional deflections, so that  
\[[\Delta^{e}(x)]=[N][\delta^{e}] \tag{27}\]  
where \([N]\) is the shape function matrix of the three-dimensional finite element, and the nodal displacement vector is  
\[[\delta^{e}]=\left[u_{1},v_{1},w_{1},\theta_{x1},\theta_{y1},\theta_{z1},u_{2},v_{2},w_{2},\theta_{x2},\theta_{y2},\theta_{z2}\right]^{T} \tag{28}\]  
Equation (27) can be written as  
\[\left\{ \begin{array}{l}u(x)\\ v(x)\\ w(x)\end{array} \right\}=[N_{T}][\delta^{e}],\,\left\{ \begin{array}{l}\theta_{ y}(x)\\ \theta_{z}(x)\end{array} \right\}=[N_{\theta}][\delta^{e}][\theta_{x}(x)]=\left[ N_{\varphi}\right][\delta^{e}] \tag{29}\]where \([N_{T}]\), \([N_{\theta}]\), and \([N_{\varphi}]\) are the translation, bending rotation, and torsional rotation shape function matrices, respectively. The shape function expressions can be found in the available references such as [28; 30; 31]. The element composed of a rigid gyroscope can be assumed as a distributed elastic beam with additional momentum. The \(i\)th gyroscope with finite length \(l_{e,\;ri}\) has the displacements  
\[u_{ri}=u,\;v_{ri}=v,\;w_{ri}=w,\;\theta_{x,ri}=\theta_{x}+\varphi,\;\theta_{y, ri}=\theta_{y},\;\theta_{z,ri}=\theta_{z} \tag{30}\]  
The gyroscope elements share the same features with beam elements except the extra gyroscope rotation angle \(\varphi\). Hence, the kinetic energy an arbitrary element is  
\[T=\frac{1}{2}\int\limits_{0}^{l_{e,i}}(\mathbf{v_{\;}}^{\mathrm{T}}m_{b} \mathbf{v}_{b}+\mathbf{\omega}_{b}^{\mathrm{T}}\mathbf{I}_{b}\mathbf{\omega}_{b})dx+ \Delta_{i,j}\frac{1}{2}\int\limits_{0}^{l_{e,ri}}(\mathbf{v_{\;}}^{\mathrm{ T}}m_{ri}\mathbf{v}_{ri}+\mathbf{\omega}_{ri}^{\mathrm{T}}\mathbf{I}_{ri}\mathbf{ \omega}_{ri})dx \tag{31}\]  
where the symbol \(\Delta_{i,j}\) denotes if the gyroscope \(i\) has been installed on the position \(j\):  
\[\Delta_{i,j}=\left\{ \begin{array}{ll}1&i=j,\\ 0&i\
eq j.\end{array} \right. \tag{32}\]  
The variables and parameters in Equation (31) are stated as follows. The mass density of the beam element and the \(i\)th gyroscope are \(m_{b}\) and \(m_{ri}\), respectively. The translational and angular velocity vectors of the beam and gyroscopes are  
\[\mathbf{v}_{b}=\left[ \begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array} \right],\;\mathbf{\omega}_{b}=\left[ \begin{array}{c}\dot{ \theta}_{x}-\theta_{y}\dot{\theta}_{z}\\ \dot{\theta}_{y}-\dot{\theta}_{z}\theta_{x}\\ \dot{\theta}_{z}+\dot{\theta}_{y}\theta_{x}\end{array} \right] \tag{33}\]  
\[\mathbf{v}_{ri}=\left[ \begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array} \right] \tag{34}\]  
Figure 3: The diagrams of the beam element and gyroscope element.