Natural Characteristics Analysis of a Dual-Rotor System with Nonparametric Uncertainty

Abstract

In order to evaluate the impacts of parameter uncertainty and nonparametric uncertainty on the natural characteristics of a dual-rotor system, a nonparametric probabilistic method based on random matrix theory is proposed. In this paper, a nonparametric Riccati whole transfer model is derived based on the maximum entropy principle and the random matrix theory. It is used to model a dual-rotor system with nonparametric uncertainty, as well as to calculate the natural characteristics of the system. Furthermore, the impacts of parameter uncertainty and nonparametric uncertainty on the natural characteristics at the intermediate support element and at the disk-shaft element are discussed using numerical simulations, and the results are compared with related references. The results show that at the same level of uncertainties, the effect of nonparametric uncertainty is often more significant than that of parameter uncertainty. The effects of uncertainties also increase with the level of uncertainties. The results of this paper provide a theoretical basis for the design of uncertain dual-rotor and multi-rotor systems.

1. Introduction

Rotor systems are widely used in aero-engines, gear transmissions, large steam wheels, and other machines [1]. In some cases, the operating conditions of these systems are pretty serious. Moreover, they will cause significant damage to economic and personal safety if an accident occurs. Therefore, it is essential to carry out a rigorous dynamic characteristics analysis for rotor systems in the design process [2,3]. So far, many studies on rotordynamics modeling methods have been published. However, there are a lot of uncertainties in the whole life cycle of a device from design, to manufacturing, to operation, to maintenance, which have great impacts on the operation of the device. These uncertainties can then be split into two classes, i.e., aleatoric uncertainty and epistemic uncertainty [4]. The former is mainly caused by changes in operating conditions and errors and is always used to consider the uncertainty of the geometric parameters and physical parameters of the system [5], also known as parameter uncertainty, which has already received great attention and sufficient research in recent years; the latter is induced from the difficulty of accurately modeling the dynamic characteristics of a system [6,7], known as nonparametric uncertainty, which is mainly introduced by modeling the actual systems abstractly and has become a hot topic in the field of stochastic rotor dynamics.

It is the stochastic parametric method that is used to deal with the parametric uncertain problem. In essence, it allows some random variables to be used to describe the uncertain parameters in the dynamic model, and then, one can combine it with the finite element method (FEM) and the Monte Carlo simulation (MCS) to study uncertainties. Nevertheless, it cannot take nonparametric uncertainty induced by model errors and calculation errors into account [8]. For instance, Fu et al. [9,10] proposed a noninvasive metamodeling method to quantify the parameter uncertainty of a dual-rotor system and calculated the steady-state nonlinear dynamic response of the system under interval uncertainty. Wang et al. [11] proposed the non-probabilistic Chebyshev convex method (CCM) and calculated critical speeds and dynamic responses. The results indicate that the response of the rotor system is no longer deterministic, but rather, it varies in an interval. For nonparametric uncertainty, a nonparametric probabilistic method based on random matrix theory (RMT) has been proposed and widely used, so it will be studied in this paper. Soize [12,13,14,15], who introduced RMT into structural dynamics and developed a large number of fundamental theories, conducted pioneering research in the study of the stochastic dynamics of nonparametric uncertainty systems. Murthy et al. [16] introduced the maximum entropy principle (MEP) into the nonparametric modeling method and obtained a nonparametric transfer model of a single rotor system, further analyzing the action of uncertainties on the eigenvalues of the system. Gan et al. [17,18] not only constructed a nonparametric model of the classic Jeffcott rotor but also studied the effects of nonparametric uncertainty on the critical speeds and unbalanced responses of the rotor with different disk-offsets via numerical simulation. Huang et al. [19] used a nonparametric method to model a high-speed motorized spindle with uncertainty and discussed vibration responses. However, the effects of the two uncertainties were not distinguished. Liu et al. [20] proposed a nonparametric model of a dual-rotor system and investigated the effects of two uncertainties on the critical speeds of the system but did not discuss the impacts on the vibration modes. The above studies show that it is feasible to use the nonparametric modeling method to analyze rotor systems with uncertainty. Nevertheless, most of the investigations focus on the critical speeds and dynamic responses of a single-rotor system. Furthermore, both types of uncertainties have an effect on vibration modes, but the current research has only considered parameter uncertainty. In this paper, the impacts of nonparametric uncertainty on the vibration modes of dual-rotor and multi-rotor systems are discussed, which makes up for the shortcomings of the current research.

In this paper, a nonparametric Riccati whole transfer model is proposed for modeling uncertain dual-rotor systems. The primary purposes are calculating the natural characteristics of the uncertain dual-rotor system, such as critical speeds and vibration modes. Furthermore, the changes in the natural characteristics with uncertainties and the effects of two types of uncertainties on the natural characteristics under the same level of uncertainty are analyzed. First, the deterministic model and the nonparametric model of a dual-rotor system are described in Section 2 and Section 3, respectively. Then, the impacts of two types of uncertainties on natural characteristics are studied using multiple numerical simulations in Section 4. Finally, in Section 5, the conclusions will be drawn.

2. Deterministic Model of a Dual-Rotor System

2.1. Calculation Model of a Dual-Rotor System

This paper cites the calculation model from Reference [21], and a structure diagram of a dual-rotor system with a single intermediary support is shown in Figure 1. The system is divided into 10 nodes according to the concentrated mass rigid disk, massless elastic shaft segment, and support. It consists of two rotors: rotor I is an inner rotor, and rotor II is an outer rotor. Rotor I is connected to the base by the first and sixth nodes, while the left end of rotor II is connected to the base by the seventh node, and the right end is connected to rotor I by the tenth node. The detailed parameters are provided in

Table 1. The parameters of the dual-rotor system.

Rotor	Speed ω/(rad·s−1)	Stiffness of Rotor J/(m4)	Station	Mass m/kg	Moment of Inertia Ip/(kg·m2)	Stiffness of Element k × 10−7	Length l/(m)
I	1047.2	2.6467 × 10−9	1	0.05770	0	0.6269	0.0762
			2	10.7023	0.0859		0.1778
			3	0.24990	0		0.1524
			4	0.15380	0	0.8756	0.0508
			5	7.08690	0.0678		0.0508
			6	0.03850	0	1.7513
II	1570.8	2.1935 × 10−3	7	0.04699	0	1.7513	0.0508
			8	7.20200	0.0429		0.1524
			9	3.69200	0.0271		0.0508
			10	0.04699	0	0.8756

.

From Figure 1, the number of nodes in rotor I and II are unequal, which is counter to constructing a whole transfer matrix. Nevertheless, virtual elements with zero mass and zero length are added at both end sections of the dual-rotor system, and the number of elements is reset [22]. The two rotors are also elastomers; the 2nd, 3rd, 5th, 6th, 7th, 8th, 10th, and 11th elements are elastomers, and the 4th and 9th elements are rigid bodies. Therefore, the elastic deformation will also have an impact on the dynamic characteristics of the system, which must be taken into account [23,24,25]. This problem will be studied in the following section. The dual-rotor system, after resetting, is shown in Figure 2. All the derivation processes described later in this paper are based on it.

The state vector of the ith node of rotors I and II can be written as ${\mathit{Z}}_i^I$ = [M^I, Q^I, x^I, θ^I]${}_i^T$ and Z${}_i^{\mathit{II}}$ = [M^II, Q^II, x^II, θ^II]${}_i^T$, where the parameters M, Q, x, and θ are the bending moment, shear force, displacement, and rotation angle, respectively. To facilitate the construction of the transfer model in the overall elements, Z${}_i^I$ and Z${}_i^{\mathit{II}}$ are combined in the state vector, Z_i.

$${\mathit{Z}}_i=\left[\begin{array}{c}{\mathit{Z}}_i^I\\ {\mathit{Z}}_i^{II}\end{array}\right]$$

(1)

Since this system is a complex rotor system composed of two rotors, the traditional transfer matrix method for a single-rotor system is no longer applicable. Therefore, the whole transfer matrix method is introduced to build the whole transfer model of this dual-rotor system. In the deterministic modeling of this system, the coupling actions at the coupling element need to be considered. In contrast, such effects need not be considered in the uncoupled element. The mean transfer model is usually used to describe a deterministic dynamic system. From transfer matrix theory [22], it is known that the state vectors of adjacent nodes agree with the following deterministic transfer model.

$${\mathit{Z}}_{i+1}={\underset{\_}{\mathit{T}}}_i{\mathit{Z}}_i=\{\begin{array}{l}{\mathit{C}}_i{\underset{\_}{\mathit{U}}}_i{\mathit{Z}}_i\mathrm{coupling}\mathrm{element}\\ {\underset{\_}{U}}_i{Z}_i\mathrm{uncoupled}\mathrm{element}\end{array}$$

(2)

where T_i is the whole mean transfer matrix of each element, and U_i is that of the uncoupled element. At the same time, C_i is the coupling matrix.

2.2. The Riccati Whole Mean Transfer Model of the Uncoupled Element

Reference [26] points out that a maximum entropy random matrix can be constructed by combining RMT and MEP. Nevertheless, the whole mean transfer matrix has to be known in this process. As a consequence, this Section 2.2 and Section 2.3 will focus on how to construct the Riccati whole mean transfer matrix of any element. Meanwhile, the construction method of the maximum entropy random matrix will be provided in detail in Section 3.1 and Section 3.2.

There are no intermediary supports in the uncoupled elements. Consequently, the transfer matrices need not consider the action of the coupling matrix. That is, the coupling term in the whole mean transfer matrices is zero. Therefore, the U_i can be expressed as

$${\underset{\_}{\mathit{U}}}_i={\left[\begin{array}{cc}{\underset{\_}{\mathit{U}}}^I& \mathbf{0}\\ \mathbf{0}& {\underset{\_}{\mathit{U}}}^{\mathit{II}}\end{array}\right]}_i$$

(3)

where U ^j_i can be written as

$${\underset{\_}{\mathit{U}}}_i^j={\left[\begin{array}{cccc}1& l^j& \left(m^j{\omega }^2-k_s^j\right)l^j& \left(J_p^j-J_d^j\frac{{\Omega }^j}{\omega }\right){\omega }^2\\ 0& 1& m^j{\omega }^2-k_s^j& 0\\ \frac{{\left(l^j\right)}^2}{2{\left(EI\right)}^j}& \frac{{\left(l^j\right)}^3(1-\gamma )}{6{\left(EI\right)}^j}& 1+\frac{\left(m^j{\omega }^2-k_s^j\right){\left(l^j\right)}^3(1-\gamma )}{6{\left(EI\right)}^j}& l^j+\frac{\left(J_p^j-J_d^j\frac{{\Omega }^j}{\omega }\right){\left(l^j\right)}^2{\omega }^2}{2{\left(EI\right)}^j}\\ \frac{l^j}{{\left(EI\right)}^j}& \frac{{\left(l^j\right)}^2}{2{\left(EI\right)}^j}& \frac{\left(m^j{\omega }^2-k_s^j\right){\left(l^j\right)}^2}{2{\left(EI\right)}^j}& 1+\frac{\left(J_p^j-J_d^j\frac{{\Omega }^j}{\omega }\right)l^j{\omega }^2}{{\left(EI\right)}^j}\end{array}\right]}_i$$

(4)

where m ^j_i, l ^j_i, E ^j_i, I ^j_i, and Ω^j_i represent the concentrated mass, shaft segment length, elastic modulus, section moment of inertia, and rotation angular velocity of the jth rotor at the ith element, respectively; J_p ^j_i and J_d ^j_i are the polar moment of inertia and the diametric moment of inertia at the rigid thin disk, respectively; k_s^j_,i is the total stiffness of the connection between the shaft segment and base; ꞷ is the angular velocity of precession; γ = 6 EJ/(ktGAl2) is the shear influence coefficient; G and A are the material shear modulus and cross-section area; and k_t is the cross-section coefficient (0.886 for solid shafts and 2/3 for thin-walled hollow shafts).

From Equations (3) and (4), the whole mean transfer matrices of the uncoupled elements can be derived. Yet, with a large number of nodes in the system, a computer value overflow will occur; furthermore, it may appear that a large number subtracts another large number in calculating the vibration modes, which will decrease the calculation accuracy and result in the vibration mode distortion [27]. To avoid the above matters, the Riccati whole mean transfer matrices, S_i, of the uncoupled elements are constructed by using the method in Reference [21], and the calculation formula is as follows:

$$\begin{array}{c}{\underset{\_}{\mathit{S}}}_{i+1}=\left({\left[\begin{array}{cc}{\underset{\_}{\mathit{U}}}_{11}^I& \mathbf{0}\\ \mathbf{0}& {\underset{\_}{\mathit{U}}}_{11}^{II}\end{array}\right]}_i{\underset{\_}{\mathit{S}}}_i+{\left[\begin{array}{cc}{\underset{\_}{\mathit{U}}}_{12}^I& \mathbf{0}\\ \mathbf{0}& {\underset{\_}{\mathit{U}}}_{12}^{II}\end{array}\right]}_i\right)\cdot \\ {\left({\left[\begin{array}{cc}{\underset{\_}{\mathit{U}}}_{21}^I& \mathbf{0}\\ \mathbf{0}& {\underset{\_}{\mathit{U}}}_{21}^{II}\end{array}\right]}_i{\underset{\_}{\mathit{S}}}_i+{\left[\begin{array}{cc}{\underset{\_}{\mathit{U}}}_{22}^I& \mathbf{0}\\ \mathbf{0}& {\underset{\_}{\mathit{U}}}_{22}^{II}\end{array}\right]}_i\right)}^{-1}\end{array}$$

(5)

where ${\underset{\_}{\mathit{U}}}_{11,i}^j,{\underset{\_}{\mathit{U}}}_{12,i}^j,{\underset{\_}{\mathit{U}}}_{21,i}^j,{\underset{\_}{\mathit{U}}}_{22,i}^j$ represent the two-order block matrix obtained by dividing the whole mean transfer matrix, U_i, and S₁ is a four-order zero matrix.

2.3. The Riccati Whole Mean Transfer Model of the Coupling Element

The transfer relationship between state vectors Z_i+1 and Z_i on the left and right nodes at the coupling elements is provided in Equation (2), which can be expressed as

$${\mathit{Z}}_{i+1}={\underset{\_}{\mathit{U}}}_{C,i}{\mathit{Z}}_i$$

(6)

$${\underset{\_}{\mathit{U}}}_{C,i}={\mathit{C}}_i{\underset{\_}{\mathit{U}}}_i$$

(7)

where U_C,i represents the whole mean transfer matrix at the coupling elements, and the coupling matrix C_i can be expressed as [21]

$${\mathit{C}}_i={\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& -k_c& 0& 0& 0& k_c& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& k_c& 0& 0& 1& -k_c& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]}_i$$

(8)

where k_c represents the linear stiffness of the intermediate supports.

The whole mean transfer matrix at the coupling element is transformed into the Riccati whole mean transfer matrix, S_C,i, and then, the process can be calculated as follows:

$${\underset{\_}{\mathit{S}}}_{C,i}^R={\underset{\_}{\mathit{S}}}_{C,i}^L+{\underset{\_}{\mathit{R}}}_i$$

(9)

where the superscripts R and L represent the right and left end sections at the coupling element, and R_i is the stiffness matrix at the intermediary support element, which can be expressed as

$${\underset{\_}{\mathit{R}}}_i={\left[\begin{array}{cccc}0& 0& 0& 0\\ -k_c& 0& k_c& 0\\ 0& 0& 0& 0\\ k_c& 0& -k_c& 0\end{array}\right]}_i$$

(10)

3. Nonparametric Model and Eigenvalue Solution

3.1. Probabilistic and Statistical Characteristics of Maximum Entropy Stochastic Models

The RMT was first proposed by Wishart [28] in statistics, and then it caught wide attention in the field of nuclear physics and other fields. In recent years, Soize 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, B, into a positive definite form. The singular values decomposition (SVD) of B is performed as follows:

$$\mathit{B}=\mathit{M}\sum {\mathit{V}}^T$$

(11)

where M and V are both unitary matrices, and V^TV = I is an identity matrix. According to this property, we can obtain the following expression:

$$\mathit{B}=\mathit{D}\mathit{A}$$

(12)

$$\{\begin{array}{l}\mathit{D}=\mathit{M}{\mathit{V}}^T\\ \mathit{A}=\mathit{V}\sum {\mathit{V}}^T\end{array}$$

(13)

where D is a unitary matrix, and A is a symmetric positive definite random matrix.

From Equations (11)–(13), it can be known that when constructing the random matrix, B, it can be transformed into the symmetric positive definite random matrix, A, through matrix transformation. Assuming p_A is the probability density function of A, it must satisfy the three constraints of a stochastic system.

$${\displaystyle \int p_{\mathit{A}}}\left(\mathit{A}\right)\tilde{d}\mathit{A}=1$$

(14)

$${{\displaystyle \int \mathit{A}p}}_{\mathit{A}}\tilde{d}\mathit{A}=\underset{\_}{\mathit{A}}$$

(15)

$${\displaystyle \int \ln \left(\det \left(\mathit{A}\right)\right)}{p}_{\mathit{A}}\left(\mathit{A}\right)\tilde{d}\mathit{A}=\nu ,\left|\nu \right|<+\infty$$

(16)

where E[A] = A, and ν is a gamma random variable.

$$\tilde{d}(\mathit{A})=2^{n(n-1)/4}{\displaystyle \prod _{1\le i\le j\le n}d{\mathit{A}}_{ij}}$$

(17)

Under the constraints of Equations (14)–(16), the MEP is used to construct the Lagrangian function to search for the optimal solution, and p_A can be calculated with

$$p_{\mathit{A}}(\mathit{A})=c_A\times {\left(\det \mathit{A}\right)}^{\lambda -1}\times \exp \left(-\frac{n-1+2\lambda }{2}\mathrm{tr}\left\{{\underset{\_}{\mathit{A}}}^{-1}{\mathit{A}}^T\right\}\right)$$

(18)

where c_A is a positive constant, which can be defined by

$$c_A=\frac{{(2\mathsf{\pi })}^{-n(n-1)/4}{((n+1)/2{\delta }^2)}^{n(n-1+2\lambda )/2{\delta }^2}}{\left\{{\displaystyle \prod _{l=1}^n\Gamma ((n+1)/2{\delta }^2+(1-j)/2)}\right\}}$$

(19)

where Γ(x) represents the Gamma function; tr(·) represents the trace of the matrix; λ is the Lagrangian multiplier; and δ is the dispersion control parameter, and it can be calculated with the following formula.

$$\lambda =\frac{1}{2{\delta }^2}\left(1-{\delta }^2\left(N-1\right)+\frac{{\left(tr\underset{\_}{A}\right)}^2}{tr{\left(\underset{\_}{A}\right)}^2}\right)$$

(20)

As can be seen from Equation (20), for a deterministic n, when λ approaches infinity, and δ approaches an infinitesimal value, A approaches A by probability. In actual calculations, δ is an artificially given parameter, and thus, λ can be obtained through Equation (20).

3.2. Generation of Maximum Entropy Stochastic Models

This section will focus on how to generate A. The n-order matrix, A, is symmetric and positive, so the rank of matrix A is n, and it is convenient to use Cholesky decomposition in mathematical software. According to Reference [7], other decomposition methods will cause the two-factor matrix elements after decomposition to not have the properties of Equations (22) and (23). Therefore, Cholesky decomposition was chosen, and it can be calculated as follows:

$$\mathit{A}={\mathit{L}}_A{}^T{\mathit{L}}_A$$

(21)

where the off-diagonal elements, L_n,ij (i < j), of A are calculated with

$${\mathit{L}}_{n,ij}={\sigma }_n{\mathit{U}}_{ij}$$

(22)

where σ_n = δ(n + 1)^−1/2, and U_ij obeys the standard Gaussian distribution. The diagonal elements of A are calculated with

$${\mathit{L}}_n{}_{,ii}={\sigma }_n\sqrt{2\nu }$$

(23)

3.3. Construction of Nonparametric Riccati Whole Transfer Model

The Riccati whole mean transfer model is constructed in Section 2.2 and Section 2.3; furthermore, the construction method of the maximum entropy random matrix is provided in Section 3.1 and Section 3.2. Therefore, the nonparametric Riccati whole transfer model can be obtained by introducing a maximum entropy random matrix to the Riccati whole mean transfer model. At the same time, the following formula can be obtained using Equations (11) and (13):

$${\mathit{B}}^T\mathit{B}={\mathit{A}}^2$$

(24)

The whole random transfer matrix, U_qr (q, r ∈ {1,2}), does not have strict positive definiteness like the random matrix, B; thus, U_qr satisfies the following constraints from Equation (25):

$${\left({\mathit{U}}_{qr}\right)}^T{\mathit{U}}_{qr}={\mathit{A}}^2$$

(25)

where U_qr is the maximum entropy random matrix corresponding to the mean matrix, U_qr, in Equation (5). Similarly, the maximum entropy random matrices, U_C,qr, at the coupling elements can be calculated as follows:

$${\left({\mathit{U}}_{C,qr}\right)}^T{\mathit{U}}_{C,qr}={\mathit{A}}^2$$

(26)

From Equations (25) and (26), U_qr and U_C,qr can be obtained and then substituted into Equations (5) and (9), respectively, to obtain the nonparametric Riccati whole transfer model of each element.

$${\mathit{S}}_{i+1}=\left({\left[\begin{array}{cc}{\mathit{U}}_{11}^I& 0\\ 0& {\mathit{U}}_{11}^{\mathit{II}}\end{array}\right]}_i{\mathit{S}}_i+{\left[\begin{array}{cc}{\mathit{U}}_{12}^I& 0\\ 0& {\mathit{U}}_{12}^{II}\end{array}\right]}_i\right)\cdot {\left({\left[\begin{array}{cc}{\mathit{U}}_{21}^I& 0\\ 0& {\mathit{U}}_{21}^{II}\end{array}\right]}_i{\mathit{S}}_i+{\left[\begin{array}{cc}{\mathit{U}}_{22}^I& 0\\ 0& {\mathit{U}}_{22}^{II}\end{array}\right]}_i\right)}^{-1}$$

(27)

$${\mathit{S}}_{\mathit{C},i+1}={\mathit{S}}_{\mathit{C},i}+{\left[\begin{array}{cc}{\mathit{T}}_{\mathit{C},12}& {\mathit{T}}_{\mathit{C},14}\\ {\mathit{T}}_{\mathit{C},32}& {\mathit{T}}_{\mathit{C},34}\end{array}\right]}_i$$

(28)

From Equations (27) and (28), the nonparametric Riccati whole transfer matrix, S₂ … S_C,_i … S_N+1, of each element in the system can be successively obtained. As such, a nonparametric Riccati whole transfer model of a dual-rotor system has been constructed successfully.

3.4. Solving the Critical Speeds and Vibration Modes

The start and end sections of the dual-rotor system shown in Figure 2 are both free ends, i.e., free boundary conditions.

$${\left[M^j,Q^j\right]}_{N+1}^T={\left[M^j,Q^j\right]}_1^T={\left[\begin{array}{cc}0& 0\end{array}\right]}^T$$

(29)

From the free boundary conditions and the nonparametric Riccati whole transfer matrices, [S]_N+1, the synchronous precession equation of the uncertain dual-rotor system can be obtained.

$$\left|{\mathit{S}}_{N+1}\right|=0$$

(30)

Although Equation (30) satisfies the above boundary conditions, the result includes not only the critical speeds of the system but also many singular points. These singular points are the zero points of the residual ∆(ω) from negative infinity to positive infinity, not the critical speed points of the system. As a consequence, the following formula, which can eliminate these singular points, is used to calculate the critical speeds.

$$\Delta \left(\omega \right)={\left|\mathit{S}\right|}_{N+1}{\displaystyle \prod _{i=1}^N\left|{\left[u_{21}\mathit{S}+u_{22}\right]}_i\right|}=0$$

(31)

The frequency scanning method is used to solve Equation (31), and the obtained roots are each order critical speeds of the system.

$${\left\{E\right\}}_i={[{\mathit{U}}_{21}(\mathit{S}+\mathit{R})+{\mathit{U}}_{22}]}_i^{-1}{\left\{E\right\}}_{i+1}$$

(32)

The above formula is the recursive formula of the deformation state vector, where {E}_i is the deformation state vector. U₂₁,_i and U₂₂,_i can be defined by

$$\{\begin{array}{l}{\mathit{U}}_{21,i}={\left[\begin{array}{cc}{\mathit{U}}_{21}^I& 0\\ 0& {\mathit{U}}_{21}^{II}\end{array}\right]}_i\\ {\mathit{U}}_{22,i}={\left[\begin{array}{cc}{\mathit{U}}_{22}^I& 0\\ 0& {\mathit{U}}_{22}^{II}\end{array}\right]}_i\end{array}$$

(33)

According to the properties of Riccati transfer matrix, the proportional solution {E}_i can be obtained. Let ${\theta }_{N+1}^{II}=1$, the normalized state vector Z_N+1 of the end section in the system can be obtained. From Equation (32), the state vectors of each element in the system can be obtained, thus each order vibration mode of the dual-rotor system can be obtained.

3.5. Calculation Process

As shown in Figure 3, the flow chart for calculating the dynamic characteristics of an uncertain dual-rotor system is provided. The specific steps are as follows: ① The structural parameters, material parameters, and operating parameters of the dual-rotor system are input. ② According to the structural characteristics of the system, the calculation model of the system is determined and the elements are divided. ③ A deterministic transfer model is constructed. ④ Through matrix transformation, the random asymmetric positive definite matrix, B, is decomposed into the product of a strict positive definite matrix, A, and a unitary matrix, D. ⑤ Based on MEP and RMT, a nonparametric transfer model is constructed. ⑥ A dynamic model with nonparametric uncertainty is obtained. ⑦ The natural characteristics of the system are calculated.

4. Impacts of Two Uncertainties on Natural Characteristics

In this section, the nonparametric Riccati whole transfer model derived in Section 2 and Section 3 is used to model the uncertain dual-rotor system shown in Figure 2. Through a great number of numerical simulations, under the same level of uncertainty, the effects of two types of uncertainties on the natural characteristics of the system are explored.

4.1. Impacts of Two Uncertainties at Intermediate Support Elements

In this section, the effects of parameter uncertainty and nonparametric uncertainty on the natural characteristics of the dual-rotor system are studied via numerical simulations. As shown in Figure 2, the ninth element is the intermediate support element connecting two rotors, and its uncertainties will greatly affect the whole transfer model. Therefore, the uncertainties at the intermediate support elements have to be analyzed under the same level of uncertainty so that one can compare the contribution of two types of uncertainties in the system. Due to the significant amount of calculation, only 100 numerical simulations utilizing the nonparametric model (NM) were finished. In comparison, 100 MCS were executed based on the parameter model (PM) proposed in Reference [29]. Furthermore, the results calculated via the mean model (MM) are provided.

In actual engineering applications, engineers often focus only on the low-order modes, because they have large amplitude, large equivalent mass, and are the most destructive. Therefore, only the first three-order modes of the system are analyzed in this paper.

Figure 4 shows the first three-order critical speeds of the system with a different C_ov,k. C_ov is the coefficient of variation, which is equivalent to δ in Equation (19). It is often set as 0.1 in practice [11]. In this paper, C_ov is set as 0–0.20 (0, 0.05, 0.10, 0.15, and 0.20) to study the impacts of two types of uncertainties on natural characteristics. In order to evaluate the impacts of uncertainties on the natural characteristics of the system, relative errors were introduced to characterize the changes in the first three-order critical speeds. It can be seen that the two types of uncertainties at the intermediary support element have similar impacts on the first three-order critical speeds from

Table 2. Relative errors of the first three-order critical speeds of the system with two uncertainties at the intermediate support elements.

	Parameter Uncertainty			Nonparametric Uncertainty
Cov, k	nc1	nc2	nc3	nc1	nc2	nc3
0.05	0.010%	0.040%	0.591%	1.292%	0.624%	2.067%
0.10	0.014%	0.084%	1.154%	2.727%	1.331%	4.135%
0.15	0.037%	0.143%	1.589%	3.924%	1.816%	5.892%
0.20	0.035%	0.217%	3.343%	6.268%	2.245%	9.596%

; that is, the relative errors of the critical speeds will increase with two uncertainties. At the same time, there are some differences. On the one hand, at the same level of uncertainty, the errors caused by nonparametric uncertainty are significantly larger than those caused by parameter uncertainty, indicating that the first three-order critical speeds are more sensitive to nonparametric uncertainty. Capiez et al. [29] reached a similar conclusion when conducting a quantitative study on the uncertainty of complex aircraft. On the other hand, at the same level of uncertainty, the impacts will increase with the orders of critical speed, indicating that the high-order critical speeds are more sensitive to the parameter uncertainty than the low-order critical speeds. At the same level of nonparametric uncertainty, this rule cannot be established. The errors of n_c2 are the smallest, followed by n_c1, and n_c3 is the largest, which indicates that the impacts of nonparametric uncertainty on different-order critical speeds are different. In short, the critical speeds of the system will vary with its own orders and the type of uncertainty.

The critical speeds reflect the speed range at which the rotor system resonates, but the vibration modes reflect the vibration form of the system, which is also an important natural characteristic of the rotor system. Therefore, it is necessary to analyze the uncertainty of the vibration modes. The modal assurance criteria (MAC) are used to analyze the effects of two types of uncertainties on the first three-order vibration modes [30], which can be calculated as follows:

$$MAC=\frac{{\left|{\left[{\phi }_a\right]}^T\left[{\phi }_b\right]\right|}^2}{{\left[{\phi }_a\right]}^T\left[{\phi }_a\right]{\left[{\phi }_b\right]}^T\left[{\phi }_b\right]}$$

(34)

where [φ_a] and [φ_b] represent the vibration mode vectors obtained from the stochastic models (PM and NM) and the MM, respectively. From a mathematical point of view, the MAC represents the cosine value of two vectors in a specific space. The smaller the MAC value, the greater the angle between the two vectors, and it is easier to distinguish the two vectors. In the field of modal analysis, the smaller the MAC value is, the greater the difference between the vibration modes of the stochastic model and the MM, and the greater the impacts of uncertainties on the system’s natural characteristics. On the contrary, if the MAC value is closer to one, it means that the vibration modes of the stochastic model and MM are more similar, and the effects of uncertainty are lesser.

Figure 5 shows the upper and lower boundaries of the first three-order vibration modes of the system with two uncertainties for C_ov,k = 0.20. In order to show the floating degree of the rotor I and rotor II vibration modes more intuitively, the same order vibration modes of the two rotors were placed in one figure. The long lines represent the vibration modes of rotor I, and the short lines represent that of rotor II. The solid blue lines represent the vibration modes calculated with NM, the solid red lines represent those calculated with MCS, and the dotted green line represents those calculated with MM. These descriptions also apply to the following figures. The impacts of two uncertainties on the vibration modes can be easily observed in Figure 6. The MAC values obtained by MCS are almost identical to one, indicating that the parameter uncertainty has little impact on the first three-order vibration modes of two rotors, which means that the vibration modes are insensitive to parameter uncertainty. At the same time, the MAC values of the first three-order vibration modes of rotor I decreased significantly under nonparametric uncertainty, indicating that the vibration modes of rotor I are sensitive to nonparametric uncertainty.

In short, the natural characteristics of the system are more sensitive to nonparametric uncertainty than to the parameter uncertainty at the intermediate supports.

4.2. Impacts of Two Uncertainties at Disk-Shaft Elements

In an actual dual-rotor system, the mass will also have a significantly large effect on the natural characteristics. In addition, it can be found from Equation (4) that the mass of elements is needed to construct the transfer matrices, so the mass parameter uncertainty will definitely affect the natural characteristics of the system. Therefore, this section will focus on the impacts of mass parameter uncertainty and nonparametric uncertainty at the 10th element on the natural characteristics of the dual-rotor system.

Figure 7 and

Table 3. Relative errors of the first three-order critical speeds of a system with two uncertainties at the disk-shaft element.

	Parameter Uncertainty			Nonparametric Uncertainty
Cov, m	nc1	nc2	nc3	nc1	nc2	nc3
0.05	0.048%	0.157%	0.050%	1.486%	0.413%	0.100%
0.10	0.096%	0.283%	0.135%	4.025%	1.002%	0.189%
0.15	0.168%	0.363%	0.181%	5.319%	1.385%	0.340%
0.20	0.240%	0.659%	0.236%	8.554%	1.831%	0.391%

show the first three-order critical speeds of the system and their relative errors for C_ov,k = 0–0.20, respectively. For n_c1 and n_c2, the results are similar to those in Section 4.1; i.e., the critical speeds of NM envelope those of PM. However, the results have changed for n_c3: the results of PM envelop those of NM, which indicates that the n_c3 of this system is more sensitive to mass parameter uncertainty. Liu [20] drew a similar conclusion in studying the effects of uncertainties on the critical speeds of the rotating system.

As shown in Figure 8, under the impacts of mass parameter uncertainty and nonparametric uncertainty at the disk-shaft element, the first three-order vibration modes change; that is, they are no longer fixed in shape, instead constantly varying within a certain interval. It should be noted that both the second-order and third-order vibration modes display a phenomenon where the vibration modes under parameter uncertainty envelope those under nonparametric uncertainty, which is different from Figure 5, indicating that the mass uncertainty of the disk-shaft elements and the stiffness uncertainty of the intermediate supports have different impacts on the vibration modes. The MAC variation in Figure 9 also confirms the existence of this phenomenon. It may be due to the large mass of the 10th element, resulting in greater contributions from mass parameter uncertainty than the corresponding nonparametric uncertainty.

The above analyses reveal that the mass parameter uncertainty, stiffness parameter uncertainty, and the corresponding nonparametric uncertainty will alter the natural characteristics of the system. It can be concluded that, at the same level of uncertainty, the impacts of parameter uncertainty on the natural characteristics are smaller than that of nonparametric uncertainty in most cases. Moreover, the natural characteristics of the system will always strongly alter with the increase in uncertainties.

5. Conclusions

A nonparametric Riccati whole transfer model of an uncertain dual-rotor system is constructed based on Riccati transfer matrix theory, RMT, and MEP. Via matrix transformation, it is used to model an uncertain dual-rotor system or a multi-rotor system and to calculate the critical speeds and vibration modes of this system. Through multiple numerical simulations, the behavior of stiffness parameter uncertainty at the intermediate supports, mass parameter uncertainty at the disk-shaft elements, and the impact of corresponding nonparametric uncertainties on the natural characteristics of the system are analyzed respectively. The results show that both parameter uncertainty and nonparametric uncertainty will cause critical speeds to transform from deterministic values to interval values, and the vibration modes will also vary in intervals. In most cases, the natural characteristics are more sensitive to nonparametric uncertainty than to parameter uncertainty at the intermediate support elements and disk-shaft elements. However, the mass parameter uncertainty of the disk-shaft elements has a greater impact on the natural characteristics than the nonparametric uncertainty. This may be due to the large mass of the 10th element, thus reinforcing the role of mass parameter uncertainty. The results of this paper will be of great help to researchers in designing reliable dual-rotor and multi-rotor systems.