Nonlinear dynamics of a dual-rotor-bearing system with active elastic support dry friction dampers

: In present study, the modified harmonic balance - alternating frequency/time domain (HB - AFT) method with embedded arc - length continuation method is used to study the nonlinear dynamic characteristics of the dual - rotor - bearing system with active elastic support dry friction damper (ESDFD). The friction on the contact interface between ESDFD moving and stationary disks is described by a two - dimensional (2D) friction contact model. The dynamic model of the dual - rotor is established by the conical Timoshenko beam element and the rigid disk element, while the inter - shaft bearing force is obtained by the Hertz contact model. The reduced order model (ROM) of dual - rotor system is constructed by the Craig - Bampton method. Based on the ROM, the modified HB - AFT method with embedded arc - length continuation procedure is used to solve the periodic solutions of the dual - rotor system under unbalanced excitation. The Floquet theory is employed to determine the stability of periodic solutions. The impact of key parameters such as Hertz contact stiffness and radial clearance of inter - shaft bearing, eccentricity of disk, and modal damping ratio on the primary resonance characteristics and inter - shaft bearing dynamic load of the dual - rotor system are revealed without considering the ESDFD. With considering the ESDFD, the influence of ESDFD normal force on the primary resonance peak and inter - shaft bearing dynamic load of the dual - rotor system is investigated. The optimal normal force and controllable region for ESDFD to control the vibration of the dual - rotor system under the target mode is determined. A control strategy based on altering the normal force within the controllable region is designed. Results show that under the proposed control strategy, the damping effect provided by ESDFD significantly reduces the vibration amplitude of the dual - rotor system and mitigates the dynamic load of the inter - shaft bearing when passing through the resonance region, completely suppresses the bi - stable phenomenon and vibration jump behavior of the dual - rotor system. Thereby reducing the structural damage caused by excessive vibration. This demonstrates promising engineering applications for ESDFD.


Introduction
Dual-rotor-bearing system is the core components of aero engines.Due to the rotary speed of the low-pressure (LP) rotor and the high-pressure (HP) rotor are generally different, the system is subjected to dual-frequency excitation caused by the imbalance of the two rotors.The LP and HP rotors are usually connected by the inter-shaft bearing.The inter-shaft bearing contains nonlinear characteristics such as radial clearance, variable stiffness excitation and fractional exponential nonlinear restoring force [1][2][3].In the working rotary speed range of the modern aero engines, the rotor system exhibits several modes and frequently passes critical speeds during operation.This makes the vibration problem of the rotor system more prominent and the vibration control more complicated [4].As a passive damper, the squeeze film damper (SFD) generates viscous damping through squeeze film, which has good vibration reduction effect and reliability [5].It is a commonly used vibration reduction mechanism for aero engines.However, under the influence of processing and installation errors, large load and sudden imbalance, the SFD cannot effectively control the vibration of all modes in the working rotary speed range [6,7].In addition, accurately predicting the dynamic characteristics of the SFD-rotor system can be challenging [8] due to the strong nonlinear stiffness generated by the oil film [7,9].On the other hand, the active elastic support dry friction damper (ESDFD) can realize vibration reduction through the friction energy dissipation of the contact interface between the moving and stationary disks, as well as adjust its vibration reduction characteristics in real time according to the change of working conditions of the rotor system.It has the advantages of simple structure, high reliability, and insensitivity to temperature [10,11], which provides a technical path for the vibration control of the rotor system where the SFD is difficult to realize.
According to the characteristics of elastic support and the vibration reduction principle of dry friction damper, Fan et al. [12] proposed a technical approach to apply ESDFD to rotor vibration reduction.It is a passive ESDFD with a mechanical spring as the adjustment mechanism, which is difficult to meet the needs of vibration control of the rotor system under complex conditions.In order to optimize the vibration reduction ability of ESDFD, Liao et al. [13][14][15] studied the feasibility of electromagnets and piezoelectric ceramics as ESDFD actuators.The results show that the vibration reduction effect of active ESDFD is remarkable, which is convenient for active control by changing the normal force between the moving and stationary disks.Recently, Wang et al. [4] designed a new type of piezoelectric ceramic ESDFD structure and established a design method of related dynamic parameters.The damper has a significant vibration reduction effect on the first two modes of the HP single-rotor simulator.Compared with piezoelectric ceramics, electromagnets can work at higher temperatures and are more suitable for vibration reduction of aero engine rotor structures.Therefore, Zhu et al. [10] proposed an adaptive control strategy for the active control of the rotor system by the ESDFD based on the electromagnetic elastic ring active ESDFD ( Chinese patent 201911015735.0,201921794557.1).Liu et al. [16] proposed a control strategy based on the concept of complex nonlinear mode to expand the effective vibration control region of rotor system controlled by ESDFD.Zhu et al. [11] carried out experimental research about the influence of ESDFD at different support on rotor vibration characteristics.The above research on the control of rotor vibration by ESDFD is mainly based on single rotor, and there few studies on the vibration control of dual-rotor system by ESDFD.
The nonlinear nature of friction and the arbitrary two-dimensional (2D) in plane motion on the contact interface between the ESDFD moving and stationary disks bring challenges to accurately and efficiently predict the forced response of the ESDFD-rotor system.The nonlinear friction force on the contact surface is characterized by the contact model.The consistency between the contact model and the physical phenomena on the friction interface directly determines the accuracy of the nonlinear response analysis.Fan et al. [12] simplified the rotor as a single degree of freedom (DOF) system, and revealed the vibration reduction mechanism of the dry friction damper based on the classical one-dimensional (1D) hysteresis friction model.Since the relative motion between the moving and stationary disks of the active ESDFD that controls the vibration of the rotor is 2D, Liu et al. [16] applied a 2D dry friction model with hysteresis characteristics to describe the friction on the contact interface.The model is quasi-three-dimensional (3D) and ignores the normal motion [17,18].In order to improve the solving efficiency, Firrone et al. [19] and Afzal et al. [20] derived the analytical Jacobian matrix of the quasi-3D contact model.However, due to the independence of the hysteresis springs in two directions, the hysteresis characteristics on the contact interface cannot be accurately reflected.Song et al. [14] described the friction on the contact surface between the moving and stationary disks based on the ball/plate 2D dry friction model.The model is full 3D and ignores the normal motion [21,22].The problem of this model is that the expression is complex and the computational efficiency is low.Sanliturk et al. [23] simplified the full 3D contact model.The core idea is to track the trajectory of the contact point, and the corresponding friction evaluation process is called trajectory tracking method (TTM).Based on the full 3D contact model, Wu et al. [24] modified the TTM based on a full 3D contact model and derived the Jacobian matrices required for calculating the response of dry friction systems using the harmonic balance-alternating frequency/time domain (HB-AFT) method, significantly enhancing computational efficiency.Based on the work of Wu et al. [24], ignoring the normal deformation of the full 3D contact model, a 2D friction contact model can be established to evaluate the friction force on the contact interface between the moving and stationary disks and solve it based on the HB-AFT method.
In order to ensure high modeling accuracy, the finite element method (FEM) is used to establish a dual-rotor dynamic model, but the model established by this method has a high DOF.In addition, due to the nonlinear dry friction and inter-shaft bearing force contained in the ESDFD-rotor-bearing system, the numerical integration method takes a long time and has a low computational efficiency.For this reason, model reduction and [25] semi-analytical solution method HB-AFT method [26] are often used to improve the computational efficiency of high-dimensional nonlinear dynamic models.Lu et al. [27] reduced the dimension of a dual-rotor-bearing system model based on the proper orthogonal decomposition (POD) method, which reduced the time of calculating the nonlinear dynamic response under the misalignment fault of the coupling by using the numerical integration method.Based on the fixed interface component modal synthesis (CMS) method, known as the Craig-Bampton method [28], Jin et al. [29] and Prabith et al. [30] established a reduced order model (ROM) of a complex dual-rotor-bearing system, and studied the dynamic characteristics under the nonlinear inter-shaft bearing force and rub-impact force respectively, which improved the calculation efficiency while ensuring the calculation accuracy.Considering the local contact nonlinearity of the dry friction system, Petrov et al. [31] and Wu et al. [32] developed a multi-step dimension reduction method, which greatly reduces the DOF of the model and improves the computational efficiency.The main idea of the HB-AFT method is as follows.Firstly, the governing equation is transformed from time domain to frequency domain by Fourier transform.Then, the nonlinear force is expressed by time/frequency transform technique.Finally, the nonlinear algebraic equations are solved by Newton iteration.The HB-AFT with embedded arc-length continuation method can track the frequency response curve and obtain periodic solutions including unstable solutions [2,33].Kim et al. [34,35] first applied the HB-AFT method to analyze nonlinear dynamic characteristics of rotor systems, and obtained quasi-periodic responses.The research of Zhang et al. [36] shows the advantages of HB-AFT method in solving nonlinear ball bearing system problems, and studies the phenomena of chaos, period-doubling, varying compliance resonances.Hou et al. [2] applied the HB-AFT method to a dual-rotor-inter-shaft bearing system and studied the nonlinear phenomena such as bi-stable and resonance hysteresis.Chen et al. [37] used the HB-AFT method to study the vibration reduction effect of the SFD on the dual-rotor system under the base excitation.Chen et al. [33] established a rotor-bearing-casing model based on the FEM, and studied the bi-stable and vibration jump behavior caused by the nonlinearity of the inter-shaft bearing based on the modified HB-AFT method.Chang et al. [38] used the modified incremental harmonic balance-alternating frequency/time domain (IHB-AFT) method to study the nonlinear dynamics and thermal coupling characteristics of the rotor-bearing system.The results show that the modified method has higher solution efficiency.Based on the ROM established by CMS method, Han et al. [39] studied the nonlinear dynamic characteristics of the rotor with SFD by using the IHB-AFT method.
At present, the research on the dynamic characteristics of the ESDFD-rotor system is mainly focused on the single rotor system, and the research on the dual-rotor system is rarely reported.The solution method is mainly based on the numerical integration with low computational efficiency.Therefore, in this paper, the nonlinear vibration characteristics of the active ESDFD controlled dual-rotor-bearing system under the target mode are studied.Aiming at the problem of low efficiency of solving the high-dimensional system containing nonlinear dry friction and inter-shaft bearing force, a semi-analytical solution method is applied to solve the dynamic response of the system based on the ROM.Firstly, the dynamic model of active ESDFD-dual-rotor-inter-shaft bearing system is established based on 2D friction contact model, FEM and Hertz contact model.Then, based on the ROM established by Craig-Bampton method, the modified HB-AFT method with embedded arc-length continuation method is used to solve the periodic solution including unstable solution.Finally, the stability of the periodic solution is determined by Floquet theory.Aiming at whether the bi-stable phenomenon caused by the inter-shaft bearing occurs in the primary resonance region of the dual-rotor system, the influence of the ESDFD normal force on the vibration amplitude of the dual-rotor system and the dynamic load of the inter-shaft bearing passing through the resonance region is studied respectively.The optimal normal force and controllable region of the ESDFD control dual-rotor system under the vibration of the target mode are determined, and the control strategy of the active ESDFD based on the controllable region to change the normal force to control the rotor vibration is given, so as to achieve better control effect at a lower cost.

Dynamic model of active ESDFD-dual-rotor-bearing system
Fig. 1 is the schematic diagram of the active ESDFD-dual-rotor-bearing system structure.The system consists of LP rotor, HP rotor, inter-shaft, supports and active ESDFD.The LP rotor speed is 1, the HP and LP rotor speed ratio is , and the HP rotor speed is 2 = 1.The LP rotor includes LP compressor (LPC) disk, LP turbine (LPT) disk and LP shaft.The HP rotor includes HP compressor (HPC) disk, HP turbine (HPT) disk and HP shaft.In the principle of modal controllability of the rotor system [4], ESDFD is designed to be installed at support 1.

Fig. 1.
Active ESDFD-dual-rotor-bearing system.Fig. 2 is the schematic diagram of the active electromagnetic elastic ring ESDFD structure, which is mainly composed of a bearing seat, an elastic ring [40], moving disks fixed on the inner ring of the elastic ring, stationary disks fixed on the axially moving electromagnet and force sensors.Under the action of the limit steel ball, the movement of the electromagnet and the stationary disk is limited in the axial direction.In order to reduce the influence of the ESDFD on the elastic ring, the left and right sides of the ESDED structure are completely symmetrical.A magnetic field will be formed after the control current is introduced into the coil of the electromagnet.Under the action of the magnetic field, the electromagnet will axially squeeze the stationary disk to contact the moving disk to generate a uniform normal force on the contact surface.The rotor drives the motion of the moving disk in the radial plane, which makes the relative motion between the moving and stationary disks.Under the action of normal force, the friction force generated between the moving and stationary disks will dissipate the vibration energy of the rotor to achieve vibration reduction.The normal force on the contact surfaces between the moving and stationary disks will change with the change of the control current and be output by the force sensor in real time.When the current is disconnected, the electromagnet will return to the original position under the action of the reset spring, and the moving and stationary disks will be separated.At this time, the damper does not work.For a more detailed introduction of the active electromagnetic elastic ring ESDFD, please refer to the Chinese patents CN201911015735.0 and 201921794557.1.

Fig. 2. Configuration of active electromagnetic elastic ring ESDFD.
The FEM is used to model the dynamics of the dual-rotor.The Timoshenko beam element considering the moment of inertia, shear factor and gyroscopic effect is mainly used to discretize the shaft.According to the section type, cylindrical and conical beam elements are used.The disk is considered as a rigid disk element with rotation effect.The inter-shaft bearing force is established by Hertz contact model.The support is simplified to linear stiffness and damping.The friction force on the contact interface between the moving and stationary disks is described by 2D friction contact model that can describe arbitrary 2D in plane motion of the moving disk with the rotor motion.

Dynamic equation of conical beam element
Fig. 3 (a) is a schematic diagram of the conical beam element, which only considers the transverse vibration of the shaft element.Each element contains two nodes.Each node contains two translational displacements and two rotational displacements, a total of four DOF.So the element displacement vector is where i and j are the two endpoints of the beam element, respectively.The dynamic equation of the conical beam element is in which

K
, and   e con F be the mass matrix, moment of inertia matrix, gyroscopic matrix, stiffness matrix, and generalized force vector of the cylindrical beam element, respectively.Then the dynamic equation of the cylindrical beam element can be expressed as

Dynamic equation of rigid disk element
Fig. 4 is the schematic diagram of the disk element.The displacement vector of the single-node disk element is The dynamic equation is The mass matrix of rigid disk is The gyroscopic matrix is

Inter-shaft bearing model
Fig. 5 depicts a schematic diagram of the inter-shaft bearing.The outer race of the inter-shaft bearing is connected to the LP rotor (node 28), while the inner race is connected to the HP rotor (node 39).As a result, the outer race of the inter-shaft bearing rotates at the speed 1, the inner race at the speed 2, and the cage rotates at where Rb and rb represent the outer and inner radius of the inter-shaft bearing, respectively.At any given time t, the angular position of the jth roller is denoted as , Nb represents the number of roller.Assuming pure rolling of the roller between the inner and outer races of the inter-shaft bearing, and considering small deformations, the deformation between the jth roller and the races can be expressed as where 0 represents the radial clearance.xH, yH, xL, and yL represent the displacements of the inner and outer races of the bearing along the horizontal and vertical directions.Based on the Hertz contact model, the force in the inter-shaft bearing can be expressed as in which cbm represents the Hertz contact stiffness, and n is a coefficient associated with the type of inter-shaft bearing.When the rolling element is a ball, n = 3/2, and when the rolling element is a cylinder, n = 10/9.And ( ) ( ) ( ) is the Heaviside function, indicating the contact status between the rolling element of the inter-shaft bearing and the inner and outer races.

The modified TTM for 2D friction contact model
Fig. 6 illustrates a schematic diagram of a friction contact model capable of describing arbitrary 2D motion.The stationary disk, connected to the bearing housing, remains absolutely stationary in the radial direction.The radial absolute displacement of the moving disk corresponds to its relative displacement with respect to the stationary disk.Considering the moving and stationary disks as concentrated masses [10], let the contact surface of the stationary disk be denoted as 1, and that of the moving disk as 2. Point a is rigidly connected to surface 1, while point b is connected to surface 2 via a tangential spring with stiffness kt.The tangential spring can rotate with the direction of relative motion.When the tangential spring force does not exceed the maximum static friction force, there is no relative sliding between contact point b and surface 2.
Otherwise, contact point b slides in the direction of ba towards point a.The normal spring force represents the normal force FN between the moving and stationary disks.When the normal force is greater than 0, the moving and stationary disks is in contact; otherwise, they are separated.It is assumed that the friction coefficient  between the moving and stationary disks remains constant [16].Fig. 6. 2D friction contact model.Utilizing a modified TTM [24] to describe the friction forces between the moving and stationary disks.As shown in Fig. 7, at time tj-1, the positions of contact surface 2 and contact point b are denoted as 2' and b', respectively.At time tj, the stick state boundary is represented by Circle-A, with point a as its center and r j as its radius, where r j is determined by the normal force  , respectively, exhibit the following relationship ( ) ( ) At time tj, contact surface 2 moves in the direction of 2'2, and contact point b transitions from point b' to point b* along the 2'2 direction.In the absence of relative sliding, the position of b* corresponds to the location reached by contact point b, and the line segment b'b* is parallel and equal in length to the line segment 2'2.The position of b* can be calculated using the following equation And the following assumptions are made at the initial time When the position of b* lies within the stick boundary, it indicates that the tangential spring force does not exceed the maximum static friction force.In this case, contact point b is in a stick state, and the position of b* coincides with the position of contact point b with the relation of On the contrary, when the position of b* lies outside the stick boundary, it indicates that the tangential spring force exceeds the maximum static friction force.In this case, contact point b transitions from the stick state to the slide state.The position of contact point b determined by the intersection between line segment b*a and Circle-A can be expressed as Based on the friction contact model capable of describing arbitrary 2D motion, the friction forces acting on the moving disk at any given moment are

Motion equations of the system
As shown in Fig. 1, the LP rotor comprises nodes 1 to 29, with the LPC disk and LPT disk fixed at nodes 2 and 29, respectively.The HP rotor encompasses nodes 30 to 31, with the HPC disk and HPT disk anchored at nodes 34 and 37, respectively.The inter-shaft bearing connects the LP rotor and HP rotor via nodes 28 and 39.The LP rotor supports 1, 2, and 3 are connected to the foundation through nodes 5, 9, and 23, respectively.The HP rotor support 4 is connected to the foundation through node 31.Taking into account the ESDFD, the moving disk are modeled as concentrated mass fixed at node 5.By assembling disc elements, conical beam elements, and cylindrical beam elements, and incorporating the nonlinear inter-shaft bearing forces and nonlinear friction forces acting on the moving disk, the nonlinear dynamic equations of the 160DOF active ESDFD-dual-rotor-bearing system are derived as follows where M, C, G, and K represent the overall mass matrix, damping, gyroscopic, and stiffness matrices of the dual-rotor system, respectively.q, Fb, Ff, Fub, and Fg are column vectors representing displacement, bearing forces, friction forces, unbalance forces, and gravity forces, respectively.The damping matrix comprises both Rayleigh damping and bearing damping components.The formula for Rayleigh damping is as follows [16,29] f1 andf2 are the first and second critical speeds of the dual-rotor system, and  represents the modal damping ratio.

Methodology formulation
The model of the dual-rotor system established based on the FEM has a large number of DOF and contains nonlinear inter-shaft bearing forces and dry friction forces caused by ESDFD, leading to a decrease in computational efficiency.In order to improve the solving efficiency while ensuring computational accuracy, it is necessary to employ a reasonable model reduction method to reduce the DOF of the high-dimensional nonlinear system.In this study, the Craig-Bampton method is used to reduce the DOF of the dynamic model of the system.Based on the ROM, a modified HB-AFT method with embedded arc-length continuation is applied to compute all periodic solution branches of the system, including unstable solutions.The stability of periodic solutions is determined by Floquet theory.

The Craig-Bampton method
Based on the Craig-Bampton method [28,30], the dimension reduction process for the full order model (FOM) of the dual-rotor system involves separately reducing the dimension of the LP and HP rotors, and then assembling them together to obtain the ROM of the dual-rotor system.The dynamic equation for a single rotor FOM is as follows ( ) Considering that the dual-rotor system is primarily subjected to unbalance excitations from the compressor and turbine disks, we designate the nodes corresponding to disks, supports, and inter-shaft bearing as primary nodes, while the remaining nodes are treated as secondary nodes in partitioning the dynamic model of single rotor system.Therefore, the dynamic equations described in terms of the DOF corresponding to the primary nodes and the DOF corresponding to the secondary nodes are as follows where qm and qs represent the DOF for the primary nodes and secondary nodes, respectively.Fm is the external force vector corresponding to the DOF of the primary nodes, consisting of unbalance forces, dry friction forces, and bearing forces.
Neglecting the gyroscopic matrix and damping matrix [41], Eq. ( 26) is simplified as Substituting qm = 0 into Eq.( 27) yields The eigenvalue problem for Eq. ( 30) is given by ( ) Extracting the first l normal modes from Eq. ( 31) as b  , and the corresponding modal coordinates is b q .The retained normal modes can be expressed as Neglecting the inertia term in Eq. ( 27) yields That is mm m ms s m Based on Eq. ( 36), we obtain That is in which The static displacement can be expressed as where According to the Craig-Bampton method, the displacement of the single rotor can be expressed as in which Substituting Eq. ( 42) into Eq.( 25) and left-multiplying by T d  , we get the single rotor ROM as ( ) where The dynamic equations of the LP and HP rotors ROM can be obtained by the aforementioned dimension reduction process.By assembling the ROM of the LP and HP rotors and incorporating the properties of the inter-shaft bearing into the corresponding DOF, we obtain the ROM of the dual-rotor system.The number of DOF of the dual-rotor ROM is determined through convergence studies.This results in a 28 DOF ROM from the original 160 DOF FOM of dual-rotor system The accuracy of the dual-rotor ROM will be validated in section 4.1.

The modified HB-AFT method
The traditional HB-AFT method involves harmonic expansion of nonlinear forces [36], while the modified HB-AFT method expands the residual [33,38].Compared to the traditional HB-AFT method, the modified HB-AFT method can efficiently compute the Jacobian matrix in a programmatic manner.This makes it better suited for handling dual-rotor dynamic systems that simultaneously contain nonlinear dry friction and inter-shaft bearing forces.
Performing a transform of the motion equations for the dual-rotor system in terms of time scale, let = t, which leads to where ( )   and ( )   represent the first and second derivatives with respect to . Substituting Eq. (48) into Eq.(47) yields ( ) The periodic solution of the ith DOF of the motion equation is expanded as an s order Fourier series ) where p represents the harmonic order of the Fourier series expansion.The number of harmonics in the Fourier series expansion is related to factors such as the type of nonlinear terms, the frequency and amplitude of external excitation, etc.The value of s is typically determined through the study of solution convergence.Define the column vector of trigonometric basis functions as ( ) The Fourier coefficient vector corresponding to the ith DOF is given by Therefore, the harmonic coefficient matrix for the periodic solution u can be expressed as: ( ) Defining the residual vector as R, Eq. ( 49) can be expressed as ( ) Under the influence of unbalance periodic excitation, the external force vector can be expanded using a Fourier series.Therefore, the residual vector R can also be expressed in terms of s level harmonic coefficients as Therefore, Eq. ( 57) is equivalent to Eq. ( 61) represents a fixed-point problem that can be solved using the Newton-Raphson iteration method.
( ) ( ) The Jacobian matrix can be expressed as follows The alternating frequency/time domain method can be used to obtain the Jacobian matrix J. Since the coefficient matrix of the Fourier series expansion contains nonlinear inter-shaft bearing and dry friction forces, obtaining the Jacobian matrix using inverse Fourier transformation involves the following detailed steps.
(1) Discrete sampling After time scale normalization, the system's non-dimensional excitation frequencies are 1 and , with corresponding non-dimensional periods T1 = 2 and T2 = 2/.To ensure that the total duration in the time domain is an integer multiple of each excitation frequency's respective period, we choose the sampling period T as the least common multiple of T1 = 2 and T2 = 2/.We divide the period T into N equal intervals and define the sampling time sequence as follows (2) Obtaining B through inverse Fourier transform The components of B can be calculated in the time domain as follows based on the inverse Fourier transform ( ) ( ) ( ) where Ri represents the residual corresponding to the ith DOF.
(3) Compute the Jacobian matrix Jacobian matrix in the time domain is calculated as where ( ) ( ) ( ) ( ) ( ) where i u and i u represent the acceleration and velocity, respectively, of the ith DOF in the rotor system.After obtaining the Jacobian matrix J and substituting it into Eq.( 62), the results converge when the norm of B is less than the allowable accuracy, Based on Eq. (55) periodic solutions of Eq. (47) are obtained.Additionally, all branches of the solution are obtained using the arc-length continuation method, and Floquet theory is utilized to determine the stability of solutions obtained with the modified HB-AFT method [2].

Arc-length continuation method
The modified HB-AFT method with embedded arc-length continuation can avoid the failure of Newton-Raphson iterations at turning points or bifurcation points.The arc-length continuation consists of prediction and correction procedure.The tangential prediction is given by where 0 u represents the first predicted point, u m is the solution vector at the mth speed, h (h > 0) is the step size.m is the normalized tangential vector at the u m point.If Eq. (49) written as Due to the introduction of the new variable m, an additional condition should be added for correcting the predicted point, as follows Where g(u) represents the hyperplane passing through 0 u and orthogonal to m, i.e.
( ) The Newton-Raphson iteration method is employed to solve Eq. (78) and Eq.(80) ( ) ( ) When the step size h is sufficiently small, and the curve Q(u) is smooth, the Newton-Raphson iteration converges to the solution vector u m+1 .For the tangential vector m+1 corresponding to the (m+1)th speed, it must be normalized and satisfy ( ) Furthermore, the direction of the curve Q(u) is maintained as

Validation of the effectiveness of the modified HB-AFT method based on the ROM
In order to efficiently and accurately calculate the dynamic response of the 160 DOF active ESDFD-dual-rotor-inter-shaft bearing system, which includes nonlinear inter-shaft bearing forces and dry friction forces, this study applies a modified HB-AFT method based on the ROM.Firstly, the accuracy of the 160 DOF FOM of the dual-rotor established in this paper is validated using FEM software.Then, the accuracy of the ROM established using the Craig-Bampton method is validated from the perspective of critical speeds and mode shapes.Finally, based on the ROM, the accuracy of the modified HB-AFT method is verified using numerical integration method.The structural parameters of the dual-rotor, inter-shaft bearing parameters, and ESDFD parameters are provided in Appendix B. Error 0.18% 0.78% -0.49% -0.13%

Validation of the ROM
Comparisons of the first three order mode shapes and critical speeds obtained from the ROM and those obtained from the FEM software model are made.The accuracy of the ROM is verified in terms of mode shapes and critical speeds.Fig. 8 shows the comparison of the first three order mode shapes of the dual-rotor obtained from the ROM and the FEM software model.The modal assurance criterion (MAC) values for the mode shapes obtained based on the ROM are all greater than 0.98.Since the MAC values close to 1 indicate a higher similarity between two mode shapes, this confirms the accuracy of the ROM from the perspective of mode shapes.2, and they are consistent between the two models.This further validates the effectiveness of the ROM from the perspective of critical speeds.

Comparison between the modified HB-AFT method and numerical integration method based on the ROM
This paper primarily focuses on the unbalance response characteristics of the first two order modes of the dual-rotor system.In the speed range of 150rad/s to 300rad/s without considering the ESDFD, the modified HB-AFT method is employed to solve the dynamic Eq. ( 49).After conducting multiple tests, it is observed that in the vicinity of the critical speeds for the first and second modes of the dual-rotor system, the excitation frequencies 1 and 2 of the LP rotor and HP rotor dominate the vibrations, while the contributions of other harmonic components in this range are negligible.Therefore, in the subsequent analysis, Eq. ( 51) and Eq. ( 53) are only considered for 1 and 2, where C and Ai are defined as follows [33,42] ( ) The preset error tolerance for Newton-Raphson iteration is 1e-10.The formula for calculating the amplitude of vibration response at the nodes of the dual-rotor system is as follows where N is the number of discrete sample points within one period, and y are the displacements of nodes in the horizontal and vertical directions at the jth instant, and x and y are the mean values of displacements in the horizontal and vertical directions within one period.The 3D unbalance response of the dual-rotor system is plotted by taking the rotary rotor speed, the axial position of nodes, and the rotor amplitude as shown in Fig. 10.As the LP rotor speed 1 and HP rotor speed 2 increase, the dual-rotor system exhibits four resonance regions.The first resonance region is near speed 1 = 176rad/s (2 = 211rad/s ), the second resonance region is near speed 1 = 213rad/s (2 = 256rad/s), the third resonance region is near speed 1 = 228rad/s (2 = 273rad/s), and the fourth resonance region is near speed 1 = 274rad/s (2 = 329rad/s).The first and third resonance regions are excited by the HP rotor, while the second and fourth resonance regions are excited by the LP rotor.In these resonance regions, the vibration amplitudes of the system increase significantly.In the first and second resonance regions, the unbalance response corresponds to the first order mode, with node 16 of the LP rotor having the maximum vibration, and node 40 of the HP rotor having the maximum vibration.In the third and fourth resonance regions, the unbalance response corresponds to the second order mode, with node 18 of the LP rotor having the maximum vibration, and node 40 of the HP rotor having the maximum vibration.Since the LP and HP rotors are connected by the inter-shaft bearing, resonance occurs synchronously.The vibration of the inter-shaft bearing is as severe as that of the rotors, leading to the inter-shaft bearing operating in harsh dynamic conditions.Hence, investigating the resonance mechanism and uncovering the impact of critical parameters of the inter-shaft bearing on the system's resonance characteristics are of utmost importance for the system's healthy operation.
To validate the accuracy of the modified HB-AFT method, both the modified HB-AFT method and the fourth-order Runge-Kutta method are used to calculate the unbalance response of the rotor system.As shown in Fig. 11, we take the example of the LPC, the time histories of the y-direction for a common period of 10 are compared.Fig. 11 (a) corresponds to rotary speed within the non-resonance region of the rotor system, with 1= 200rad/s (2 = 240rad/s).Fig. 11 (b) corresponds to rotary speed within the resonance region of the rotor system, with 1= 228rad/s (2 = 274rad/s).Fig. 12 and Fig. 13 show the corresponding orbits frequency spectrums for Fig. 11 (a) and Fig. 11 (b).In Fig. 11-13, the solid lines represent the results obtained using the modified HB-AFT method, while the dashed lines represent the results obtained using the fourth-order Runge-Kutta method.It can be observed from Fig. 11-13 that the time histories, rotor's orbits, and frequency spectrums obtained by the fourth-order Runge-Kutta method and the modified HB-AFT method are in good agreement, indicating that the two methods have similar accuracy.This validates the effectiveness and accuracy of the modified HB-AFT method based on the ROM.In the subsequent discussions, this paper focuses only on the unbalance response of the first two order modes of the dual-rotor system excited by the LP rotor.The average computation time for steady-state response at a single speed, calculated using the FOM and ROM with the modified HB-AFT and fourth-order Runge-Kutta methods, is listed in Table 3.Furthermore, the initial values for iteration in both methods are based on the calculation results from the preceding points.The were performed on a computer equipped with an Intel Core i9-12900K CPU and 16 GB of RAM.The results indicate that the Craig-Bampton and the modified HB-AFT methods significantly enhance the computational efficiency of high-dimensional nonlinear dynamic equations.
Table3 Comparison of computation time.

Method
Model Calculation time fourth-order Runge-Kutta FOM 85.8h

Parameter discussion
The impact of key parameters such as Hertz contact stiffness and radial clearance of inter-shaft bearing, eccentricity of disk, and modal damping ratio on the primary resonance characteristics and inter-shaft bearing dynamic load of the dual-rotor system are revealed without considering the ESDFD.Furthermore, the effects of ESDFD normal force on the resonance amplitude and the inter-shaft bearing dynamic load is investigated.The study identifies controllable region and optimal normal force for the ESDFD to control rotor vibrations through the first and second resonance region.A control strategy is devised based on altering the normal force within the controllable region to mitigate rotor vibration effectively.

Dynamic characteristics of the dual-rotor system without active ESDFD
radial clearance of the inter-shaft bearing is one of the causes of nonlinear vibration phenomena in the dual-rotor system.Analyzing the influence of the inter-shaft bearing radial clearance is crucial for revealing the mechanism of nonlinear dynamic behavior.Periodic solutions of the dual-rotor system are obtained using the modified HB-AFT method for different radial clearances ( = 4m, 8m, 10m, and 12m).The stability of periodic solutions is determined using the Floquet theory.Fig. 14 (a) illustrates the unbalance response of the LPC under various radial clearances.The response curve exhibits a hardening behavior, bending to the right side, which is attributed to the 3/2 fractional exponential nonlinearity inter-shaft bearing force.As the radial clearance increases, the changes in the resonance peak are relatively small, and the response curve shifts slightly to the left side.This is due to the larger radial clearance altering the contact conditions of the inter-shaft bearing's rolling elements, thereby reducing its stiffness.The second order resonance is more sensitive to changes in radial clearance.When the radial clearance increases, the appearance of the bi-stable phenomenon becomes evident, as indicated by the dashed lines.In practical systems, unstable solutions are not observed, resulting in vibration jump behavior within the bi-stable region.
To analyze the vibration characteristics of the inter-shaft bearing, the unbalance response of the nodes corresponding to the inner and outer rings of the bearing is calculated.Based on Eq. ( 11), the periodic variations in the inter-shaft bearing forces at different speeds can be obtained.The effective value calculation formula for the inter-shaft bearing force is as follows The variation of the effective value of inter-shaft bearing force with speed varying for different radial clearance is shown in Fig. 14 (b).Similarly, there are two resonance regions in the curve of inter-shaft bearing force with speed varying as the radial clearance increases.With the increase in radial clearance, the inter-shaft bearing force also exhibits a jump phenomenon in the second order resonance region, with the jump value of the inter-shaft bearing force reaching up to 300N.This harsh loading condition is highly unfavorable for the healthy operation of the inter-shaft bearing.The effect of radial clearance on inter-shaft bearing force aligns with its influence on unbalance response, as the inter-shaft bearing force is determined by the deformation of the inner and outer rings of the inter-shaft bearing.In the subsequent discussion, a radial clearance of 12μm is considered.eccentricities, e2 = 18m, 20m, 22m, and 25m, and the corresponding inter-shaft bearing force variation with speed varying.As the eccentricity increases, the resonance peak becomes larger, shifts to the right side, and the critical speed of the dual-rotor system increases.In the second order resonance region, the bi-stable range decreases, and both the vibration jump amplitude and inter-shaft bearing force jump amplitude decrease.

Dynamic characteristics of the dual-rotor system with active ESDFD
Fig. 14 demonstrates that when a radial clearance of 4m is employed for the inter-shaft bearing, the dual-rotor system exhibits no bi-stable within the resonance regions of the first two order modes under LP excitation.Using both the modified HB-AFT method and the fourth-order Runge-Kutta method, we compute the unbalance response of the dual-rotor system under LP excitation at normal force of 0N, 50N, 100N, 150N, and 200N.As depicted in Fig. 18 (a), the unbalance response curves from both methods remain consistent within rotary speed range 180rad/s to 300rad/s.This confirms the accuracy and effectiveness of the modified HB-AFT method for solving systems involving nonlinear dry friction and inter-shaft bearing forces.With increasing normal force, the resonance peaks shifts to the right side due to the additional stiffness of the ESDFD.Fig. 18 (b) further analyzes the impact of normal force on the resonance peaks of the first two order modes.The resonance peaks initially decrease and then increase.Consequently, an optimal normal force exists for ESDFD to control rotor vibration, consistent with the previous results in references [14,43].The optimal normal force for controlling the first order resonance is 100N, while for the second order resonance, it is 140N.order resonance peaks with normal force.In Fig. 19 (a), we observe unbalance response curves near the first modal resonance region (180rad/s <  < 230rad/s) under two conditions: 0N normal force and optimal normal force 100N.The intersection of the curves at point A corresponds to rotary speed of  = rad/s.Fig. 19 (b) shows unbalance response curves around the second modal resonance region (250rad/s <  < 300rad/s) under two conditions: 0N normal force and the optimal normal force 140N.The intersection of the curves at point B corresponds to rotary speed of  = rad/s.These results demonstrate that under the optimal normal force for controlling the first two order modes with ESDFD, the resonance peaks are significantly reduced.However, it's important to note that the effect of ESDFD on reducing rotor vibration is not consistent in the resonance region.Specifically, in the vicinity of the first modal resonance, when the rotor speed exceeds , the unbalance response under optimal normal force exceeds that under 0N normal force.Similarly, in the vicinity of the second modal resonance, when the rotor speed exceeds , the unbalance response under optimal normal force surpasses that under 0N normal force.Consequently, the controllable region for controlling the first two modal resonances with active ESDFD are defined as [180, ] rad/s and [250, ] rad/s, respectively.The control strategy is designed to maintain the optimal normal force on the contact surface between the ESDFD's moving and stationary disks within the controllable region, while keeping the normal force at 0N outside this region.This approach achieves superior control effectiveness at a lower cost.normal force of 0N and optimal normal force.Fig. 20 displays the variation of bearing forces near the resonance regions of the first two order modes under normal force of 0N and the optimal normal force.Under the optimal normal force, the inter-shaft bearing forces significantly decrease.At the first resonance peak, the inter-shaft bearing dynamic load reduced by 54%, while at the second resonance peak, it decreased by 56%.This indicates that active ESDFD serves a protective role for the inter-shaft bearing, promoting its stable and healthy operation.under normal force of 0N and optimal normal force.Fig. 21 illustrates the unbalance response amplitudes at various nodes of the dual-rotor system when passing through the first two critical speeds under normal force of 0N and the optimal normal force.It is evident from the graph that under the optimal normal force, both the LP and HP rotors exhibit lower resonance amplitudes compared to the 0N normal force condition.At the first critical speed, the amplitude at node 16 (associated with the maximum vibration of the LP rotor) decreased by 60%, while at node 40 (associated with the maximum vibration of the HP rotor) it decreased by 56%.At the second critical speed, the amplitude at node 18 (associated with the maximum vibration of the LP rotor) decreased by 53%, and at node 40 (associated with the maximum vibration of the HP rotor) it decreased by 49%.Under the optimal normal force for ESDFD to control the first two order modes, the additional damping provided by ESDFD significantly reduces the vibration amplitudes of the dual-rotor system when passing through critical speeds, demonstrating effective vibration reduction capabilities.and optimal normal force.The above study demonstrates that when the dual-rotor system operates without bi-stable, ESDFD exhibits effective vibration reduction capabilities.As shown in Fig. 14, when the radial clearance of the inter-shaft bearing  = 12m, the dual-rotor system exhibits clear bi-stable when passing through the second modal resonance under LP excitation, resulting in vibration jump behavior in both unbalance response amplitudes and inter-shaft bearing force.Fig. 22 investigates the unbalance responses and inter-shaft bearing dynamic load of the dual-rotor system under LP excitation in the vicinity of the second modal resonance (250rad/s <  < 290rad/s) for ESDFD normal force of 0N, 50N, 100N, 150N, and 200N.Within this resonance region, there are two bifurcation points, labeled as C and D. These points define the bi-stable region, with point C marking the lower jump in unbalance response amplitudes and inter-shaft bearing forces, and point D marking the upper jump in unbalance response amplitudes and inter-shaft bearing forces.23 (a), with increasing the normal force, the resonance peaks of the dual-rotor system initially decrease and then increase, and the bi-stable region becomes zero and remains unchanged.The optimal normal force for ESDFD to control the second modal resonance is 130N.The minimum normal force required to suppress bi-stable completely is 70N.Fig. 23 (b) shows that with increasing normal force, the jump amplitudes of unbalance response and inter-shaft bearing forces decrease to zero.This indicates that the additional damping provided by ESDFD can entirely suppress bi-stable caused by nonlinear inter-shaft bearing forces, preventing the occurrence of vibration jump behavior.This aligns with the principle demonstrated in Fig. 17, where increasing modal damping effectively suppresses bi-stable.24, to suppress bi-stable and prevent vibration jumps and inter-shaft bearing force jumps of the dual-rotor system, ESDFD requires a minimum normal force of 70N.To reduce the resonance peak when the dual-rotor system passes through critical speed, the optimal normal force required by ESDFD is 130N.The intersection point of the curves under the minimum normal force and optimal normal force is labeled as point B, corresponding to rotary speed of  = rad/s.Consequently, we define two controllable regions near the second modal resonance: controllable region-Ⅰ i.e. [250rad/s,  rad/s and controllable region-Ⅱ i.e. [ 290] rad/s.From the graph, it's evident that within controllable region-Ⅱ, applying the optimal normal force results in larger amplitudes compared to the minimum normal force.Therefore, the strategy is to apply the optimal normal force of 130N within controllable region-Ⅰ and the minimum normal force of 70N within controllable region-Ⅱ.This allows ESDFD to suppress vibration jump phenomena while passing through the second modal resonance region with smaller vibration amplitudes.

Conclusions
This paper applies the FEM, 2D friction contact model, and Hertz contact theory to establish the dynamic model of an active ESDFD-dual-rotor-bearing system.The ROM of the system is established using the Craig-Bampton method.Based on the ROM, a modified HB-AFT method with embedded the arc-length continuation procedure is used to efficiently and accurately solve the dynamic equations of the dual-rotor system with nonlinear dry friction force and inter-shaft bearing force.The Floquet theory is employed to determine the stability of periodic solutions.The following conclusions are obtained: (1) With increasing normal force, the resonance peaks of the dual-rotor system passing through critical speed first decrease and then increase.Consequently, an optimal normal force exists for minimizing the resonance amplitude of the system.At this optimal force, the active ESDFD effectively mitigates vibration amplitude and inter-shaft bearing dynamic load by up to 60% and 56%, respectively, when the rotor system passes through critical speed.
(2) When the dual-rotor system exhibits bi-stable and vibration jumping behavior caused by the inter-shaft bearing, with the increase of normal force, the bi-stable region decreases to zero and remains unchanged.Therefore, there exists a minimum normal force for the ESDFD to suppress vibration jumping behavior.Under the minimum normal force, the damping effect provided by the ESDFD can completely suppress bi-stable behavior, prevent vibration amplitude and dynamic load of the inter-shaft bearing from jumping, so protect the inter-shaft bearing and reduce its damage.
(3) Based on the optimal and minimum normal forces, the controllable region for ESDFD to control the vibration of the dual-rotor system under the target mode is defined.The control strategy for the active ESDFD to change the normal force based on the controllable region to control rotor vibration is given, so as to achieve better vibration reduction effects with less cost near the resonance region.
The above work provides a theoretical basis for the active ESDFD to control the vibration of the dual-rotor-bearing system.The active ESDFD has a significant effect on reducing the vibration amplitude and dynamic load of the inter-shaft bearing when the dual-rotor-bearing system passes through the critical speed, and has certain engineering application prospects.In addition, experimental verification is necessary.In this paper, theoretical research on the vibration control of the dual-rotor system by the active ESDFD is carried out.In the future, we will build a test rig for the active ESDFD-dual-rotor system to conduct experimental research on the vibration control of the dual-rotor system with the active ESDFD.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
( ) According to the FEM and Lagrange's principle, the dynamic matrix for the conical beam element can be expressed as follows ( )

Fig. 3
Fig.3 (b) is the schematic diagram of the cylindrical beam element.Compared with the conical beam element, the section parameters of the cylindrical beam element are constant along the axial direction.The dynamic matrix of the cylindrical beam element can be obtained by substituting the section parameters of the cylindrical beam element into the dynamic matrix of the conical beam element.Let

Fig. 5 .
Fig. 5. Schematic diagram of the inter-shaft bearing.Assuming pure rolling of the roller between the inner and outer races of the inter-shaft bearing, and considering small deformations, the deformation between the jth roller and the races can be expressed as

FN
and the friction coefficient .Here, b* signifies the movement trend of contact point b.When b* resides within Circle-A, contact point b is in a stick state, coinciding with b*.When b* moves outside of Circle-A, contact point b initiates sliding, separating from b*.In the slide state, the position of contact point b is determined by the intersection between b*a and Circle-A.

Fig. 7 .
Fig. 7. Schematic diagram of the modified TTM for the 2D friction contact model: (a) stick state; (b) slide state.The formula for calculating the maximum displacement radius at time tj is ( ) for calculating the tangential vector m+1 is as follows

Fig. 8 .
Fig. 8.The first three order mode shapes of the dual-rotor computed based on the ROM and FEM software model: (a) first order mode shape; (b) second order mode shape; (c) third order mode shape.Fig. 9 shows the Campbell diagrams for the dual-rotor system excited by the LP and HP rotors obtained based on the ROM and the FOM.The corresponding backward whirl (BW) and forward whirl (FW) critical speeds excited by the LP and HP rotors are listed in Table2, and they are consistent between the two models.This further validates the effectiveness of the ROM from the perspective of critical speeds.

Fig. 9 . 2
Fig. 9. Campbell diagrams for the dual-rotor system excited by (a) the LP rotor, and (b) the HP rotor, computed based on the ROM and the FOM.Table 2 Critical speeds of the dual-rotor model.Excited by the LP rotor Excited by the HP rotor Order BW (rpm) FW (rpm) BW (rpm) FW (rpm) 1st 1800 2040 1914 2040

Fig. 14 .
Fig. 14.The influence of inter-shaft bearing radial clearance  on (a) unbalance response, and (b) inter-shaft bearing dynamic load.Fig. 15 (a) and (b) depict the unbalance response and the variation of inter-shaft bearing forces with speed for different inter-shaft bearing contact stiffness values, cbm = 2.5e8N/m 3/2 , 5.0e8N/m 3/2 , 7.5e8N/m 3/2 and 10e8N/m 3/2 .As the inter-shaft bearing contact stiffness increases, the resonance peak shifts to the right side, and the critical speed increases.The bi-stable region in the second order resonance area decreases, and both vibration jump amplitude and inter-shaft bearing force jump amplitude decrease, indicating enhanced system stability.

Fig. 15 .
Fig. 15.The influence of inter-shaft bearing contact stiffness cbm on (a) unbalance response, and (b) inter-shaft bearing dynamic load.Fig. 16 (a) and (b) display the unbalance response curves of the LPC for different

Fig. 16 .
Fig. 16.The influence of LPC eccentricity e2 on (a) unbalance response, and (b) inter-shaft bearing dynamic load.Fig. 17(a) and (b) depict the unbalance response curve of the LPC for modal damping ratios Fig. 16.The influence of LPC eccentricity e2 on (a) unbalance response, and (b) inter-shaft bearing dynamic load.Fig. 17(a) and (b) depict the unbalance response curve of the LPC for modal damping ratiosof  = 0.01, 0.02, 0.03, and 0.04, as well as the curve of inter-shaft bearing force variation with speed varying.As the system modal damping ratio increases, the resonance peaks shift to the left side, resulting in a decrease in critical speed and a reduction in the resonance peaks.The bi-stable region of the second order resonance region decrease and the vibration jump behavior in both unbalance response and inter-shaft bearing force vanishes.

Fig. 17 .
Fig. 17.The influence of modal damping ratio  on (a) unbalance response, and (b) inter-shaft bearing dynamic load.

Fig. 18 .
Fig. 18.(a) Unbalance responses under different normal force, and (b) variation of the first two order resonance peaks with normal force.In Fig. 19 (a), we observe unbalance response curves near the first modal resonance region

Fig. 19 .
Fig. 19.Unbalance responses at (a) the first, and (b) the second modal resonance regions under normal force of 0N and optimal normal force.Fig. 20 displays the variation of bearing forces near the resonance regions of the first two

Fig. 20 .
Fig. 20.Dynamic load of inter-shaft bearing at (a) the first, and (b) the second resonance regionunder normal force of 0N and optimal normal force.Fig.21illustrates the unbalance response amplitudes at various nodes of the dual-rotor system when passing through the first two critical speeds under normal force of 0N and the optimal normal force.It is evident from the graph that under the optimal normal force, both the LP and HP rotors exhibit lower resonance amplitudes compared to the 0N normal force condition.At the first critical speed, the amplitude at node 16 (associated with the maximum vibration of the LP rotor) decreased by 60%, while at node 40 (associated with the maximum vibration of the HP rotor) it decreased by 56%.At the second critical speed, the amplitude at node 18 (associated with the maximum vibration of the LP rotor) decreased by 53%, and at node 40 (associated with the maximum vibration of the HP rotor) it decreased by 49%.Under the optimal normal force for ESDFD to control the first two order modes, the additional damping provided by ESDFD significantly reduces the vibration amplitudes of the dual-rotor system when passing through critical speeds, demonstrating effective vibration reduction capabilities.

Fig. 21 .
Fig. 21.The (a) first, and (b) second modal resonance peak of each node under normal force of 0Nand optimal normal force.The above study demonstrates that when the dual-rotor system operates without bi-stable, ESDFD exhibits effective vibration reduction capabilities.As shown in Fig.14, when the radial

Fig. 22 .
Fig. 22.Under different normal force, (a) unbalance response, (b) inter-shaft bearing dynamic load at the second modal resonance regions.As depicted in Fig.23(a), with increasing the normal force, the resonance peaks of the dual-rotor system initially decrease and then increase, and the bi-stable region becomes zero and remains unchanged.The optimal normal force for ESDFD to control the second modal resonance is 130N.The minimum normal force required to suppress bi-stable completely is 70N.Fig.23 (b)shows that with increasing normal force, the jump amplitudes of unbalance response and inter-shaft bearing forces decrease to zero.This indicates that the additional damping provided by ESDFD can entirely suppress bi-stable caused by nonlinear inter-shaft bearing forces, preventing the occurrence of vibration jump behavior.This aligns with the principle demonstrated in Fig.17, where increasing modal damping effectively suppresses bi-stable.

Fig. 23 .
Fig. 23.Under various normal force, (a) resonance peaks and bi-stable regions, and (b) vibration jump amplitudes and inter-shaft bearing force jump amplitudes.As shown in Fig.24, to suppress bi-stable and prevent vibration jumps and inter-shaft bearing force jumps of the dual-rotor system, ESDFD requires a minimum normal force of 70N.To reduce the resonance peak when the dual-rotor system passes through critical speed, the

Table 1 ,
the first four natural frequencies of the dual-rotor are calculated based on the FOM established in this paper and the model established in FEM software.The errors are listed, and the analysis of the calculation results indicates that the errors are all less than 1%, confirming the accuracy of the FOM established in this paper.

Table 1
The natural frequencies of the dual rotor model in the non-rotating state.