Model dimension reduction and nonlinear analysis of the sprag clutch-flexible rotor system considering local nonlinearity

 Abstract: In order to study the influence of local nonlinear factors such as bearing clearance and Hertzian contact on the nonlinear behavior of the sprag clutch-rotor system during high-speed steady-state engaging, a nonlinear force model of rolling bearing considering radial clearance and Hertzian contact is established. Regarding the inner and outer rotors as substructures, the fixed interface modal synthesis method is used to reduce the dimension of the substructure model, and the nonlinear dimension reduction model of the sprag clutch-flexible rotor system (SC-FRS) is obtained. To improve computational efficiency, the Newmark method combined with the fixed interface synthesis method is presented to calculate the system response. The response results of the reduced model with a different number of dominant modes and the unreduced model are compared, and the number of reserved dominant modes of the inner and outer rotors is selected based on the calculation efficiency and accuracy. The effect of the radial clearance of different bearings on the nonlinear vibration response and the dynamic load of the inter-shaft bearing is discussed. It is found that the sensitivity of different radial clearances to the system's amplitude-frequency response is different, and the amplitude jump and the frequency hysteresis appear at the resonance peak.


Introduction
As a special inter-shaft bearing, the sprag clutch can be equivalent to the linear stiffness of each degree in the steady engaging state.However, due to the existence of nonlinear factors such as bearing clearance, the rotor system will exhibit nonlinear characteristics.As the speed of the rotor system increases, the impact of bearing nonlinearity will be more obvious.The existing research on the influence of bearing nonlinearity on the vibration characteristics of the rotor system can be utilized as an important reference for the research of the nonlinear vibration properties of the sprag clutch-rotor system.
The nonlinear vibration research of the rolling bearing-single rotor system was discussed firstly.Some scholars have presented the surface waviness expression of support bearing balls, inner ring and outer ring, and calculated contact nonlinear force based on Hertz contact theory, and studied the influence of surface waviness on nonlinear vibration response [1][2][3][4].The nonlinear force model with bearing radial clearance was proposed in references [5][6][7][8].It was found that the radial clearance directly affects the stability of the system, and the larger the clearance, the easier it is to lose stability.A dynamic model with local surface defects is established to study the impact of bearing surface defects on vibration.The existence of surface defects will increase the vibration of the rotor system, and the fault frequency corresponding to the defect type will appear in the dynamic response spectrum [9][10][11].
Compared with the rolling bearing-single rotor system, it is more difficult to calculate and analyze the nonlinear vibration characteristics of the inter-shaft bearing-double rotor system.Some scholars have simplified the rotor flexibility by establishing a low-dimensional simplified model, which greatly reduces the calculation time to reveal the law and mechanism of vibration characteristics and form the guiding principle.
For the dual-rotor system of aero-engine, Hu et al. [12] proposed a quasi-static nonlinear force calculation model considering the five degrees of freedom of the bearing.Through the time domain response analysis of the inter-shaft bearing-rigid rotor system, it was pointed out that the nonlinearity of the inter-shaft bearing has a significant impact on the nonlinear vibration response of the whole rotor system.Hou et al. [13] established a vibration model of the double rotor system based on the Lagrange equation in consideration of the waviness of the inter-shaft bearing, and analyzed the influence of speed and waviness on the nonlinear vibration response.Gao et al. [14] presented a nonlinear mechanical model of inter-shaft bearing with inner and outer ring defects, and established a nonlinear vibration model of inter-shaft bearing-rigid rotor system, and analyzed the influence of surface defects on dynamic characteristics.For changing the low precision of low-dimensional model analysis, an inter-shaft bearing-flexible rotor system model based on finite element theory is built to study the nonlinear vibration characteristics of the rotor system.
Deng et al. [15] considered the radial clearance of the inter-shaft bearing to discretize the rotor by the finite element method, and proposed a nonlinear dynamic model of the dual rotor-rolling bearing system to analyze the influence of the radial clearance and structural parameters of the inter-shaft bearing on the dynamics.The results of dynamic analysis were verified by comparison with the experiments.Payyoor et al. [16] used Timoshenko beam elements to discretize the rotor of a dual-shaft turbine rotor system, and considered the inter-shaft bearing clearance and nonlinear Hertz contact to establish a nonlinear vibration model of the whole machine, and analyzed the influence of rotating speed, inter-shaft bearing clearance and unbalance on the stability.Lu et al. [17] proposed a nonlinear vibration model of the aero-engine dual-rotor system based on the finite element theory, and analyzed the influence of the radial clearance of the inter-shaft bearing, the speed ratio of the inner and outer rotors, and the vertical tension on the unbalanced response.
Using the finite element theory to model the inter-shaft bearing-rotor system with complex structure, the degree of freedom of the model is often tens of thousands, hundreds of thousands, or even millions, especially when bearing nonlinearity is considered in the model, the analysis will become very difficult.So the model reduction technology is chosen to reduce the dimension of the model.It can greatly reduce the degree of freedom of the model and improve the calculation efficiency, while ensuring the accuracy and precision of the analysis results.
Shanmugam et al. [18] proposed a fixed-free hybrid interface synthesis method by combining the advantages of the fixed interface synthesis method and the free interface synthesis method, and pointed out that this method is very suitable for the analysis of the nonlinear vibration characteristics of a dual-rotor system.Yang et al. [19] adopted the fixed interface modal synthesis method to reduce the model.Considering the nonlinear force of the squeeze film damper and the inter-shaft bearing, the nonlinear vibration characteristics of a double unbalanced excitation-reverse double rotor system were studied.Jin [20] utilized the fixed interface modal synthesis method to reduce the linear dimension of the model, and expanded the modes to improve the calculation accuracy, and applied eigen-orthogonal decomposition to reduce the nonlinear dimension.
It can be known from the existing research that due to the radial clearance of the support bearing, the rotor system will have obvious nonlinear vibration characteristics, which will affect the stability of operation.Because of the large amount of calculation in nonlinear analysis, under the premise of ensuring the accuracy of the calculation results, model condensation is often adopted to reduce the dimension of the model to improve computational efficiency.In the above research, there are no reports on the model dimension reduction and nonlinear analysis of SC-FRS.Therefore, the purpose of this paper is to reduce the dimension of the finite element model of the system and establish the SC-FRS vibration model considering the local nonlinearity of the rolling bearing.Using the proposed vibration model, the influence of the radial clearance of different bearings on the amplitude-frequency response (AFR), as well as the influence of the radial clearance and unbalance of different bearings on the dynamic load of the inter-shaft bearing are studied.

Nonlinear Vibration Model of SC-FRS
The finite element vibration model of SC-FRS is established by adopting the same system structure and finite element unit division method as in reference [21].Taking into account the local nonlinearity such as bearing radial clearance, it is introduced into the system vibration model in the form of dynamic nonlinear force.As shown in Figure 1, considering the local nonlinearity of the bearing 2 and the inter-shaft bearing 4, they are expressed as the nonlinear force with the clearance, while the bearing 1, the inter-shaft bearing 3, the sprag clutch, and the bearing 5 are still expressed as stiffness matrix with six degrees of freedom, and the calculation method of their stiffness is shown in reference [21].
Because of the displacement of the outer ring, the calculation method of the nonlinear force of the inter-shaft bearing is more representative.Next, the calculation process of the nonlinear force of the bearing is discussed by taking the inter-shaft bearing as an example.Figure 1 The sprag clutch-rotor system with bearing nonlinearity

Nonlinear Force of Inter-shaft Bearing
The inter-shaft bearing is a key component connecting the inner and outer rotors.Its inner and outer rings are respectively fixedly connected with the inner and outer rotors.Therefore, the nonlinear force of the inter-shaft bearing is a function of the vibration displacement of the inner and outer rotors.At the same time, the nonlinearity of the inter-shaft bearing will directly affect the vibration of the inner and outer rotors.
Figure 2 shows the schematic diagram of the inter-shaft bearing.The linear velocity of the inner and outer rings of the inter-shaft bearing can be expressed as: where i v ,  i and b i r are the velocity, the rotating angular velocity, and the radius of the inner ring.Assuming that there is pure rolling between the rolling elements and the inner and outer rings, and the rotation speed of the cage is the same as the revolution speed of the rolling elements, the linear velocity and angular velocity of the cage are respectively 1 ( ) The location of the kth roller is assumed as where Nb is the number of the roller.The normal contact deformation between the kth roller and the rings is given by 0 (( ) cos ( )sin ) cos where α is the contact angle of angular contact bearing, γ0 is the bearing radial clearance.The contact force between the kth rolling element and the races can be calculated by where kn is the Hertz contact stiffness, ( ) The components of the nonlinear force of the inter-shaft bearing in X and Y directions are

Dimension Reduction Based on Fixed Interface Synthesis Method
The Craig-Bampton fixed interface synthesis method is applied to the dimension reduction calculation of the sprag clutch-rotor model, so as to improve the calculation efficiency of the nonlinear analysis.
The total degree of freedom is divided into the set of interface degrees of freedom and the set of internal degrees of freedom.The nodes where the support bearing, inter-shaft bearing and the sprag clutch are located belong to the set of interface degrees of freedom, and all the degrees of freedom of these nodes will be retained, which is called the reserved master nodes.The rest of the nodes belong to the set of internal degrees of freedom, and the degrees of freedom of these nodes will be transformed into a few degrees of freedom of modal coordinates, so they are called subordinate nodes.
The sprag clutch-rotor system shown in Figure 1 is divided into two single-rotor systems connected by a boundary support bearing and inter-shaft bearing.For each single-rotor system, the system dynamic equation can be written as: ( ) where M, C, G, and K are the mass matrix, damping matrix, gyro matrix, and stiffness matrix of the dynamic equation of the single rotor system.Ω is the rotor angular velocity, F e is the unbalanced force vector.By constructing the coordinate transformation matrix Φd, the transformation relationship between the original coordinates and the new coordinates after condensation can be obtained where um and us are the displacement vectors of the master nodes and the subordinate nodes, Φk is the preserved normal mode matrix of the subordinate nodes, Φc is the interface constraint modal matrix, pk is the normal modal coordinates of the subordinate nodes.The calculation method of Φk and Φc is given in reference [22].
According to formula (10), [um pk] T is denoted as the new modal coordinate pd, and formula ( 9) is left multiplied by Φd T .The dynamic equation is obtained as follows: ( ) Taking into account the supporting force of the boundary bearing and unbalanced force on the master nodes of the inner and outer rotors, the dynamic equation including the elastic force of the inter-shaft bearing, the elastic force of the sprag clutch, and the nonlinear force of the inter-shaft bearing is established.
where the subscripts A and B denote the outer rotor and the inner rotor, ins K is the stiffness matrix at the interface degree of freedom of the sprag clutch, inb K is the stiffness matrix at the interface degree of freedom of inter-shaft bearing, inb F is the non-linear force of the inter-shaft bearing acting on the master nodes of the inner and outer rotors connected with the inter-shaft bearing.
To facilitate the numerical solution by the Newmark method, the generalized degrees of freedom of the system are rearranged in the order of the master nodes and the subordinate nodes to obtain the transformation relationship between the new coordinate {q} and the original coordinate {p}.

    
Substituting equation ( 13) into equation (12), the system dynamic equation to be solved is obtained as

Newmark Algorithm for Solving Dimension Reduction Model
Equation ( 14) is rewritten into the form of master-subordinate node degrees of freedom.
The time interval of the Newmark numerical method is [tn, tn+1], where tn+1= tn+Δt, Δt is the time step.
represent the acceleration vector of the system response at time tn+1 and tn, respectively. 1 q & represent the velocity vector of the system response at time tn+1 and tn, respectively. 1 and n m q 、 n s q represent the displacement vector of the system response at time tn+1 and tn, respectively.Based on the assumption of the Newmark method, the expressions of the system response velocity and displacement vector at time tn+1 are as follows [22]     After obtaining the displacement, velocity, and acceleration vectors of the master and subordinate degrees of freedom of the reduced model at time tn+1, they are transformed into the responses of the actual nodes through the conversion from modal coordinates to physical coordinates.Take the response value at time tn+1 as the known condition of the next time interval [tn+1, tn+2], and recalculate according to the above process until the specified calculation time is reached.

Implementation and Verification of Dimension Reduction Model
As shown in Figure 1, taking the sprag clutch-rotor system with eccentric masses at node 4 and node 19 as the object, considering the clearance of boundary bearing 2, the dimension reduction of the finite element model of the system is carried out.By comparing the calculation results of the reduced dimension model with the original system model, the appropriate number of retained dominant modes of the inner and outer rotors is selected and the accuracy of the reduced dimension model is verified.

Dimension reduction model of SC-FRS
Figure 3 shows the condensation structure of the outer rotor system.Node 2, node 6, node 9, node 10, and node 11 are selected as master nodes, and the remaining nodes are subordinate nodes.Among them, the unbalanced force is located at the subordinate node 4, and it will be transformed into the modal coordinates together with the degrees of freedom of node 4.
Similarly, Figure 4 shows a schematic diagram of the condensation structure of the inner rotor system.Node 14, node 15, node 16, and node 21 are selected as master nodes, and the remaining nodes are subordinate nodes.The unbalanced force is located at the subordinate node 19, and it will be transformed into the modal coordinates together with the degrees of freedom of node 19. Figure 4 The master nodes of the inner rotor After obtaining the condensation dynamic equations of the inner and outer rotors, the bearing support force and nonlinear force are added to the master degrees of freedom of the corresponding structures.By connecting the stiffness matrix of the inter-shaft bearing and the sprag clutch, the dynamic equations of the inner and outer rotors are synthesized and written into the overall system dynamic equations as shown in equation ( 14).Using the calculation method combining the Newmark algorithm and the fixed interface, the response of each node can be obtained.

Influence analysis of the different number of retained dominant modes
Taking the sprag clutch-rotor system with a radial clearance of bearing 2 as the object, the first 15 orders, the first 10 orders, and the first 5 orders dominant modes are reserved respectively.According to the dynamic equation shown in formula ( 14), the time domain response and amplitude-frequency response of specific nodes are calculated by Newmark method combined with a fixed interface mode synthesis method to compare with the original system responses calculated by Newmark algorithm.
The structural parameters of inner and outer rotors used in the calculation are shown in Table 1.
Bearings 1 and 2 are angular contact bearings, the model is 7202C.Inter-shaft bearings 3 and 4 are angular contact bearings, model 7004A.Bearing 5 is a deep groove ball bearing, and its stiffness coefficient is calculated as an angular contact ball bearing with a zero contact angle, but only the stiffness in the x and y directions is considered.The axial preload is 500N, and the bearing stiffness coefficient is shown in Table 2.The torsional stiffness of the sprag clutch is 352 Nm/rad under the preload torque of 30 Nm. Figure 5 shows the time-domain responses of node 4 of the outer rotor and node 19 of the inner rotor in the y direction.It can be seen from the figure that the time-domain responses calculated by each model are highly consistent.At the same time, in terms of calculation time, the original system model takes 168.6s, while the reduced dimension models with the first 15 orders, the first 10 orders, and the first 5 orders dominant modes take 113.7s, 95.3s, and 74.2s respectively, saving 54.9s, 73.3s, and 94.4s respectively.Figure 6 shows the amplitude-frequency responses of node 19 of the inner rotor in the x direction and the torsion direction.As shown in the figure, the calculation results of the reduced dimension model with the first 15 orders and the first 10 orders dominant modes are highly consistent with the original system model, while for the reduced dimension model with the first five dominant modes preserved, the amplitude-frequency responses have obvious low amplitude near the resonance peak.Therefore, considering the calculation accuracy and time, the reduced dimension model with the first 10 dominant modes is selected for subsequent analysis.

Influence Analysis of Bearing Radial Clearance
Considering the radial clearance of bearing 1, bearing 2, the inter-shaft bearing 4, and bearing 5, the frequency sweep calculation is performed within the speed range of 1-40000 r/min to obtain the amplitude-frequency response of the system.Figure 7 shows the The figure shows that there are three resonance peaks in the region corresponding to the first three-order critical speeds considering the radial clearance of different bearings.
For the amplitude-frequency curve considering the bearing 2 clearance, the speeds of the first three-order resonance peak are obviously lower than the critical speed, and a new resonance peak appears between the first and second resonance peaks.For the amplitude-frequency curve considering the radial clearance of the inter-shaft bearing 4, the corresponding speeds of the first three-order resonance peak are almost the same as the first three-order critical speed.It can be seen that the sensitivity of the radial clearance of different bearings to the response is different, and the reason is that the stiffness nonlinearity of the bearings at different positions has a different sensitivity to the response, which is related to the load distribution between the bearings.For example, for the inter-shaft bearing 4, because it is very close to the inter-shaft bearing 3 and the sprag clutch, and shares the load in the x and y directions at this position, the stiffness changes in the x and y directions have little effect on the system response.
The amplitude-frequency response of the system considering the radial clearance of different bearings has the phenomenon of amplitude jump at the resonance peak, especially for the amplitude-frequency response of bearing 2, with the increase of the radial clearance, the vibration jump phenomenon becomes more obvious.At the same time, compared with the critical speed, the frequency corresponding to the resonance peak still has a hysteresis, and as the radial clearance of the bearing increases, the frequency lag corresponding to the resonance peak is more obvious.To more specifically show the amplitude jump and frequency hysteresis near the critical speed, the vibration response of node 4 and node 19 considering the radial clearance of the bearing 2 near the first-order resonance peak are separately displayed, and the amplitude-frequency response with a bearing radial clearance of 5μm is used for comparison, as shown in Figure 8.
Under the different radial clearances of bearing 2, the resonance peaks of the amplitude-frequency curves of node 4 and node 19 correspond to the same speed, and the relative change trend of the resonance peak is also the same.In the forced resonance region, both of them have obvious amplitude jumps and frequency hysteresis.The larger the radial clearance of the bearing, the more obvious the amplitude jump at the resonance peak and the more lagging the frequency, which fully shows that the bearing with large radial clearance has stronger nonlinearity.In addition, in the speed range before the first resonance peak, with the increase of the bearing radial clearance, the vibration amplitude of the system under the non-resonant speed also increases.As shown in Figure 9, the amplitude jump and frequency hysteresis are quite obvious in the first and third resonant peaks, and they become more significant with the increase of the radial clearance.However, in the second resonant peak, the bearing radial clearance hardly affects the formant frequency and its amplitude.
Figure 10 shows the amplitude-frequency curve of the dynamic load of the inter-shaft bearings 3 and 4 under the unbalance of the inner and outer rotors, and the radial clearance of bearing 2 is 10μm.
rotating angular velocity, and the radius of the outer ring.

Figure 2 
Figure 2 The schematic diagram of the inter-shaft bearing As shown in Figure 2, ui and ue are the center movement distances of the inner ring and outer ring, respectively.For the inter-shaft bearing 4, the components of ui in the X and Y directions are i x  and  i y , and the components of ue in the X and Y directions are e x

Figure 3
Figure 3 The master nodes of the outer rotor

Figure 5
Figure 5 Time-domain responses of nodes at 4000rpm: a Node 4 in the y direction; b Node 19 in the y directionFigure6shows the amplitude-frequency responses of node 19 of the inner rotor in the x direction and the torsion direction.As shown in the figure, the calculation results of the reduced dimension model with the first 15 orders and the first 10 orders dominant modes are highly consistent with the original system model, while for the reduced dimension model with the first five dominant modes preserved, the amplitude-frequency responses have obvious low amplitude near the resonance peak.Therefore, considering the calculation accuracy and time, the reduced dimension model with the first 10 dominant modes is selected for subsequent analysis.

Figure 6
Figure 6 AFR of node 19 under the 20um radial clearance of bearing 2: a AFR in the x direction; b AFR in the torsional direction curves of bearing 1, bearing 2, inter-shaft bearing 4, and bearing 5 under unbalanced excitation with a radial clearance of 0μm, 10μm, and 20μm, respectively.

Figure 7
Figure 7 AFR of node 19 under radial clearances of different bearings: a Under clearances of bearing 1; b Under clearances of bearing 2; c Under clearances of inter-shaft bearing 4; d Under clearances of bearing 5To more specifically show the amplitude jump and frequency hysteresis near the critical speed, the vibration response of node 4 and node 19 considering the radial clearance of the bearing 2 near the first-order resonance peak are separately displayed, and the amplitude-frequency response with a bearing radial clearance of 5μm is used for comparison, as shown in Figure8.Under the different radial clearances of bearing 2, the resonance peaks of the amplitude-frequency curves of node 4 and node 19 correspond to the same speed, and the relative change trend of the resonance peak is also the same.In the forced resonance region, both of them have obvious amplitude jumps and frequency hysteresis.The larger the radial clearance of the bearing, the more obvious the amplitude jump at the resonance peak and the more lagging the frequency, which fully shows that the bearing with large radial clearance has stronger nonlinearity.In addition, in the speed range before the first resonance peak, with the increase of the bearing radial clearance, the vibration amplitude of the system under the non-resonant speed also increases.

dFigure 8 19 3. 2
Figure 8 AFR of nodes at the first-order formant under the radial clearances of bearings 2: a AFR of node 4; b AFR of node 19

Figure 9
Figure 9 AFR of inter-shaft bearings under the radial clearances of bearing 2: a AFR of inter-shaft bearing 3; b AFR of inter-shaft bearing4As shown in Figure9, the amplitude jump and frequency hysteresis are quite obvious in the first and third resonant peaks, and they become more significant with the increase of the radial clearance.However, in the second resonant peak, the bearing radial clearance hardly affects the formant frequency and its amplitude.Figure10shows the amplitude-frequency curve of the dynamic load of the inter-shaft bearings 3 and 4 under the unbalance of the inner and outer rotors, and the radial clearance of bearing 2 is 10μm.

Figure 10
Figure 10 AFR of inter-shaft bearings under the action of unbalance: a AFR of inter-shaft bearing 3; b AFR of inter-shaft bearing4It can be seen from the figure that at the first and third-order resonance peaks, with the decrease of the unbalance, the dynamic load Huang et al.

Table 1
Parameters of the rotors

Table 2
The stiffness coefficients of bearings