## 3 Nonparametric Model and Eigenvalue Solution  
### Probabilistic and Statistical Characteristics of Maximum Entropy Stochastic Models  
The RMT was first proposed by [PERSON] [28] in statistics, and then it caught wide attention in the field of nuclear physics and other fields. In recent years, [PERSON] introduced it to structural dynamics and induced the nonparametric probabilistic model to describe the dynamic characteristics of systems with nonparametric uncertainty. After continuous development, the nonparametric modeling method has gradually become the most widely used method to solve uncertainty problems in structural dynamics, vibration noise, rotordynamics, and so on. In RMT, entropy represents the uncertainty degree of a random matrix under known constraints [16]. The MEP indicates that the one with the most entropy is closest to the actual systems among all probability density functions that satisfy the constraints. Reference [26] shows that the key to constructing nonparametric probabilistic models is to propose the maximum entropy random matrix corresponding to the mean matrix. The obtained maximum entropy random matrix is a positive definite matrix; nevertheless, the transfer matrix in the real uncertain dual-rotor system is not always a positive definite matrix, so it is essential to transform the random matrix, \(\mathbf{B}\), into a positive definite form. The singular values decomposition (SVD) of \(\mathbf{B}\) is performed as follows:  
\[\mathbf{B}=\mathbf{M}\sum\mathbf{V}^{T} \tag{11}\]  
where \(\mathbf{M}\) and \(\mathbf{V}\) are both unitary matrices, and \(\mathbf{V}^{T}\mathbf{V}=\mathbf{I}\) is an identity matrix. According to this property, we can obtain the following expression:  
\[\mathbf{B}=\mathbf{D}\mathbf{A} \tag{12}\]  
\[\left\{ \begin{array}{l}\mathbf{D}=\mathbf{M}\mathbf{V}^{T}\\ \mathbf{A}=\mathbf{V}\sum\mathbf{V}^{T}\end{array} \right. \tag{13}\]  
where \(\mathbf{D}\) is a unitary matrix, and \(\mathbf{A}\) is a symmetric positive definite random matrix. From Equations (11)-(13), it can be known that when constructing the random matrix, \(\mathbf{B}\), it can be transformed into the symmetric positive definite random matrix, \(\mathbf{A}\), through matrix transformation. Assuming \(\mathbf{p}_{\mathbf{A}}\) is the probability density function of \(\mathbf{A}\), it must satisfy the three constraints of a stochastic system. \[\int p_{\mathbf{A}}(\mathbf{A})\widetilde{\mathbf{A}}=1 \tag{14}\]  
\[\int\mathbf{A}p_{\mathbf{A}}\widetilde{\mathbf{A}}=\mathbf{\underline{A}} \tag{15}\]  
\[\int\ln(\det(\mathbf{A}))p_{\mathbf{A}}(\mathbf{A})\widetilde{\mathbf{A}}=\
u,\;|\
u|<+\infty \tag{16}\]  
where \(E[\mathbf{A}]=\underline{\mathbf{A}}\), and \(\
u\) is a gamma random variable. \[\widetilde{d}(\mathbf{A})=2^{n(n-1)/4}\prod_{1\leq i\leq j\leq n}d\mathbf{A}_{ij} \tag{17}\]  
Under the constraints of Equations (14)-(16), the MEP is used to construct the Lagrangian function to search for the optimal solution, and \(\mathbf{p_{\mathbf{A}}}\) can be calculated with  
\[p_{\mathbf{A}}(\mathbf{A})=c_{\mathbf{A}}\times(\det\mathbf{A})^{\lambda-1}\times\exp\left(- \frac{n-1+2\lambda}{2}\text{tr}\Big{\{}\underline{\mathbf{A}}^{-1}\mathbf{A}^{T}\Big{\}}\right) \tag{18}\]  
where \(\mathbf{c_{\mathbf{A}}}\) is a positive constant, which can be defined by  
\[c_{\mathbf{A}}=\frac{(2\pi)^{-n(n-1)/4}((n+1)/2\delta^{2})^{n(n-1+2\lambda)/2 \delta^{2}}}{\left\{\prod\limits_{l=1}^{n}\Gamma((n+1)/2\delta^{2}+(1-j)/2) \right\}} \tag{19}\]  
where \(\Gamma(\mathbf{x})\) represents the Gamma function; \(tr(\cdot)\) represents the trace of the matrix; \(\lambda\) is the Lagrangian multiplier; and \(\delta\) is the dispersion control parameter, and it can be calculated with the following formula.