# Dynamical Effect Investigations of Component’s Internal Interface by Using Techniques of Rigid-Flex Coupling Simulation

Authors: Te-te Li; Wei Du; Ming-wei Piao; Yong-zheng Guo; Shi-ying Jin; Chun-ge Nie; Ji Fang; Ya-jun Cheng; Jun Fan
Journal: Shock and Vibration
 (2021)

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## Abstract

As a component of servicing car body, the internal interfaces of aluminum alloy carbody include all connections of equipments hanged under floor and mounted on roof, which are expected to form the weak coupling relationship. For an imported prototype with primary hunting phenomenon, a dynamical design methodology of speeding-up bogies was proposed. The analysis graph of full-vehicle stability properties and variation patterns is used to clarify a self-adaptive improvement direction, i.e.,λeN ≥ λemin, and λemin = (0.03–0.05). Therefore, the central hollow tread wear can be self-cleaned in time or regularly by crossing over the dedicated lines of different speed-grades. The modified strategy with strong/weak internal interface transaction of servicing car body was furthermore formulated based on the dynamical condensation method of component interface displacements. The causal relationship between bogie vibration alarm and car body fluttering phenomenon was then demonstrated by using techniques of rigid-flex coupling simulation. The self-excited vibration of traction converter intersects with the unstable hunting oscillation, ca. 9.2/9.3 Hz, which is consistent with the conclusions of tracking-test investigations on two car body fluttering formations. The technical space to promote the construction speed is thereby lost completely because of ride comfort decline, unsafe vibration of onboard electrical equipments, and weld fatigue damage of aluminum alloy car body. However, the rigid-flex coupling simulation analyses of trailer TC02/07 confirm that the safety threshold of bogie vibration warning can be appropriately increased as long as the lateral modal frequency of traction converters is greater than 12 Hz, preferably close to 14 Hz.


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## Body

## 1. Introduction

The servicing car body is taking the aluminium alloy car body as a component, the internal interface of which includes the connection relationships between the aluminium alloy car body and all the equipments hanged under floor and mounted on roof. By using the techniques of rigid-flex coupling simulation, the dynamical effect investigations of component’s internal interface will be helpful to promote scientifically the construction speed of High-Speed Rolling Stock (HSRS) on the reasonable wheel-rail matching conditions.Similar to the coupling interface between spacecraft and launch vehicle, the internal interface of servicing car body is expected to form a weak coupling relationship. Different from the situations of carbon or stainless steel car body, the manufacture of aluminium alloy car body adopts the new technological measures, like extrusion forming, longitudinal welding, and the cylinder wall effect of which makes then the modal frequency of car body 1st vertical bending in servicing state close to or more than 11 Hz. However, due to the inherent defects, like floor without longitudinal beams and side walls without frameworks, the dynamical effects of internal interface may have substantial impacts on the evaluations of vibration comfort and structural fatigue. So, in reviewing the MBS (Multi-Body System) approaches for rail and road vehicles, Bruni pointed out that the higher frequency response analysis is a new task or challenge [1].Same as the wing fluttering of servicing fighters, the lateral elastic vibration of servicing car body is the primary mechanical problem to be solved in the design process of HSRS rigid-flex coupling system. Considering the influences of first car body fluttering phenomenon on the relevant equipments, e.g., traction converter or pantograph is self-excited [2, 3], the formation mechanism of lateral coupling vibrations was proposed on the basis of three mechanical criteria of excitation source, transmission medium, and resonance condition [4], i.e., the lateral coupling relationship can be formed between the servicing car body and the running gear when the anti-hunting high-frequency impedance interaction is taken as the correlative excitation.Different from the speeding-up situations on European existing rails [5], the central hollow tread wear is a unique form of detrimental wear in Chinese HSR (High-Speed Rails) practices [6–8], which cannot be removed only by improving the wheel-rail relationship [9–13]. Unlike the dual-point contact on rail shoulder, the dynamical effect caused by the dual-point contact on rail top will become more and more intense, which becomes a strong enough input excitation.Once the local conformal or bad contact is formed occasionally between worn wheel and rail in some specific sections, the dual-point contact on the rail top will force the wheelset to produce an unstable hunting oscillation, the dominant frequency of which is ca. (7.0–8.0) Hz. Since the anti-hunting phase lag is attenuated close to zero, the anti-hunting high-frequency impedance interaction [14] will be introduced between servicing car body and running gear, the dynamical impact of which is the same as the spring with higher rigidity.If the lateral modal frequency of any equipment hanged under floor or mounted on roof intersects with or is close to the above dominant frequency of unstable hunting oscillation, the coupling resonance of servicing car body will take place, which is referred to as the car body fluttering phenomenon. And, even the conventional fatigue is forced to be transformed into the difficult problem of threshold crossing [15], i.e., the peak coefficient increases continuously and exceeds the safety threshold due to the roof resonances.Chinese HSR practices should be traced back to the essential issue of HSRS operation and maintenance, so as to correctly recognize the formation mechanism of central hollow tread wear and its possible negative influences from the viewpoint of wheel-rail lateral dynamical equilibrium relationship [16, 17]. If the different kinds of foreign HSRS are used as the imported prototypes, they should be gradually absorbed and transformed in combination with our own particularities.The original design defects of an imported prototype were pointed out on the basis of relevant theoretical and experimental researches, i.e., the primary hunting phenomenon and its negative feedback impacts on wheel creepage and wear [18–20]. The self-adaptive technical improvements and their technical effects are then formulated and evaluated for these reasons [21], which provides the necessary technical preparation for scientifically promoting the construction speed under the reasonable wheel-rail matching condition. According to the requirements of standards UIC518 or EN14363, the maximum value of actual equivalent conicity is required by λemax ≤ 0.15, when the speed v > 280 km/h. Therefore, the nominal equivalent conicity shall be greater than or equal to its allowable minimum value, λeN ≥ λemin and λemin = (0.03–0.05).The two tracking-test results were reported recently in [22, 23], either the high-speed carbody shaking phenomenon happens again at lower track conicity because of the renewal design of LMB-10 tread or the carbody fluttering phenomenon occurs more seriously due to changed to use the new design of rubber hanging elements. So the following proposition is necessary to be demonstrated, i.e., whether there is a causal relationship between the bogie vibration warning and carbody fluttering phenomenon, which is the main motivation in this research.This motivation comes from the enlightenment of interdisciplinary researches, i.e., Bionic Flapping Wing. The rigid-flex wing consists of rigid and flexible wings, and the coupling interaction between them depends mainly on the backlash nonlinearity. A boundary controller with input backlash was proposed in Refs. [24, 25] based on the Partial Differential Equations (PDEs), and the wing system was proven to be stable by using Lyapunov’s direct method, i.e., weak coupling interaction between rigid and flexible wings.Considering the integration and complexity of HSRS systems, the rigid-flex coupling simulation techniques are adopted in this research to demonstrate the above causal relationship. The transaction strategy is modified further with the strong/weak coupling interfaces of aluminum alloy car body to full-vehicle MBS, so as to weaken as much as possible the coupling interaction produced on the internal interfaces of servicing car body, and to satisfy with reliability, economy, and practicability the vibration safety of onboard electrical equipments required by standard IEC61373 – 2010.The proposition demonstration of above causal relationship has the three important meanings as follows:(1)
Under the premise of adhering to the RAMS (Reliability, Availability, Maintainability, and Safety) management system, the condition-based maintenance can be established in the analysis of fault formation mechanism, to avoid as far as possible the human arbitrariness, and make the maintenance work from passive to active, and then gradually move towards the intelligent operation and maintenance.(2)
The initial HSRS is based on the different kinds of imported prototypes. Nevertheless, we must not make the HSR practices become “the water of no source” or “the end of no origin” by relying on the wisdom of Chinese people, instead of blindly copying the foreign experiences such as railhead grinding and its profile modification.(3)
The rigid-flex coupling system design for Chinese HSRS should consolidate the foundation and cultivate the talents, to promote scientifically the construction speed under the reasonable wheel-rail matching condition, and make the sufficient technical preparation for breaking the operating mode of HSR dedicated lines.For these reasons, this research first briefly discussed the HSRS rigid-flex coupling system with the main nonlinear factors in Section2. For an imported prototype with primary hunting phenomenon, a dynamical design methodology was proposed with three key techniques to clarify the self-adaptive improvement direction in Section 3. For scientifically promoting the construction speed, the above causal relationship was demonstrated by using the modified strategy of strong/weak coupling interface transaction in Section (4–5). And the conclusions and prospects were summarized in Section 6.The following two innovative works are attempted to be accomplished:(1)
To further modify the interface transaction strategy of flexible body to full-vehicle MBS with strong/weak coupling relationship;(2)
To further introduce the singularities of complex constraints into the HSRS rigid-flex coupling systems, so as to explore the dual mechanical attributes of vibration fatigue.


## 2. HSRS Rigid-Flex Coupling System with Nonlinear Influencing Factors

The basic reason for the linear elastic vibrations of flexible bodies lies in the nonlinearity of complex constrained internal forces, which is the main task of this research on HSRS rigid-flex coupling system. Therefore, we must firmly grasp the two main nonlinear influencing factors of wheel-rail contact and bogie suspension, and use the analysis graphs of full-vehicle stability properties and variation patterns to guide instructively the optimal parameter configuration of self-adaptive high-speed bogies, so as to return the bogie nominal model to the regular perturbation problem in the sense of asymptotical stability.


### 2.1. Singularities of Complex Constraints

The basic equations of HSRS rigid-flex coupling system are as follows:(1)Mξ¨+M˙−12∂M∂ξξ˙T+Dξ˙+Kξ+fg+∂C∂ξTμ=Q,in which ξ = [qq¯]T is spanning the generalized and modal spaces, including MBS's generalized space variables and flexible body's elastic modal space variable q¯. M is the mass matrix of MBS, including flexible bodies (e.g., hypothesis of nine invariants [26]), which is nonsymmetric, nonpositive definite, and noninvertible; K and D are the stiffness and damping matrices of the coupling system, respectively; fgis the gravity obtained through the gravitational potential energies of rigid- and flex- entities by the partial derivation of ξ; Q is the external resultant force; μ is an undetermined factor, which can form the constrained inner force term including Coriolis forces by multiplying the transposition of formula (2).(2)Cq≡:∂C∂ξ.In the generalized space, the following two constraints must be satisfied:Holonomic constraints:C1=0,C˙1=0andC¨1=0Nonholonomic constraints:C2=0andC˙2=0。Therefore,q can be further divided into two subsets: the independent and dependent variables, i.e., q=y∪xFor the lightweight aluminium alloy car body, formula (1) gives the following two engineering meanings to the HSRS rigid-flex coupling system:(1)
Construction Speed. As shown in the first red box, the generalized mass has the positive or negative effects on the damping term of the coupling system, which leads to the mutual transformation of (non-) conservative mechanical properties, which is mainly evaluated by indexes of vibration comfort or assessments of structural fatigue damage.(2)
The Nonlinearities of Constrained Inner Forces. As shown in the second red box, the techniques of rigid-flex coupling simulation can use the improved integral algorithm of generalized augmentation to obtain the accurate analysis results of complex constrained inner forces and guarantee the MSR (Modal Stress Recovery) accuracy.The complex constrained inner forces have the quasi-statical and dynamical components, and the latter represents the nonlinear effects such as Coriolis acceleration, friction, and so on. The SSR (State Space Reduction) method usually needs to satisfy the following conditions(3)detCx≠0⋅Non Singularity.However, the improved integral algorithm of generalized augmentation can deal with this difficult problem successfully; details seen in references [27–29].


### 2.2. Nonlinearities of Wheel-Rail Contact

For the MBS approach of HSRS, the singularities of complex constraints can make the system structure of MBS change thereby, e.g., the primary hunting phenomenon makes the contribution enhance of anti-rolling torsion rod device to carbody rolling stiffness, and forces the carbody lateral or yaw become the minimum resistance motion direction. Under this new mechanism of lateral vibration coupling, the two different forms of detrimental wear are inevitable.For the speeding-up practices of European present rails or Japanese Shinkansen rails, the lateral span of nominal rolling circles,L = 1500 mm, the wheel tread has the narrow low-wear zone. Nevertheless, when considering the negotiations of smaller-radius curve, the actual RRD (Radial Radius Deference) curve forms the negative slope close to the zero-crossing point and the dual-point contact always occurs on the rail shoulder [5]. So, the grinding of railhead is necessary with profile modifications.However, the central hollow tread wear is one special form of detrimental wear in Chinese HSR practices of HSRS operation and maintenance. When L = 1493 mm, the low-wear zone of wheel-rail contact becomes wider than one when L = 1500 mm. The dual-point contact on rail top can be proven an exclusive occurrence in the long-term operations of HSRS on the dedicated lines, which has the three prominent characteristics: (1) The actual RRD curve has a discontinuity close to the zero-crossing point; (2) the equivalent conicity curve forms the negative slope variation near the amplitudeA = 3 mm of wheelset lateral displacement; and (3) the corresponding flange side wear is thereby very slight.Different from the speeding-up situations of European present rails and Japanese Shinkansen rails, the HSRS operation practices on Chinese newly built HSR dedicated lines have the following three particularities: (1) The lateral span of wheelset’s nominal rolling circles,L = 1493 mm, the wheel-rail clearance is increased then by 3.5 mm on each side; (2) the proportion of tangent line running and large radius curving is relatively higher in the dedicated lines of rapid or high-speed grades; and (3) especially in some mountainous HSR-dedicated lines, bridges and tunnels account for more than 90% of the line.As far as Chinese HSR practices are concerned, the wheel-rail relationship improvement is very important, but it is impossible to remove the central hollow tread wear. So the re-innovations of imported prototypes should be combined with the above particularities of our own in order to establish the reasonable wheel-rail matching conditions, i.e.,λeN ≥ λemin and λemin = (0.03–0.05).In view of the above reasons, the excitation samples borne by the important components do not meet the two basic conditions of stationarity and ergodicity. Since the reliable load spectrum is difficult to be formulated, the interchangeability or universality of spare parts cannot be promoted. These become the radical reasons for the higher cost of Chinese HSRS operation and maintenance.


## 2.1. Singularities of Complex Constraints

The basic equations of HSRS rigid-flex coupling system are as follows:(1)Mξ¨+M˙−12∂M∂ξξ˙T+Dξ˙+Kξ+fg+∂C∂ξTμ=Q,in which ξ = [qq¯]T is spanning the generalized and modal spaces, including MBS's generalized space variables and flexible body's elastic modal space variable q¯. M is the mass matrix of MBS, including flexible bodies (e.g., hypothesis of nine invariants [26]), which is nonsymmetric, nonpositive definite, and noninvertible; K and D are the stiffness and damping matrices of the coupling system, respectively; fgis the gravity obtained through the gravitational potential energies of rigid- and flex- entities by the partial derivation of ξ; Q is the external resultant force; μ is an undetermined factor, which can form the constrained inner force term including Coriolis forces by multiplying the transposition of formula (2).(2)Cq≡:∂C∂ξ.In the generalized space, the following two constraints must be satisfied:Holonomic constraints:C1=0,C˙1=0andC¨1=0Nonholonomic constraints:C2=0andC˙2=0。Therefore,q can be further divided into two subsets: the independent and dependent variables, i.e., q=y∪xFor the lightweight aluminium alloy car body, formula (1) gives the following two engineering meanings to the HSRS rigid-flex coupling system:(1)
Construction Speed. As shown in the first red box, the generalized mass has the positive or negative effects on the damping term of the coupling system, which leads to the mutual transformation of (non-) conservative mechanical properties, which is mainly evaluated by indexes of vibration comfort or assessments of structural fatigue damage.(2)
The Nonlinearities of Constrained Inner Forces. As shown in the second red box, the techniques of rigid-flex coupling simulation can use the improved integral algorithm of generalized augmentation to obtain the accurate analysis results of complex constrained inner forces and guarantee the MSR (Modal Stress Recovery) accuracy.The complex constrained inner forces have the quasi-statical and dynamical components, and the latter represents the nonlinear effects such as Coriolis acceleration, friction, and so on. The SSR (State Space Reduction) method usually needs to satisfy the following conditions(3)detCx≠0⋅Non Singularity.However, the improved integral algorithm of generalized augmentation can deal with this difficult problem successfully; details seen in references [27–29].


## 2.2. Nonlinearities of Wheel-Rail Contact

For the MBS approach of HSRS, the singularities of complex constraints can make the system structure of MBS change thereby, e.g., the primary hunting phenomenon makes the contribution enhance of anti-rolling torsion rod device to carbody rolling stiffness, and forces the carbody lateral or yaw become the minimum resistance motion direction. Under this new mechanism of lateral vibration coupling, the two different forms of detrimental wear are inevitable.For the speeding-up practices of European present rails or Japanese Shinkansen rails, the lateral span of nominal rolling circles,L = 1500 mm, the wheel tread has the narrow low-wear zone. Nevertheless, when considering the negotiations of smaller-radius curve, the actual RRD (Radial Radius Deference) curve forms the negative slope close to the zero-crossing point and the dual-point contact always occurs on the rail shoulder [5]. So, the grinding of railhead is necessary with profile modifications.However, the central hollow tread wear is one special form of detrimental wear in Chinese HSR practices of HSRS operation and maintenance. When L = 1493 mm, the low-wear zone of wheel-rail contact becomes wider than one when L = 1500 mm. The dual-point contact on rail top can be proven an exclusive occurrence in the long-term operations of HSRS on the dedicated lines, which has the three prominent characteristics: (1) The actual RRD curve has a discontinuity close to the zero-crossing point; (2) the equivalent conicity curve forms the negative slope variation near the amplitudeA = 3 mm of wheelset lateral displacement; and (3) the corresponding flange side wear is thereby very slight.Different from the speeding-up situations of European present rails and Japanese Shinkansen rails, the HSRS operation practices on Chinese newly built HSR dedicated lines have the following three particularities: (1) The lateral span of wheelset’s nominal rolling circles,L = 1493 mm, the wheel-rail clearance is increased then by 3.5 mm on each side; (2) the proportion of tangent line running and large radius curving is relatively higher in the dedicated lines of rapid or high-speed grades; and (3) especially in some mountainous HSR-dedicated lines, bridges and tunnels account for more than 90% of the line.As far as Chinese HSR practices are concerned, the wheel-rail relationship improvement is very important, but it is impossible to remove the central hollow tread wear. So the re-innovations of imported prototypes should be combined with the above particularities of our own in order to establish the reasonable wheel-rail matching conditions, i.e.,λeN ≥ λemin and λemin = (0.03–0.05).In view of the above reasons, the excitation samples borne by the important components do not meet the two basic conditions of stationarity and ergodicity. Since the reliable load spectrum is difficult to be formulated, the interchangeability or universality of spare parts cannot be promoted. These become the radical reasons for the higher cost of Chinese HSRS operation and maintenance.


## 3. Dynamical Design Methodology of Speeding-Up Bogies

For this end, as shown in Figure1, a dynamical design methodology was proposed in this research. As far as the speeding-up bogies are concerned, the dynamical design methodology refers to a set of method systems based on the accurate analyses of complex constrained inner forces, so as to avoid as far as possible the strong coupling interface formations of wheel-rail contact and/or bogie to servicing car body.Figure 1
Dynamical design methodologies for speeding-up bogies with software analysis flows based on full-vehicle stability properties and variation patterns.The dynamical design methodology of speeding-up bogies is composed of the following three key techniques: (1) The analysis graph of full-vehicle stability properties and variation patterns; (2) the improved interface transaction strategy of flexible car body to full-vehicle MBS; and (3) the accurate analyses of complex constrained inner forces. The case investigation of car body fluttering phenomenon is directed by the dynamical design methodology.


### 3.1. Analysis Graph of Full-Vehicle Stability Properties and Variation Patterns

The complicated calculation is one of the main defects of improved integral algorithm, i.e., the variable-step integral algorithm with three stages of prediction, correction, and evaluation. However, the successful applications can be proven in solving effectively the practical engineering problems [30, 31], like longitudinal and vertical coupling resonance.In spite of changing to use Newmark second-order difference algorithm, the complicated simulation calculations make it impossible to optimize the parameters of large-scale MBS for the higher speed advanced researches on the next generation development of HSRS. For these reasons, some optimal optimizing tools are to be urgently constructed by means of big data mining such as orthogonal decomposition or modal design.Negrut proposed the concept of virtual augmentation variables, and the minimum resistance motion direction can be determined by cleverly reorganizing the independent and dependent variables, so as to avoid the ill-condition of Jacobian matrix. By using this improved integral algorithm, we can get the Jacobian matrixJλe,v of full-vehicle MBS with constant speed v, in which the actual wheel-rail contact is replaced with the linear equivalent unit with mono-curvature, including the equivalent parameters such as λe.For the MBS approach of speeding-up rail vehicles, assuminguandq˙=u, we get the-first order difference formula(4)Mu˙+∂C∂qTμ=Q.In each steady state, the quasi-equilibrium feature is assumed to be the following equations for the rail vehicle system.(5)Fy,y˙,t=0,in which, y represents all steady-state variables.At the working point,y=ykand y˙=y˙k, the first-order difference formula of equation (5) is given as follows:(6)∂F∂y|yk,y˙k+1hβ0∂F∂y˙|yk,y˙kΔy=−Fy,y˙,t,where β0 is the scalar constant, which is related to the integrator order; Δy is the corrected difference direction; h is the integration step size; and −F is the residual part of equation (5), which represents the imbalance degree of the force systems.The left matrix of equation (6) is the F Jacobean matrix to represent the required quasi-equilibrium features, i.e., the Jacobean matrix of formula (4).(7)Jλe,v=Mhβ0−CuMqu˙+CqqTμ−QqCqTI−1hβ000Cq0,in the matrix, I is the unit matrix; u,q,qq are subscripts, i.e., the first- or second-order partial derivatives of (non-) holonomic constraints C to u/q.By using the orthogonal decomposition technique, i.e., solving the eigenvalue problem,eigJλe,v, the closed-loop poles can be obtained, so as to draw the root-locus graph, which is referred to as the analysis graph of full-vehicle stability properties and variation patterns. Different from the conventional root-locus graph, the analysis graph of full-vehicle stability properties and variation patterns has the following three stability characteristics, i.e., closed-loop pole, critical damping, and convected motion relationship.Jacobian matrixJλe,v is the implicit expression of closed-loop transfer function. At each specific speed and equivalent conicity, the solution of eigenvalue problem can give all the motion modes of full-vehicle system, including modal frequency and damping. By using the bilinear transformation technique, the real and imaginary axes of the complex plane can be transformed into modal damping and frequency.Considering the singularities and impacts of complex constraints, the convected motion relationship has become one of the important stability features in the analysis graph of full-vehicle stability properties and variation patterns. The convected motion relationship refers to the convected motion that may be formed between the relevant modes in the generalized space, and constitutes thereby the energy exchange or transformation mutually. The convected motion relationship may then change with the equivalent conicity increase, which resolves the variation patterns of full-vehicle stability properties.By applying the mode design technique, the analysis graph of full-vehicle stability properties and variation patterns is used to guide instructively the optimal parameter configuration of self-adaptive bogies, making the beneficial convected movement relationships more robust and eliminating the unfavourable ones.


### 3.2. Interface Transaction Modified Strategy of Flexible Car body to Full-Vehicle MBS

The analysis softwares of mechanical system dynamics like ADAMS have thereby become an integrated platform for multidisciplinary collaborative design and simulation, e.g., the rigid-flex coupling simulation techniques, which are integrated with the structural dynamics and the multi-body system dynamics.The interface transaction strategy of flexible body to MBS is formulated by applying the dynamical condensation method of component interface displacements [32], including constrained mode definition and high-order inherent mode truncation with FEM (Finite Element Mesh) models, and constraint reconstruction of flexible body in full-vehicle MBS.All the master nodes of the interface are given by them of constrained DoFs (Degrees of Freedom). Generally, the 6 constrained DoFs are defined per master node, while the s is set for the other unimportant DoFs, which depend on the required truncation of higher order inherent modes. The required truncation is determined according to the frequency response analysis requirements of the engineering problem to be solved. The total displacement vector is formed by the components, i.e., x = [xsxm]T. In the light of the static condensation method (or the Guyan method), the statical equation shown in equation (8) can be reduced to the stiffness matrix reduction equation shown in equation (9) by the Gauss elimination method.(8)KssKsmKmsKmmxsxm=0Fm,(9)I−T¯0K¯xsxm=0Fm,in which Fm is the total load to be borne on the constrained DoF of the master nodes. Therefore, the reduced stiffness matrix shown in equation (10) is weakened(10)K¯=Kmm−KmsKss−1Ksm,and −T¯=Kss−1Ksm.In the same way, the stiffness matrixK is replaced by the dynamical matrixM−Kω2, and the dynamical condensation method (or the Kuhar method) can also obtain the similar influencing relationship of the constrained DoFs on the master nodes to the dynamical effects of the component’s interface.When the aluminum alloy car body is taken as a component, the interface transaction strategy is modified further with the strong/weak coupling relationship as follows:(1)
Weak Coupling Internal Interface. Each master node of the internal interface is given by the 0 constrained DoF, the dynamical effect is considered weak, and the corresponding reduced stiffness matrix is not weakened.(2)
Strong Coupling Internal Interface. Each master node of the internal interface is given by the 6 constrained DoFs, and the dynamical interaction is considered to be enhanced, and the corresponding reduced stiffness matrix is weakened.The dynamical condensation of interface displacements can also be regarded as a special RSM (Response Surface Model) approach of components, but there are still two points worthy of special attention as follows: (1) The characteristic constrained modes [33] are used to grasp the elastic deformation under quasi-statical states, which improve the calculation efficiency on the premise of ensuring the response analysis accuracy of the reduced finite element model; (2) This special RSM model can be used to carry out the multidisciplinary or multi-professional collaborative designs, such as the robust design of lightweight structure optimization [34], so as to achieve the better efforts like vibration and noise reduction.By applying the dynamical condensation method, the flexible car body can be formulated by solving the constrained eigenvalue problem. Furthermore, the modal analysis of full-vehicle rigid-flex coupling system will be calculated again based on the linear equivalent unit of wheel-rail contact, including the motion and elastic modes. The conclusions of modal orthogonality and modal shape (anti-) symmetry can only be established under the specific premise of positive definite matrix which can be inversed. However, the Jacobian matrixΓλe,v does not always satisfy the above premise in the generalized space. In other words, the radical reason of flexible car body producing linear elastic vibration lies in the nonlinear variations of relevant constrained inner forces due to singularities.


### 3.3. Accurate Analysis of Complex Constrained Inner Forces

The MSR (Modal Stress Recovery) is useful information provided by rigid-flex coupling simulation techniques, which depends on the accurate analysis of complex constrained inner forces.For Newmark second-order difference algorithm, the following three points are very important to guarantee the accurate analysis of complex constrained inner forces:(1)
Correct preload. Before preloading, remove the redundant constraints, and use the auxiliary constraints to obtain the correct vertical preload. After the preloading is completed, restore the corresponding redundant constraints, and remove the auxiliary constraints.(2)
Quasi-Statical Deformation. If the deflection span of servicing car body is larger, or the quasi-statical deformation is prominent, some special skills should be used to make a smooth transition to the quasi-statical steady state. Otherwise, if the initial conditions are not consistent, their conflicts will impact the flexible car body.(3)
Higher Frequency Range of Response Analyses. The maximum frequency fmax is determined by the output steps N in the simulating time T, i.e., fmax = N/2T. As mentioned above, this kind of integrated simulations depend mainly on the calculation capacity of computing stations like CPU dominant frequency and cache size, fast storage of hard disks and thread number. Using the rigid-flex coupling simulation techniques proposed in this research, the frequency response analysis can reach 50 Hz or higher with the general configurations of computing stations.For the constrained and inherent modes,q¯=q¯1q¯2⋯q¯m+s, the corresponding second-order differential equations are given in formula (11) with their modal factors Γ=Γ1Γ2⋯Γm+s, and Γ=−ΦTM¯.(11)μiq¯¨i+2ξiμiωiq¯˙+μiωi2=Γiδ¨i.Furthermore, we can obtain the elastic deformationz of flexible car body under the multiaxial excitations of full-vehicle MBS.(12)M¯z¨+C¯z˙+K¯z=Γδ¨,in which, ΦTM¯Φ=diagμi, ΦTK¯Φ=diagμiωi2, and ΦTC¯Φ=diag2ξiμiωi; Φ is the modal matrix; M¯, K¯, and C¯are the mass, stiffness, and damping matrices, respectively; and μi, ωi, and ξi are the modal mass, frequency, and damping of the i-th order mode, respectively.However, the corresponding effective modal excitationsδ=δ1δ2⋯δm+s reflect better the dual mechanical attributes of rigid-flex coupling vibrations, i.e., the nonlinear variations of complex constrained internal forces and the linear elastic vibrations of flexible car body. The constrained inner forces fq, including the external forces acting on flexible car body, can be rewritten as follows:(13)fq=M^qqq¨−ΓTδ¨,in which, M^qq is the Guyan mass matrix [15], representing the kinetic energy of flexible car body motions in the generalized space; another partial of the constrained inner forces causes the elastic deformation of flexible car body, including the two components of quasi-statical and dynamical deformations.In a specific response direction1r, it can be proven [15] that 1rTM^qq1r=mT, i.e., the mass/moment of inertia for flexible car body motions in the generalized space. So fT and mT are, respectively, substituting for fq and Mqq, And, the other partial can be rewritten as −ΓTδ¨=meδ¨.When the harmonic excitations are assumed as follows, i.e.,δ¨=Δ0ejωt, fT=FTejωt, z=Z0ejωt, as shown in formula (14), the dynamical mass can be obtained, from which the three important conclusions can be drawn thereby:(14)FTωΔ0ω=mT+meω,(15)mT=∑i=1m+sΓi2μi,meω≈∑i=1m+sΓi2μiω2ωi2+2jξiωiω−ω2.(1)
In order to follow the mass conservation, the sum of all the effective modal masses is equal to the mass/moment (i.e.,mT) of inertia for the flexible car body motions in the generalized space.(2)
And, the elastic dynamical massme is determined by the effective modal mass Γi2/μi and the external excitation frequency ω, when ω⟶ωi,i=1,…m, the self-excited vibration of the i-th mode is then transformed into resonance, and the corresponding effective modal mass becomes the dominant component of elastic dynamical mass.(3)
For the constrained modes, the corresponding effective modal mass is very small, i.e.,Γi2/μi≈0,i=m+1,…m+s, only taking account of the influences of quasi-statical deformations, e.g., statical deflection of flexible car body.By using the mode superposition method, i.e.,z=Φq¯, we can obtain the displacements/forces at every nodal point of flexible car body, the modal stress recovery process is shown in Figure 2 based on effective modal excitations.Figure 2
Modal stress recovery process based on effective modal excitations: the effective modal excitations are calculated in rigid-flex coupling simulations on the left side; the structural stress at nodes along weld line can be recovered in the constrained modal solution of original FEM on the right side.Especially, the structural stress at every node along weld line can also be calculated by the Dong’s method based on the nodal forces [35, 36]. Compared with the conventional frequency domain method [37], like Dirlik, this time domain method can be assessing the fatigue safety on the critical weld lines based on the HSRS rigid-flex coupling simulations.In this way, the condition-based maintenance is established in the analyses of resonance formation mechanisms to formulate further the reliable load spectrum of important components. Therefore, the HSRS rigid-flex coupling simulations should grasp the essential problems and cultivate interdisciplinary talents. Otherwise, if there are a lot of conceptual confusions, they will mislead the solution of engineering problems.


## 3.1. Analysis Graph of Full-Vehicle Stability Properties and Variation Patterns

The complicated calculation is one of the main defects of improved integral algorithm, i.e., the variable-step integral algorithm with three stages of prediction, correction, and evaluation. However, the successful applications can be proven in solving effectively the practical engineering problems [30, 31], like longitudinal and vertical coupling resonance.In spite of changing to use Newmark second-order difference algorithm, the complicated simulation calculations make it impossible to optimize the parameters of large-scale MBS for the higher speed advanced researches on the next generation development of HSRS. For these reasons, some optimal optimizing tools are to be urgently constructed by means of big data mining such as orthogonal decomposition or modal design.Negrut proposed the concept of virtual augmentation variables, and the minimum resistance motion direction can be determined by cleverly reorganizing the independent and dependent variables, so as to avoid the ill-condition of Jacobian matrix. By using this improved integral algorithm, we can get the Jacobian matrixJλe,v of full-vehicle MBS with constant speed v, in which the actual wheel-rail contact is replaced with the linear equivalent unit with mono-curvature, including the equivalent parameters such as λe.For the MBS approach of speeding-up rail vehicles, assuminguandq˙=u, we get the-first order difference formula(4)Mu˙+∂C∂qTμ=Q.In each steady state, the quasi-equilibrium feature is assumed to be the following equations for the rail vehicle system.(5)Fy,y˙,t=0,in which, y represents all steady-state variables.At the working point,y=ykand y˙=y˙k, the first-order difference formula of equation (5) is given as follows:(6)∂F∂y|yk,y˙k+1hβ0∂F∂y˙|yk,y˙kΔy=−Fy,y˙,t,where β0 is the scalar constant, which is related to the integrator order; Δy is the corrected difference direction; h is the integration step size; and −F is the residual part of equation (5), which represents the imbalance degree of the force systems.The left matrix of equation (6) is the F Jacobean matrix to represent the required quasi-equilibrium features, i.e., the Jacobean matrix of formula (4).(7)Jλe,v=Mhβ0−CuMqu˙+CqqTμ−QqCqTI−1hβ000Cq0,in the matrix, I is the unit matrix; u,q,qq are subscripts, i.e., the first- or second-order partial derivatives of (non-) holonomic constraints C to u/q.By using the orthogonal decomposition technique, i.e., solving the eigenvalue problem,eigJλe,v, the closed-loop poles can be obtained, so as to draw the root-locus graph, which is referred to as the analysis graph of full-vehicle stability properties and variation patterns. Different from the conventional root-locus graph, the analysis graph of full-vehicle stability properties and variation patterns has the following three stability characteristics, i.e., closed-loop pole, critical damping, and convected motion relationship.Jacobian matrixJλe,v is the implicit expression of closed-loop transfer function. At each specific speed and equivalent conicity, the solution of eigenvalue problem can give all the motion modes of full-vehicle system, including modal frequency and damping. By using the bilinear transformation technique, the real and imaginary axes of the complex plane can be transformed into modal damping and frequency.Considering the singularities and impacts of complex constraints, the convected motion relationship has become one of the important stability features in the analysis graph of full-vehicle stability properties and variation patterns. The convected motion relationship refers to the convected motion that may be formed between the relevant modes in the generalized space, and constitutes thereby the energy exchange or transformation mutually. The convected motion relationship may then change with the equivalent conicity increase, which resolves the variation patterns of full-vehicle stability properties.By applying the mode design technique, the analysis graph of full-vehicle stability properties and variation patterns is used to guide instructively the optimal parameter configuration of self-adaptive bogies, making the beneficial convected movement relationships more robust and eliminating the unfavourable ones.


## 3.2. Interface Transaction Modified Strategy of Flexible Car body to Full-Vehicle MBS

The analysis softwares of mechanical system dynamics like ADAMS have thereby become an integrated platform for multidisciplinary collaborative design and simulation, e.g., the rigid-flex coupling simulation techniques, which are integrated with the structural dynamics and the multi-body system dynamics.The interface transaction strategy of flexible body to MBS is formulated by applying the dynamical condensation method of component interface displacements [32], including constrained mode definition and high-order inherent mode truncation with FEM (Finite Element Mesh) models, and constraint reconstruction of flexible body in full-vehicle MBS.All the master nodes of the interface are given by them of constrained DoFs (Degrees of Freedom). Generally, the 6 constrained DoFs are defined per master node, while the s is set for the other unimportant DoFs, which depend on the required truncation of higher order inherent modes. The required truncation is determined according to the frequency response analysis requirements of the engineering problem to be solved. The total displacement vector is formed by the components, i.e., x = [xsxm]T. In the light of the static condensation method (or the Guyan method), the statical equation shown in equation (8) can be reduced to the stiffness matrix reduction equation shown in equation (9) by the Gauss elimination method.(8)KssKsmKmsKmmxsxm=0Fm,(9)I−T¯0K¯xsxm=0Fm,in which Fm is the total load to be borne on the constrained DoF of the master nodes. Therefore, the reduced stiffness matrix shown in equation (10) is weakened(10)K¯=Kmm−KmsKss−1Ksm,and −T¯=Kss−1Ksm.In the same way, the stiffness matrixK is replaced by the dynamical matrixM−Kω2, and the dynamical condensation method (or the Kuhar method) can also obtain the similar influencing relationship of the constrained DoFs on the master nodes to the dynamical effects of the component’s interface.When the aluminum alloy car body is taken as a component, the interface transaction strategy is modified further with the strong/weak coupling relationship as follows:(1)
Weak Coupling Internal Interface. Each master node of the internal interface is given by the 0 constrained DoF, the dynamical effect is considered weak, and the corresponding reduced stiffness matrix is not weakened.(2)
Strong Coupling Internal Interface. Each master node of the internal interface is given by the 6 constrained DoFs, and the dynamical interaction is considered to be enhanced, and the corresponding reduced stiffness matrix is weakened.The dynamical condensation of interface displacements can also be regarded as a special RSM (Response Surface Model) approach of components, but there are still two points worthy of special attention as follows: (1) The characteristic constrained modes [33] are used to grasp the elastic deformation under quasi-statical states, which improve the calculation efficiency on the premise of ensuring the response analysis accuracy of the reduced finite element model; (2) This special RSM model can be used to carry out the multidisciplinary or multi-professional collaborative designs, such as the robust design of lightweight structure optimization [34], so as to achieve the better efforts like vibration and noise reduction.By applying the dynamical condensation method, the flexible car body can be formulated by solving the constrained eigenvalue problem. Furthermore, the modal analysis of full-vehicle rigid-flex coupling system will be calculated again based on the linear equivalent unit of wheel-rail contact, including the motion and elastic modes. The conclusions of modal orthogonality and modal shape (anti-) symmetry can only be established under the specific premise of positive definite matrix which can be inversed. However, the Jacobian matrixΓλe,v does not always satisfy the above premise in the generalized space. In other words, the radical reason of flexible car body producing linear elastic vibration lies in the nonlinear variations of relevant constrained inner forces due to singularities.


## 3.3. Accurate Analysis of Complex Constrained Inner Forces

The MSR (Modal Stress Recovery) is useful information provided by rigid-flex coupling simulation techniques, which depends on the accurate analysis of complex constrained inner forces.For Newmark second-order difference algorithm, the following three points are very important to guarantee the accurate analysis of complex constrained inner forces:(1)
Correct preload. Before preloading, remove the redundant constraints, and use the auxiliary constraints to obtain the correct vertical preload. After the preloading is completed, restore the corresponding redundant constraints, and remove the auxiliary constraints.(2)
Quasi-Statical Deformation. If the deflection span of servicing car body is larger, or the quasi-statical deformation is prominent, some special skills should be used to make a smooth transition to the quasi-statical steady state. Otherwise, if the initial conditions are not consistent, their conflicts will impact the flexible car body.(3)
Higher Frequency Range of Response Analyses. The maximum frequency fmax is determined by the output steps N in the simulating time T, i.e., fmax = N/2T. As mentioned above, this kind of integrated simulations depend mainly on the calculation capacity of computing stations like CPU dominant frequency and cache size, fast storage of hard disks and thread number. Using the rigid-flex coupling simulation techniques proposed in this research, the frequency response analysis can reach 50 Hz or higher with the general configurations of computing stations.For the constrained and inherent modes,q¯=q¯1q¯2⋯q¯m+s, the corresponding second-order differential equations are given in formula (11) with their modal factors Γ=Γ1Γ2⋯Γm+s, and Γ=−ΦTM¯.(11)μiq¯¨i+2ξiμiωiq¯˙+μiωi2=Γiδ¨i.Furthermore, we can obtain the elastic deformationz of flexible car body under the multiaxial excitations of full-vehicle MBS.(12)M¯z¨+C¯z˙+K¯z=Γδ¨,in which, ΦTM¯Φ=diagμi, ΦTK¯Φ=diagμiωi2, and ΦTC¯Φ=diag2ξiμiωi; Φ is the modal matrix; M¯, K¯, and C¯are the mass, stiffness, and damping matrices, respectively; and μi, ωi, and ξi are the modal mass, frequency, and damping of the i-th order mode, respectively.However, the corresponding effective modal excitationsδ=δ1δ2⋯δm+s reflect better the dual mechanical attributes of rigid-flex coupling vibrations, i.e., the nonlinear variations of complex constrained internal forces and the linear elastic vibrations of flexible car body. The constrained inner forces fq, including the external forces acting on flexible car body, can be rewritten as follows:(13)fq=M^qqq¨−ΓTδ¨,in which, M^qq is the Guyan mass matrix [15], representing the kinetic energy of flexible car body motions in the generalized space; another partial of the constrained inner forces causes the elastic deformation of flexible car body, including the two components of quasi-statical and dynamical deformations.In a specific response direction1r, it can be proven [15] that 1rTM^qq1r=mT, i.e., the mass/moment of inertia for flexible car body motions in the generalized space. So fT and mT are, respectively, substituting for fq and Mqq, And, the other partial can be rewritten as −ΓTδ¨=meδ¨.When the harmonic excitations are assumed as follows, i.e.,δ¨=Δ0ejωt, fT=FTejωt, z=Z0ejωt, as shown in formula (14), the dynamical mass can be obtained, from which the three important conclusions can be drawn thereby:(14)FTωΔ0ω=mT+meω,(15)mT=∑i=1m+sΓi2μi,meω≈∑i=1m+sΓi2μiω2ωi2+2jξiωiω−ω2.(1)
In order to follow the mass conservation, the sum of all the effective modal masses is equal to the mass/moment (i.e.,mT) of inertia for the flexible car body motions in the generalized space.(2)
And, the elastic dynamical massme is determined by the effective modal mass Γi2/μi and the external excitation frequency ω, when ω⟶ωi,i=1,…m, the self-excited vibration of the i-th mode is then transformed into resonance, and the corresponding effective modal mass becomes the dominant component of elastic dynamical mass.(3)
For the constrained modes, the corresponding effective modal mass is very small, i.e.,Γi2/μi≈0,i=m+1,…m+s, only taking account of the influences of quasi-statical deformations, e.g., statical deflection of flexible car body.By using the mode superposition method, i.e.,z=Φq¯, we can obtain the displacements/forces at every nodal point of flexible car body, the modal stress recovery process is shown in Figure 2 based on effective modal excitations.Figure 2
Modal stress recovery process based on effective modal excitations: the effective modal excitations are calculated in rigid-flex coupling simulations on the left side; the structural stress at nodes along weld line can be recovered in the constrained modal solution of original FEM on the right side.Especially, the structural stress at every node along weld line can also be calculated by the Dong’s method based on the nodal forces [35, 36]. Compared with the conventional frequency domain method [37], like Dirlik, this time domain method can be assessing the fatigue safety on the critical weld lines based on the HSRS rigid-flex coupling simulations.In this way, the condition-based maintenance is established in the analyses of resonance formation mechanisms to formulate further the reliable load spectrum of important components. Therefore, the HSRS rigid-flex coupling simulations should grasp the essential problems and cultivate interdisciplinary talents. Otherwise, if there are a lot of conceptual confusions, they will mislead the solution of engineering problems.


## 4. Self-Adaptive Improvement of an Imported Bogies

### 4.1. An Imported Prototype of High-Speed Bogies

For an imported prototype of high-speed trailer TC02/07, as shown in Figure3(a), the design defect of primary hunting phenomenon had been proven by using the analysis graph of full-vehicle stability properties and variation patterns, which restricts the wheel-rail matching conditions, i.e., λemin ≥ 0.10. Otherwise, if λemin < 0.10, primary hunting will change into the secondary hunting phenomenon.Figure 3
Design defects of an imported trailer prototype and self-adaptive improvement. (a) Primary hunting phenomenon in an imported prototype. (b) Simple car body instability in self-adaptive improvement.
(a)(b)Since the strong rigid constraints of wheelset positioning, i.e., the longitudinal/lateral stiffness of wheelset positioning isKx/y = 120/12.5 MN/m, an imported prototype has the car body yaw overdamped characteristics, which forces the yaw phase margin of rear bogie to decrease, so that the primary hunting phenomenon is then formed with the car body rolling mode. Therefore, the primary hunting phenomenon has the following constraints on the wheel-rail matching conditions: (1) The high-speed car body shaking phenomenon occurs when λe < 0.10; (2) If λe ≥ 0.10, the long-term operations on the HSR-dedicated lines pave way for the central hollow tread wear to be formed gradually, ca. 10 × 104 km.As mentioned above, the novel anti-yaw dampers are widely applied to the speeding-up bogies, and effectively suppressing the rapid attenuation of bogie yaw phase margin. However, the anti-hunting high-frequency impedance interactions force the bogie frames form the lateral vibration disturbances, so that the contributions of anti-rolling stiffness are enhanced due to the primary hunting phenomenon, and the lateral movement of servicing car body becomes the minimum resistance one.In this situation, i.e., the singularities of complex constraints lead to the structural variations of full-vehicle MBS, the long-term operations confirm that the central hollow tread wear is the exclusive form of detrimental wear particular to Chinese HSR practices. Meanwhile, the dynamical simulation analyses on rails CN60KG with cant 1/40 indicate the following:(1)
Whenλe = 0.06, speed 300 km/h in tangent line, the ride comfort is acceptable, but if speed is increased to 350 km/h or higher, the ride comfort is unacceptable, details seen in Ref. [22], which is referred to as the high-speed car body shaking phenomenon at low track conicity.(2)
Whenλe = 0.10, speed 300 km/h in tangent line, the wheel wear index distribution presents the V-shaped characteristics, the tread contact light band becomes narrow, and the Root Mean Square of wheelset lateral displacement amplitude under the occurrence possibility of 95%, (RMS)2.2σ = (4.5–4.8) mm. When LMB-10 tread is used instead, the central hollow tread wear is almost unchanged [13].(3)
Like the LM tread, moderately enhancing the wheelset gravity stiffness through the RRD curve can have a positive effect, but this positive factor is quickly worn away. The higher the equivalent conicityλe, e.g., S1002G tread (flange thickening by 3.5 mm), λeN = 0.166, the more prominent the V-shaped characteristics of wheel wear index distribution.Even for the slight central hollow tread wear, when the local conformal or bad contact between worn wheel and rail takes place occasionally in some specific sections, the contrast analyses of dynamical simulations with speed 350 km/h or higher in tangent line confirm that:(1)
The lateral acceleration PSD (Power Spectrum Density) responses of bogie frames have the (7.0–8.0) Hz resonant peak value, which is more than three times that of normal wear with the same or similar equivalent conicity.(2)
The maximum exciting frequency cannot reach more than 10 Hz due to the overload protection of safety valves in the bottom of novel anti-hunting dampers, e.g., ZF Sachs T60/T70 with mono-/dual-circulation.(3)
The dominant frequencyf0 of bogie frame lateral acceleration depends mainly on the anti-hunting high-frequency impedance interaction and wheelbase l0, e.g., another imported prototype, wheelbase l0 is extended to 2.7 m, and the number of anti-yaw dampers is reduced to 2 per bogie, the dominant frequency f0 is lowered to ca. 5.0 Hz.


### 4.2. Self-Adaptive Improvement

The self-adaptive improvement makes the intractable primary hunting phenomenon change into the simple problem of car body instability, as shown in Figure3(b), when the speed range of (160–200) km/h at λe < 0.10. The influences of car body instability on ride comfort can be proven to have reduced moderately by applying the practical and reliable methods, e.g., the solenoid valves at both end joints are used to (in-) activate the anti-yaw dampers of ZF Sachs T70, or the semi-active damping techniques are adopted between car bodies.As a result, the reasonable wheel-rail matching condition is defined as follows,λeN = 0.06 when the commended tread XP55 or other similar tread is selected for wheels, and the initial contact point on rail deviates to the gauge corner side and is ca. (5–8) mm from the centreline of the railhead. Therefore, self-adaptive improvement has the potentiality of crossing over the dedicated lines of different speed-grades, so as to achieve timely or regular self-cleaning of central hollow tread wear.The self-adaptive improvement selects the best configuration between the constrained stiffness of wheelset positioning and anti-hunting dynamical features, i.e.,Kx/y = 15/6 MN/m, and the parallel connection of ZF Sachs T60 and T70, corresponding parameters seen in Figure 3(b). The constrained stiffness reduction of wheelset positioning promotes the HSRS adaptability to track lines with servicing conditions. However, the dedicated lines of 380 km/h grade also need to satisfy the following quality requirements, i.e., λe =(0.25–0.35) with the occurrence possibility of less than 5%, so as to control the longitudinal uneven wear degree of wheel-rail rolling.According to the results of current comprehensive evaluations, the self-adaptive improvement is expected to achieve the following two important objectives, i.e., approaching to or breaking through the highest speed of 575 km/h in the worldwide HSR practices and the higher speed of 450 km/h required for the next generation design.For a mileage of 5 km in tangent line, the UK small defect spectrum（ERRI B176）is taken as the input excitation of rail irregularities with the maximum wavelength Ωmax = 100 m. When considering the regular transaction of preventive rail grinding, the irregularities are moderately weakened with the wavelength Ω ≤ 3 m, and λe = 0.10. The above simulation conditions are no more related again.


## 4.1. An Imported Prototype of High-Speed Bogies

For an imported prototype of high-speed trailer TC02/07, as shown in Figure3(a), the design defect of primary hunting phenomenon had been proven by using the analysis graph of full-vehicle stability properties and variation patterns, which restricts the wheel-rail matching conditions, i.e., λemin ≥ 0.10. Otherwise, if λemin < 0.10, primary hunting will change into the secondary hunting phenomenon.Figure 3
Design defects of an imported trailer prototype and self-adaptive improvement. (a) Primary hunting phenomenon in an imported prototype. (b) Simple car body instability in self-adaptive improvement.
(a)(b)Since the strong rigid constraints of wheelset positioning, i.e., the longitudinal/lateral stiffness of wheelset positioning isKx/y = 120/12.5 MN/m, an imported prototype has the car body yaw overdamped characteristics, which forces the yaw phase margin of rear bogie to decrease, so that the primary hunting phenomenon is then formed with the car body rolling mode. Therefore, the primary hunting phenomenon has the following constraints on the wheel-rail matching conditions: (1) The high-speed car body shaking phenomenon occurs when λe < 0.10; (2) If λe ≥ 0.10, the long-term operations on the HSR-dedicated lines pave way for the central hollow tread wear to be formed gradually, ca. 10 × 104 km.As mentioned above, the novel anti-yaw dampers are widely applied to the speeding-up bogies, and effectively suppressing the rapid attenuation of bogie yaw phase margin. However, the anti-hunting high-frequency impedance interactions force the bogie frames form the lateral vibration disturbances, so that the contributions of anti-rolling stiffness are enhanced due to the primary hunting phenomenon, and the lateral movement of servicing car body becomes the minimum resistance one.In this situation, i.e., the singularities of complex constraints lead to the structural variations of full-vehicle MBS, the long-term operations confirm that the central hollow tread wear is the exclusive form of detrimental wear particular to Chinese HSR practices. Meanwhile, the dynamical simulation analyses on rails CN60KG with cant 1/40 indicate the following:(1)
Whenλe = 0.06, speed 300 km/h in tangent line, the ride comfort is acceptable, but if speed is increased to 350 km/h or higher, the ride comfort is unacceptable, details seen in Ref. [22], which is referred to as the high-speed car body shaking phenomenon at low track conicity.(2)
Whenλe = 0.10, speed 300 km/h in tangent line, the wheel wear index distribution presents the V-shaped characteristics, the tread contact light band becomes narrow, and the Root Mean Square of wheelset lateral displacement amplitude under the occurrence possibility of 95%, (RMS)2.2σ = (4.5–4.8) mm. When LMB-10 tread is used instead, the central hollow tread wear is almost unchanged [13].(3)
Like the LM tread, moderately enhancing the wheelset gravity stiffness through the RRD curve can have a positive effect, but this positive factor is quickly worn away. The higher the equivalent conicityλe, e.g., S1002G tread (flange thickening by 3.5 mm), λeN = 0.166, the more prominent the V-shaped characteristics of wheel wear index distribution.Even for the slight central hollow tread wear, when the local conformal or bad contact between worn wheel and rail takes place occasionally in some specific sections, the contrast analyses of dynamical simulations with speed 350 km/h or higher in tangent line confirm that:(1)
The lateral acceleration PSD (Power Spectrum Density) responses of bogie frames have the (7.0–8.0) Hz resonant peak value, which is more than three times that of normal wear with the same or similar equivalent conicity.(2)
The maximum exciting frequency cannot reach more than 10 Hz due to the overload protection of safety valves in the bottom of novel anti-hunting dampers, e.g., ZF Sachs T60/T70 with mono-/dual-circulation.(3)
The dominant frequencyf0 of bogie frame lateral acceleration depends mainly on the anti-hunting high-frequency impedance interaction and wheelbase l0, e.g., another imported prototype, wheelbase l0 is extended to 2.7 m, and the number of anti-yaw dampers is reduced to 2 per bogie, the dominant frequency f0 is lowered to ca. 5.0 Hz.


## 4.2. Self-Adaptive Improvement

The self-adaptive improvement makes the intractable primary hunting phenomenon change into the simple problem of car body instability, as shown in Figure3(b), when the speed range of (160–200) km/h at λe < 0.10. The influences of car body instability on ride comfort can be proven to have reduced moderately by applying the practical and reliable methods, e.g., the solenoid valves at both end joints are used to (in-) activate the anti-yaw dampers of ZF Sachs T70, or the semi-active damping techniques are adopted between car bodies.As a result, the reasonable wheel-rail matching condition is defined as follows,λeN = 0.06 when the commended tread XP55 or other similar tread is selected for wheels, and the initial contact point on rail deviates to the gauge corner side and is ca. (5–8) mm from the centreline of the railhead. Therefore, self-adaptive improvement has the potentiality of crossing over the dedicated lines of different speed-grades, so as to achieve timely or regular self-cleaning of central hollow tread wear.The self-adaptive improvement selects the best configuration between the constrained stiffness of wheelset positioning and anti-hunting dynamical features, i.e.,Kx/y = 15/6 MN/m, and the parallel connection of ZF Sachs T60 and T70, corresponding parameters seen in Figure 3(b). The constrained stiffness reduction of wheelset positioning promotes the HSRS adaptability to track lines with servicing conditions. However, the dedicated lines of 380 km/h grade also need to satisfy the following quality requirements, i.e., λe =(0.25–0.35) with the occurrence possibility of less than 5%, so as to control the longitudinal uneven wear degree of wheel-rail rolling.According to the results of current comprehensive evaluations, the self-adaptive improvement is expected to achieve the following two important objectives, i.e., approaching to or breaking through the highest speed of 575 km/h in the worldwide HSR practices and the higher speed of 450 km/h required for the next generation design.For a mileage of 5 km in tangent line, the UK small defect spectrum（ERRI B176）is taken as the input excitation of rail irregularities with the maximum wavelength Ωmax = 100 m. When considering the regular transaction of preventive rail grinding, the irregularities are moderately weakened with the wavelength Ω ≤ 3 m, and λe = 0.10. The above simulation conditions are no more related again.


## 5. Causal Relationship Demonstration

The local conformal contact or bad contact between worn wheel and rail has the occasional conditions, which can only occur when the above conditions are met in some specific rail section. So it is understandable to appropriately increase the safety threshold of bogie vibration warning. However, a causal relationship is necessary to be demonstrated between bogie frame vibration alarm and car body fluttering phenomenon, so as to scientifically promote the construction speed under the reasonable wheel-rail matching condition.In accordance with the modified strategy of strong/weak interface transactions, as shown in Figure4, the external/internal interface of servicing car body is transacted, respectively. There are 6 master nodes in the external interface of servicing car body, as shown in Figure 4(a), including the connection relationships of the front and rear couplers with buffers and the bolsters to aluminium alloy car body. Each master node definition must be given 6 constrained DoFs. The internal interface, as shown in Figures 4(b) and 4(c), consists of the roof and the floor of aluminium alloy car body with a total of 12 master nodes.Figure 4
Modified transaction strategy with strong/weak interfaces of servicing car body. (a) External interface with 6 master nodes. (b) Internal interface on roof with 4 master nodes. (c) Internal interface under floor frame with 8 master nodes.
(a)(b)(c)


### 5.1. Strong Coupling Relationship on Internal Interface

If a strong coupling relationship is adopted on the internal interface of servicing car body, as shown in Figure5, the rigid-flex coupling simulation analysis of servicing trailer TC02/07 shows that the coupling resonances of the middle rhombic and the 1st lateral bending mode become the main constraining factors to promote the construction speed when the equipments are rigidly hanged under floor.Figure 5
Evolution of middle rhombic and 1st lateral bending modes under strong coupling relationship. (a) Car body middle rhombic inherent mode. (b) Servicing car body middle rhombic elastic mode. (c) Car body 1st lateral bending inherent mode. (d) Servicing car body 1st lateral bending elastic mode.When the lateral modal frequency is 9.71 Hz for the traction converter suspended by the rubber hanging elements, the rigid-flex coupling simulation can prove that it can effectively suppress the above coupling resonances. However, considering the negative influences of slight central hollow tread wear, both conclusions of the dynamical simulation analyses and the tracking test investigation are consistent on the first fluttering formation [4]. As shown in Figure 6, the serious impacts are caused by the lateral resonance of traction converter intersecting with the unstable hunting oscillation, ca. 9.2/9.3 Hz.Figure 6
Safety evaluation on lateral resonances of traction converter under strong coupling relationship based on standard IEC61373 – 2010.The rigid-flex coupling simulation analyses indicate that the self-excited lateral vibration of hanged traction converter is transformed into the coupling resonance, which makes some wedge rubber stacks to failure with wedging by self-weight. According to standard IEC61373–2010, the vibration will be unsafe for onboard electrical equipments. And, the traction converter with a dead weight of 6.6 t produces the lateral play, by which the car body fluttering phenomenon occurs and a lot of skirt supporting frames is then caused to crack, details seen in Ref. [4].The following two different technical measures have been taken respectively: (1) The reinforcement design of skirt supporting frames, which introduces high-frequency vibrations and causes the resin glue of threaded connection to looseness and failure, such as the cover plate of water inlet falling off; (2) The new design of rubber hanging elements are used instead of the original ones. Consequently, the second car body fluttering phenomenon occurs again [21].Due to the lack of wedging by self-weight or bolt pre-tightening, the second car body fluttering occurs seriously when changed to use the new design of rubber hanging elements, the self-excited vibration frequency of which is also ca. 9.2 Hz. But the maximum of lateral/vertical accelerations is 1.4/1.7 g, respectively, due to the unstable vibration of transformer (ca. 1.6 t). Consequently, the resonances of the ceiling interior are caused thereby.Correspondingly, the rigid-flex coupling simulation analyses also indicated that the middle floor produces the high-frequency elastic vibration with dominant frequency ca. (30–40) Hz, which has serious impacts on vibration comfort, e.g., the vertical comfort evaluation,Wz = 2.76/2.80/2.89 when speed 350/380/420 km/h, which means the poor ride comfort. Especially, when speeding up to 420 km/h, the torsion resonance on front and rear roof will occur, ca. 17.4 Hz, under strong coupling relationship of internal interface.Obviously, the fillet weld line located at connection basement of pantograph and wind deflector becomes one of the critical weld lines. Because the stress concentration effects are not exactly the same at both ends of fillet weld line, as shown in Figure7, the geometric nonlinear influences may be further exposed on the connection basement of the pantograph and wind deflector. And, the fatigue life on the right end, as shown in rectangular box, is reduced by 10−2 to ca. 2730 × 104 km, which may be transformed into the notch effect problem in fracture mechanics [38].Figure 7
Fatigue life assessments of critical weld line under strong/weak coupling interfaces. (a) Fillet weld line location at connection basement of pantograph and wind deflector. (b) Fatigue life assessment contrast.
(a)(b)


### 5.2. Weak coupling Relationship on Internal Interface

If a weak coupling relationship is formed on the internal interface of servicing car body, the rigid-flex coupling simulation analysis of servicing trailer TC02/07 shows that, as shown in Figure8, the coupling resonances of servicing car body 1st lateral bending mode on the lower/upper part becomes the main constraining factor for construction speed promotion when the equipments are rigidly hanged under the floor.Figure 8
Evolution of 1st lateral bending modes on lower/upper parts of servicing car body under strong coupling relationship. (a) Car body 1st lateral bending inherent mode. (b) Severing car body 1st lateral bending elastic mode on the lower part. (c) Severing car body 1st lateral bending elastic mode on the upper part.
(a)(b)(c)Under the premise of defining a weak coupling relationship, when the lateral modal frequency is 14 Hz, there is a weak correlation between the slight central hollow tread wear and the middle floor vibration under the speed of 350/380 km/h running on tangent lines, and the vertical comfort evaluationWz = 1.93/1.97, which means excellent ride comfort. Even when speeding up to 420 km/h, as shown in Figure 9, the onboard equipments are vibrating safely. However, the higher frequency elastic vibration occurs above fore and rear bogies, the dominant frequency ca. 36 Hz, and Wz = 2.23, which still means good ride comfort. This is in accord with the trial operation of HSR-dedicated line from Beijing to Shanghai, such as the cracking of wooden floor and its upper rubber plate when the maximum test speed exceeds 380 km/h.Figure 9
Lateral acceleration response contrast of traction converter under weak/strong coupling relationship with safety evaluation based on standard IEC61373 – 2010.The lateral modal frequency of traction converter is raised or decreased to 13/14/15 Hz, respectively, by changing the stiffness of rubber hanging elements. Considering the negative influences of slight central hollow tread wear, the response output of servicing trailer rigid-flex coupling system does not vary significantly. Therefore, only under the premise of weak coupling internal interface can the aluminum alloy car body satisfy the 30-year life requirement of servicing car body.Thus, it can be seen that the resonance condition is not established, the internal interface is weakly coupled, and the reduced stiffness matrix is not weakened thereby, which maybe an effect of floating plate. Therefore, the safety threshold of bogie vibration warning can be raised moderately in the premise of weak coupling relationship.


## 5.1. Strong Coupling Relationship on Internal Interface

If a strong coupling relationship is adopted on the internal interface of servicing car body, as shown in Figure5, the rigid-flex coupling simulation analysis of servicing trailer TC02/07 shows that the coupling resonances of the middle rhombic and the 1st lateral bending mode become the main constraining factors to promote the construction speed when the equipments are rigidly hanged under floor.Figure 5
Evolution of middle rhombic and 1st lateral bending modes under strong coupling relationship. (a) Car body middle rhombic inherent mode. (b) Servicing car body middle rhombic elastic mode. (c) Car body 1st lateral bending inherent mode. (d) Servicing car body 1st lateral bending elastic mode.When the lateral modal frequency is 9.71 Hz for the traction converter suspended by the rubber hanging elements, the rigid-flex coupling simulation can prove that it can effectively suppress the above coupling resonances. However, considering the negative influences of slight central hollow tread wear, both conclusions of the dynamical simulation analyses and the tracking test investigation are consistent on the first fluttering formation [4]. As shown in Figure 6, the serious impacts are caused by the lateral resonance of traction converter intersecting with the unstable hunting oscillation, ca. 9.2/9.3 Hz.Figure 6
Safety evaluation on lateral resonances of traction converter under strong coupling relationship based on standard IEC61373 – 2010.The rigid-flex coupling simulation analyses indicate that the self-excited lateral vibration of hanged traction converter is transformed into the coupling resonance, which makes some wedge rubber stacks to failure with wedging by self-weight. According to standard IEC61373–2010, the vibration will be unsafe for onboard electrical equipments. And, the traction converter with a dead weight of 6.6 t produces the lateral play, by which the car body fluttering phenomenon occurs and a lot of skirt supporting frames is then caused to crack, details seen in Ref. [4].The following two different technical measures have been taken respectively: (1) The reinforcement design of skirt supporting frames, which introduces high-frequency vibrations and causes the resin glue of threaded connection to looseness and failure, such as the cover plate of water inlet falling off; (2) The new design of rubber hanging elements are used instead of the original ones. Consequently, the second car body fluttering phenomenon occurs again [21].Due to the lack of wedging by self-weight or bolt pre-tightening, the second car body fluttering occurs seriously when changed to use the new design of rubber hanging elements, the self-excited vibration frequency of which is also ca. 9.2 Hz. But the maximum of lateral/vertical accelerations is 1.4/1.7 g, respectively, due to the unstable vibration of transformer (ca. 1.6 t). Consequently, the resonances of the ceiling interior are caused thereby.Correspondingly, the rigid-flex coupling simulation analyses also indicated that the middle floor produces the high-frequency elastic vibration with dominant frequency ca. (30–40) Hz, which has serious impacts on vibration comfort, e.g., the vertical comfort evaluation,Wz = 2.76/2.80/2.89 when speed 350/380/420 km/h, which means the poor ride comfort. Especially, when speeding up to 420 km/h, the torsion resonance on front and rear roof will occur, ca. 17.4 Hz, under strong coupling relationship of internal interface.Obviously, the fillet weld line located at connection basement of pantograph and wind deflector becomes one of the critical weld lines. Because the stress concentration effects are not exactly the same at both ends of fillet weld line, as shown in Figure7, the geometric nonlinear influences may be further exposed on the connection basement of the pantograph and wind deflector. And, the fatigue life on the right end, as shown in rectangular box, is reduced by 10−2 to ca. 2730 × 104 km, which may be transformed into the notch effect problem in fracture mechanics [38].Figure 7
Fatigue life assessments of critical weld line under strong/weak coupling interfaces. (a) Fillet weld line location at connection basement of pantograph and wind deflector. (b) Fatigue life assessment contrast.
(a)(b)


## 5.2. Weak coupling Relationship on Internal Interface

If a weak coupling relationship is formed on the internal interface of servicing car body, the rigid-flex coupling simulation analysis of servicing trailer TC02/07 shows that, as shown in Figure8, the coupling resonances of servicing car body 1st lateral bending mode on the lower/upper part becomes the main constraining factor for construction speed promotion when the equipments are rigidly hanged under the floor.Figure 8
Evolution of 1st lateral bending modes on lower/upper parts of servicing car body under strong coupling relationship. (a) Car body 1st lateral bending inherent mode. (b) Severing car body 1st lateral bending elastic mode on the lower part. (c) Severing car body 1st lateral bending elastic mode on the upper part.
(a)(b)(c)Under the premise of defining a weak coupling relationship, when the lateral modal frequency is 14 Hz, there is a weak correlation between the slight central hollow tread wear and the middle floor vibration under the speed of 350/380 km/h running on tangent lines, and the vertical comfort evaluationWz = 1.93/1.97, which means excellent ride comfort. Even when speeding up to 420 km/h, as shown in Figure 9, the onboard equipments are vibrating safely. However, the higher frequency elastic vibration occurs above fore and rear bogies, the dominant frequency ca. 36 Hz, and Wz = 2.23, which still means good ride comfort. This is in accord with the trial operation of HSR-dedicated line from Beijing to Shanghai, such as the cracking of wooden floor and its upper rubber plate when the maximum test speed exceeds 380 km/h.Figure 9
Lateral acceleration response contrast of traction converter under weak/strong coupling relationship with safety evaluation based on standard IEC61373 – 2010.The lateral modal frequency of traction converter is raised or decreased to 13/14/15 Hz, respectively, by changing the stiffness of rubber hanging elements. Considering the negative influences of slight central hollow tread wear, the response output of servicing trailer rigid-flex coupling system does not vary significantly. Therefore, only under the premise of weak coupling internal interface can the aluminum alloy car body satisfy the 30-year life requirement of servicing car body.Thus, it can be seen that the resonance condition is not established, the internal interface is weakly coupled, and the reduced stiffness matrix is not weakened thereby, which maybe an effect of floating plate. Therefore, the safety threshold of bogie vibration warning can be raised moderately in the premise of weak coupling relationship.


## 6. Conclusions and Prospects

Compared with the profound basic research of Bionic Flapping Wing, the car body fluttering phenomenon is one of the most complex engineering problems in the HSR practices. However, according to the lateral vibration coupling mechanism, the weak/strong coupling internal interface is one of the important factors to determine whether the coupling resonance takes place. Aiming at the primary hunting design defect existing in an imported prototype, the dynamical design methodology of speeding-up bogies was proposed in this research. And, the analysis graph of full-vehicle stability properties and variation patterns is used to clarify the self-adaptive improvement direction, i.e.,λeN ≥ λemin and λemin = (0.03–0.05). So, the central hollow tread wear can be self-cleaned in time or regularly by crossing over the dedicated lines of different speed-grades. To promote scientifically the construction speed, the causal relationship between bogie vibration alarm and car body fluttering phenomenon was demonstrated by applying the modified strategy with strong/weak internal interface transaction of servicing car body.Through these research works, the three main conclusions are as follows:(1)
The HSRS rigid-flex coupling vibrations depend on two important nonlinear factors of wheel-rail contact and bogie suspension, both of which facilitate the establishment of a new mechanism of lateral vibration coupling below the excitation frequency of 10 Hz. Under this new mechanism, the central hollow tread wear becomes the exclusive form of detrimental wear, particular to Chinese HSR practices.(2)
Both conclusions of the dynamical simulation analyses and the tracking-test investigations are consistent on the formations of twice car body fluttering phenomena, i.e., the self-excited vibration of traction converter intersecting with the unstable hunting oscillation, ca. 9.2/9.3 Hz. As a result, the technical space to promote the construction speed is also lost completely because of ride comfort decline, unsafe vibration of onboard electrical equipments, and weld fatigue damage of aluminum alloy car body. However, the rigid-flex coupling simulation analyses of trailer TC02/07 confirm that the safety threshold of bogie frame vibration warning can be appropriately raised as long as the lateral modal frequency of traction converters is greater than 12 Hz, preferably close to 14 Hz.(3)
The analysis graph of full-vehicle stability properties and variation patterns is used to guide instructively the optimal parameter configuration of high-speed bogies by means of big data mining such as orthogonal decomposition or modal design. And, the modified strategy with strong/weak internal interface transaction of servicing car body is formulated based on the dynamical condensation method of component interface displacements.For the self-adaptive improvement of an imported prototype, the determination of maximum construction speed will be one of the important future investigations in this research. When considering the two influencing factors when speeding up, i.e., the phase lag nonlinearities between four axle box suspensions and the dynamical features of secondary airspring suspensions, the vibration transmission rate is thereby enhanced from running gear to servicing car body and causes the local floor vibrations. And, the ca 36 Hz vertical resonance occurs locally due to the self-excited pitching vibration of traction converter when the speed is near 650 km/h. If the fatigue safety can be guaranteed for aluminum alloy car body, a foam material can be used to achieve the vibration isolation. And, the next generation development of HSRS can approach or break 575 km/h, i.e., the highest speed record in worldwide HSR practices.



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*Source: 6509950-2021-11-08.xml*