Frequency veering of railway vehicle systems and its mapping to vibration characteristics

: Frequency veering is a phenomenon that occurs during the process of modal parameter changes, which is closely related to the response characteristics of the system. Firstly, take the system with simple DOFs as the research object, the variations of modal damping ratio and modal shape in the process of frequency veering are analyzed, and the criterion for identifying frequency veering is preliminarily proposed. Then, to explore the modal evolution of complex vehicle systems with multiple DOFs, an adaptive modal continuous tracking algorithm based on local search algorithm is proposed using the Euclidean closeness of complex modal shapes as an index. The frequency veering phenomenon is analyzed with the established vehicle system dynamics model (Model Ⅰ) and reproduced through the SIMPACK model for multi-body dynamics simulation (Model Ⅱ). The perturbation method is used to analyze the mechanism of the vehicle system eigenvectors being prone to mutations during frequency veering, and the abnormal changes of the modal shapes in the process of frequency veering are further verified. In addition, two quantitative indexes for identifying frequency veering are proposed based on the modal assurance criterion (MAC) and modal shape similarity (MSS). Finally, the mapping relationship between frequency veering and vehicle system response characteristics is explored. The results indicate that before and after frequency veering, the modal shapes interchange, and in the frequency veering zone, the damping-hopping phenomenon also occurs, resulting in a significant decrease in system stability. Corresponding to the phenomena of modal damping ratios and modal shapes, the motion morphology of the vehicle system is clearly observable, and the DOF response of the car body and bogie is obviously increased, which is also manifested in the increase of the vibration of the car body and the bogie and the deterioration of the vehicle ride quality.


Introduction
In engineering applications, changes in structural parameters are closely related to their vibration response and even the system's stability [1].Especially for the systems with coupled high-density modes, the system response characteristics are susceptible to small parameter changes [2].Frequency veering is a typical phenomenon that occurs in the process of system modal parameter changes.Changes in structural parameters and other factors can cause changes in modal frequency, leading to phenomena such as two modal frequency curves approaching, intersecting, merging, and separating with changes in system parameters [3].From an apparent perspective, sometimes two frequency curves only appear to intersect, but in reality, the trajectories of the two curves veer with a large curvature and extend along the previous trajectory trend of each other, which is the phenomenon of frequency veering.
As early as the 1960s, Claassen et al. [4] and Webster [5] discovered the phenomenon of frequency curve veering in the study of modal parameters in structural dynamics systems, but they did not define it.Leissa, who first defined the phenomenon of frequency veering, described the frequency veering in detail based on the vibration of a rectangular membrane [6].Frequency veering was initially believed to be caused by computational errors, but Kuttler et al. subsequently used strict error bounds to confirm the existence of frequency veering in precise mathematical models [7].
In recent years, frequency veering has received extensive attention from scholars in the field of vibration, especially in aeronautical engineering.Chan et al. [8] discussed the mode localization and frequency veering in disordered engineering structures based on the simplified aircraft model.Li et al. [9] studied the frequency veering of drum-disk-shaft structures with elastic connections in the aero-engine rotor system.Modal coupling is a vital issue in studying the vibration characteristics of bladed disks in the aero-engine.References [10][11][12][13][14] have analyzed the frequency veering of bladed disks during mode coupling.Among them, Kenyon et al. [10] found that the system modes in the frequency veering zone are highly dense, and more modes are involved in the vibration of the bladed disk, which significantly increases the detuning sensitivity of the bladed disk.
Zhang et al. [13] further analyzed the effect of frequency veering on the vibration localization of a mistunedbladed disk.She et al. [14] emphatically analyzed the veering and merging phenomena of the nonlinear resonance frequency in bladed disk systems and discussed the sensitivity of these phenomena to some key parameters.
In the study of various other engineering problems, the analysis of frequency veering is also relatively extensive.For beam structures, a multi-span beam with irregular spacing of the supports [15], the elastically connected double-beam system [16], cable-stayed bridge and suspension bridge considering aeroelastic effects [17], beam on elastic foundation considering nonlinear effects [18][19][20], and elastically supported double span structure system [21] have all been found to have frequency veering.
For the coupled centralized mass system, Stephen [22] studied the frequency veering phenomenon of the centralized mass system coupled by springs.Even though this system was mentioned in the studies on frequency veering by Perkins et al. [23] and Liu [24], the difference is that Stephen used a way to describe the frequency veering phenomenon through asymptote rather than the derivative of eigenvalues and eigenvectors used by previous researchers.Giannini et al. [25] took the simple machine system as the research object and conducted experimental analysis on frequency veering, crossing, and lock-in phenomena.Bonisoli et al. [26] also made a numerical-experimental comparison of a parametric test rig for crossing and veering phenomena.
For plate structures, Zhao et al. [27] used the absolute nodal coordinate formulation to describe thin plate elements and discussed the veering and crossing phenomena of eigenvalue trajectories and corresponding mode shape changes when conducting modal analysis on rotating thin plates.Saito et al. [28] analyzed the frequency veering phenomenon in the nonlinear vibration of cantilever-cracked plates and proposed a method to estimate these frequencies.This phenomenon also exists in rotating functionally graded rectangular plate systems [29] and Sandwich Plates with Honeycomb Cores [30].
In addition, for planetary gears, Lin et al. [31] observed veering phenomena of the eigenvalue trajectory when studying the relationship between natural frequency and planetary gear parameters, and further calculated the coupling factor between two similar eigenvalue trajectories.For tires, frequency veering can be caused by the angular velocity [32] and deformation [33] of rotating tires.Furthermore, frequency veering phenomena have been found in studies on rotating pre-twisted blades [34], centrifugal pendulum vibration absorbers [35,36], automotive drum brakes [37], human body mode [38], wave propagation [39,40].
Recently, relevant research has also been conducted on the frequency veering of railway systems or components.Gong et al. [2] found there is frequency veering between the car body and under-chassis equipment system of railway vehicles, and research shows that when the natural frequency of the under-chassis equipment is equal to that at the veering point, the vibration transmissibility near the car body first-order bending motion is the lowest.Xia et al. [41] studied the mechanism of abnormal changes in the modal damping of vehicle systems and found that the modal frequency curve veers while abnormal changes in damping occur.
Huang et al. [42] also found that frequency veering occurs between the bogie frame and the motor mode under specific wheel equivalent conicity.Baeza et al. [43] captured the frequency veering of the rotating wheel when analyzing high-frequency railway dynamics based on the Eulerian models of the rotating flexible wheelset.
Modal frequency veering may also lead to thorny problems.Research shows that unstable coupling between modes in the veering area can lead to railhead corrugation [44].Similarly, when Nam et al. [45] analyzed the causes and mechanism of a rail pad brake squeal in a railway brake system, they found that the system showed dynamic instability when frequency veering occurred in the rail pad system.So far, there is still little research on modal frequency veering of railway vehicle systems and even less on the mapping between frequency veering and system response characteristics.However, under the external excitations of vehicle systems, changes in modal parameters are closely related to vehicle dynamics performance [46,47].Therefore, this paper researches the frequency veering phenomenon during the modal evolution of vehicle systems and explores the mapping relationship between the frequency veering and the multi-mode coupling response of vehicle systems.
The rest of this paper is organised as follows: In Section 2, the modal frequency veering phenomenon of the two-DOF spring-mass-damping system is analyzed, and the changes in modal damping ratio and modal shape during the process are explored.Moreover, the criterion for identifying frequency veering is preliminarily proposed from the perspective of MAC.In order to explore the modal evolution of complex vehicle systems with multiple DOFs, section 3 introduces the method of obtaining eigenvalues and eigenvectors of complex modes of vehicle systems.Furthermore, based on the Euclidean closeness of complex modal shapes, an adaptive modal continuous tracking algorithm based on local search algorithm was proposed.
In section 4, the Model Ⅰ is established, and the frequency veering was analyzed using the adaptive modal continuous tracking algorithm.The frequency veering was reproduced using Model Ⅱ, and then the perturbation method was used to analyze the mechanism of eigenvectors being prone to mutations during frequency veering.Furthermore, Model Ⅰ and Model Ⅱ were used to verify the phenomenon of modal shapes interchange during veering, and two quantitative indexes for identifying veering are proposed based on the MAC and MSS.Section 5 is based on the pseudo excitation method to solve the response of the vehicle system, analyzing the motion morphology, response characteristics of related DOFs, and ride quality of the vehicle system in the frequency veering zone.Section 6 concludes the paper.Fig. 1 illustrates the research flowchart of the vehicle system frequency veering presented in this paper.

Frequency veering of simple mechancial system
The two-DOF spring-mass-damping system is a typical and simple mechanical system, as shown in Fig. 2.
The differential equation of motion for this system is: k  on the system modal parameters is analyzed.For the two-DOF system, the modal frequency, modal damping ratio, and modal shape are easily solved.The modal parameters of the system are shown in Fig. 3. From Fig. 3 (a), it can be seen that the modal frequency curves veer, and the frequency veering occurs near 1 1 k = , which is called the frequency veering zone.It can be observed from Fig. 3 (b) that the modal damping ratios of the system change in the frequency veering zone, in which the reduction of the modal damping ratio means a decrease of the system stability.In the frequency veering zone, the modal shape of the system also changes, which is easily manifested as the coupling form of the two modal shapes before or after veering [2,26].In order to quantify the changes of the two modal shapes in the frequency veering process, MAC between the initial modal shape and the modal shape during frequency veering can be measured [48]: where i、j are the modal order, 1 k is the stiffness parameter variable, and 10 k is the initial stiffness value For a two-DOF system with only two eigenvalues interacting, the modal parameters are easy to obtain, and the frequency veering is easy to identify.However, for complex systems with more DOFs, the modal coupling effect is more robust, and there is more interaction between eigenvalues.In other words, obtaining modal parameters and identifying frequency veering is also more difficult.In addition, for complex systems with excitation energy inputs, the change of modal damping during frequency veering may lead to stability problems in the system.Next, the frequency veering phenomenon of the railway vehicle system and its impact on vehicle dynamics characteristics will be studied.

Eigenvalues and eigenvectors of complex mode
In the railway vehicle system, wheel-rail contact has general viscous damping, and the damping value is large, which makes the complex modal analysis theory applied to the modal analysis of the vehicle system.For a vehicle system with n DOFs, the linear differential equation of the free vibration of the whole vehicle can be expressed as: where M, C and K are respectively the mass matrix, damping matrix and stiffness matrix of the vehicle system, wr C and wr K are matrices related to wheel-rail contact parameters, q is the generalized coordinate column vector of the system, and v is the vehicle's velocity.
Set the state vector y: Then the system state equation can be obtained from equation ( 3): = + Gy Ry 0 (5) where, The particular solution of Eq. ( 5) has t e  = y φ .That is, the problem can be transformed into a general eigenvalue problem about the real value matrix.The particular solution is substituted into Eq.( 5): where  , φ are respectively eigenvalues and eigenvectors of state equation (6).For n th-order system, it has 2n complex mode conjugate eigenvalues: where  i ,  i are the i th-order modal damping and modal circular frequency of the system, respectively.Further, the system modal frequency and damping ratio can be obtained:

/ ( )
Substituting 2n complex modal conjugate eigenvalues i  and i  into Eq. ( 6 yields n eigenvectors and n complex conjugate eigenvectors: where it is worth noting that i  φ and i  φ are eigenvectors corresponding to the eigenvalue problem of state equation ( 5), so they are not called modal vectors, but only eigenvectors, while φ i and φ i can be called modal vectors.That is to say, φ i and φ i are the i th-order complex conjugate modal shape vectors of the vehicle system: ( =1,2,..., )

Euclidean closeness of complex modal shapes
In fuzzy mathematical analysis, to find subsets of the same type as the sample set in the fuzzy set, it is often necessary to analyze the similarity between the sample set and each subset in the entire set.Moreover, in fuzzy pattern recognition, when two fuzzy sets are compared according to specific characteristics, closeness can be used as an index of the similarity between two fuzzy sets.
Commonly, euclidean distance is a method of analyzing similarity between sets, and the larger the Euclidean distance, the more significant the difference; On the contrary, the higher the similarity.For two fuzzy sets B and H, their Euclidean distances can be expressed as: , m is the cardinal number of the fuzzy set, for k bB  and k hH  , Eq. ( 12) can be expressed as: To calculate the Euclidean closeness of fuzzy sets B and H, it is necessary to further process ( , ) D B H : where k b and k h are the maximum normalized elements of fuzzy sets B and H respectively.That is, ( ) Therefore, the Euclidean closeness of fuzzy sets B and H can be expressed as: Similarly, for n-DOF vehicle system in Eq. ( 11), the Euclidean closeness between the i th-order modal shape vector i φ (or i φ ) and the sample modal shape vector j φ (or j φ ) can be expressed as: where ki  and ki  are the maximum normalized elements of the i th-order modal shape vector, kj  and kj  are the maximum normalized elements of the sample modal shape vector.The larger the Euclidean of complex modal shape vector, the higher the similarity between the modal shape vector and the sample modal shape vector.Conversely, the more significant the difference.

Adaptive modal continuous tracking based on the local search algorithm
In order to master the variation law of the modal frequency, modal shape, and modal damping of the vehicle with the change of velocity, it is necessary to clarify the ascription relationship of the modal parameters under different velocities.At any velocity, the modal frequency, modal shape, and modal damping of the vehicle system all correspond one-to-one.That is, by identifying one modal parameter through the modal continuous tracking, the ascription relationship of the other two modal parameters can be identified.
Which modal parameter is used as the tracked object?Although the dimensions of the modal frequency vector and modal damping vector are low at different velocities and the numerical calculation is small, it has been found in modal continuous tracking that the modal frequency and damping of different modes are easily equal within a specific velocity range, which can lead to errors in modal tracking.Therefore, the modal parameters ascription of each order mode is identified by the modal shape vectors at different velocities, and the dimension of the modal shape vectors is high, which also ensures the accuracy and robustness in the modal continuous tracking process.
For the concerned vehicle's velocity v E  , E is the domain and v is the initial analysis velocity, and f v is the cut-off analysis velocity.In the process of tracking the modal parameters, due to the strong continuity requirement of the incremental velocity v , it is unnecessary to conduct global search analysis directly in the global domain E, which will generate too much computation.As a heuristic algorithm to solve optimization problems, local search algorithm has both high computing speed and reliability, that is, search for optimization in the right neighborhood ( ) ( ) v , where  is the neighborhood action function variable.Furthermore, considering that the modal parameters of the vehicle system do not vary linearly with velocity, the optimal velocity solutions to achieve modal continuous tracking are not uniformly distributed.Therefore, to achieve modal continuous tracking in the velocity domain E, this paper adopts a local search algorithm to adaptively obtain the optimal velocity solutions with variable step sizes, and then identify the ascription relationship of modal parameters at different velocities.
The process of adaptive modal continuous tracking based on the local search algorithm combined with Euclidean closeness of complex modal shapes is shown in Fig. 5, and the details are as follows: In the velocity domain , for any i th-order mode of the vehicle system, the modal shape pi φ , modal frequency pi f and modal damping pi σ at the initial velocity 1 v can be calculated, and then the neighborhood action function can be set: where r is a natural number, t is a positive integer and v is the set of optimal velocity solutions for the i th-order mode, and p =1 for initial velocity.
The local domain

(
) ( ) of the corresponding velocity v can be generated by the neighborhood action function Eq. ( 17).The search value of the parameter r according to the gradient of r =0,1,2… can realize the exponential change of  the n th-order modal shape p φ of the system can be obtained: Then, perform Euclidean closeness analysis on the obtained modal shape pi φ (sample set) and any k thorder modal shape vector pk φ in Eq. ( 18), and take the maximum value for Euclidean closeness threshold verification to ensure the convergence of the iterative velocity solutions: (19) where U  is the European closeness threshold is close to 100%.The closer this threshold is to 100%, the higher the reliability of the modal continuous tracking process.In this paper, the threshold is set as 99.9%.
to the modal shape vector pk φ are obtained.Until the velocity v in the local domain meets f vv  , the iteration ends.So far, the optimal velocity solutions for non-uniform distribution i v , modal shape i φ , modal frequency i f and modal damping i σ for the i th-order mode of the vehicle system are obtained.

Vehicle system dynamics model (Model Ⅰ)
To obtain the modal parameters of the railway vehicle system, a vehicle system dynamics model is established, as shown in Fig. 6.In the model, the primary and secondary suspension parameters are linearized, and the creep coefficient is set as a constant, using a linear wheel-rail contact relationship [49].There are 23 DOFs considered as shown in   The motion equations of the vehicle system are as follows: For the car body, where the suspension forces and moments can be expressed as: ( ( ... ( 2 For the bogies ( i = 1 and i = 2 correspond to the front and rear bogies, respectively; i = 1 at = 1, 2; where the suspension forces and moments in Eq. ( 22) are:

M k h y h y h l c h y h y
For the wheelsets, where the suspension forces and moments in Eq. ( 24) are derived as follows:

Frequency and damping in the process of frequency veering
When the velocity is low, the bogie hunting frequency is low, but the hunting frequency gradually increases as the velocity increases.During this process, it is inevitable that the hunting frequency is close to other modal frequencies at a certain specific velocity.Our team's research has found the presence of frequency veering in the vehicle system for the railway vehicle system with dynamic parameters shown in Table A1 of Appendix A.
As for the established vehicle system dynamics model with 23 DOFs, Eq. ( 20)-Eq.( 26) can be expressed in the form of Eq. ( 3), so the modal parameters of the vehicle at any velocity can be obtained through the complex modal analysis theory described in section 3.1.Fig. 7(a) and Fig. 7(b) show the modal frequencies and damping ratios of the vehicle system directly obtained through complex modal analysis theory, further, Fig. 7(c) and Fig. 7(d) show the modal parameters identified by using a constant velocity step size (similar to the analysis method of simple mechancial system shown in Fig. 3).It can be observed that, unlike simple mechancial system, the vehicle system has a large number of modal feature roots and some of them are relatively close.
The use of constant velocity step size can easily lead to a divergence in the identification of modal parameters.
Although this can be converged by infinitely reducing the step size, it will greatly increase the computation and cannot guarantee confidence.Therefore, this paper proposes an adaptive variable step size algorithm, that is, the selection of modal parameters at which speeds is determined by the adaptive modal continuous tracking algorithm described in section 3.3, and the obtained velocity solutions and modal parameters are shown in Fig. 8.
According to Fig. 8(a) and Fig. 8(b), the different densities of velocity solutions obtained in the frequency and non-frequency veering zones indicate that the tracking algorithm can adaptively obtain velocity solutions with variable step sizes.Moreover, it can be seen from the frequency and damping ratio curves that the algorithm can clearly track the changes in modal frequency and modal damping ratio with velocity.It can be seen from Fig. 8(a) that the frequency curves of the 9th-order and 11th-order modes and the 10th-order and 15th-order modes have veered around at 90 km/h and 200 km/h.Fig. 8(b) shows that there is a dampinghopping phenomenon in the frequency veering zone corresponding to the mode in which frequency veering occurs.That is, the modal damping ratio curves undergo sudden changes in the veering zone.
In addition, when the vehicle's velocity increases to 400 km/h, the 11th-order modal damping ratio, mainly the hunting motion of the bogie, has decreased to nearly 0, indicating that the hunting stability margin of the vehicle system is significantly insufficient.

Reproduction of frequency veering (Model Ⅱ)
In order to verify the frequency veering phenomenon found, based on the dynamic parameters in Table A1 of Appendix A, the multi-body dynamics software SIMPACK is used to establish the vehicle dynamics model The modal frequency parameters of the vehicle system at different velocities can be directly obtained through Model Ⅱ, and the frequency curves are identified as shown in Fig. 9(b).It can be found that frequency veering occurs in the vehicle system, and its modal frequency curve is in good agreement with the modal frequency curve results in Fig. 8(a).Furthermore, the simulation results of Model Ⅱ also verify the correctness of Model Ⅰ.

Analysis of eigenvector mutation
The perturbation method is an effective tool for analyzing the characteristics of systems with changing parameters.For any i th-order mode of the vehicle system, the state equation can be obtained: where, 0  C is the damping matrix of the initial system,  C is the first-order disturbance caused by velocity, and the dimensionless parameter , then we can get: where, i  and i φ can be expanded into power series by  : (30) where 0 i  and 0 i φ are the original characteristic parameters of the i th-order mode and satisfy the orthogonal condition.Substituting Eq. ( 29) and Eq. ( 30) into Eq.( 27), considering only first-order perturbations and setting  to zero, we can obtain: (31) For any eigenvector, it can be represented by a linear combination of orthogonal bases.Similarly, for the n-DOF vehicle system, 1 i φ can be represented as: (32) where m  is the expansion coefficient of the first-order perturbation of the i th-order eigenvector corresponding to the m th-order original eigenvector base Substituting Eq. ( 32) into Eq.( 31 It can be seen from Eq. ( 34) that when the system feature root  may become large.
That is, the first-order disturbance of the eigenvector in Eq. ( 32) becomes large, resulting in a significant change in modal shape.Therefore, for the vehicle system, with the increase of velocity, the hunting frequency of the bogie increases to close to other modal frequencies of the vehicle system, which may cause a mutation in the modal shape.So next, the modal shapes of the vehicle system in the process of frequency veering will be analyzed.

Modal shapes in the process of frequency veering
Similar to the two-DOF spring-mass-damping system, the modal shapes that undergo frequency veering will also change before and after the veering zone.To analyze the changes in modal shapes before and after two frequency veerings (90 km/h and 200 km/h), the modal shapes at 50 km/h, 125 km/h, and 250 km/h can be selected for research.On the one hand, the modal shapes can be obtained through Model Ⅰ; on the other hand, they can also be reproduced by Model Ⅱ, both of which coincide and are simultaneously plotted in Table .2. .1, it can be seen that before the frequency veering of the 9th-order and 11thorder modes (50 km/h), the 9th-order modal shape is mainly manifested as the hunting of the wheelset driving lateral displacement of the bogie frame and car body, and the 11th-order modal shape is mainly manifested as lower center roll of the car body (car body lateral displacement + in-phase roll).While after the frequency veering (125 km/h), the 9th-order modal shape is dominated by lower center roll of the car body, and the 11thorder modal shape is mainly manifested as the hunting of the wheelset driving lateral displacement of the bogie frame and car body, that is, the two modal shapes are interchanged before and after veering.

Based on Table. 2 and Table
Similarly, combining Table .3 and Table .1, it can be found that the 10th-order and 15th-order modal shapes before frequency veering (125 km/h) are mainly the hunting of the wheelset driving lateral displacement of bogie frame and car body, and upper center roll of the car body (car body lateral displacement + anti-phase roll), respectively.When the two modes pass through the frequency veering zone (250 km/h), the modal shapes also undergo an interchange phenomenon.
Table .2 The 9th-order and 11th-order modal shapes before and after frequency veering.The first quantitative index is MAC： where, i and j are the modal order numbers, 0 v is the initial velocity before the frequency veering, and v is the velocity variable.
Fig. 10 (a) and Fig. 10 (b) show the MAC of the 9th-order and 11th-order modes, 10th-order and 15th-order modes of the vehicle system.The values of MAC (9,9), MAC (11,11), MAC (10,10), and MAC (15,15) decrease continuously after passing through the frequency veering zone, indicating that their modal shapes no longer exist as similar to the original modal shapes.On the contrary, the values of MAC (9,11), MAC (11,9), MAC (10,15), and MAC (15,10) continue to increase after passing through the veering zone, which indicates that for the two modes that undergo frequency veering, the modal shape of one mode before veering has high similarity with the modal shape of the other mode after veering.In other words, there is an interchange between the 9th-order and 11th-order modal shapes, as well as between the 10th-order and 15th-order modal shapes.In addition, it can be found that the frequency veering of certain two modes not only affects their own modal shapes, but also causes fluctuations in the MAC values of other modes of the system.That is, the frequency veering of certain two modes will also have a certain degree of impact on the multi-mode shapes of the entire system.
Considering that the modal coupling of the vehicle system with multiple DOFs is complicated, another quantitative index MSS was proposed based on a similarity analysis method for pharmaceuticals [51].The MSS between the i th-order and j th-order modes is defined as: ) where, i φ is the n-dimensional modal shape vector, i  and j  are respectively the mean values of modal shape i φ and j φ .In addition, MSS(i, j) is a real number between [0,1], and the larger the value, the higher the similarity between the two modal shapes.If and only if two modal shapes are the same, MSS(i, j) =1.
The MSS of the 9th-order and 11th-order modes, as well as the 10th-order and 15th-order modes, are shown in Fig. 10 (c) and Fig. 10 (d).For the two modes in which frequency veering occurs, the MSS significantly increases when approaching the veering zone.Therefore, the MSS index can also serve as a criterion for frequency veering. ( Fig. 10.Two criteria for frequency veering: MAC of (a) 9th-order and 11th-order modes, (b) 10th-order and 15th-order modes; MSS of (c) 9th-order and 11th-order modes, (d) 10th-order and 15th-order modes.

Mapping between frequency veering and dynamic response
Frequency veering is a phenomenon that occurs in modal parameter change, and it still belongs to the analysis in the modal space.The parameter change in the modal space is closely related to the dynamic characteristics in the physical space.Based on the above frequency veering analysis, it is found that there is an interchange in the modal shapes coupled with the damping-hopping phenomenon.Will these modal parameter changes impact on the dynamic characteristics of the vehicle system?Therefore, section 5 explores the mapping relationship between a series of phenomena of frequency veering in modal space and dynamic responses in physical space.
In section 5, for solving the response of the vehicle system, the input excitation of Model Ⅱ is the highspeed train track irregularity spectrum, as shown in Fig. 9 (a).However, Model Ⅰ adopts a solution method of response in the frequency domain based on the pseudo excitation method, as shown in Appendix B, where the input excitation is also the high-speed train track irregularity, and the parameters of the track irregularity are shown in Table .B1.

Motion morphology of the vehicle system
The pseudo excitation method can obtain the PSD of the vehicle system response, that is, the power spectral density () S  , through the input power spectrum of the system and the basic theory of spectral analysis.Fig. 11 shows the lateral vibration PSD at the center of the car body and the end of the bogie frame at five velocity levels before and after frequency veering based on Model Ⅰ.It can be seen that at the velocities corresponding to the frequency veering zone, namely 90 km/h and 200 km/h, the amplitudes of the lateral vibration PSD at the low frequencies significantly increase, with corresponding frequencies of 0.61 Hz and 1.12 Hz, respectively, which are close to the lower center roll frequency of car body at 0.63Hz and the upper center roll frequency of car body at 1.13 Hz; the amplitudes of lateral vibration PSD at the end of the bogie frame increase significantly at 0.62 Hz and 1.13 Hz, which are respectively close to the hunting frequency of the bogie.
Therefore, in the frequency veering zone, the response of the lower (upper) center roll motion of the car body and the hunting motion of the bogie both significantly increase.
Consistent modal shapes are obtained based on Model Ⅰ and Model Ⅱ. Table .4 and Table .5 show the modal shapes of the 9th-order and 11th-order modes, as well as the 10th-order and 15th-order modes of the vehicle system in the frequency veering zone, respectively.It can be found that in the frequency veering zone at 90 km/h, the 9th-order and 11th-order modal shapes both exist in the coupling form of "wheelset and frame lateral displacement + car body lower roll"; in the frequency veering zone at 200 km/h, the 10th-order and 15th-order modal shapes both exist in the coupling form of "wheelset and frame lateral displacement + car body upper roll".That is to say, with the increase of the velocity, when the bogie hunting frequencies (9th-order and 10thorder) increase to near the frequencies of lower center roll (10th-order) and upper center roll (15th-order), the frequency veering occurs.In addition, in the frequency veering zone, the modal damping ratio of the bogie hunting decreases, which intensifies the coupling motion between the bogie hunting and the car body's lower (upper) center roll.

Responses of the vehicle's DOF
To explore the response characteristics of the vehicle system in the frequency veering zone, the responses of the vehicle's DOF are solved.The PSD of the vehicle's DOF at different velocities can be obtained through the pseudo excitation method.Further, based on Model Ⅰ, in order to evaluate the energy of PSD at different velocities, the second-order central moment 2  of the power spectral density () S  is introduced:

Vibration in the time domain and ride quality
Fig. 13 shows the lateral time-domain vibration at the center of the car body floor and the end of the front bogie frame based on Model Ⅱ.It can be seen that as the velocity increases, the vibration amplitudes of the car body and the bogie frame do not monotonically increase, but corresponding to the frequency veering zone, the vibration amplitudes increase significantly at 90km/h and 200km/h.That is, the time-domain vibration also indicates that the response of the coupling mode between the car body and the bogie is excited when frequency veering occurs.
The response amplitude k a in the frequency domain can be obtained by Eq. ( 8) in Appendix B based on Model Ⅰ.In addition, the time-domain vibration can be obtained directly based on Model Ⅱ, and vibration k a in the frequency domain can be obtained further through Fast Fourier transform.Therefore, the Sperling index s W [52] for evaluating the ride quality of passenger trains is shown below: where, k a (m/s 2 ) is the acceleration amplitude in the frequency domain; f is the corresponding acceleration frequency (Hz); and () Ff is the frequency-weighted function.The limit values of ride quality according to GB/T 5599-2019 are 2.50, 2.75, and 3.00 for excellent, medium, and qualified levels, respectively.

Conclusions
This study analyzes modal frequency veering in the universal mechanical system with simple DOFs, and further delves into the frequency veering of complex vehicle systems with multiple DOFs.A series of phenomena of damping-hopping and modal shapes interchange in the process of vehicle system frequency veering are explored, and two quantitative indexes for identifying frequency veering are proposed.Finally, the mapping relationship between frequency veering and response characteristics of the vehicle system is explored.
The modal damping ratio of the simple mechanical system changes obviously when frequency veering occurs, where the reduction of the modal damping ratio means a decrease in the system stability, and the two modal shapes are interchanged.However, unlike the system with simple DOFs, obtaining modal parameters and identifying frequency veering for the vehicle system is more complicated.Combined with the established vehicle system dynamics model, the proposed adaptive modal continuous tracking method can accurately and clearly track the evolution law of each modal frequency, damping ratio and modal shape of the vehicle system with adaptive velocity step.
With the increase of the velocity, when the bogie hunting frequencies increase to near the frequencies of the lower (upper) center roll of the car body, two frequency curves tend to approach and then veer with a large curvature to separate, accompanied by damping-hopping and the interchange of modal shapes.
In the frequency veering zone, the modal shape is presented as the coupling form of the two modal shapes before the frequency veering, and the proposed quantitative indexes based on MAC and MSS can effectively identify the frequency veering.In addition, the vehicle system's motion morphology can be observed in the frequency veering zone.The stability of the vehicle system decreases due to the reduction of the bogie hunting damping ratio, causing the vehicle system's DOF response related to the coupling mode to be excited: The response of the coupling mode between the bogie hunting and the lower (upper) center roll of car body significantly increases, and the amplitudes of lateral vibration PSD of car body increase at low frequencies, corresponding to 0.61Hz and 1.12Hz, respectively, which are close to the lower center roll frequency 0.63Hz and upper center roll frequency 1.13 Hz of the car body before frequency veering; the amplitude of the lateral vibration PSD at the end of the bogie frame significantly increases at 0.62Hz and 1.13Hz, which are the hunting frequencies of the bogie in the two frequency veering zones, respectively.In addition, the time-domain vibration of the car body and the bogie frame, and the ride quality of the vehicle indicate the intensification of the vehicle system response.
The frequency veering threatens the dynamic characteristics and even stability of the vehicle system, but it can be controlled through the design of modal parameters of railway vehicles and the planning of operating velocity.In subsequent research, the impact of critical parameters of the vehicle system on frequency veering and the control of frequency steering will be analyzed.
In addition, in various engineering practices, the phenomenon of changes in system parameters is common, which may lead to frequency veering and potentially affect system stability.Therefore, this research method can also provide a reference for the study of frequency veering of other engineering structures.

Appendix B
Model Ⅰ adopts the solution method of response in frequency domain based on the pseudo excitation method.
For the four-axis railway vehicle running on the track, the random excitation to the vehicle system is:  (1) where, ( 1, 2,3, 4) j a j = is the excitation strength of each wheelset, () Ft is the time history function of stationary random excitation.( 1, 2,3, 4) j t j = is the lag time of the wheelset excitation, and with the first wheelset as the reference point, there is 1 0 t = .
() F t is regarded as a generalized single point excitation, and if the self-spectral density of () Ft is () FF S  , then the corresponding pseudo excitation is: By the above method, the multi-point cross-correlation excitation can be converted into a generalized single point excitation, so that the track irregularity spectrum can be used as input.The common track irregularity is the spatial spectrum ()   S .In order to obtain the power spectrum of the system response, it needs to be converted into the frequency-domain power spectrum ( , )

 
Sv , which is related to velocity.The conversion relationship is as follows: where,  is the spatial circular frequency (rad/m), ω is the excitation circle frequency (rad/s).
According to the theory of random vibration, the response of the vehicle system under track random excitation is: where, p E is the position matrix of excitation input, Q ) are the position matrices of the displacement and velocity excitation input.Furthermore, the power spectrum of the system response can be obtained as: where,  Y and  Y represent the conjugate and transposed vectors of the vector Y , respectively.When establishing the vibration equation of a vehicle system, the coordinate system is usually chosen at a symmetrical position.Therefore, when the vibration at the asymmetric position is needed to be calculated, there is: ˆ= y Ψy (6) where, Ψ is the coordinate transformation matrix, substituting Eq. ( 6) into Eq.( 5), and the power spectrum of the system response is: ˆˆ= yy   S Ψyy Ψ (7) Due to the ride quality index using frequency-domain acceleration as the calculation element, the vibration amplitude k a at frequency  k is obtained as: where, is the sampling frequency step size.
In addition, the track irregularity ()   S in the form of space spectrum in Eq. ( 3) can be expressed as： For the vertical irregularity, For the lateral irregularity and gauge irregularity, there is a same spectral density expression : where, S is the power spectral density, 2 / (1/ ) mm .
The parameters of track irregularity are :

Fig. 1 .
Fig. 1.The research flowchart of the vehicle system frequency veering.

where 2 k and 2 c
are the parameters that characterize the coupling properties between the two DOFs, they are set to 2 0.05 k = N/m and 2 0.05 c = Ns/m.Without loss of generality, considering the case of 12 1 mm == kg and 13 1 cc == Ns/m，the influence of 1 [0.6,1.4]

of 1 k
before frequency veering.The system's MAC is calculated by arbitrarily selecting two states before veering ( 1 0.7 k = ) and after veering ( 1 1.3 k = ) , as shown in Fig. 4 (a), which indicates that the two mode shapes of the system are interchanged before and after veering.Furthermore, Fig. 4 (b) shows the four MACs in the frequency veering process.It can be seen that MAC (1,2) and MAC (2,1) increase significantly when passing through the veering zone, and when 1 1.0 k = , the four MACs are no longer close to 0 or 1, indicating that the modal shapes no longer exist in a form similar to the original modal shapes.Therefore, MAC can also serve as a criterion for identifying frequency veering.(a) MAC for 1 0.7 k = and 1 1.3 k = (b) MAC as 1 k changes Fig. 4. MAC of the two-DOF spring-mass-damping system.

1
10 r − in the neighborhood action function, thus realizing the velocity v to search along ( ) ( ) ii pp  +→ vv gradient direction.For any velocity v in the search process,

Fig. 5 .
Fig. 5.The process of adaptive modal continuous tracking based on local search algorithm.

(
Model Ⅱ), as shown in Fig.9(a).The model consists of 1 car body, two bogies, four wheelsets, and eight axle boxes, among which the car body, bogie, and wheelset all have 6 DOFs, including extension, Lateral displacement, bounce, roll, pitch, yaw, and the axle box has pitch DOF.In addition, the input excitation of the vehicle model adopts the high-speed train track irregularity spectrum in the dynamic calculations.
, the fluctuation of  C caused by the change in velocity is: 0 ) and multiplying left by 0 orthogonality, we can obtain:

3
The 10th-order and 15th-order modal shapes before and after frequency veering.are proposed to explore the changes in modal shapes in the process of frequency veering, which can serve as criteria for frequency veering.

Fig. 11 .
Fig. 11.Lateral vibration PSD at (a) the center of the car body and (b) the end of the bogie frame.

Fig. 12 Fig. 12 .
Fig.12shows the second-order central moment of PSD of the front and rear bogies' lateral displacement and yaw, and that of the car body's lateral displacement and roll.The results indicate that in the frequency veering zone, the DOF responses of lateral displacement and yaw of the frames and wheelsets increase, which means an increase in the response of the bogie hunting.Moreover, the DOF responses of the lateral

Fig. 14 Fig. 13 .Fig. 14 .
Fig.14shows the lateral Sperling index of the vehicle related to the velocity based on Model Ⅰ and Model Ⅱ.It can be seen that the Spelling index results obtained by the two models are consistent, and in the frequency veering zone, the ride quality of the vehicle obviously deteriorates.In addition, when the velocity reaches 400 km/h, due to the hunting instability of the bogie, the Sperling index sharply increases, and the ride quality is no longer excellent level.Therefore, the operating velocities of vehicles should try to avoid the velocities in the frequency veering zone.
T is the time lag matrix, and for the four-axle vehicle,

Table . 1
DOF order number for vehicle system dynamics model. ) )

Table . 4
Modal shapes of the 9th-order and 11th-order modes in frequency veering.

Table . 5
Modal shapes of the 10th-order and 15th-order modes in frequency veering.

Table B1 .
The parameters of track irregularity.