# Uncertainty Representation of Natural Frequency for
## 2 Problem Description
\tag{9}\]  
where \(U_{m}\), \(V_{m}\) and \(W_{m}\) are the amplitude in the \(r\), \(\theta\) and \(x\) directions, \(k_{ns}\) is the axial wave number, \(m\) and \(n\) are the axial and circumferential mode orders, \(\omega\) is the circular frequency. For the different boundary conditions, there are different ways to calculate the axial wave number \(k_{ns}\), which can use the calculation method of the axial wave number in the beam model to approximate the axial wave number in composite layered cylindrical shells [51]. The axial wave numbers under different boundary conditions are given in the Table 1.  
By substituting Equation (9) into Equation (1), the differential equation of free vibration of the laminated composite cylindrical shells can be expressed as:  
\[[T_{3\times 3}]\big{[} \begin{array}{cc}U_{ns}&V_{ns}&W_{ns}\end{array}  \big{]}^{T}=0, \tag{10}\]  
The concrete expressions of the matrix \([T_{3\times 3}]\) are described as:  
\[\left\{ \begin{array}{l}T_{11}=-\frac{1}{R^{2}}\Big{(}A_{11}\lambda^{2}+A_{6 6}n^{2}-\Omega^{2}hE_{2}\Big{)},\\ T_{12}=-\frac{1}{R^{2}}(A_{12}+A_{66})\lambda n,\\ T_{13}=-\frac{1}{R^{2}}A_{12}\lambda i,\\ T_{21}=T_{12},\\ T_{22}=-\frac{\lambda^{2}}{R^{2}}\Big{(}A_{66}+\frac{2D_{66}}{R^{2}}\Big{)}- \frac{n^{2}}{R^{2}}\Big{(}A_{22}+\frac{D_{22}}{R^{2}}\Big{)}+\frac{E_{2}h \Omega^{2}}{R^{2}},\\ T_{23}=-\frac{1}{R^{2}}A_{22}ni-\frac{1}{R^{4}}(D_{12}+2D_{66})\lambda^{2}ni- \frac{D_{22}n^{3}i}{R^{4}},\\ T_{31}=-T_{13},\\ T_{32}=\frac{1}{R^{2}}A_{22}ni+\frac{1}{R^{4}}(D_{12}+4D_{66})\lambda^{2}ni+ \frac{D_{22}n^{3}i}{R^{4}},\\ T_{33}=-\frac{1}{R^{4}}D_{11}\lambda^{4}-\frac{2}{R^{4}}(D_{12}+2D_{66}) \lambda^{2}n^{2}-\frac{D_{22}n^{4}}{R^{4}}-\frac{A_{22}}{R^{2}}+\frac{E_{2}h \Omega^{2}}{R^{2}},\end{array} \right. \tag{11}\]  
where \(\Omega\) represents the dimensionless natural frequency (ONF) of the laminated composite cylindrical shells, which can be calculated by \(\Omega=\omega R\sqrt{\frac{\rho}{E_{2}}}\). When the composite material is determined, there is only a proportional relationship between the DNF and the natural \begin{table}
\begin{tabular}{c c} \hline
**Boundary Condition** & **Axial Wave Number \(k_{ns}\)** \\ \hline Free–free & \(k_{ns}X\)\(=(2\mathrm{m}+1)\pi/2\) \\ Clamped–clamped & \(k_{ns}X\)\(=(2\mathrm{m}+1)\pi/2\) \\ Free–clamped & \(k_{ns}X\)\(=(2\mathrm{m}-1)\pi/2\) \\ Free–Supported & \(k_{ns}X\)\(=(4\mathrm{m}+1)\pi/2\) \\ Clamped–supported & \(k_{ns}X\)\(=(4\mathrm{m}+1)\pi/2\) \\ Supported–supported & \(k_{ns}X\)\(=\mathrm{m}\pi\) \\ \hline \end{tabular}
\end{table} 
Table 1: The calculation method of the axial wave number under different boundary conditions. frequency. Therefore, the DNF can be used to replace the natural frequency to represent. To give a non-zero solution, the determinant of coefficient matrix must be 0. \[\left| \begin{array}{ccc}T_{11}&T_{12}&T_{13}\\ T_{21}&T_{22}&T_{23}\\ T_{31}&T_{32}&T_{33}\end{array} \right|=0, \tag{12}\]  
The DNF \(\Omega\) of laminated composite shells with finite length can be obtained by solving the coefficient matrix, and the determinant is transformed into an algebraic equation of DNF, which can be expressed as  
\[\Omega^{6}+\alpha_{1}\Omega^{4}+\alpha_{2}\Omega^{2}+\alpha_{3}=0, \tag{13}\]  
where \(\alpha_{i}(i=1,2,3)\) are the coefficients of an algebraic equation of DNF, which are the functions of radius, thickness, longitudinal elastic modulus, transverse elastic modulus, Poisson's ratio, ply angle and other parameters. By observing Equation (13), it can be found that there are three pairs of characteristic roots, including three positive roots and three negative roots, for each natural frequency. The three positive roots, respectively, represent the free vibration frequency of the axial, circumferential and radial of the laminated composite cylindrical shells, where the minimum positive root corresponds to bending vibration, and the other two positive roots correspond to in-plane vibration [52].