# Determination and Comparative Analysis of Critical Velocity for Five Objects of Railway Vehicle Class

^{*}

## Abstract

**:**

## 1. Introduction

_{n}) for five railway vehicle objects: three bogies (25TN bogie of a freight car, a bogie with averaged parameters, and a bogie of the MKIII passenger car) and two 2-axle freight cars (the car with averaged parameters and the hsfv1 car). Moreover, the results of a 4-axle passenger car MKIII from [1] are considered as well. Note in this context that very often researchers focus on a single object and try to learn about or find a solution for it. The present paper is different, the authors are not attached to and restricted by such a single object. They consider six objects of different classes that cover a wide range of real railway vehicles. Then they look for similarities and differences between these objects rather than for any restricted solution for a particular object. The authors are on their way to hopefully finding some generalizations or showing honestly that they do not exist or at the current stage cannot be formulated. Papers with such an approach are rather exceptional in the field studied. The paper’s purpose and scope just formulated are also different from the authors’ earlier papers. For example, in [1], just two objects were of interest and just one was identical to discussed here. In [2], the objects’ non-linear behaviour was of interest. The critical velocity was neither of the main interest nor compared for different objects. Determining the critical velocity in the context of the dynamics of rail vehicles in TCs and their vicinity, especially at velocities higher than the critical velocity, is important for modern railways. Everyone expects that rail vehicles will travel at higher and higher velocities. Railway vehicles as a rule are built so that their range of exploitation velocities is below the critical velocity. On the other hand, exceeding critical velocity does not at once means some unacceptable or dangerous state. It is usually just a less favourable state than motion below the critical velocity. However, the higher the velocity above the critical one, the higher amplitudes of vehicle hunting motion occur. At some stage, these are so high that a real danger of vehicle derailment can appear. These basic features of critical velocity connected with motion below and above the critical value of velocity explain its importance both for vehicle construction, modernization, and exploitation as well as safety of motion.

## 2. The Basics of the Motion Stability Analysis

_{n}.

_{c}and v

_{n}and called the linear and non-linear critical velocity, respectively. They are related to Hopf’s bifurcation and saddle-node bifurcation (e.g., [8,13]), respectively. This in fact means they are different physical quantities. The linear speed v

_{c}is defined by the place where the stable stationary solution bifurcates. For the subcritical system, it bifurcates into unstable periodic and unstable stationary solutions (Figure 1). The stable solutions are presented in Figure 1 by a solid line and the unstable solutions by a dashed line. For a supercritical system, the stable stationary solution bifurcates at velocity v

_{c}into unstable stationary and stable periodic solutions, while v

_{n}= v

_{c}, e.g., [13]. The linear velocity v

_{c}can be calculated with the use of linear stability analysis methods, analytically (e.g., by testing the eigenvalues of the system) or numerically, applying simulation of the motion of linear rail vehicles models (in particular the linear description of the wheel–rail contact). In the last case, velocity v

_{c}is considered the one where the hunting motion no longer tends to disappear. On the other hand, non-linear velocity v

_{n}is the lowest velocity at which in the mechanical system represented by a non-linear dynamic model of a rail vehicle (in particular with a non-linear description of the wheel–rail contact), stable periodic solutions may occur, i.e., at the state where the vehicle begins hunting motion reaching the limit cycle. Velocities v

_{n}cannot be determined using linear models of rail vehicles, especially with linear wheel–rail contact. Moreover, there are no general, effective analytical methods for testing the stability of non-linear systems, especially those with large dimensions (large number of DOFs). Thus velocity v

_{n}is most often determined by simulation (numerical) methods.

_{s}. This velocity, understood just as in [5], represents the value at which numerical simulations stop for whatever reason. Such a stop may be arbitrary by the software operator, e.g., when the velocity values reach unrealistically large values. The calculation may also be stopped for computational reasons. The latter situation is sometimes referred to as “numerical derailment” and the velocity v

_{s}can then be referred to as “velocity of numerical derailment” [5].

_{n}.

## 3. Methods of Determining the Value of Critical Velocity

_{n}, based on the papers related to this issue. Four available methods are mentioned.

_{n}. The method is based on formulating the stability problem as a problem of stability for a stable periodic solution. It requires, for each velocity v being considered, to sweep over the values of the initial conditions and check whether the same solution was obtained despite their different values. Approximate values of v

_{n}in this method can result from a non-complete search for sets of solutions over the initial conditions, and application of not necessarily small the velocity v interval. This results in a relatively fast calculation of velocity v

_{n}.

_{n}, refines the results of the first method and leads to exact results. The procedure applied therein is the same as in the first method, but without any simplifications. Here, the solution sets over the initial conditions are carefully searched for and a small velocity interval is used.

_{n}during a single simulation with variable velocity. This method may lead to results that are approximate or even inaccurate as compared to other methods. Basically, when increasing and decreasing the velocity v the obtained values of the critical velocity are different. In such circumstances, for the right action for determining v

_{n}, there are simulations for the decreasing velocity v. The use of this method requires experience, and in a technical sense, it may sometimes be necessary to perform several simulations to ensure that the result is correct. Moreover, the serious practical trouble with the use of this method in CCs is described in [5].

_{n}. This accuracy depends on the velocity v sampling step (interval) for subsequent simulations. The accuracy of this method and the second method are potentially the same. The authors have not found the publication where the fourth method was used for v

_{n}determination in CCs so far.

## 4. Critical Velocity Determined for the Objects Tested

#### 4.1. The Method Used

_{n}in ST and CC with different radii R was primarily realized with the first method. This is due to obtaining the result relatively quickly and generally small differences in v

_{n}between the first (approximate) and the second (exact) method shown in [9]. However, some elements of the second method were utilized as well. First, the second method was used occasionally when in doubt about the character and accuracy of the results. Second, the element regularly used was the dense variation of the velocity while approaching the critical value v

_{n}(with an interval of up to 0.1 m/s). Searching the range of solutions with sweeping over the initial conditions has been significantly limited in the present paper. The justification for this limitation is a very thorough study of the stability properties of the objects (vehicles) of interest in the present paper, carried out e.g., in [1,3,5,8,9,14,15,16,21] by their authors. This practically means that features of these objects related to critical velocity determination are already known to the authors of the present paper.

_{n}for three tested bogies (25TN, with averaged parameters, and of the MKIII car) and two 2-axle wagons (with averaged parameters and hsfv1). Additionally, the data on the velocity v

_{n}for the 4-axle vehicle (passenger MKIII car) is quoted further on basing on the results presented in [1].

#### 4.2. Models of the Considered Objects

#### 4.3. Conditions of the Critical Velocity Determination and Example Simulation Results

_{n}has been determined are presented in Table 1 and Table 2. Table 1 and Table 2 show the route parameters for which the tests were performed, i.e., for the bogies and 2-axle vehicles, respectively. The route consisted of ST only in the case of determining the critical velocity for ST. On the other hand, in the case of determining the critical velocity for a CC, that section was preceded by ST and TC.

_{i}(0) = 0.0045 m, a value imposed on lateral displacements of wheelsets, bogie frames, and vehicle body. This concerns both v

_{n}determination in ST and CCs. The choice of such value arises from the already stated experience and knowledge of the objects’ properties gathered in the earlier studies. Results in [1,3,5,8,9,14,15,16,21] showed that such a value is enough high to start periodic motion if it exists. The higher values of the initial conditions give a bigger chance for the system to take periodic behaviour than the smaller ones that could lead to stationary behaviour in the range v

_{n}< v < v

_{c}. As explained in Section 4.1, variation of the initial condition was performed in case of doubts, too. The length of ST was always l = 500 m. The length of CC was always l = 500 m. The radii R of the curve ranging from R = 300 m to R = 10,000 m were tested, with exact discrete values R = 300, 600, 900, 1200, 2000, 4000, 6000, and 10,000 m. The superelevation h was chosen so that the equilibrium between lateral components in the track plane of gravity and centrifugal forces was provided at velocity v, being the maximum velocity allowed by regulations in the CC of R = 600 m. If the value of h calculated this way exceeded the maximum allowed value (h

_{max}= 150 mm) then h

_{max}was applied. The interesting and justifying result in this context by the lead author is presented in [14]. It was shown that taking the variable h guarantees the equilibrium in each simulation, i.e., for each value of velocity v does not influence values of the critical velocities in ST and CC of different radii R. It was actually done for the hsfv1 car also studied in the present paper. Moreover, such h adoption leads to unnaturally high values, exceeding those allowed by regulations several times. This fact triggered strong criticism from railway practitioners. Indeed, the superelevation of the real track cannot be freely changed during exploitation. All these caused the authors to resign from this idea and take the superelevation value as it is done in this paper.

_{n}are decisive. Similar graphs were made for all the objects for ST and CCs sections of different R for conditions and parameters given in Table 1 and Table 2. They were also made for the range of the object’s velocity v. However, due to paper volume limitation, just three velocities for the above-mentioned graphs were selected for ST and three for CC. In each tire, in some of the figures, the velocities are selected to be in order lower, equal, and higher than the critical velocity. Moreover, Figure 8, Figure 9 and Figure 10 have a simplified form. They represent motion in CC only, while in fact the compound route of ST, TC, and CC was used when v

_{n}was determined in CC. The horizontal lines in Figure 6 and Figure 9 make it possible to recognize the stable periodic solution (limit cycle). On that occasion, one can note that solutions in Figure 5, Figure 6 and Figure 7 (ST), including limit cycle, are symmetrical relative to the track centre line, while in Figure 8, Figure 9 and Figure 10 (CC) are shifted laterally i.e., are asymmetric relative to the track centre line.

_{n}determination. According to the definition in Section 2, the value of v

_{n}is indicated by the figure in Figure 6 or Figure 9, obtained at the smallest velocity for which a stable periodic solution still exists. In Figure 6, Figure 7, Figure 9 and Figure 10 the limit cycles (stable periodic solutions) can be observed. Their amplitudes are higher for the higher velocities. In Figure 5 and Figure 8, stable stationary solutions appear resulting from the decaying vibrations.

#### 4.4. Results of the Critical Velocity Determination

_{n}determined in the studies have been presented in Table 3 and Table 4. These tables show the results for five tested vehicles, i.e., for three bogies and two 2-axle vehicles, respectively. The values of v

_{n}in Table 3 and Table 4 obviously match the conditions specified in Table 1 and Table 2, correspondingly.

_{zx}= 206.7 kN/m. For the nominal stiffness, the k

_{zx}= 2067 kN/m value of the critical velocity in the ST v

_{n}= 64.5 m/s [3].

_{n}easier, Figure 11 and Figure 12 were elaborated on based on data from Table 3 and Table 4, respectively. The changes in the value of v

_{n}depending on the CC radius R are illustrated in Figure 11 and Figure 12 for the bogies and 2-axle vehicles, respectively.

_{n}depending on the radius R shown. To make the information complete it has to be noted that this result was obtained for the higher than the nominal value of the longitudinal stiffness in the secondary suspension, k

_{pzx}= 1000 kN/m. For the nominal value, k

_{pzx}= 10 kN/m, while [1] gives the critical velocity v

_{n}for the ST only. It is v

_{n}= 19.1 m/s. This result was confirmed by these authors’ own study. Moreover, the study by the authors determined the value of v

_{n}of the MKIII vehicle with nominal parameters in the CC of R = 600 m as v

_{n}= 40 m/s.

## 5. Discussion of the Results and Conclusions

#### 5.1. Detailed Discussion of the Results

_{n}for five objects (three bogies and two 2-axle wagons). As a result, it was possible to determine these velocities relatively quickly, and the method turned out to be very effective. On the other hand, the results of v

_{n}in [1] for the MKIII passenger car invoked in the present paper were also obtained based on the combination of the first and second methods of v

_{n}determination. A similar conclusion can be formulated from [1], then.

_{n}= 35 m/s. As the radius increases, the values of this velocity also increase. The increase is initially faster, and then at the larger curve radii, it is slower to finally reach the value of v

_{n}= 45.2 m/s for the largest curve radius R. However, in the range of the radii of the curve from R = 1200 m to R = 10,000 m, the difference in the values of the critical velocities is small and amounts up to only 1.2 m/s (see also Figure 11). The velocity v

_{n}for ST was also successfully found as v

_{n}= 45.5 m/s. It is visible both in Table 3 and Figure 11. One should also realize that result v = v

_{n}= 44.0 m/s for R = 2000 m shown in Figure 9 as well as result v = v

_{n}= 45.3 m/s for ST shown in the Figure 6 do coincide with the results in Table 3 and Figure 11. One can conclude from all these sources that critical velocity for bogie of MKIII car is in ST higher (Figure 6, Table 3, and Figure 11) than v

_{n}values for all curve radii R (Figure 9, Table 3, and Figure 11).

_{n}was not reached. It might be a bit surprising that for the next radius R = 2000 m the critical velocity v

_{n}= 47.1 m/s was obtained while it is lower than the derailment velocity for the radius R = 1200 m. The successive values of critical velocities obtained for successively increasing radii are decreasing, which is still strange (see also Figure 11). One could rather expect that the greater the radius R, the higher the value of the critical velocity v

_{n}. A similar rather unexpected result was also obtained in the group of 2-axle vehicles. The potential for such a feature was also revealed in [2] for the 25TN bogie. The reasons, or rather conditions, favourable to such a feature have not been explained by the authors yet. So, for R = 4000 m—v

_{n}= 46.3 m/s, R = 6000 m—v

_{n}= 45.8 m/s, and for R = 10,000 m—v

_{n}= 45.2 m/s. For the ST, the critical velocity v

_{n}= 45.8 m/s and is equal to the critical velocity for a CC with the R = 6000 m and lower than the critical velocity for the largest CC of R = 10,000 m (Figure 11). Again, such a result is not obvious but rather surprising. On the other hand, the differences between v

_{n}for ST and CC of R = 6000 and 10,000 are not big. In addition, the critical velocity values for this object can be described as relatively high for the whole range of the periodic solutions’ existence.

_{n}= 29.1 m/s or 29.2 m/s (Table 3). Considering this feature, the critical velocity for the ST does not seem to be a surprise. Its value is v

_{n}= 29.2 m/s (see also Figure 11).

_{n}= 40 m/s and then v

_{n}grows with the increase of the radius R, which is a rather obvious feature. The difference in the values of the critical velocities is small and amounts up to 2.6 m/s for the entire range of the tested circular curves (see also Figure 12). The critical velocity for ST is v

_{n}42.6 m/s, which is 0.2 m/s higher than the largest radius tested R = 10,000 m (see Figure 12). For this vehicle, the behaviour is described as predictable and the graph of the critical velocity values is similar to that obtained for the bogie of the MKIII car (compare Figure 12 to Figure 11).

_{n}= 41.0 m/s. The same critical velocity is obtained for two successive circular curve radii R = 900 and 1200 m. Then, starting from CC with the radius of R= 2000 m, the critical velocity decreases with the increase of the radius. The lowest value of the critical velocity is for the radius R = 10,000 m and it is v

_{n}= 40.0 m/s (see also Figure 12). The same value of the critical velocity was obtained for ST. These results are rather surprising, because for this vehicle, by analogy to the hsfv1 car, the highest velocity v

_{n}in ST was expected. Thus, that result is opposite to the generally expected situation. Moreover, an increase of v

_{n}with R increase in the range R = 2000 to 10,000 m would also be expected, by the same analogy to the hsfv1 car.

_{n}increases with the R increase, reaching the highest value for the radius R = 2000 m. Then it begins to decrease with the increase of the radius R. The lowest velocity v

_{n}= 36.5 m/s was recorded for ST i.e., for R = ∞ (see Figure 13).

#### 5.2. General Conclusions

_{n}versus R has got convex shape for the bogie of the MKIII passenger car. Opposite it, the shape of the bogie with averaged parameters is convex. To make the matter even more complicated, the shape of the 25TN bogie of a freight car is a horizontal line, which means a constant v

_{n}value in the whole range of R, including ST. The 25TN bogie has a slightly different structure than the other two bogies, which could potentially explain the difference. On the other hand, the most surprising is the difference between the bogie of the MKIII car and the bogie with averaged parameters. They both have the same structure and differ in their parameters only. Generally, these differences might be subjectively recognized as rather small.

_{n}with the R increase is unique among all objects of the railway vehicle class considered in the present paper. Interesting, here, might be looking at the results of the isolated bogie of the MKIII car (Figure 11) and the complete MKIII car possessing two such bogies (Figure 13). Unfortunately, the shapes of courses in these objects are entirely different. Even critical velocities in ST vary for both these objects considerably, i.e., equal 45.3 and 36.5 m/s, respectively. This confirms the importance of the secondary suspension in the case of 4-axle vehicles equipped with bogies.

_{n}increasing with R increase was also obtained in the second group of objects (2-axle vehicles) for the hsfv1 car. In turn, unexpected results were obtained for the bogie with average parameters (bogie) and vehicle with average parameters (car). In both cases, with the increase of the radius R, the critical velocity v

_{n}decreased and corresponding graphs follow a similar course, they are just decreasing. Some but a slight exception, here, is the result for bogie with averaged parameters in ST. The third case was obtained for one object, namely the 25TN bogie. There, the graph resembles a straight line, parallel to the x-axis, which means the results are almost identical. The fourth type of plot is obtained for the 4-axle MKIII car. It can be described as two-stage one, initially increasing and then decreasing with the R increase. Summing up, it is unfortunately not possible to generalize these test results to all vehicles or a group of vehicles, because the results are too varied for individual vehicles and for vehicles within the groups as well.

_{n}or in conjunction with wider stability studies, could bring some explanation and, thus, a better understanding of the diversity of the result obtained in the present paper.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Typical bifurcation graph for a wheelset in CC [2].

**Figure 5.**Lateral coordinates y and ψ of the wheelset of the bogie of the MKIII car on the ST section; v = 40.0 m/s < v

_{n}; y

_{i}(0) = 0.0045 m.

**Figure 6.**Lateral coordinates y and ψ of the wheelset of the bogie of the MKIII car on the ST section; v

_{n}= 45.3 m/s; y

_{i}(0) = 0.0045 m.

**Figure 7.**Lateral coordinates y and ψ of the wheelset of the bogie of the MKIII car on the ST section; v = 50.0 m/s > v

_{n}; y

_{i}(0) = 0.0045 m.

**Figure 8.**Lateral coordinates y and ψ of the wheelset of the bogie of the MKIII car on the CC section; R = 2000 m; h = 0.0450 m; v = 40.0 m/s < v

_{n}; y

_{i}(0) = 0.0045 m.

**Figure 9.**Lateral coordinates y and ψ of the wheelset of the bogie of the MKIII car on the CC section; R = 2000 m; h = 0.0450 m; v

_{n}= 44.0 m/s; y

_{i}(0) = 0.0045 m.

**Figure 10.**Lateral coordinates y and ψ of the wheelset of the bogie of the MKIII car on the CC section; R = 2000 m; h = 0.0450 m; v = 46.0 m/s > v

_{n}; y

_{i}(0) = 0.0045 m.

**Figure 11.**Change of the value of the critical velocity v

_{n}of the bogies depending on the curve radius.

**Figure 12.**Change of the value of the critical velocity v

_{n}of 2-axle vehicles depending on the curve radius; for the hsfv1 car reduced value of longitudinal stiffness in the primary suspension k

_{zx}= 206.7 kN/m.

**Figure 13.**Change of the value of the critical velocity v

_{n}of 4-axle vehicle depending on the curve radius; the reduced value [14] of longitudinal stiffness of the secondary suspension k

_{pzx}= 1000 kN/m.

Object | Initial Conditions y _{i}(0) (m) | ST Length; l (m) | CC Length; l (m) | CC Radius; R (m) | Superelevation h (m) |
---|---|---|---|---|---|

Bogie of MKIII car, Bogie with averaged parameters | 0.0045 | 500 | - | - | 0 |

0.0045 | - | 500 | 600 | 0.1500 | |

0.0045 | - | 500 | 1200 | 0.0750 | |

0.0045 | - | 500 | 2000 | 0.0450 | |

0.0045 | - | 500 | 4000 | 0.0225 | |

0.0045 | - | 500 | 6000 | 0.0150 | |

0.0045 | - | 500 | 10,000 | 0.0090 | |

25TN bogie | 0.0045 | 500 | - | - | 0 |

0.0045 | - | 500 | 300 | 0.1500 | |

0.0045 | - | 500 | 600 | 0.1500 | |

0.0045 | - | 500 | 900 | 0.1420 | |

0.0045 | - | 500 | 1200 | 0.0750 | |

0.0045 | - | 500 | 2000 | 0.0450 | |

0.0045 | - | 500 | 4000 | 0.0225 | |

0.0045 | - | 500 | 6000 | 0.0150 | |

0.0045 | - | 500 | 10,000 | 0.0090 |

Object | Initial Conditions y _{i}(0) (m) | ST Length l (m) | CC Length l (m) | CC Radius R (m) | Superelevation h (m) |
---|---|---|---|---|---|

hsfv1 car, Vehicle with averaged parameters | 0.0045 | 500 | - | - | 0 |

0.0045 | - | 500 | 300 | 0.1500 | |

0.0045 | - | 500 | 600 | 0.1500 | |

0.0045 | - | 500 | 900 | 0.1420 | |

0.0045 | - | 500 | 1200 | 0.0750 | |

0.0045 | - | 500 | 2000 | 0.0450 | |

0.0045 | - | 500 | 4000 | 0.0225 | |

0.0045 | - | 500 | 6000 | 0.0150 | |

0.0045 | - | 500 | 10,000 | 0.0090 |

Object | ST; v _{n} (m/s) | CC; R (m) | CC; v _{n} (m/s) |
---|---|---|---|

Bogie of MKIII car | 45.3 | - | - |

600 | 35.0 | ||

900 | 41.0 | ||

1200 | 44.0 | ||

2000 | 44.0 | ||

4000 | 45.0 | ||

6000 | 45.1 | ||

10,000 | 45.2 | ||

Bogie with averaged parameters | 45.8 | - | - |

600 | n. derailment—at 42 * | ||

1200 | n. derailment—at 59 * | ||

2000 | 47.1 | ||

4000 | 46.3 | ||

6000 | 45.8 | ||

10,000 | 45.2 | ||

25TN bogie | 29.2 | - | - |

300 | 29.1 | ||

600 | 29.1 | ||

900 | 29.2 | ||

1200 | 29.1 | ||

2000 | 29.1 | ||

4000 | 29.2 | ||

6000 | 29.1 | ||

10,000 | 29.2 |

_{n.}

Object | ST v _{n} (m/s) | CC R (m) | CC v _{n} (m/s) |
---|---|---|---|

hsfv1 car ** | 42.8 | - | - |

300 | n. derailment—at 39.0 * | ||

600 | 40.0 | ||

900 | 41.5 | ||

1200 | 42.0 | ||

2000 | 42.3 | ||

4000 | 42.3 | ||

6000 | 42.4 | ||

10,000 | 42.6 | ||

Vehicle with averaged parameters | 40.0 | - | - |

300 | n. derailment—at 10.2 * | ||

600 | 41.0 | ||

900 | 41.0 | ||

1200 | 41.0 | ||

2000 | 40.9 | ||

4000 | 40.3 | ||

6000 | 40.1 | ||

10,000 | 40.0 |

_{n}. ** reduced value of the longitudinal stiffness of in the primary suspension k

_{zx}= 206.7 kN/m.

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**MDPI and ACS Style**

Zboinski, K.; Golofit-Stawinska, M.
Determination and Comparative Analysis of Critical Velocity for Five Objects of Railway Vehicle Class. *Sustainability* **2022**, *14*, 6649.
https://doi.org/10.3390/su14116649

**AMA Style**

Zboinski K, Golofit-Stawinska M.
Determination and Comparative Analysis of Critical Velocity for Five Objects of Railway Vehicle Class. *Sustainability*. 2022; 14(11):6649.
https://doi.org/10.3390/su14116649

**Chicago/Turabian Style**

Zboinski, Krzysztof, and Milena Golofit-Stawinska.
2022. "Determination and Comparative Analysis of Critical Velocity for Five Objects of Railway Vehicle Class" *Sustainability* 14, no. 11: 6649.
https://doi.org/10.3390/su14116649