Analysis of the Gyroscopic Stability of the Wheelset

Abstract

The wheelset of the railway vehicle is a rotor which itself has gyroscopic effect. Nowadays, the rolling stock has entered the era of high speed, and the wheel rotates faster than in the past. The influence of gyroscopic effect on stability is little understood. Metelitsyn’s inequality theorem for asymptotic stability has some advantages to analyze this problem although this method is sufficient but not necessary condition. Based on its deduction, the extremal eigenvalues criterion and compared with Routh-Hurwitz criterion, both are applied to solve the critical value of speed. Further, according to the instability criterion, gyroscopic contributory ratio is derived to study how the role the gyroscopic effect plays in stability. Moreover, the effect of gyroscopic matrix or gyroscopic terms pitch rotor inertia Iy on stability coefficient is investigated. The results show that Iy is a key factor to wheelset gyroscopic stability. The gyroscopic effect becomes significant, and the stability increases with increasing Iy. The results also indicate that the critical value of speed solved by Metelitsyn theorem is more conservative than the one it solved by Hurwitz criterion, which proves that Metelitsyn inequality theorem for asymptotic stability is a sufficient but not necessary condition in the way of attaining the numerical simulation result. Finally, the test for the influence of gyroscopic effect on stability needs to be further studied.

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 [1]. 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 [2]. 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 [3]. Chen et al. investigated the stability of high speed trains passing curved rail in crosswind [4]. 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 [8]). 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 [9].

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 [10]. He also deduced the analytic solution of the linear critical speed for a simple wheelset with primary suspension [11]. Knudsen et al. applied Poincare sections and Poincare maps to analyze the bifurcation and chaos phenomenon of a single wheelset [12]. 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 [15]. 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 [16].

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 [17]. They also investigated the influence of parameters on linear critical speed [18]. He et al. used Hurwitz criterion and SQP algorithm to solve the nonlinear optimization stability problem of car [19]; see also the study by Law and Brand [20]. 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 [21]. 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 [22]. 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 [23]. 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 [24]. 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 [25]. He also summarized a complete overview on the subject of nonconservative stability from the modern point of view [26].

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 [29] 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 [22]: [𝐌]𝑥̈+[𝐃+𝐆]𝑥̇+[𝐏+𝐄]𝑥=0, (1) where the dot denotes time differentiation, 𝑥∈𝑅^𝑚, and [𝐏] and [𝐃] are symmetric matrices. [𝐆] and [𝐄] are skew-symmetric matrices. These matrices [𝐏]=[𝐏]^𝑇, [𝐃]=[𝐃]^𝑇, [𝐆]=−[𝐆]^𝑇, [𝐄]=−[𝐄]^𝑇 are related to potential forces matrix, damping matrix, gyroscopic matrix, and circulant matrix [30].

The eigenvalue problem of (1) is 𝜆²[𝐌]+𝜆[𝐃+𝐆]+[𝐏+𝐄]𝑢=0,𝑢≠0. (2)

With normalized eigenvectors 𝑢^𝑇𝑢=1, we obtain the relation 𝑀𝜆²+(𝐷+𝑖𝐺)𝜆+(𝑃+𝑖𝐸)=0, (3) where 𝑀, 𝐷, 𝑃, 𝐺, and 𝐸 are real quantities (known as Rayleigh quotients): 𝑀=𝑢^𝑇[𝑀]𝑢,𝐷=𝑢^𝑇[𝐷]𝑢,𝑃=𝑢^𝑇[𝑃]𝑢,𝑖𝐺=𝑢^𝑇[𝐺]𝑢,𝑖𝐸=𝑢^𝑇[𝐸]𝑢. (4)

Theorem 1. If linearized system (1) is asymptotic stable, all the eigenvalues of (2) have negative parts Re𝜆<0 [28].

Besides, one of the sufficient and necessary conditions of all the eigenvalues Re𝜆<0 is Hurwitz criterion. Meanwhile, Metelitsyn was the first to derive an inequality theorem for all the eigenvalues Re𝜆<0. His theorem is depicted as follows [25, 26]: 𝐷>0,𝑀𝐸²−𝐺𝐷𝐸<𝐷²𝑃. (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 [31], the corresponding quantities 𝑀, 𝐷, 𝑃 (Rayleigh quotients) are therefore limited by their eigenvalues: 𝑀_min=𝜆_min([𝐌])≤𝑀≤𝜆_max([𝐌])=𝑀_max,𝐷_min=𝜆_min([𝐃])≤𝐷≤𝜆_max([𝐃])=𝐷_max,𝑃_min=𝜆_min([𝐏])≤𝑃≤𝜆_max([𝐏])=𝑃_max, (6)

with 𝐺_max=𝜆_max(𝑖[𝐆]),𝐸_max=𝜆_max(𝑖[𝐄]). (7)

So we have −𝐺_max≤𝐺≤𝐺_max,−𝐸_max≤𝐸≤𝐸_max,𝐺_min=−𝐺_max,𝐸_min=−𝐸_max. (8)(9) Therefore, inequality (5) yields a more sufficient stability condition: 𝐷_min𝐷_min𝑃_min+𝐺_min𝐸_min−𝑀_max𝐸²_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 𝑦̈^𝑀𝜑̈+𝑚00𝐽^𝐶2𝑓₂₂𝑉2𝑓₂₃𝑉−𝐼_𝑦𝑉𝜆𝑟₀𝑏𝐼_𝑦𝑉𝜆𝑟₀𝑏−2𝑓₂₃𝑉2𝑓₁₁𝑙²𝑉𝑦̇𝜑̇+^𝐾2𝑘_𝑦+𝑊𝜆𝑏−2𝑓₂₂2𝑓₁₁𝑏𝜆𝑟₀2𝑙²₀𝑘_𝑥−𝑊𝜆𝑏+2𝑓₂₃𝑦𝜑=0. (11)

Figure 1 

Dynamic model of wheelset with primary suspension.

Therefore, [𝐃]=𝐂+𝐂^𝐓2=2𝑓₂₂𝑉002𝑓₁₁𝑙²𝑉,[𝐆]=𝐂−𝐂^𝐓2=02𝑓₂₃𝑉−𝐼_𝑦𝑉𝜆𝑟₀𝑏𝐼_𝑦𝑉𝜆𝑟₀𝑏−2𝑓₂₃𝑉0,[𝐏]=𝐊+𝐊^𝐓2=2𝑘_𝑦+𝑊𝜆𝑏𝑓₁₁𝜆𝑏𝑟₀−𝑓₂₂𝑓₁₁𝜆𝑏𝑟₀−𝑓₂₂2𝑙²₀𝑘_𝑥−𝑊𝜆𝑏+2𝑓₂₃,[𝐄]=𝐊−𝐊^𝐓2=0−𝑓₂₂−𝑓₁₁𝜆𝑏𝑟₀𝑓₂₂+𝑓₁₁𝜆𝑏𝑟₀0. (12)(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 𝑀_min=𝜆_min([𝐌])=𝐽,𝑀_max=𝜆_max𝐌𝐷([])=𝑚,_min=𝜆_min([𝐃])=𝐷_max=𝜆_max([𝐃])=2𝑓₂₂𝑉=2𝑓₁₁𝑙²𝑉,𝐺_max=𝜆_max(𝑖[𝐆])=2𝑓₂₃𝑉−𝐼_𝑦𝑉𝜆𝑟₀𝑏,𝐸_max=𝜆_max(𝑖[𝐄])=𝑓₂₂+𝑓₁₁𝜆𝑏𝑟₀,𝑃_min=𝜆_min([𝐏])=𝐴+𝐴²+4𝑏𝑟²₀𝐵−2𝑏𝑟²₀,𝑃_max=𝜆_max([𝐏])=𝐴−𝐴²+4𝑏𝑟²₀𝐵−2𝑏𝑟²₀, (14) where 𝐴=−2𝑏𝑓₂₃𝑟²₀−2𝑏𝑘_𝑦𝑟²₀−2𝑏𝑘_𝑥𝑙²𝑟²₀−𝜆𝑟²₀𝑊+𝑏²𝜆𝑟²₀𝑊,𝐵=𝑏²𝑓²₁₁𝜆²−2𝑏²𝑓₁₁𝑓₂₂𝜆𝑟₀+𝑏𝑓²₂₂𝑟²₀−4𝑏𝑓₂₃𝑘_𝑦𝑟²₀−4𝑏𝑘_𝑥𝑘_𝑦𝑙²𝑟²₀−2𝑓₂₃𝜆𝑟²₀𝑊+2𝑏²𝑘_𝑦𝜆𝑟²₀𝑊−2𝑘_𝑥𝑙²𝜆𝑟²₀𝑊+𝑏𝜆²𝑟²₀𝑊². (15)

Table 1 

Parameters of wheelset.

Substituting (14) into inequality (10) yields 2𝑓₂₂𝑉2𝑓₂₂𝑉×𝐴+𝐴²+4𝑏𝑟²₀𝐵−2𝑏𝑟²₀+2𝑓₂₃𝑉−𝐼_𝑦𝑉𝜆𝑟₀𝑏𝑓₂₂+𝑓₁₁𝜆𝑏𝑟₀−𝑚𝑓₂₂+𝑓₁₁𝜆𝑏𝑟₀²>0⟹𝑉<−2𝐴𝑓²₂₂−4𝑏²𝑓₁₁𝑓₂₂𝑓₂₃𝜆𝑟₀−4𝑏𝑓²₂₂𝑓₂₃𝑟²₀−2𝑓²₂₂𝐴²+4𝑏𝐵𝑟²×−2𝑏𝑓₁₁𝑓₂₂𝐼_𝑦𝜆²+𝑏³𝑓²₁₁𝜆²𝑚−2𝑓²₂₂𝐼_𝑦𝜆𝑟₀+2𝑏²𝑓₁₁𝑓₂₂𝜆𝑚𝑟₀+𝑏𝑓²₂₂𝑚𝑟^2−11/2⟹𝑉_𝐶𝑀=−2𝐴𝑓²₂₂−4𝑏²𝑓₁₁𝑓₂₂𝑓₂₃𝜆𝑟₀−4𝑏𝑓²₂₂𝑓₂₃𝑟²₀−2𝑓²₂₂𝐴²+4𝑏𝐵𝑟²×−2𝑏𝑓₁₁𝑓₂₂𝐼_𝑦𝜆²+𝑏³𝑓²₁₁𝜆²𝑚−2𝑓²₂₂𝐼_𝑦𝜆𝑟₀+2𝑏²𝑓₁₁𝑓₂₂𝜆𝑚𝑟₀+𝑏𝑓²₂₂𝑚𝑟^2−11/2, (16)(17)(18) where 𝑉_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_𝐶𝑀 and 𝑉_cr. With parameters in Table 1, it yields 𝑉_𝐶𝑀=88.36m/s,𝑉_cr=126.38m/s. (19)

As we know, Hurwitz criterion is the sufficient and necessary condition for all eigenvalues having negative part and the result 𝑉_𝐶𝑀<𝑉_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_𝐶𝑀 and 𝑉_cr as shown in Figures 2–4.

Figure 2 

Influence of 𝑘𝑥 on critical value 𝑉.

Figure 3 

Influence of 𝑘𝑦 on critical value 𝑉.

Figure 4 

Influence of 𝑘𝑥 and 𝑘𝑦 on 𝑉𝐶𝑀.

From Figures 2 and 3, we can also find 𝑉_𝐶𝑀<𝑉_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_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 [12]. 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_𝑡 with gyroscopic term and 𝑎_𝑛𝑡 without gyroscopic term: 𝑎_𝑡=𝐷_min𝐷_min𝑃_min+𝐺_min𝐸_min𝑀_max𝐸²_max,𝑎_𝑛𝑡=𝐷²_min𝑃_min𝑀_max𝐸²_max. (20) Thus, the sufficient condition for asymptotical stability is 𝑎_𝑡>1 or 𝑎_𝑛𝑡>1.

Furthermore, gyroscopic contributory ratio (GCR) can be defined as 𝑎GOR=^𝑡−𝑎_𝑛𝑡𝑎_𝑡=𝐺_max𝐸_max𝐷_min𝐾_min+𝐺_min𝐸_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.

Figure 5 

Gyroscopic contributory ratio.

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

Figure 6 

Influence of 𝑘𝑥 on 𝑉𝐶𝑀 and GOR.

Figure 7 

Influence of 𝑘𝑦 on 𝑉𝐶𝑀 and GOR.

As shown in Figures 6 and 7, the results show that GOR for 𝑉_𝐶𝑀 increases slightly along with increasing 𝑘_𝑥, and, further, with 𝑘_𝑥 increasing, both GOR for 𝑉_𝐶𝑀 and 𝑉_𝐶𝑀 itself tend to be levelled off. Hence, overall, the change of GOR for 𝑉_𝐶𝑀 is small, and the value of GOR for 𝑉_𝐶𝑀 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_𝑥 and further with 𝑘_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_𝐶𝑀. But when 𝑉_𝐶𝑀 is low and the operation speed gets much higher than 𝑉_𝐶𝑀, GOR will become very significant.

5.2. The Influence of Gyroscopic Term on Stability

We reconsider the gyroscopic matrix [𝐆]=02𝑓₂₃𝑉−𝐼_𝑦𝑉𝜆𝑟₀𝑏𝐼_𝑦𝑉𝜆𝑟₀𝑏−2𝑓₂₃𝑉0. (22)

It contains the terms 𝑓₂₃, 𝐼_𝑦, 𝑟₀, 𝑏, 𝜆. Furthermore, among the above five terms, we can find that the pitch rotor inertia 𝐼_𝑦 is the only factor which just affects [G] but does not affect the other matrices. For this reason, the influence of univariate 𝐼_𝑦, which means the gyroscopic effect, on stability needs to be studied. The influence of 𝐼_𝑦 on 𝜆([G]) and critical value of 𝑉are shown in Figure 9.

In Figures 8 and 9, the results indicate that, with increasing 𝐼_𝑦, the critical values of 𝑉_𝐶𝑀 and 𝑉_cr both increase but 𝑉_𝐶𝑀 is lower than 𝑉_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_𝑦, |𝜆([𝐆])| and GOR also increase. This illustrates that 𝐼_𝑦 is a key factor in wheelset gyroscopic stability. If it is greater, the gyroscopic effect becomes more significant and 𝑉_𝐶𝑀 gets higher.

Figure 8 

Influence of 𝐼𝑦 on |𝜆([𝐆])| and critical value of 𝑉.

Figure 9 

Influence of 𝐼𝑦 on 𝑉𝐶𝑀 and GOR.

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_𝑦 plays a key role in gyroscopic matrix. The gyroscopic effect becomes significant, and the 𝑉_𝐶𝑀 gets higher with increasing 𝐼_𝑦. However, when 𝑉_𝐶𝑀 is low, but with high operation speed, gyroscopic contributory ratio is large. Therefore, to ensure high 𝑉_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_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 [34], we mainly focused on the influence of suspension parameters such as 𝑘_𝑥 and 𝑘_𝑦 on stability but ignored 𝐼_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.

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.