## 2 Composite Flywheel Rotor Modeling  
### Analytical Model Description  
The analytical model has been discussed in several publications, therefore only a brief description will be provided here. While the present study focuses on the solution of a single-rim rotor, the analysis can be generalized to multi-rim rotors as described in [20], variable thickness rotors [21], and functionally graded materials [22]. The stress development in the thick composite rotor is assumed to be axisymmetric, meaning the resultant stresses and strains are independent of the circumferential coordinate. The material used for these rotors is a unidirectional filament-wound FRP composite where the winding angle is taken to be circumferential, i.e., 90\({}^{\circ}\). Hence, the composite is assumed to be transversely isotropic. Additionally, it was assumed the aluminum hub and composite rim are permanently bonded, that is, the model is unable to simulate separation between hub and rim. However, the latter condition is indicated by interfacial radial stress being greater than or equal to zero. Due to axisymmetry, the rotor response must only satisfy the governing equation in the radial direction [23]. The stress equilibrium equation in cylindrical coordinates is given as [24]  
\[\frac{\partial\sigma_{r}}{\partial r}+\frac{\sigma_{r}-\sigma_{\theta}}{r}+ \rho r\omega^{2}=0, \tag{1}\]where \(\sigma_{r}\) and \(\sigma_{\theta}\) are the radial and circumferential hoop stresses, which are the only non-trivial terms in the stress matrix; \(\rho\) is the material density; and \(\omega\) is the rotor angular velocity. The stress-strain relationship is defined as  
\[\left\{ \begin{array}{c}\sigma_{\theta}\\ \sigma_{z}\\ \sigma_{r}\\ \sigma_{\theta z}\end{array} \right\}=\left[ \begin{array}{cccc}Q_{11}&Q_{12 }&Q_{13}&0\\ Q_{21}&Q_{22}&Q_{23}&0\\ Q_{31}&Q_{32}&Q_{33}&0\\ 0&0&0&Q_{66}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{\theta}\\ \varepsilon_{z}\\ \varepsilon_{r}\\ \varepsilon_{\theta z}\end{array} \right\}, \tag{2}\]  
where \([Q]\) is the stiffness matrix and \(\{\varepsilon\}\) is the strain vector. Note that the \(z\)-coordinate is associated with the rotor axial direction. The \([Q]\) matrix is the inverse of the compliance matrix \([S]\)[25], such that  
\[[Q]=[S]^{-1}=\left[ \begin{array}{cccc}S_{11}&S_{12}&S_{13}&0\\ S_{21}&S_{22}&S_{23}&0\\ S_{31}&S_{32}&S_{33}&0\\ 0&0&0&S_{66}\end{array} \right]^{-1}. \tag{3}\]  
Considering Equation (2), the compliance matrix must define the behavior in the circumferential (parallel to fibers), radial (transverse to fibers), axial (transverse to fibers), and shear directions. Since the rotor material is assumed to be transversely isotropic with no applied shear forces, the symmetric matrix simplifies from 10 unique terms to seven. The strain in the circumferential and radial directions can be written as, respectively,  
\[\varepsilon_{\theta}=\frac{u_{r}}{r}\text{ and }\varepsilon_{r}=\frac{ \partial u_{r}}{\partial r}, \tag{4}\]  
where \(u_{r}\) is the displacement in the radial direction and \(r\) is an arbitrary location along the rotor radial direction. Invoking a plane strain assumption, strain in the axial and shear directions is defined correspondingly by Equation (5). The appropriateness of this assumption will be discussed later in this text. \[\varepsilon_{z}=0\text{ and }\varepsilon_{\theta z}=0. \tag{5}\]  
Combining Equations (1), (2), and (4) yields a second order inhomogeneous ordinary differential equation. Solving this equation gives the local displacement and local stress at an arbitrary radius defined as  
\[u_{r}=-\rho\omega^{2}\varphi_{0}r^{3}+C_{1}\varphi_{1}r^{\kappa}+C_{2} \varphi_{2}r^{-\kappa}, \tag{6}\]  
\[\sigma_{r}=-\rho\omega^{2}\varphi_{3}r^{2}+C_{1}r^{\kappa-1}+C_{2}r^{-\kappa- 1}. \tag{7}\]  
The \(C\) parameters are integration constants dependent on the boundary conditions and material properties. The \(\kappa\) and \(\varphi\) coefficients are intermediate terms dependent on the stiffness matrix, defined as follows:  
\[