## 3 Modal Discretization  
To describe the displacements of the beam elements and gyroscope elements, two reference frames are used: the inertial frame \(F_{b}\) with the origin on one end of the beam on which the displacement of the beam is measured, and the non-inertial frame \(F_{ri}\) on which the rigid rotors are described (Figure 2). The undeformed position vector of an arbitrary small element \(dm\) in the beam is \(\overrightarrow{l}_{m}\) measured in \(F_{b}\), the displacement vector is \(\overrightarrow{u}_{m}\), and the rotational angular vector is \(\overrightarrow{\beta}_{m}\). Similarly, the undeformed position vector and the displacement vector of the element \(dm^{ri}\) on the \(i\)th rotor are \(\overrightarrow{l}_{ri}\), \(\overrightarrow{u}_{ri}\), and \(\overrightarrow{\beta}_{ri}\), respectively. Measured on the non-inertial frame \(F_{ri}\), the position vector of the rotor element is \(\overrightarrow{r}_{ri}\). The rotating velocity of the rotor is \(\overrightarrow{\omega}_{ri}=\mathbf{\Omega}_{i}=\left[\Omega_{i},0,0\right]^{ \mathrm{T}}\) with respect to the frame \(F_{ri}\). Figure 1: Diagram of an Euler beam with distributed gyroscopes. The translational and rotational displacements of the element \(dm\) can be cast into the generalized coordinates by using the beam's modal functions without gyroscopes:  
\[\overrightarrow{u}_{m}=\mathbf{f}_{b}^{T}\mathbf{T}_{m}\boldsymbol{\tau}_{b} \tag{1}\]  
\[\overrightarrow{\beta}_{m}=\mathbf{f}_{b}^{T}\mathbf{R}_{m}\boldsymbol{\tau}_{b} \tag{2}\]  
where \(\mathbf{f}_{b}\) is the matrix of unit vectors of the \(F_{b}\) basis vectors; \(\mathbf{T}_{m}\) and \(\mathbf{R}_{m}\) are the translational and rotational displacement vectors, respectively, the values of which are given by the sine functions of the supported beam modes on the element position; and \(\boldsymbol{\tau}_{b}\) is the generalized modal coordinate variable vector. By the geometry of the elements shown in Figure 2, the total displacements of the beam and the \(i\)th rotor measured in \(F_{b}\) are  
\[\overrightarrow{r}_{m,b}=\overrightarrow{l}_{m}+\overrightarrow{u}_{m} \tag{3}\]  
\[\overrightarrow{r}_{m,ri}=\overrightarrow{l}_{ri}+\overrightarrow{u}_{ri}+ \overrightarrow{r}_{ri} \tag{4}\]  
The corresponding velocities are  
\[\overrightarrow{v}_{m,b}=\dot{\vec{u}}_{m} \tag{5}\]  
\[\overrightarrow{v}_{m,ri}=\dot{\vec{u}}_{ri}+\dot{\vec{\beta}}_{ri}\times \overrightarrow{r}_{ri}+\dot{\omega}_{ri}\times\overrightarrow{r}_{ri} \tag{6}\]  
and the accelerations are  
\[\overrightarrow{a}_{m,b}=\ddot{\vec{u}}_{m,b} \tag{7}\]  
\[\dot{\vec{a}}_{m,ri}=\ddot{\vec{u}}_{ri}+\dot{\vec{\beta}}_{ri}\times(\dot{ \vec{\beta}}_{ri}\times\overrightarrow{r}_{m,ri})+\dot{\vec{\beta}}_{ri} \times(\dot{\omega}_{ri}\times\overrightarrow{r}_{m,ri})+\ddot{\vec{\beta}}_{ ri}\times\overrightarrow{r}_{m,ri}+\dot{\vec{\omega}}_{ri}\times\overrightarrow{r}_{m,ri} \tag{8}\]  
where \(\dot{\vec{\beta}}_{ri}\times(\dot{\vec{\beta}}_{ri}\times\overrightarrow{r}_ {m,ri})\) is small and ignored in Equation (8). The velocities and accelerations can be expressed in the modal discretized variables by substituting Equations (1) and (2) into Equations (5)-(8):  
\[\overrightarrow{v}_{m,b}=\mathbf{f}_{b}^{T}\mathbf{T}_{m,b}\dot{\boldsymbol{ \tau}}_{b} \tag{9}\]  
\[\overrightarrow{u}_{m}=\mathbf{f}_{b}^{T}\mathbf{T}_{m}\boldsymbol{\tau}_{b} \tag{1}\]0  
\[\overrightarrow{u}_{m}=\mathbf{f}_{b}^{T}\mathbf{T}_{m}\boldsymbol{\tau}_{b} \tag{1}\]1 The displacements of the beam and the gyroscopes.