# Analysis of the gyroscopic stability of the wheelset.

1. Introduction

Dynamic stability problem of railway vehicle has been studied since hunting motion phenomenon was discovered by Stephenson from the United Kingdom 100 years ago . For security operation of railway vehicle, stability problem is of primary importance. Instability may increase the derailment risk and lead to disastrous consequences. Guan investigated derailment caused by instability . Goodall and Iwnicki proposed a new method to assess the vehicle stability which mainly refers to the equivalent conicity between the wheel and rail contact . Chen et al. investigated the stability of high speed trains passing curved rail in crosswind . Zeng et al. presented the random energy theorem to investigate the phenomenon of train derail, which gives an effect way to reveal the principle of the world's problem "derailment" [5-7].

Hunting phenomenon mentioned before always exists in the railway vehicle running process because the lateral and yaw motions of the vehicle are coupled. Specifically, the unconverged hunting means instability. As the vehicle system is usually considered a nonlinear system, the critical speed corresponds to the point of limit-cycle oscillation disappearing and the linear critical speed corresponds to the Hopf bifurcation point (the more detailed analysis is given by ). Furthermore, the former is usually lower than the latter. In this case, it means that if the running speed is lower than the former one, the stationary motion is global stability. Otherwise, sometimes the former equals the latter and it means that the stationary motion will lose its stability if the operation speed is higher than the critical speed. True gives the whole process to determine the critical speed in .

Moreover, the wheelset is the key component of the whole vehicle. Many researchers have focused on the wheelset stability to clarify the mechanism of instability. Wickens investigated the nonlinear stability of railway wheelset and bogie with worn wheel treads . He also deduced the analytic solution of the linear critical speed for a simple wheelset with primary suspension . Knudsen et al. applied Poincare sections and Poincare maps to analyze the bifurcation and chaos phenomenon of a single wheelset . Ahmadian and Yang used asymptotic method to investigate the hunting stability of a wheelset with primary suspension [13,14]. Liu et al. studied the random stability of single wheelset with Zhu's stochastic bifurcation theorem . Compared with previous study in deterministic systems, they yielded the critical speed range of wheelset in stochastic system and firstly found the first passage failure phenomenon of wheelset .

Many researchers solved the linear critical speed with using traditional Hurwitz criterion before. For example, Sedighi and Shirazi proposed equivalent functions method for complex nonlinearities to analyze the limit cycle of wheelset based on Yang and Ahmadian's asymptotic method, where the Hopf bifurcation point is solved also by Hurwitz criterion . They also investigated the influence of parameters on linear critical speed . He et al. used Hurwitz criterion and SQP algorithm to solve the nonlinear optimization stability problem of car ; see also the study by Law and Brand . As for the nonlinear critical speed, it refers to the nonlinear bifurcation theory. Nevertheless, neither the nonlinear stability nor the traditional linear stability theorem is the concern of this paper. What we are concerned with is the gyroscopic effect analysis, and somehow the traditional linear stability theory is utilized for comparison.

Nowadays, we step into the age of high speed railway. China's high speed electric motor units (EMU) CRH2 and CRH3 run as fast as 300 km/h. Furthermore, in the near future, the operation speed of China's EMU CRH500 is possibly to get up to 500 km/h. With the operation speed increasing, the wheelset is a rotor, whose gyroscopic effect on stability must be further studied. In the study of rotor stability, Kirillov investigated the asymptotic stability domain of a linear autonomous nonconservative system with Lineard and Chipart criterion . He found that the stability boundary was governed by Whitney umbrella singularity. Junfeng and Zhaolin investigated the stability of linear nonconservative systems subjected to potential, gyroscopic, and circulatory forces and Rayleigh damping with three stability theorems. They proved that the three theorems are convenient and useful . Stephen C. Spry et al. discussed the stabilization of unstable vehicle, which is characterized by single wheel. Differing from railway vehicle of wheelset, the single wheel vehicle is stabilized only by gyroscopic effect . In view of further development of the nonconservative stability theory of gyroscopic systems, Pommer and Kliem offer an overview of some of the recent researches on the stability and response bounds of linear system . Kirillov analyzed the stability of a linear autonomous nonconservative system with an even number of DOF. In his study, an approximation of the boundary of the asymptotic stability domain near the singularity is explicitly found, and an analytical estimate of the critical gyroscopic parameter is obtained . He also summarized a complete overview on the subject of nonconservative stability from the modern point of view .

Metelitsyn built his inequality theorem for asymptotic stability criterion [27,28]. Compared with Hurwitz criterion, his criterion is only sufficient but not necessary. Not only did Kliem and Seyranian  give a counter example to prove the defect of Metelitsyn's criterion, but also they developed Metelitsyn's criterion. Therefore, Metelitsyn's theorem has some advantages for clear physical meanings and mechanism in derivation process. What is more is that the basic principles of Metelitsyn theorem and Hurwitz criterion are somehow the same. Specifically, they are both inequality criteria and based on stationary motion (the details are presented in Section 2). As we mentioned above, Metelitsyn theorem is only sufficient but not necessary for inequality. This leads to the derived results differing from Hurwitz criterion as shown in Section 3. It is not the traditional linear critical speed that we focus on, but it is the value solved by Metelitsyn's criterion and the derived gyroscopic effect analysis that we investigate in this paper.

2. Metelitsyn's Inequality Theorem for Asymptotic Stability

Consider the following linear differential equation of the second order :

[M] x + [D+G] x + [P+E] x = 0, (1)

where the dot denotes time differentiation, x [member of] [R.sup.m], and [P] and [D] are symmetric matrices. [G] and [E] are skew-symmetric matrices. These matrices [P] = [[P].sup.T], [D] = [[D].sup.T], [G] = -[[G].sup.T], [E] = -[[E].sup.T] are related to potential forces matrix, damping matrix, gyroscopic matrix, and circulant matrix .

The eigenvalue problem of (1) is

([[lambda].sub.2] [M] + [lambda][D + G] + [P + E]) u = 0, u [not equal to] 0. (2)

With normalized eigenvectors [u.sup.T]u = 1, we obtain the relation

M[[lambda].sup.2] + (D + iG) [lambda] + (P + iE) = 0, (3)

where M, D, P, G, and E are real quantities (known as Rayleigh quotients):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Theorem 1. If linearized system (1) is asymptotic stable, all the eigenvalues of (2) have negative parts Re [lambda] < 0 .

Besides, one of the sufficient and necessary conditions of all the eigenvalues Re [lambda] < 0 is Hurwitz criterion. Meanwhile, Metelitsyn was the first to derive an inequality theorem for all the eigenvalues Re [lambda] < 0. His theorem is depicted as follows [25, 26]:

D > 0,

[ME.sup.2] - GDE < [D.sup.2] P. (5)

Compared with Hurwitz criterion, Metelitsyn theorem is a sufficient but not necessary condition. But its matrices terms have physical meanings. This theorem provides a way to study the influence of gyroscopic effect on stability.

Moreover, based on Lancaster and Tismenetsky's theorem , the corresponding quantities M, D, P (Rayleigh quotients) are therefore limited by their eigenvalues:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

with

[G.sub.max] = [[lambda].sub.max] (i [G]), [E.sub.max] = [[lambda].sub.max] (i[E]). (7)

So we have

-[G.sub.max] [less than or equal to] G [less than or equal to] [G.sub.max], -[E.sub.max] [less than or equal to] E [less than or equal to] [E.sub.max, (8)

[G.sub.min] = -[G.sub.max], [E.sub.min] = -[E.sub.max]. (9)

Therefore, inequality (5) yields a more sufficient stability condition:

[D.sub.min] ([D.sub.min][P.sub.min] + [G.sub.min][E.sub.min]) - [M.sub.max] [E.sup.2.sub.max] > 0. (10)

Compared with inequality (5), inequality (10) no longer depends on the eigenvectors u. This makes it suitable for engineering applications.

3. Wheelset Lateral Equation

Consider a wheelset model which contains the primary suspension. Its dynamic model is depicted in Figure 1. And (9) presents its lateral motion [13, 32]. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

4. Asymptotic Domain of Railway Wheelset by Metelitsyn's Theorem

Firstly, we yield the limited eigenvalues of the matrices M, D, P. With parameters in Table 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Substituting (14) into inequality 10) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

where [V.sub.CM] represents the critical value of V solved by Metelitsyn theorem.

We can solve the linear critical speed according to (16) and Hurwitz criterion [17-20, 33], hereinafter referred to as [V.sub.CM] and [V.sub.cr]. With parameters in Table 1, it yields

[V.sub.CM] = 88.36 m/s, [V.sub.cr] = 126.38 m/s. (19)

As we know, Hurwitz criterion is the sufficient and necessary condition for all eigenvalues having negative part and the result [V.sub.CM] < [V.sub.cr] proves that Metelitsyn theorem is the sufficient but not necessary condition for asymptotic stability. Besides, based on equality (16) and Hurwitz criterion, we can solve the influence of parameters on [V.sub.CM] and [V.sub.cr] as shown in Figures 2-4.

From Figures 2 and 3, we can also find [V.sub.CM] < [V.sub.cr]. It has been again proved that Metelitsyn theorem is the sufficient but not necessary condition for asymptotic stability. Therefore, when this theorem is used for analysis, the corresponding [V.sub.CM] is safe but conservative.

5. Gyroscopic Effect on Wheelset Stability

5.1. Gyroscopic Contributory Ratio. As mentioned in Section 1, the single wheel vehicle is stabilized only by gyroscopic effect . But it is the railway wheelset that we discuss in this paper, with primary suspension, whose main factors for stabilization are the primary suspension. However, how is the gyroscopic effect for stabilization of railway wheelset represented?

To represent the gyroscopic effect, from inequality (8), we can define stability coefficient [a.sub.t] with gyroscopic term and ant without gyroscopic term:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Thus, the sufficient condition for asymptotical stability is [a.sub.t] > 1 or [a.sub.nt] > 1.

Furthermore, gyroscopic contributory ratio (GCR) can be defined as

GOR = [a.sub.t] - [a.sub.nt]/[a.sub.t] = [G.sub.max][E.sub.max]/[D.sub.min][K.sub.min] + [G.sub.min][E.sub.min]. (21)

With wheelset parameters, we can obtain the following.

It can be seen in Figure 5 that, with the equivalent conicity increasing, the gyroscopic contributory ratio for stabilization increases. As we know, the larger equivalent conicity leads to the worse stability. Meanwhile, the gyroscopic effect on stability becomes more obvious.

Next, we continue to discuss the gyroscopic effect on stability under V = 300 km/h (83m/s) of high speed railway operation speed and [V.sub.CM] condition. The influences of positioning stiffness [k.sub.x] or [k.sub.y] on [V.sub.CM] and on GOR for both [V.sub.CM] and high speed railway operation speed V = 300 km/h are presented in Figures 6-7.

As shown in Figures 6 and 7, the results show that GOR for [V.sub.CM] increases slightly along with increasing [k.sub.x], and, further, with [k.sub.x] increasing, both GOR for [V.sub.CM] and [V.sub.CM] itself tend to be levelled off. Hence, overall, the change of GOR for [V.sub.CM] is small, and the value of GOR for VCM is as low as less than 5%. If, however, GOR for high operation speed 300 km/h firstly decreases a lot along with increasing [k.sub.x] and further with [k.sub.x] increasing, it also tends to be levelled off. That is, GOR for high operation speed 300 km/h drops significantly from 80% to less than 5%. This phenomenon indicates that GOR is insignificant when the operation speed is low and even if it gets to [V.sub.CM]. But when [V.sub.CM] is low and the operation speed gets much higher than [V.sub.CM], GOR will become very significant.

5.2. The Influence of Gyroscopic Term on Stability. We reconsider the gyroscopic matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

It contains the terms [f.sub.23], [I.sub.y], [r.sub.0], b, [lambda]. Furthermore, among the above five terms, we can find that the pitch rotor inertia [I.sub.y] is the only factor which just affects [G] but does not affect the other matrices. For this reason, the influence of univariate [I.sub.y], which means the gyroscopic effect, on stability needs to be studied. The influence of [I.sub.y] on [lambda]([G]) and critical value of V are shown in Figure 9.

In Figures 8 and 9, the results indicate that, with increasing [I.sub.y], the critical values of [V.sub.CM] and [V.sub.cr] both increase but [V.sub.CM] is lower than [V.sub.cr]. This again proves that the result attained by Metelitsyn theorem is more conservative than the one attained by Hurwitz criterion. And further, with increasing [I.sub.y], [absolute value of [lambda]([G])] and GOR also increase. This illustrates that [I.sub.y] is a key factor in wheelset gyroscopic stability. If it is greater, the gyroscopic effect becomes more significant and [V.sub.CM] gets higher.

6. Conclusion and Outlook

We derive the critical value of V and gyroscopic contributory ratio of wheelset equation with Metelitsyn inequality theorem. Analytical and numerical result shows that pitch rotor inertia [I.sub.y] plays a key role in gyroscopic matrix. The gyroscopic effect becomes significant, and the VCM gets higher with increasing [I.sub.y]. However, when [V.sub.CM] is low, but with high operation speed, gyroscopic contributory ratio is large. Therefore, to ensure high [V.sub.CM] of vehicle as well, increasing pitch rotor inertia can further increase the stability. In practice, to get larger value of pitch rotor inertia, one way is to increase the mass of the wheelset, and another way is to increase the radius of the wheelset, both of which can get higher linear critical speed as generally realized.

What is mentioned above just refers to the theoretical analysis. As we know, the pitch rotor inertia [I.sub.y] is the main factor in the gyroscopic system, and it would provide a method for testing verification. In the past EMU test on roller rig , we mainly focused on the influence of suspension parameters such as [k.sub.x] and [k.sub.y] on stability but ignored [I.sub.y]. Nowadays, in the era of high speed EMU development, therefore, the new test item "gyroscopic effect on stability" is further required. Since China's roller rig has the condition of software and hardware, the following work of how to test the influence of gyroscopic effect on stability needs to be conducted.

http://dx.doi.org/10.1155/2014/151625

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Basic Research Program of China ("973" Program) (Grant no. 2011CB711106), the National High Technology Research and Development Program of China ("863" Program) (Grant no. 2012AA112001), the National Key Technology Support Program of China (Grant no. 2009BAG12A01), the Southwest Jiaotong University 1st Outstanding Innovative Talents Fund, and the Doctoral Student Innovation Fund of Southwest Jiaotong University.

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Hao Dong, Jing Zeng, Liang Wu, and Huanyun Dai

Traction Power State Key Laboratory, Southwest Jiaotong University, Chengdu 0086 610031, China

Correspondence should be addressed to Hao Dong; fusiji@my.swjtu.edu.cn

Received 20 February 2014; Revised 9 May 2014; Accepted 15 May 2014; Published 2 June 2014

Academic Editor: Mickael Lallart
```
Table 1: Parameters of wheelset.

Parameters            Value           Commet

m                    1500 kg          Wheelset mass
J             825.7 kg x [m.sup.2]    Yaw inertia
[f.sub.11]          6.728 MN          Longitudinal creep coefficient
[f.sub.22]          6.728 MN          Lateral creep coefficient
[f.sub.23]         1500 N x m         Spin creep coefficient
[I.sub.y]       70 kg x [m.sup.2]     Pitch rotor inertia
[lambda]              0.28            Equivalent conicity
[r.sub.0]            0.533 m          Wheel radius
b                   0.7175 m          Half of rolling cycle gauge
[l.sub.0]              1m             Half of primary suspension gauge
[k.sub.x]             5MN/m           Primary longitudinal stiffness
[k.sub.y]             5MN/m           Primary lateral stiffness
W                      m*g            Axle load
V                      --             Operation speed
```
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