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The critical speed and hunting frequency are two basic research objects of vehicle system dynamics and have a significant influence on the dynamic performance. A lateral dynamic model with 17 degrees of freedom was established in this study to investigate the critical speed and hunting frequency of a high-speed railway vehicle. The nonlinearities of wheel/rail contact geometry, creep forces, and yaw damper were all considered. A heuristic nonlinear creep model was employed to estimate the contact force between the wheel and the rail. The Maxwell model, which covers the influence of the stiffness characteristic, is used to simulate the yaw damper. To reflect the blow-off of the yaw damper, the damping coefficient is described by stages. Based on the mathematical model, the combined effects of vehicle parameters on the critical speed in the straight line and hunting frequency of the wheelset were investigated innovatively. The novel phenomenon that the hunting frequency exhibits a sudden increase from a smaller value to a larger value when the blow-off of the yaw damper occurs was discovered during the calculations. The extents to which various parameters affect the critical speed and hunting frequency are clear on the basis of the numerical results. Moreover, all of the parameter values were divided into three sections to determine the sensitive range for the critical speed and hunting frequency. The results show that the first section of values plays the decisive role on both the critical speed and the hunting frequency for all parameters analyzed. The investigation in this paper enriches the study of hunting stability and gives some ideas to probably solve the abnormal vibrations during the actual operation.

Stability analysis, referred to as one of the classical research areas in vehicle system dynamics, is studied throughout the development of railway vehicles and plays an important role in the development of high-speed trains in China [

Since Stephenson found the hunting motion phenomenon, researchers have conducted extensive studies on this subject from different aspects [

The bogie hunting motion, regarded as a special model in the vehicle system, has a significant influence on the vehicle stability. The damping ratio of this model largely determines whether the hunting motion converges to the equilibrium or diverges to a stable limit cycle. In addition to the damping ratio, the frequency of the hunting motion should also be given enough attention, as it could affect the abnormal vibration of the vehicle on many situations. Both car body hunting [

In this paper, a lateral dynamic model of the high-speed vehicle was established, and the nonlinearities of wheel/rail contact geometry, creep forces, and yaw damper forces were taken into consideration. The parameters involved in the nonlinear wheel/rail contact relationship were attained using polynomial interpolation. The nonlinear relationship between creepage and creep force based on a heuristic nonlinear creep model was employed for estimating the contact forces between the wheel and the rail. To integrate the stiffness behavior, the yaw damper was modeled as a Maxwell element, whose damping coefficient is described by stages. The influences of the vehicle parameters on the critical speed and hunting frequency were studied together. The ranges of the parameters studied were divided into three sections and the influence proportion of each section was subsequently compared.

Figure

Model of a high-speed vehicle. (a) Top view. (b) Front view.

The subscripts

This study used the common way of representing a yaw damper: a linear spring in series with a viscous damper._{d} denotes the damping generated as a result of the resistance of the fluid passing through the various damping valves, and_{d} denotes the series stiffness representing an elastic characteristic which includes the flexibility related to the connection joints of the damper and oil compressibility.

It is convenient to derive the equations of motion for the lateral dynamic vehicle system by applying the d’ Alembert-Lagrange principle.

The governing equations of motion for the lateral displacement_{c}, yaw angle_{c}, and roll angle_{c} of the car body are given, respectively, by the following:

As shown in Figure _{sy1} and_{sy2}, the suspension moments acting on the car body in the vertical direction,_{sz1} and_{sz2}, and the suspension moments acting on the car body in the longitudinal direction,_{sx1},_{sx2},_{sy1},_{sy2}, and_{gc}, are given by_{d1} and_{d2} are the damping forces produced by the yaw dampers of the front and rear bogie frames, and they will be listed in the next section. Consider

The governing equations of motion for the lateral displacement_{ti}, yaw angle_{ti}, and roll angle_{ti} of the bogie frames are given, respectively, by

Meanwhile, with respect to the bogie frames, the suspension forces acting in the lateral direction,_{py1} and_{py2}, the suspension moments acting in the vertical direction,_{pz1} and_{pz2}, the suspension moments acting in the longitudinal direction,_{pxi},_{si}, and_{gti}, are given by

The governing equations of motion for the lateral displacement _{ryij}, the gravity restoring force_{gij}, the moment in the vertical direction produced by the longitudinal creep force and spin moment_{rzij}, and the moment in the vertical direction produced by the gravity restoring force_{gij}are listed below.

A single wheelset model consisting of 2 DOFs is established, and the front view of the wheelset is depicted in Figure _{L} and_{R} denote the radius of the left and right wheels at the contact points on the rail, respectively. The creepages of the left and right wheels in the longitudinal, lateral, and spin directions of the contact plane can be derived as follows:

Front view of the wheelset.

By using Kalker’s linear creep theory [

The forces and moments acting on the track coordinates can be obtained as follows:

With respect to the rolling radii_{L} and_{R} and the contact angles_{L} and_{R}, they can be taken from tables as functions of excursion_{0} is the distance between the contact point and the center of the wheel without displacement,_{1} is the linear gradient on the conical tread, and

The relationship between the contact angles and the lateral displacement is given by

One more prerequisite mentioned in [

The sine and cosine functions are expanded in the Taylor series, and only the first two terms are considered and all higher order derivatives are ignored. Similarly, the tangent function in the Taylor series is expanded to the first two terms in the calculation of_{g}.

The wheel profile type of S1002CN and rail profile type of UIC60 are selected in this study, and the parameters of _{L} or_{R}, the values of_{0}=0.0544,_{1}=-9.45,_{4}=3.8e7, and_{5}=-3.58e9 can be obtained using fifth-order polynomial interpolation. A comparison between the real data and the result of polynomial interpolation is given in Figure

Relationship between lateral displacement

The yaw damper considered here is modeled as a Maxwell element, which is used as a part model introduced in Pracny et al. [_{d} denotes the damping force,_{d} denotes the displacement of the damper. To consider the curve negotiating property of the vehicle and protect the damper, a yaw damper possessing the blow-off characteristic is applied, and the damping can be described by stages, as shown in Figure _{u} is the blow-off velocity, and_{u} is the blow-off force.

Model and nonlinearity of yaw damper. (a) Maxwell model. (b) Damping characteristic curve.

The relationship for the spring force can be written as follows:

Thus far, all the terms on the right-hand side of Equations (

A bifurcation diagram is a powerful tool for analyzing railway nonlinear dynamics. The railway system expresses the typical Hopf bifurcation, whose diagram is illustrated in Figure _{lin} at point

Typical types of Hopf bifurcations of a railway vehicle system.

For the subcritical Hopf bifurcation, system vibration will result in a stable equilibrium position for any external excitations when the vehicle speed_{nl}. When_{lin}, the system will exhibit a stable periodic motion with a large amplitude regardless of the amplitude of the disturbance. In the interval_{nl} <_{lin}, however, the vibration of the system depends on the amplitude of the external excitation, and the interval is an uncertain region. For the supercritical Hopf bifurcation (b), the system always exhibits a stable periodic motion when_{lin}. With the increase in_{nl} ~_{c}, in which the solution will converge to equilibrium or a limit cycle with a small amplitude at a small track excitation and diverge to a limit cycle with a large amplitude at a large track excitation. The bifurcation diagram can be calculated using a path following method or a set of numerical simulations [

To investigate the stability of a railway vehicle system in practical engineering, the nonlinear critical hunting speed_{nl} is generally considered. From a mechanical viewpoint, a system can be regarded as stable if the oscillation of the system decays after a discontinuation excitation. Increasing the vehicle speed slowly, when the system displays a periodic motion at a constant amplitude, the corresponding speed can be regarded as the critical speed_{cr}, namely,_{nl}. The lateral displacement of the front wheelset is used as the output value to judge the critical speed in this study. With respect to excitations, three types of excitations are applied widely to railway vehicle system [

No excitation, running on an ideal track, starting from the limit cycle, and reducing the speed until a stable wheelset motion is achieved

Excitation by a singular irregularity, followed by an ideal track (or with a short irregularity sequence followed by an ideal track), with or without variation in the excitation amplitude

Excitation by a stochastic (measured) track irregularity, and applying the criteria used during the vehicle acceptance test.

The hunting frequency of the wheelset is an important factor in a vehicle dynamic system, whose value can significantly affect the occurrences of car body hunting and car body shaking phenomena. According to the Klingel theory [

In this study, the calculation of hunting frequency is divided into two parts. When the vehicle speed is less than_{cr}, the system vibration will result in a stable equilibrium position and the nonlinearities will have no influences on hunting frequency. For a linear system, the hunting frequency can be determined using the roots of the characteristic equation, whose imaginary part is the angular frequency of the wheelset hunting motion. When the vehicle speed exceeds_{cr}, the system vibration will diverge to a limit cycle with a large amplitude and the nonlinearities of the system should be taken into consideration. In this way, a numerical method based on the characteristics of time histories of vehicle components is proposed to calculate the hunting frequency of the wheelset.

The nonlinear governing equations of motion of the vehicle can be solved using the fourth-order Runge–Kutta method. The linear interpolation is subsequently used to change the variable step time series into a fixed step. The wheelset motion signals are collected over a finite time_{c} and consist of a discrete number of points obtained at a selected sampling frequency. The data can then be modeled as a sum of the sine and cosine functions of time_{c}. The fast Fourier transform (FFT) is then used to calculate the frequency spectrum, and the hunting frequency can thus be obtained by selecting the main frequency in the spectrum. During the calculation of hunting frequency, the track irregularity is removed and the wheelset is set to an initial displacement.

Simulation studies of self-excitation vibrations and stability of the nonlinear vehicle model were carried out based on the fourth-order Runge–Kutta method [

To determine the critical speed, the Dichotomy method was applied to the simulation. The simulation was started with a speed of 300 km/h, and the speed was increased or decreased in intervals of 10 km/h to determine another boundary of the interval. The calculation precision was 1 km/h, and the time series of the lateral displacement and the state-space plot of the wheelset lateral shift velocity versus displacement were determined. In the state-space plot, the data of the initial 3 s are removed. The results obtained for the last two speeds are shown in Figure

Calculated wheelset lateral movements. (a) Time history at 321 km/h. (b) Shift velocity versus displacement at 321 km/h. (c) Time history at 322 km/h. (d) Shift velocity versus displacement at 322 km/h.

The bifurcation diagram of this vehicle is shown in Figure

Calculation results. (a) Bifurcation diagram. (b) Hunting frequency.

Figure _{1} to point_{2} at 375 km/h which just corresponds to point_{d}. The red line represents the blow-off velocity of the yaw damper. It can be seen that, when_{d} is lower than the blow-off velocity, the cycle of the wheelset displacement is 0.25 s, namely, a frequency of 4 Hz. When_{d} is greater than the blow-off velocity, the cycle of the wheelset displacement is 0.19 s, namely, a frequency of 5.3 Hz. The influence of the damping coefficient on the hunting frequency needs to be explained, and it is described below. Thus, it can be concluded that the hunting frequency exhibits a sudden increase once the blow-off of the yaw damper occurs, and the hunting frequency can be divided into two parts: before blow-off and after blow-off.

Time histories of (a) wheelset lateral displacement and (b) yaw damper velocity.

To illustrate the consequences of this phenomenon, the time-frequency analysis for a section of the measured accelerations on the car body floor is given in Figure

Measured signal in time and frequency domains. (a) STFT spectrum of lateral acceleration on the back floor. (b) Time history of local lateral acceleration of vehicle floor.

Predictably, the choice of key parameters largely determines the critical speed of the vehicle. Furthermore, the resonance problem such as those pertaining to the hunting and shaking of the car body, which are closely related to the hunting frequency, should be focused on during the design of the vehicle. For this reason, the combined effects of key parameters in the vehicle system on critical speed and hunting frequency are analyzed immediately. These parameters include the first stage damping _{d}, the coefficient of friction_{px}, creep force coefficients_{11} and_{22}, and the lateral damping of secondary suspension_{sy}.

Figure _{cr}, the blue solid line represents the hunting frequency_{cr} increases logarithmically with the increase in _{cr} reaches 300 km/h. Before that,_{cr} virtually increases linearly in two parts with the increase in _{cr} reaches 300 km/h. The difference is that_{cr} firstly increases with increasing_{d}and then decreases and finally tends to a stable value. The_{d} overall, and the jump of

Change in critical speed and hunting frequency with (a) increase of_{d1}, (b) increase of_{d2}, and (c) increase of_{d}.

Figure _{cr} and_{11},_{px}, and_{sy}. During the calculation of_{22} is set equal to_{11}. They both affect_{cr} and_{cr} decreases with the increase in_{cr} is less than 350 km/h, and_{cr} is larger than 350 km/h. It indicates that_{11} (_{22}), and it can be seen that, with the increasing of_{11} (_{22}),_{cr} decreases gradually,_{px} and_{sy} are another two important parameters in the vehicle suspension system.

Change in critical speed and hunting frequency with (a) increase of_{11}, (c) increase of_{px}, and (d) increase of_{sy}.

Figure _{px} on_{cr} and_{cr} increases logarithmically with the increase in_{px} and finally stabilizes at 322 km/h. The_{px} and finally stabilizes at 4.9 Hz. As for_{sy}, the influences on both_{cr} and_{px} in the trend, as shown in Figure

To further compare the influence of the same parameter whose value is taken in different sections on the critical speed and hunting frequency, the parameters analyzed above were divided into three sections, as shown in Table _{11}, and _{px} and_{sy}, by contrast, have less influence on the critical speed, and _{d1},_{px}, and_{sy}. The first section is close to the second section in_{11}, and the third section accounts for a small proportion in all parameters. Figure _{d}, and_{11} come next. However, it is worth noting that the parameters of_{d2} only work when the vehicle loses stability, and parameters of _{px} and_{sy}, by contrast, have less influence on the hunting frequency, and Δ

Value division of analyzed parameters.

Parameter | First section | Second section | Third section | Unit |
---|---|---|---|---|

| 20–300 | 300–600 | 600~1000 | kN·s/m |

| 0.1–6 | 6–12 | 12–20 | MN/m |

| 0–40 | 40–80 | 80–120 | kN·s/m |

| 0.05–0.2 | 0.2–0.35 | 0.35–0.5 | - |

| 10–30 | 30–50 | 50–80 | MN/m |

| 2–5 | 5–8 | 8–11 | MN |

| 0–20 | 20–40 | 40–60 | kN·s/m |

Comparison of the influences of different parameters and different sections on (a) critical speed and (b) hunting frequency.

The parameters analyzed in the previous section significantly influence_{cr} without changing the bifurcation type. However, the blow-off force_{u} affects not only_{cr} but also the bifurcation type, as shown in Figure _{u}, the type of Hopf bifurcation transforms from subcritical to supercritical Hopf bifurcation of type (a), and, when_{u} tends to infinity, namely, with a linear damping coefficient, the bifurcation type transforms to supercritical Hopf bifurcation of type (b). Furthermore,_{u} does not affect the linear critical speed, which is decided by _{u} can increase_{cr} until the system reaches the linear critical speed.

Variation of bifurcation type with increase in_{u}.

In this study, the lateral dynamic model of a vehicle with 17 degrees of freedom was established, and the nonlinearities of wheel/rail contact geometry, creep forces, and yaw damper were considered. The wheel profile type of S1002CN and the rail profile type of UIC 60 were selected, and the bifurcation type was supercritical Hopf bifurcation of type (a). In addition to the critical speed, the hunting frequency of the wheelset was considered together during the calculation. The following conclusions can be drawn:

Owing to the discontinuity of the damping coefficient of the yaw damper, the hunting frequency of the wheelset increases suddenly when blow-off of the yaw damper occurs.

Parameters_{d} have the greatest influence on the critical speed, followed by parameters _{11}, and parameters_{px} and_{sy} have the least influence. Parameters

As for the influence proportion of each section on the critical speed and hunting frequency, all of the parameters analyzed show the same regularity that the first section comes first, the second section comes next, and the third section is the least.

_{u} affects not only the critical speed but also the bifurcation type. With the increase in the blow-off force, the Hopf bifurcation transforms from subcritical to supercritical of type (a), and then to supercritical of type (b).

It is noted that the results presented herein are based on calculation performed considering the straight line, and further research regarding the influence of vehicle parameters on the stability when a vehicle traverses a curve is needed. This will be a topic in our future work.

See Table

Notation | Parameter | Value | Unit |
---|---|---|---|

| Car body mass | 28000 | kg |

| Bogie frame mass | 3000 | kg |

| Wheelset mass | 2400 | kg |

| Roll moment of inertia of car body | 84560 | kg·m^{2} |

| Yaw moment of inertia of car body | 1102730 | kg·m^{2} |

| Roll moment of inertia of bogie frame | 2106 | kg·m^{2} |

| Yaw moment of inertia of bogie frame | 2600 | kg·m^{2} |

| Yaw moment of inertia of wheelset | 1029 | kg·m^{2} |

| Longitudinal stiffness of primary suspension | 85 | MN/m |

| Lateral stiffness of primary suspension | 15 | MN/m |

| Vertical stiffness of primary suspension | 1.1 | MN/m |

| Longitudinal stiffness of secondary suspension | 0.2 | MN/m |

| Lateral stiffness of secondary suspension | 0.2 | MN/m |

| Vertical stiffness of second suspension | 0.2 | MN/m |

| Lateral damping of primary suspension | 10 | kN·s/m |

| Vertical damping of primary suspension | 20 | kN·s/m |

| Lateral damping of secondary suspension | 30 | kN·s/m |

| Vertical damping of secondary suspension | 10 | kN·s/m |

| Series stiffness of yaw damper | 10 | MN/m |

| First stage damping of yaw damper | 300 | kN·s/m |

| Second stage damping of yaw damper | 20 | kN·s/m |

| Blow-off force of yaw damper | 3000 | N |

| Blow-off velocity of yaw damper | 0.01 | m/s |

| Half of track gauge | 0.7465 | m |

| Half of primary spring arm | 1 | m |

| Half of secondary spring arm | 1.23 | m |

| Half of secondary vertical damping arm | 1.323 | m |

| Half of yaw damper arm | 1.35 | m |

| Vertical distance from car body center of gravity to secondary suspension | 0.852 | m |

| Vertical distance from car body center of gravity to secondary lateral damping | 0.97 | m |

| Vertical distance from bogie frame center of gravity to secondary suspension | 0.162 | m |

| Vertical distance from bogie frame center of gravity to wheelset center | 0.148 | m |

| Vertical distance from bogie frame center of gravity to secondary lateral damping | 0.044 | m |

| Longitudinal distance from bogie frame center to car body center | 8.75 | m |

| Half of wheelbase | 1.25 | m |

| Longitudinal creep force coefficient | 8.4 | MN |

| Lateral creep force coefficient | 8.4 | MN |

| Lateral/spin creep force coefficient | 3000 | N·m^{2} |

| Spin creep force coefficient | 16 | N |

| Axle load | 10900 | kg |

| Normal force acting on wheelset | 53410 | N |

| Coefficient of friction | 0.3 | — |

| Acceleration of gravity | 9.8 | m/s^{2} |

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

This research has been supported by the National Science Foundation for Young Scholars (Grant no. 51805450), by the Joint Key Fund Projects (Grant no. U1734201), and by the Independent Subject of State Key Laboratory of Traction Power (Grant no. 2018TPL_T04). The authors wish to express their many thanks to the reviewers and editors of the present paper, whose help was invaluable in revising and improving the English language in this paper.