Uncertainty Quantification for Nonlinear Vibration of Supercritical Drive Shaft With a Dry Friction Damper

Uncertainty Quantification for Nonlinear Vibration of Supercritical Drive Shaft With a Dry Friction Damper

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Liyao Song¹, Meijun Liao¹^,², Weifang Chen¹, Rupeng Zhu¹, Dan Wang¹

International Journal of Mechanical System Dynamics | 2025, 5(3) : 463 - 480

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International Journal of Mechanical System Dynamics | 2025, 5(3): 463-480

• RESEARCH ARTICLE •

Uncertainty Quantification for Nonlinear Vibration of Supercritical Drive Shaft With a Dry Friction Damper

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Liyao Song¹, Meijun Liao¹^,², Weifang Chen¹, Rupeng Zhu¹, Dan Wang¹

Affiliations

¹National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing, People's Republic of China

²National Key Laboratory of Science and Technology on Helicopter Transmission, AECC Hunan Aviation Powerplant Research Institute, Zhuzhou, People's Republic of China

doi: 10.1002/msd2.70028

Outline

Abstract

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The supercritical drive shaft is becoming increasingly popular in helicopter transmission system. Dry friction dampers are specially employed to ensure the supercritical shafts crossing the critical speed safely. Due to design tolerances, manufacturing errors and time-varying factors, the parameters of the damper are inherently uncertain, affecting the safety performance of the rotor system. This paper incorporates these parameter uncertainties to investigate the dynamic response uncertainties of a supercritical shaft and dry friction damper system, which is characterized by its high dimensionality and nonlinear behaviors of rub-impact and dry friction. The nonintrusive Polynomial Chaos Expansion (PCE) is adopted to achieve the propagation of uncertainties in the rotorsystem. To achieve efficient uncertainty quantification for this high-dimensional nonlinear system, a double-layer dimensionality reduction algorithm combining modal superposition with sparse grid technique has been applied. In the computational workflow, the inner layer uses modal superposition and the outer layer uses sparse grid techniques. The stochastic dynamic response of the rotorsystem is analyzed considering the uncertainty of five design parameters of the damper. Furthermore, as a post-processing of the PCE coefficients, the Sobol global sensitivity analysis is conveniently conducted. The influence of individual parameters or groups of parameters on the dynamic response is studied. A multi-objective optimization design for the key parameters is then carried out based on the established PCE model. The dynamic model and optimization design method are verified by experiments. The results will benefit uncertainty quantification analysis of high-dimensional nonlinear rotorsystem.

Key words

modal superposition / rub-impact / sparse grid / supercritical shaft / uncertainty quantification

Cite this Article

Liyao Song, Meijun Liao, Weifang Chen, Rupeng Zhu, Dan Wang. Uncertainty Quantification for Nonlinear Vibration of Supercritical Drive Shaft With a Dry Friction Damper[J]. International Journal of Mechanical System Dynamics, 2025 , 5 (3) : 463 -480 . DOI: 10.1002/msd2.70028

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1 | Introduction

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The supercritical transmission system, with its advantages of high operating speed, small torque transmission, and fewer supports, can better meet the development needs of aircraft transmission systems for high efficiency, reliability, and safety. A typical example is the supercritical tail rotor drive shaft system of helicopters [1, 2]. However, due to their large span and high flexibility, supercritical transmission shafts experience significant lateral vibration issues, which may result in bending or even failure. To ensure the supercritical shafts crossing the critical speed safely, dry friction dampers are specially designed for mitigating the lateral vibrations, as shown in Figure 1.

The performance of dry friction dampers is crucial in determining whether the rotor can safely traverse its critical speed. Therefore, understanding the dynamics of these dampers and optimizing their design is essential for the safe operation of supercritical rotors. There exists rub-impact coupling with stick-slip dry friction forces in the supercritical rotor system with dry friction damper causing strong nonlinearity and complex dynamic behaviors. A series of research achievements have been made on the nonlinear vibration of the supercritical rotor system with dry friction damper. Dżygadło and Perkowski [3-5] first verified the effectiveness of the dry friction damper in damping transcritical vibrations of a helicopter supercritical drive shaft. Based on a macroslip friction model, Özaydın et al. [6, 7] established three low-dimensional dynamical models: a one-dimensional model without clearance, a one-dimensional model with clearance, and a two-dimensional model without clearance. Huang et al. [8] mainly focus on the coupling effect between misalignment and rub-impact, and further studied the effects of viscous internal damping and gyroscopic moment on stability and phase difference of a supercritical drive shaft [9]. Wang et al. [10, 11] built a two-dimensional Jeffcott rotor model to conveniently apply the harmonic balance method and Floquet theory to analyze the stability of the supercritical drive shaft. Zhu et al. [12] studied the self-excited vibration and rub-impact occurring in a supercritical helicopter tail transmission system equipped with floating spline and dry friction damper. In previous studies, the parameters of dry friction dampers are treated as deterministic variables, with a focus on analyzing their impact on dynamic response, lacking parameter design methodologies. However, the parameters are actually often subjected to many potential uncertainties that may rise from design tolerances, manufacturing errors and time-varying factors, such as installation clearance errors, wear between contact interfaces, and so on. These uncertainties will inevitably cause the rotor system response indeterministic, affecting the reliability design. It is crucial to acknowledge and address these uncertainties to ensure the optimal performance and reliability of rotor system. Therefore, an anticipated improvement of previous studies is to incorporate the uncertainties of the parameters for prediction of the rotorsystem response, and to further perform sensitivity analysis and optimization design on the key parameters of the damper.

The uncertainty quantification (UQ) of rotordynamics has attracted increasing attention, and several stochastic methods to rotordynamics have been applied, including classic Monte Carlo Simulations (MCS), perturbation method, Advanced Kriging model and the Polynomial Chaos Expansion (PCE), and both random and interval uncertainties of parameters has been in consideration in the past research [13]. By building statistics from responses obtained by sampling uncertain inputs through a large number of runs, MCS is the most robust but computationally expensive tools. To avoid this, the perturbation method is proposed based on the expansion of random quantities into Taylor series and the Neumann method based on Neumann series expansion [14-16]. But the perturbation method provides acceptable results only for small random fluctuations, and it is an intrusive method not applicable for large freedom and complexity rotor system. Kriging model is also popularly applied to UQ of rotordynamics. Denimal and Sinou [17] proposed a hybrid surrogate-model for UQ and global sensitivity analysis (GSA) of rotordynamics, which combines Kriging and PCE. Ma et al. [18] adopted an advanced Kriging model for propagation of uncertainties in fixed-point rub-impact rotor system. The PCE, which can work in a nonintrusive way, is an efficient uncertainty propagation method by expanding the response onto a particular basis of the probability space [19, 20]. Specifically, by combining harmonic balance method and PCE, Didier et al. proposed a stochastic harmonic balance method [21] and applied it to nonlinear rotor systems with unbalance, misalignment and initial bending to obtain the uncertain nonlinear response [22]. After that, they further proposed multidimensional stochastic harmonic balance method [23] and studied on the stochastic response of rotor with non-regular nonlinearities [24] and local non-linearities [25]. Fu et al. proposed the PCE in combination with the Chebyshev Surrogate Method (CSM) [26] on the uncertainty quantification of rotor systems with both random uncertain parameters and interval uncertain parameters, and the Chebyshev Convex Method [27] on interval uncertain quantification. They further studied the stochastic response of the uncertain notched rotor systems [28] and dual-rotor systems [29, 30]. Zhang et al. [31] analyzed the nonlinear stochastic dynamics of a fixed-point rub-impact rotor system based on harmonic balance method combined with PCE, in which the rotor is simplified as a Jeffcott rotor model. Ma et al. [32] adopted PCE to the UQ and GSA of the self-excited vibration which occurs in the spline-shafting system.

With the advantage of substantial mathematical foundation and excellent performance in uncertainty propagation, PCE is widely applied in UQ of rotor system. In this paper, the PCE method is employed to contribute a stochastic model for the uncertain supercritical rotor and dry friction damper system. While the classic Jeffcott model is straightforward to analyze, its two-degree-of-freedom structure limits the representation of the rotor system's dynamic characteristics. In contrast, finite element models provide richer dynamic information but significantly increase the system's degrees of freedom, requiring substantial computational resources for UQ. In addition, when employing PCE methods for uncertainty quantification, there exists so-called “curse of dimensionality.” To address the challenges of high dimensionality, this paper employs a double-layer dimensionality reduction algorithm combining modal superposition with sparse grid technique. This approach is applied to effectively assess the vibration damping performance of the damper. After that a GSA is conveniently performed by computing the Sobol indices analytically as a post-processing of the PCE coefficients [33].

The rest of the paper is organized as follows. Section 2 describes the dynamic modeling of a supercritical drive shaft with a dry friction damper under uncertainty. Section 3 provides a detailed explanation of the double-layer dimensionality reduction algorithm. Section 4 presents a comprehensive uncertainty propagation analysis along with comparative discussions of various parameter cases. The results are compared with those obtained from MCS to demonstrate the validity and efficiency of the established model. Then a multi-objective optimization design to enhance the vibration damping performance of dry friction damper is carried out. A transcritical vibration experiment of a supercritical rotor with dry friction damper is conducted to verify the effectiveness of dynamic model and optimization method. Finally, the conclusions of the present work are summarized in Section 5.

2 | Dynamic Model of the Rotor System Under Uncertain Parameters

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2.1 | Dynamic Model of the Rotor System

In this study, a typical single-span supercritical transmission system with a dry friction damper is considered, as shown in Figure 2. It consists of a hollow rotating shaft, two isotropic elastic supports and a dry friction damper installed in the middle of the shaft span to ensure that the shaft pass through the first critical speed safely. The damper is mainly composed of a movable rub-impact ring made in half-half type for easy disassembling and assembling, two pre-tightening springs, four friction discs, two bolts, and a base, as illustrated in Figure 2C. There exist clearances

between the rub-impact ring and the shaft, as well as the ring and the bolts. There exists friction on the surfaces between the friction discs and the rub-impact ring, and the friction force can be set by varying the pre-tightening spring force.

In Figure 3A, an inertial coordinate system

is associated with the centroid of the support, and the cross-section of the shaft is assumed to remain in the

plane. The supports at both end of the shaft are modeled ideally by isotropic linear springs with stiffness

. The shaft is modeled by the finite element method with Timoshenko beam theory, divided into N elements, N + 1 nodes. Each node has 4 degrees of freedom, including translation along the y, z directions and rotation around the y, z directions. Considering that the rotor in the system is a slender flexible shaft without a disk, the influence of the gyroscopic effect is minimal, as detailed in the appendix. To facilitate the application of the modal superposition method, the gyroscopic effect is ignored accordingly.

The damper is located at the middle node of the shaft, and the rotation of the rub-impact ring, which has a mass

, is neglected. The nonlinear forces of the damper include rub-impact forces between the shaft and the rub-impact ring, as well as the ring and the bolts, and the dry friction force between rub-impact ring and friction discs. The rub-impact forces which are denoted as

and

, include normal contact forces and tangential friction forces as shown in Figure 3B. The normal and tangential rub-impact force can be expressed as

(1)

(2)

where

, represents the rub-impact between shaft and ring, ring and bolts respectively,

is Heaviside function with

for

, and

is the relative radial displacement at the contact point,

are the contact stiffness,

are the tangential friction coefficient at the interface,

are the relative tangential velocity at the contact points. And their detailed expressions are provided in the appendix. The dry friction force

which opposites to velocity of the ring

, as shown in Figure 3B, can be derived by

(3)

where

for

is the absolute value of the velocity of the ring, with the detailed expression provided in the appendix.

is the pre-tightening force from springs and

is the sliding friction coefficient.

is the critical dry friction force, representing the product of

and

, which determines the transition for the motion of the ring from stick to slip. The unbalance excitation on the shaft can be expressed as

(4)

where

is the equivalent mass of the shaft and

is the eccentric distance,

is the rotation angular of the shaft. Therefore, the equations of motion of the rotorsystem can be expressed as

(5)

where

and

are the mass, stiffness and damping matrices of the shaft, respectively, assembled with the support element. The stiffness matrix of the support node is

. The damping matrix is calculated using Rayleigh damping, with its detailed expressions provided in the appendix.

and

is the mass, stiffness and damping matrices of the ring, and the dynamic model of the ring is established by lumped mass method with

and

being ignored.

are the displacement vectors of the shaft and ring, respectively.

are the forces on shaft and ring, expressed as

(6)

(7)

where

and

are the components in y and z directions of

and

, respectively. The drive shaft is divided into 16 elements and has 64 degrees of freedom. Including the two degrees of freedom from the damper, the system has a total of 66 degrees of freedom, which greatly increases the computational complexity compared to the Jeffcott rotor model.

2.2 | Uncertainty Propagation Based on PCE

The PCE method is applied to derive the uncertain dynamic response. In this system, five parameters of the damper are modeled as random variables, including the mass of the ring

, normal rub-impact stiffness

, tangential rub-impact friction coefficients

, clearance

and critical dry friction force

. These parameters are denoted by vector

, for example,

. All the uncertain parameters are considered to be independent. These uncertain parameters can be converted to standard random variables denoted as

is the dimension of random variables and

in this model. According to the PCE method, the stochastic response can be written as a function of standard random variables, then the uncertain dynamic model of the rotorsystem can be written as

(8)

can be written as

(9)

where

are the unknown expansion coefficient and

is the n-order multi-dimensional orthogonal polynomials related to the distribution of

, for example, Hermite polynomials for Gaussian distribution, Legendre polynomials for uniform distribution, Laguerre polynomials for gamma distribution, Jacobi polynomials for beta distribution [34]. In practice, expansion shall be truncated for computational purposes. Considering all d-dimensional orthogonal polynomials of order not exceeding p, the response can be approximated as follows:

(10)

is the number of unknown coefficients in this summation, which can be derived by

(11)

The PCE coefficients can be evaluated by Galerkin projection and regression method of which the former is more robust [19]. Based on Galerkin projection, the coefficients are evaluated by

(12)

where

is expectation operator. When the coefficients are obtained, the mean and variance of the output

also can be easily derived from PCE as follows:

(13)

(14)

where

is the variance operator.

Once the PCE of rotor system is constructed, the Sobol indices can be directly derived from the PCE coefficients and the parameters most sensitive to the transcritical response can be ranked. These indices, called PC-based Sobol indices and denoted by

can be straightforwardly given as

(15)

where

respresents all polynomials related only to

. For more details, Refs. [33, 35]. can be referred to. The first-order sensitivity indices

give the effect of each parameter taken alone whereas the higher order indices account for possible interact effect of various parameters. The total sensitivity indices

evaluating the total effect of an input parameter are defined as the sum of all partial sensitivity indices

involving parameter

The primary challenge is to derive the expectations of

and

. One of the main approaches to approximate the expectation is full factorial numerical integration (FFNI) which operates over a configuration space composed of specific configuration points. The configuration space of d-dimensional random variables based on FFNI is derived by direct tensor-product [36] as follows:

(16)

where

represents the configuration space composed of

points of the jth dimensional variable, and

corresponds to algebraic accuracy of order

. It is obvious that the number of configuration points increases exponentially with the dimension number of random variables, causing large compute load. Therefore, considering sparsizing configuration points of FFNI, the sparse grid numerical integration (SGNI) method is proposed [37]. On this basis, the improved sparse grid method has been further proposed and applied in the UQ of nonlinear systems [38]. SGNI constructs the configuration points by utilizing special tensor-product operations, which will be discussed in detail below.

3 | Double-Layer Dimensionality Reduction Algorithm Combining Modal Superposition With Sparse Grid Technique

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Considering that directly quantifying uncertainties in this high-dimensional system would require significant computational resources, a double-layer dimensionality reduction algorithm combining modal superposition with sparse grid technique is applied in UQ of this system. The execution logic for UQ of the supercritical rotor system with dry friction damper is illustrated in Figure 4.

3.1 | Dynamic Modeling Through Modal Superposition

In this process, the inner layer uses modal superposition to reduce the degrees of freedom of the uncertain system. To conduct it, the vibration equation of the shaft is rewritten in the following form

(17)

Define

, Equation (17) can be written as

(18)

Let

, Equation (18) can be written as

(19)

By solving the generalized eigenvalue problem in Equation (19), the eigenvalues and eigenvectors can be obtained, which correspond to the modal frequencies and mode shapes. The mode shape matrix is denoted as

. The response of the shaft can be transformed from physical space to modal space as

, and

is the uncertain response in modal space. By substituting it to Equation (8), it is can be derived that

(20)

Multiply by matrix

, then the uncertain dynamic equation in modal space can be derived as

(21)

where

. There are 64 degrees of freedom in Equation (21) corresponding to 64 order modes. By truncating the first n order modes from Equation (21), the truncated dynamic equation of the shaft is obtained as follows:

(22)

where

represent the truncated modal response, mass matrix, stiffness matrix, and damping matrix, respectively.

denotes the truncated external forces. The dynamic equations of the damper do not need to be transformed into modal space. Therefore, the uncertain dynamic equations can be rewritten as

(23)

where there are

degrees of freedom. Due to the nonlinear forces in

and

, it is necessary to convert the response in modal space back to physical space at each step of the numerical calculation. By substituting

into Equations (1)–(3), the nonlinear forces of

and

can be derived in physical space. Then

and

can be obtained by Equations (6) and (7). Transforming

into modal space through

and performing an n order truncation,

can be finally derived.

3.2 | PCE Coefficients Through Sparse Grid Technology

In outer layer, sparse grid technology is applied to get the configuration points, namely sample pool, denoted as

. To obtain the stochastic transcritical response of the rotorsystem over time, the response derived by solving dynamic equations within the interval

is furtherly discretized into a time sample pool

. In outer layer, PCE models for each time sample point are established. The configuration space of d-dimensional random variables based on SGNI is

(24)

where

, and

. Using SGNI, the expectation operator can be expressed as

(25)

where

and

are the

-th configuration point and the corresponding quadrature weight of the j-th dimensional variable and

is the output response corresponding to configuration point. By substituting

and

into Equation (25),

and

can be derived. To calculate

, the configuration points for

need to be converted to standard random variables. When

follows the normal, uniform, exponential distribution, etc., values of

and

can be directly derived from the Gauss-Hermite, Gauss-Legendre, and Gauss-Laguerre quadrature formula [39], which can be obtained by consulting a chart. The quadrature weight corresponding to the configuration point

(26)

where

is an integral precision control parameter corresponding to algebraic accuracy of order

[40]. The truncation order

of PCE and the value of

in SGNI also need to be matched to obtain the accuracy results. Equation (24) means only points that satisfy

are needed, effectively decreasing the number of configuration points by eliminating certain unimportant quadrature points compared to FFNI, as shown in Figure 5. And the higher the dimension of random variables, the more obvious the advantage of sparse grids.

4 | Results and Discussion

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In this section, the effects of uncertainties on the transcritical response of the shaft and dry friction damper system are studied. For each stochastic variable, dispersion is taken around the mean equal to its corresponding deterministic value. The deterministic values of a supercritical drive shaft parameters are shown in Table 1.

Moreover, the parameters of the rotor system are converted into dimensionless form as follows

(27)

where

, are, respectively, the equivalent mass and equivalent stiffness of the shaft can be calculated by treating the drive shaft as a simply supported beam. The other parameters of the rotor system are also nondimensionalized following the rules specified in Equation (27), which are not presented here for brevity. The deterministic value of the dimensionless parameters of the dry friction damper are set in Table 2.

4.1 | Modal Validation

The deterministic transcritical response of the shaft and the rub-impact ring of damper are derived based on the deterministic parameters shown in Tables 1 and 2. All the transient responses are provided as the deflection of the geometric center of shaft middle node and ring, that is,

and

and dimensionless. The deterministic transcritical response of the shaft with different truncated orders are shown in Figure 6, where

and

are derived from the direct numerical calculation of the original equation. It can be seen that a convergent response can be achieved with truncation order 3. Therefore,

is selected as the final modal truncation order for the calculations that follow.

It can be seen that, with the frequency increases, the vibration of the shaft exceeds

and contact with the rub-impact ring, which keeps sticking under the pre-tightening force during a-b phase. When rub-impact force overcomes the critical dry friction force, the ring turns into sliding as shown in b-c phase, the orbits of shaft and ring during this phase are shown in Figure 6B. A jump phenomenon occurs at point c, resulting in the shaft departing from the ring. The orbit of the shaft, illustrating this departure, is clearly depicted in Figure 6C. During the jump phase, the system is unstable. Phase a-c is the called transcritical region of the system and the maximum amplitude of shaft is denoted as

. The frequency of the point c called jump frequency is denoted as

, which is an important indicator in determining the safety margin between the operating speed and the critical area as shown in Figure 6A. Therefore, both

and

are crucial parameters of concern in design.

Considering

follows a uniform distribution with a coefficient of variation of 10%, the PCE is performed to derive the uncertain transcritical response. To verify the accuracy of the PCE, a MCS is performed with 1000 samples. A convergence study of the stochastic response with the order of

and

is performed by comparing with the result from MCS. The upper and lower bounds are derived by percentile difference method [41] with 95% confidence bounds. It has been found that the PCE model with an order

which means the calculation of only seven samples, exhibits good consistency with the MCS method, as shown in Figure 7. Therefore, the PCE approach with an order

is ultimately employed in this study.

4.2 | Effect of the Uncertain Parameters

In this section, we investigate several cases with different random uncertainties. The stochastic response of the rotor system is obtained considering the uncertainty of five design parameters with coefficients of variation of 10%. The values of the deterministic physical parameters of the rotor are given in Table 2. The random parameters are summarized in Table 3 for each case, where CV means coefficient of variation.

The stochastic response has been verified by MCS for each case, which are not presented here for brevity. The stochastic responses of the shaft and ring are shown in Figures 8-12 for Case 1–Case 5. Furthermore, to illustrate the damping effect of the damper under different parameter uncertainties, the probability distribution functions (PDFs) of

and

are obtained by establishing the PCE models of

and

and then performing MCS with 10000 points on these models. Furthermore, the scatterplots with regression lines illustrating the relationships between

and the parameters, as well as

and the parameters, are presented.

With the uncertainty of

, the shaft's response exhibits slight dispersion in the b-c phase, while remaining unaffected in other phases due to the stationary state of the ring, as illustrated in Figure 8A. The response of the ring shows minimal dispersion and remains largely unaffected as shown in Figure 8B.

exhibits a certain degree of dispersion following with uniform distribution, negatively correlated with

as shown in Figure 8C. While

exhibits a little degree of dispersion following with right-skewed distribution, positively correlated with

as shown in Figure 8D. With the uncertainty of

, the shaft's response in the a-c phase exhibits a slight degree of dispersion, while the ring's response shows minimal dispersion, as shown in Figure 9A,B.

and

show a certain degree of dispersion following with left-skewed distribution and right-skewed distribution, respectively. And

negatively correlates with

while

positively correlated with

. The entire transcritical process appears to be minimally influenced by

as shown in Figure 10A,B. This is due to the ring being made of self-lubricating material, resulting in a very low value for

and ensuring the system remains stable. With the uniform distribution of

and

follow a skewed distribution as shown in Figure 10C,D. The transcritical response of shaft and ring in a-c phase are notably affected by the uncertainty of

, exhibiting significant deviation as shown in Figure 11A,B. The uncertainty of

led to significant variations in

and

both distributed uniformly in Figure 11C,D. It can be seen that the correlation between

and

is very significant, with a negative correlation shown in Figure 11C, while there is a clear positive correlation between

and

, as shown in Figure 11D. The uncertainty of

introduces a degree of dispersion in phase b-c, with the point c displaying variations, as shown in Figure 12A,B. It can be seen that

and

show a certain degree of dispersion following with uniform distribution, and

negatively correlates with

while

positively correlated with

. From Figures 8–12, it can be seen that the stochastic transcritical response of the system still holds the shape of typical transcritical process. The response far from transcritical region is unaffected by parameter uncertainty, whereas the transcritical region,

and

exhibit a certain degree of deviation. Additionally,

and

show an inverse correlation with the uncertain parameters.

4.3 | GSA and Key Parameters Optimization

A Sobol global sensitivity analysis is analytically performed as the post-processing of PCE to obtain the key parameters and combinations thereof of the damper affecting the maximum amplitude and jump frequency. The values of maximum amplitude and jump frequency with the uncertainty of five design parameters shown in Table 3 are derived from the present method with 3002 grids, while 7776 grids are need in FFNI and more than 10 000 samples are needed in MCS. The comparison of the first-order and the total sensitivity indices of

is presented in Figure 13A. It can be found that the

is most sensitive to the initial clearance between shaft and ring (

) while insensitive to other parameters. And the first-order Sobol indices of each uncertain parameter are close to its total Sobol indices, which means that the interact effects of the five parameters are minor, as shown in Figure 13B. The Sobol sensitivity indices of

are presented in Figure 14A. By comparison, it can be observed that

is much more sensitive to

, then followed by

, and insensitive to the other parameters. The interact effects of the parameters are shown in Figure 14B. It can be seen that the interact effects to

are also minor.

In conclusion,

and

are highly sensitive to variations in

and

, making them crucial design parameters. However, in Figures 11 and 12, inverse correlations can be observed between

and

, as well as

and

. This suggests that it is not feasible to decrease

and

simultaneously, a trade-off between them is necessary. Therefore, it is crucial to carefully choose the values of

and

to maintain optimal values for both

and

A key parameters optimization aiming to minimize

and

is further conducted.

and

remain unchanged shown in Table 2, further optimization is conducted for

and

with design ranges of

, respectively. The PCE is performed to establish a surrogate model with

and

as inputs and

as output, as well as a surrogate model with

and

as inputs and

as output. To do that, assuming that

follows a uniform distribution on the interval

, and

follows a uniform distribution on the interval

, convert them into standard random variables denoted as

and

. For the convenience of expression, the PCE model of

is written as

and the PCE model of

is written as

. The accuracy of the PCE models has been examined. The mathematical model for the multi-objective optimization of the damper can be written as

(28)

In this paper, NSGA-II algorithm [42] is used to solve the optimization problem, which is one of the currently popular multi-objective genetic algorithms with the advantages of fast running speed and good convergence of solution sets. In the integration of PCE models with NSGA-II optimization, the resulting optimization outcome is visually presented in Figure 15A. By analyzing the Pareto front generated, point A is identified as the optimal choice, and point B represents the outcome with the initial set of parameters before optimization. The transcritical response of the shaft with the parameters at point A, that is,

is shown in Figure 15B. It clearly shows a decrease in both the maximum amplitude and jumping frequency of the shaft compared to initial result. This indicates a notable enhancement in the vibration reduction performance of the dry friction damper. The results signify a successful optimization effort that has effectively reduced vibration levels, leading to improved overall performance.

4.4 | Experimental Verification

To verify the dynamic model and optimization design method, experiments are carried out on a supercritical rotor rig. The experimental device is shown in Figure 16, which consists of a rotor test bench, a dry friction damper and a signal acquisition system. The dry friction damper is installed on the middle of the long shaft which is a supercritical rotor. To measure the transcritical vibration response of the shaft, the displacement sensors are installed in both horizontal and vertical directions near the dry friction damper. The rotor system is derived by a 380 V three-phase asynchronous motor and the speed can be adjusted continuously from 0 to 4500 r/min. In this experiment, the rotor system is accelerated from 0 to 4000 r/min, making the long shaft cross the critical speed. The transcritical vibration response of the shaft is obtained through displacement sensors and signal acquisition system. The dynamic model of the experimental rotor system is established as shown in Figure 17 based on the modeling method proposed in Section 2. The main simulation parameters of the experimental system are shown in Table 4. To validate the accuracy of the established dynamic model, four sets of inner diameters of rings are utilized to change the initial clearance

, which are 1, 1.5, 2, and 2.5 mm, respectively, as shown in Figure 18A. The critical dry friction force

can be changed by adjusting the length of the spring, and measured by using a tension meter as shown in Figure 18B. The transcritical vibration experiments are conducted, with the peak-to-peak values of the shaft vibration as the maximum amplitude. Each experiment is repeated five times.

The comparison of amplitude variation trends under different clearances is depicted in Figure 19, showcasing a maximum error of 1.5% between the simulation results and experimental results, which verify the accuracy of the established dynamic model. Based on the optimization design method proposed above, the parameters of the dry friction damper are optimized, the resulting values of

are determined as the optimal parameters for the damper. The comparison of experimental results before and after optimization design is shown in Figure 20. It can be seen that the amplitude of the shaft is reduced, yet still maintaining a small range of transcritical region, indicating the effectiveness of the optimization method.

5 | Conclusions

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In this paper, a stochastic dynamic model for the uncertainty and sensitivity analysis of a supercritical rotor system is established, using PCE method and a double-layer dimensionality reduction algorithm. The uncertain dynamic responses of the rotor are derived under different parameter uncertainties and verified by comparison with MCS. The effects of parameter uncertainties are analyzed. A multi-objective optimization design is carried out based on PCE model combined with NSGA-II. The dynamic model of the rotorsystem and the optimization design method are then validated by experiments. The conclusions can be obtained as follows:

1.	Compared to directly performing MCS on the original dynamic equations, using modal superposition combined with sparse grid techniques can significantly improve the efficiency of uncertainty quantification.
2.	With a small variation in design parameters, there exists a certain level of variability in the transcritical response of the rotor, especially the transcritical region.
3.	The sensitivity analysis reveals that the transcritical amplitude and jump frequency are highly sensitive to variations in clearance and critical dry friction force. There exist inverse correlations between the two key parameters and damping performance, which suggests that it is not feasible to simultaneously reduce both transcritical amplitude and jump frequency; a trade-off is necessary.
4.	The multi-objective optimization design for key parameters demonstrates a clear reduction in both the maximum amplitude and jump frequency of the shaft, indicating a notable improvement in the vibration reduction performance of the dry friction damper.

Funding

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Postgraduate Research & Practice Innovation Program of Jiangsu Province(KYCX24_0552)
National Natural Science Foundation of China(52005253)

References

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Hetherington

P. L.

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M. S.

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Year 2025 volume 5 Issue 3

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doi: 10.1002/msd2.70028

Receive Date：2025-01-02
Online Date：2026-03-24

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History

Received：2025-01-02
Revised：2025-03-06
Accepted：2025-03-26

Funding

Postgraduate Research & Practice Innovation Program of Jiangsu Province(KYCX24_0552)

National Natural Science Foundation of China(52005253)

Affiliations

¹National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing, People's Republic of China

²National Key Laboratory of Science and Technology on Helicopter Transmission, AECC Hunan Aviation Powerplant Research Institute, Zhuzhou, People's Republic of China

Corresponding:

Dan Wang (wangdan_053@nuaa.edu.cn)

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表12种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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TABLE 1 Values of a helicopter supercritical drive shaft parameters.

Symbol	Description	Value
	Length	3080 mm
	Inner radius	56 mm
	Outer radius	57 mm
	Density	2700
	Young's modulus of elasticity
	Poisson's ratio	0.3
	Damping ratio	0.02
	Damping ratio	0.02
	First natural frequency	215 rad/s
	Second natural frequency	860 rad/s
	Eccentricity	0.3 mm
	Stiffness of support
	Acceleration of shaft rotation
	Operating speed	490 rad/s

TABLE 2 Deterministic value of the dry friction damper dimensionless parameters.

Symbol	Description	Value
	Lumped mass	0.07
	Normal rub-impact stiffness	40
	Normal rub-impact stiffness	40
	Tangential rub-impact friction coefficients	0.02
	Tangential rub-impact friction coefficients	0.02
	Initial clearance	4
	Initial clearance	5
	Critical dry friction force	3

TABLE 3 Statistical properties of input uncertain parameters.

Case	Distribution	Mean	Lower limit	Upper limit	CV
1	Uniform	0.07	0.0579	0.0821	0.1
2	Uniform	40	33.0718	46.9282	0.1
3	Uniform	0.02	0.0165	0.0235	0.1
4	Uniform	4	3.3072	4.6928	0.1
5	Uniform	3	2.4804	3.5196	0.1

TABLE 4 Value of the main simulation parameters of experimental system.

Symbol	Description	Value
	Length of the short shaft	810 mm
	Length of the long shaft	1635 mm
	Inner radius	34 mm
	Outer radius	38 mm
	Density	2700
	Young's modulus of elasticity
	Poisson's ratio	0.3
	Eccentricity	0.3 mm
	Stiffness of support
	Stiffness of coupling
	Lumped mass	0.036 kg
	Normal rub-impact stiffness
	Normal rub-impact stiffness
	Tangential rub-impact friction coefficients	0.04
	Tangential rub-impact friction coefficients	0.04
	Initial clearance	2 mm
	Initial clearance	2 mm
	Critical dry friction force	30 N

FIGURE 1 Supercritical tail drive shaft system of a helicopter.

FIGURE 2 A typical single-span supercritical transmission system with a dry friction damper. (A) 3-D model, (B) detailed view of dry friction damper; (C) detailed structures of the dry friction damper: 1—Half-half type rub-impact ring, 2—Pre-tightening spring, 3—Friction disc, 4—Bolt, 5—Base.

FIGURE 3 Dynamic modeling of the system: (A) finite element model of a single-span supercritical shaft with dry friction damper and (B) schematic diagram of the rub-impact forces and dry friction force.

FIGURE 4 Execution logic for UQ of the supercritical rotor system.

FIGURE 5 Configuration points through (A) FFNI and (B) SGNI with algebraic accuracy of order 5 for Gauss-Hermite in the two-dimensional standard random space.

FIGURE 6 Deterministic transcritical response of the shaft and ring: (A) transient responses of the shaft and ring, (B) orbits of the shaft and ring during b-c phase, and (C) orbit of the shaft and ring during jump phase.

FIGURE 7 Transcritical response of the system with uncertain : (A) shaft (B) ring (red = high bound, bule = lower bound, dashed = MCS, and solid = PCE.

FIGURE 8 Stochastic transcritical responses of the system for Case 1: (A) shaft, (B) ring, PDFs of (C) , and (D) .

FIGURE 9 Stochastic transcritical response of the system for Case 2: (A) shaft, (B) ring, PDFs of (C) , and (D) .

FIGURE 10 Stochastic transcritical response of the system for Case 3: (A) shaft, (B) ring, PDFs of (C) , and (D) .

FIGURE 11 Stochastic transcritical response of the system for Case 4: (A) shaft, (B) ring, PDFs of (C) , and (D) .

FIGURE 12 Stochastic transcritical response of the system for Case 5: (A) shaft, (B) ring, PDFs of (C) , and (D) .

FIGURE 13 Sobol global sensitivity analysis of : (A) first-order and total Sobol indices and (B) interact effect of the five parameters.

FIGURE 14 Sobol global sensitivity analysis of : (A) first-order and total Sobol indices and (B) interact effect of the five parameters.

FIGURE 15 Multi objective optimization results: (A) the optimization outcome based on NSGA-II combined with PCE models (B) the transcritical response of shaft with parameters at point A.

FIGURE 16 Supercritical rotor experimental rig equipped with a dry friction damper: (A) rotor test bench, (B) dry friction damper, and (C) signal acquisition system.

FIGURE 17 The dynamic model of the experimental rotor system.

FIGURE 18 Different parameters of the dry friction damper: (A) rub-impact rings with different inner diameters and (B) the measurement of critical dry friction force .

FIGURE 19 Comparison of simulation and experimental results: (A) transcritical response under parameters in Table 4 and (B) comparison of amplitude variation trends under different clearances.

FIGURE 20 Comparison of experimental results before and after the optimization design.

FIGURE B1 Campbell diagrams of the shaft: (A) considering the gyroscopic effect and (B) ignoring the gyroscopic effect.

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