Importance and recommendations for imaginary forces omission in rail vehicle dynamics
Abstract
This chapter focuses on problems related to building mathematical and numerical models of railway vehicle dynamics and then using these models in the process of vehicle dynamics simulation. Finally, the results of such simulations devoted to selected dynamical problems are presented, highlighting the importance of powerful tools such as both the modeling and the simulation. The dynamical problems selected for the presentation concern railway vehicle stability and importance of kinematics accuracy for the description of the dynamics. These selected problems focus on the vehicle dynamics in a curved track, both in the circular and transition sections. Type of the chapter should be defined as the review paper, however, based on the authors’ own results in the main.
Keywords
 Railway vehicle
 vehicle dynamics
 curved track
 transition curve
 numerical simulation
1. Introduction
The present review chapter is based on the authors’ results gathered and published in subsequent parts for many years of his work in the field of railway vehicle dynamics with focus on a curved track motion. The idea of the chapter is to combine all the results on the one hand and to select them suitably on the other hand. Both these demands are fulfilled in order to give a picture of comprehensiveness of the combined issues and not overload the reader with excessive details that could spoil the presentation of the chapter’s main aim. The aim is to present the outstanding role of modeling, simulation, and results of their use, in a shortened way, in the contemporary research questions of the railway vehicle dynamics. However, this is going to be done through exploitation of the author’s own results. The reference could be done here to [1], the comprehensive monograph in Polish (371 pages). This chapter is profiled differently, however, and results are presented in the concise way and internationally accessible form, i.e. in English.
1.1. The fundamentals in rail vehicle dynamics
It is rather a wellknown fact (e.g. Refs. [1–3]), however sometimes being forgotten, that in rail vehicle dynamics, perturbations of vehicle motion relative to vehicle general motion are of the primary interest. The general motion of vehicle is of lesser interest as it is predefined by the track alignment (shape). Hence, in the general view, the vehicle reproduces the motion imposed by the track centerline shape, as the vehicle is guided by the track (rails). Such a guided motion is already known. Instead, the perturbed motion relative to the track centerline appears to be of real importance in the rail vehicle dynamics problems.
Therefore, the description of such a perturbed motion makes the bases in the rail vehicle dynamics. As recently shown by the author in Ref. [1, 2], there are three options to perform such description.
where
where B' is the inertia forces in relation to
where
Commenting on Eqns. (1–3), one should note that the lefthand side equalities represent dynamical equations of motion. In case of Eq. (3), the equations of motion are supplemented with kinematical relations. They express absolute variables with the relative ones and need to be introduced into Eq. (31) when equations of motion are being solved. The general form of Eqns. (1–3) serves any form of the vectorial and scalar equations as well as the matrix form in certain cases. Meaning of the dynamical equations of motion is typical. It becomes obvious as B represents inertia forces relative to absolute (inertial) system, Z represents external forces, B’ represents inertia forces relative to moving (noninertial) system, and P represents imaginary forces (inertia forces depending on the transportation, i.e., motion of noninertial system relative to the inertial one). Moreover, the operators
The curly brackets represent solutions of the equations of motion. As the equations are secondorder ordinary differential equations, the solutions are displacements (coordinates) and velocities, both the linear and angular ones in general.
It is worth emphasizing that Eq. (2) is the only one that needs explicit form of the imaginary forces to be recorded. In contrast, in case of Eqns. (1) and (3), the inertia forces are also taken into account, however, in an inexplicit way. This is because
Based on the denotation explained above, one can note in the curly brackets that Eq. (2) is the only one that leads directly to the solution representing the relative coordinates and velocities. Eq. (2) is advantageous as the relative variables are those of interest in most of railvehicle dynamics problems. In case of Eq. (1), the relative solution can be obtained indirectly, whereas in the case of Eq. (3) a part of the solution represents absolute variables and another the relative ones.
Discussing briefly the practical use of particular options, it has to be stated at first that numbers and types of application differ. The approach defined with Eq. (1) is definitely used most rarely. The approach defined with Eq. (3) is the one among all three that is used most often in the commercial software packages for automatic generation of equations of motion (AGEM). This includes the software suitable for the systems in a rail vehicle type. Currently, such software (e.g., MEDYNA, VAMPIRE, SIMPACK, or VIRAIL codes) are used quite often mainly in solving many contemporary engineering problems. However, their use in scientific research is questionable. It is contested because of the socalled black box problem. Thus, following the formal scientific methodology, the researcher himself and the others should precisely know how the problem was resolved. Therefore, the assumptions and the way of modeling have to be elucidated, which allows others to repeat the study or to compare it with results for other similar but different approaches (for different assumptions and methods of modeling). Examples of use of this approach to build the models and the simulation software for the scientific purposes can also be found. However, for such purposes, the option defined with Eq. (2) seems to be used more frequently, where different levels of accuracy are used in practice. Thus, some authors neglect selected terms of the imaginary forces. The importance of these terms in the rail vehicle dynamics is comprehensively discussed by the author of the present chapter in Ref. [4].
2. Equations of motion based on dynamics of relative motion approach
In his studies, the author of the present chapter himself practices and is a supporter of option 2 described earlier. This option defined with Eq. (2) exploits direct results of the dynamics of relative motion. Both, the variables used and the equations of motion, are defined directly in relation to moving coordinate systems in type
2.1. The Newton–Euler equations
Based on the fundamental kinematical relations for relative motion, relating absolute and relative velocities and accelerations,
and vectorial forms of Newton and Euler equations, the vectorial equations of relative motion for translation and rotation of a single free rigid body can be obtained as those presented in Refs. [7, 1, 2]. Based on these vectorial equations, their matrix forms can be recorded in several ways as shown, e.g., in Refs. [1, 2, 8–10]. The forms as in Refs. [1, 2] are as follows:
The meanings of the denotations present in the above Eqns. (4–6) are as follows:
One should realize here that in order to express all matrices in Eqns. (7, 8) through their elements selection of the vector bases
Equations (7, 8) can easily be recorded in one matrix equation:
where
and
On analyzing Eq. (10), the variables
It can be shown that the equation similar to Eq. (9) can be obtained when arbitrarily chosen set of variables
and linear kinematical relations
the equation can be written as follows:
while the derivative equals
After the introduction of Eq. (13) in (9), one gets
where
As shown explicitly in Refs. [1, 2], the result represented by Eq. (14) can be generalized to the case of constraint system with holonomic and nonholonomic constraints (e.g., see Ref. [7]). Then it can be extended so that the inertia matrix becomes symmetrical one what enables to get the corresponding inverse matrix and finally to solve the equations. This can be done through the lefthand side multiplication of the equation by the transpose matrix
where
In the equations above,
When one is not interested in values of the constraint (internal) forces
After Eq. (19) is exploited in Eq. (14), the state–space equation in the independent generalized velocities takes the following form:
where
Note that inertia matrix
The final Eqns. (16, 20) are valid for a single rigid body. It was shown in Ref. [8] for inertial systems that forms valid for a single body can directly be generalized to any number of rigid bodies. As in principle the structure of equations for noninertial systems differs in additional inertia terms of correction character only (see, e.g., Eqns. (2, 3)), this result can be extended to noninertial systems. In terms of the notation, it is trivial and means that Eqns. (16) and (20) valid in noninertial systems remain unchanged for any number of rigid bodies.
2.2. Kane’s equations
Any of the formalisms of analytical mechanics valid in inertial systems can be adapted to describe the relative motion in the noninertial system. The author of the chapter performed such an adaptation for Kane’s equation. It was done in a formal manner, that is, corresponding equations of relative motion were derived as shown in Refs. [6, 2, 1].
The partial velocities are fundamental to original Kane’s approach [11]. The corresponding relative linear and angular partial velocities were introduced by the author as shown in Refs. [6, 1, 2]. Let us introduce them for the simple nonholonomic system [11] of
where
Form of the adapted Kane’s equations is as follows:
Most of the denotations in Eq. (24) are as already defined. Here, the meanings in Eqns. (4–8) are helpful. Note, however, that in Eq. (24) the
Equation (24) can be expressed in the shortened matrix form as follows [6, 2, 1]:
where scalar components are defined with
In the above Eqns. (26–28),
Equations (24–28) are not useful in AGEM approach as they are too close to the vectorial origin of Kane’s equations. Nevertheless, final forms of Eqns. (26–28) can be treated as the initial ones for the process of deriving the useful form. The process is based on the Huston results [12–14] for the inertial scleronomic systems and present author’s extensions [6, 2, 1] to noninertial and rheonomic systems. Consequently, invoking results from Refs. [6, 2, 1] one can write
where
Besides, in Refs. [6, 2, 1], the form of imaginary forces corresponding to Eqns. (30) and (31) was presented as follows:
The supplements to Eq. (29) are kinematical relations:
where in the above equations
Equations (24, 25, 29, 32, 33) serve nonholonomic, holonomic, and free systems. In Eqns. (25, 29, 32, 33), the difference between particular systems is taken into account by suitable values of partial velocities’ coefficients
It was shown by the author in Refs. [6, 2, 1] for the rail vehicle systems, taking account of their moderate dimension, nonoccurrence of nonholonomic constraints, limited number of holonomic constraints, identical coordinates and transportation for each rigid body within vehicle model, that it can be reasonable to make use of equations (or particular type of forces) for free systems. Assuming holonomic and nonholonomic constraint equations as
the following formula enables to build equations (or selected forces type) for constraint nonholonomic system based on the equations (or selected forces type) for free system [6, 2, 1]:
If the system is holonomic then Eq. (34) vanishes, while Eq. (36) is reduced to just two first addends. In Eqns. (34–36), the following so far undefined notations appear:
2.3. Discussion of the equations
Taking account of Refs. [15, 16], both the Newton–Euler and Kane’s equations are those most often used in AGEM. This is an interesting fact because major differences between both approaches exist. Therefore, the equations are worthy of discussion in this context.
It can be seen in Section 2.1 that building Eq. (16) and (20) that represent receipt useful in AGEM are operations on matrices. To perform these operations, the matrices have to be defined with their components. To do this, one needs to adopt base vectors
Equations (29–33) form the receipt useful in AGEM of Kane’s equations. Their forms might seem discouraging as they are complex, especially while comparing with Eqns. (16, 20). This impression is misleading as equations possess serious advantages as well. The advantage is the same form of either scalar (Eq. (24)) or matrix equations (Eqns. (29–33)) for free, holonomic, and nonholonomic systems. The most distinguishing advantage of Kane’s equations, appearing just for this formalism, is the moment the components of vectors and tensors are defined through the vector bases selection. This is done at the very end of the equations building. This is possible since Eqns. (24, 29–33) originate directly from Eq. (24) based on vectors. Therefore, Eqns. (24, 29–33) are valid for any vector basis. An additional advantage is the type of Kane’s equations that are always ODEs.
3. Example objects, nominal models, and numerical models
3.1. The objects and their nominal models
Equations of motion of mechanical system are referred to as its mathematical model. In order to make use of the general methods of building equations of motion as shown in Section 2, the nominal model of particular vehicle (object) has to be first determined. In case of railway vehicle, its nominal model projects its structure and selected physical features of the structure elements. The nominal models of two example objects will be presented below. Both are of British origin and possess relatively simple structure that is suitable for the basic character of the author’s research. The first model [1, 5, 17] corresponds to 2axle hsfv1 freight car and is shown in Figure 2. The second one [1, 18] corresponds to 4axle MKIII passenger car and is shown in Figure 3.
If the flexibility of track is going to be taken into consideration in the study, then some track nominal models should be adopted. In case one considers lowfrequency dynamics of the vehicle motion (not more than 50 Hz) but not the higherfrequency phenomena in the track itself, then it is reasonable to build the track model that is composed of rigid bodies. Then the whole vehicletrack system model is a multibody model. Separate models for lateral (Figure 4) and vertical (Figure 5) directions as used by the author in his studies are presented below [1, 5, 17, 18].
The denotations used in Figures 2–5 are as follows:
The whole hsfv1 cartrack system model has 18 degrees of freedom (DOFs). The whole MKIII cartrack model has as many as 38 degrees of freedom.
3.2. Numerical models
When mathematical models derived traditionally are available, based on general methods of the equations building as described in Section 2 and adopted nominal models as, for example, those shown in Section 3.1, then they have to be converted to numerical models. When AGEM approach is used to build the equations, based again on the results of Section 2 and Section 3.1, then equations automatically form the numerical model. The numerical models are indispensable, since rail vehicle systems are multidimensional ones. In addition, their nonlinear versions are mainly used at present. The only way to solve dynamical differential equations of motion of such systems is numerical integration. In order to do this, the numerical model representing the mathematical model in a form understandable to computers must be built, which means that the equations have to be coded in some of the programming languages. In the case of this author, it is Fortran. Then, the equations are solved with use of one of the numerical methods of equation integration. The author uses Gear’s method [19]. The software containing traditionally derived equations of motion or generating the equations automatically combined with the integration procedure is called the simulation software in railway vehicle dynamics.
In order to make use of the procedure, shortly described above, three additional major elements have to be provided. The first one is to adopt the model and then to build its numerical module to take account of relative kinematics arising from description in moving coordinate systems. Strictly speaking, the linear and angular velocities and accelerations of transportation, represented with functions
The next element is tangential contact forces calculation. These forces arise from wheel/rail relative slip (creepage). Consequently, the longitudinal, lateral, and spin creepages are the inputs in the existing methods of the tangential forces calculation. Generally, one can talk about the exact and simplified numerical methods of nonlinear tangential contact forces calculation. The problem with the exact methods, as, for example, Kalker’s CONTACT program [22], lies in the very slow calculations for the simulation purposes of rail vehicle dynamics. Thus, simplified methods are in use in the simulation programs (numerical models). One of them is Kalker’s FASTSIM program [23]. This is just the software used by the present author in his models and so in the studies. Usually, the adopted value of the friction coefficient, necessary for the program to run, in the author’s studies is
The last among three elements to be discussed is the problem of nonlinear contact geometry. This author uses ArgeCare RSGEO program [28] to address this. The program resolves normal contact problems and purely geometrical ones. As a result, contact areas become known and geometrical variables in the contact. The most important geometrical variables are instantaneous contact angles, rolling radii, and contact point positions on the wheel and rail for any lateral relative shift between wheel and rail. In addition, the program is capable of taking account of the influence of wheelset yaw angle on the parameters. Both the left and righthand side wheel and rail parameters are calculated in the same step. The results of ArgeCare RSGEO program (the author uses) are tabulated before simulations for a given wheel and rail pair of profiles as a function of the discrete lateral wheel/rail displacements and yaw angles. To get the contact parameters for any relative displacement and yaw rotation between wheel and rail, the tabulated data are linearly interpolated.
At the end, let us return to the issue already discussed at the beginning of Section 2. It is the way in which the equations of motion in the simulation software (numerical model) are introduced. Here, this refers to the traditionally derived and automatically generated equations of motion (AGEM approach). Now, we will provide some references corresponding to both approaches. If one is interested in samples of the equations traditionally derived by this author, then he could find them in Ref. [29]. If one is interested in details of the package ULYSSES and its core program TITAN cobuilt by this author, then he can find them in Refs. [1, 2, 6]. The software packages based on AGEM built by other authors are discussed in Refs. [15, 16].
4. Example results of the selected simulation studies
The author of the present chapter has exploited simulation in his studies for more than 20 years. There is no possibility to refer to all of them. Readers interested in earlier applications of simulation by the author can find reviews of the corresponding references in Ref. [20]. Similar review, however, also including current applications, is done in Ref. [1]. Total number of all author’s applications runs to tens. Due to the limit of space, just two applications of simulation used in currently studied problems will briefly be discussed in the following sections. Their content will be focused more on the achievements, contributions, and final results than on the details of the methods or procedures used in those studies. Interested readers will be informed about the essential publications where the details can be found. These publications can also be treated as a main base source for the results discussed further. The secondary source is Ref. [1]. The applications in view concern the stability and kinematics issues.
Two other applications extensively studied at present are connected with TC shape optimization with the use of simulation and optimization methods as well as with dynamics of railway vehicles in TCs at velocities around the critical
4.1. Results of simulation use in rail vehicle nonlinear lateral stability studies
The serious interest of the author on the stability problems started with reference [17]. The most important results were presented many years later in Refs. [33, 34]. The fundamental reference to the methods used by this author, based on bifurcation approach and on usage of the simulation, is [35]. Recently, the remarkable contributions have been by H. True and his coauthors (e.g., see Refs. [36, 37]). Works by O. Polach (e.g., see Ref. [38]) are also interesting. Much more comprehensive and detailed literature review can be found in Refs. [1, 33, 34].
The meaning of stability is fundamental in the discussed problem. This author could describe its meaning in most of his works as follows. The used method of nonlinear lateral stability analysis regards formal theory of stability. By contrast, it is not so formal as compared with the methods based strictly on the theory. Instead, one finds the tool to be more practical. The simplification is based on adoption of certain assumptions as well as on behavioral expectations for the studied system. They arise from commonly existing knowledge about the systems in the type of rail vehicle. Therefore, one can skip formal adoption of some solution as the reference in the simplified approach. Moreover, he can skip introducing some perturbations into the system to examine whether the new solution stays close enough (in narrow vicinity) to the reference solution. Such a stay is required by the formal definition of the stability theory. Taking account of that, bifurcation plot building for the system is the main task in the method. However, formal verification whether the solutions enabling to build this plot are stable in completely formal sense is not such a task. The assumption exists in the method that any typical solution of rail vehicle system (either stationary or periodic) is stable. It might be accepted with care basing on understanding that periodic solutions in rail vehicle dynamics are selfexciting vibrations being governed through the wheel–rail tangential contact forces. Following that, the theory of selfexciting vibrations can be useful in order to predict/expect typical periodic behavior of the system. The adopted assumption enables to limit the number of simulations for different initial conditions but the same given velocity. Such simplified approach is described in Ref. [29]. In case of any doubts if obtained solution is stable or about the possibility of multiple solutions existence, more formal check for the stability is a must. Then, reasonably a denser sweeping over the initial conditions, to introduce the perturbation, is performed. Such a more accurate approach is called the extended analysis and is described in Ref. [30].
The author has been especially interested in the stability of rail vehicles in a curved track from the very beginning of his interest in stability problems. The studies were initiated as a result of the observed periodic solutions in a circular curve of track. They appear for velocities
They involved curve radii
It was successively improved for years and recently published in Refs. [33, 34]. The essence of this method is included in Figure 8.
The meaning of the symbols and acronyms present in Figure 8 is as follows:
The influence of several factors on the stability was studied based on the stability map technique. Among the factors studied are accuracy of wheelset’s angle of attack determination, track superelevation, types of nominal wheel and rail profiles, type and value of the wheel and rail wear, stiffness and damping values in suspension, type of vehicle (car and bogie), rail inclination, way of mean rolling radius modeling, track gauge, and value of coefficient of friction. Figures 9 and 10 can represent part of the results for one of the factors, namely the wheel profile shape. Both maps were obtained for the same object, however, with different nominal wheel profiles. One can observe significant differences between both maps. One among the most important differences are nonlinear critical velocities
The results in Figures 9 and 10 refer to 2axle freight car hsfv1 of the model shown in Figure 2. Example results of stability studies for the model of MKIII passenger car shown in Figure 3 can be found in Ref. [18].
4.2. Use of the simulation in studying the influence of kinematics accuracy on vehicle dynamics in a curved track at variable velocity
Use of Option 2 from Section 1.1 to build mathematical and numerical models of vehicletrack systems by the present author, made it natural that he was always interested in the importance of accurate modeling relative kinematics connected with description in moving reference systems. This interest was amplified since an additional work is necessary as compared with description in absolute reference systems. Besides, some of the authors neglect additional terms (imaginary forces) in their equations of motion, without proper justification. Good examples of such works might be provided in Refs. [3, 40–42]. Author of this book chapter undertook two stage attempt to finally resolve the problem of particular inertia terms importance. The first one concerned vehicle motion with constant velocity. It finished with publication [5]. The second attempt concerned the motion with variable velocity and finished with publication [4]. Publication [4] is also verification of the results from [5], as motion with constant velocity was just a special case in Ref. [4]. Therefore, Ref. [4] is the main source for results forthcoming in the current section. These results were also published in Ref. [1]. Both Refs. [4] and [1] present the issue in a comprehensive form. Here, only samples of the results and most general conclusions will be represented.
The idea of the study was to compare the results for the vehicle model with all imaginary forces included with those for the model in which the imaginary forces were omitted. In order to precisely determine importance of particular terms types, the forces (and torques) were selectively, rather than totally, omitted. Let us now recognize the generalized imaginary forces according to Ref. [7]. Here, equation (24) will be useful. The forces’ terms are present in the second line of this equation and the torques’ terms are present in the fourth line of it. Note that all the terms are multiplied by the corresponding linear and angular partial velocities. Therefore, we can refer directly to content of the square brackets in lines 2 and 4. Making use of that idea, the 1st term in the square bracket in line 2 represents inertia forces of translation. The 2nd term in line 2 and 1st one in line 4 form inertia forces of rotation. The 3rd term in line 2 and 2nd one in line 4 correspond to centrifugal forces of inertia. The 4th term in line 2 and the 3rd in line 4 make gyroscopic forces. The abovementioned four categories of imaginary forces’ terms appear in the equations as the terms’ components corresponding to the direction a particular equation of motion describes. Finally, the omissions of all components for given directions were performed. In fact, these were longitudinal, lateral, and vertical directions. At the end, for the given direction, the analyses were performed to find the components of the terms of particular (practical) importance among all.
The variable velocity was realized in the studies with use of the uniform variable motion. The change in velocity is indirectly represented by the corresponding acceleration value
4.2.1. Scope of the studies and example results of simulation
Generally, the tests described in Ref. [4] represent seven different routes and two different vehicles. The routes included radii
Two types of simulation results are presented below. The first type is imaginary forces being omitted in the study. They are a cause for the solution differences being of interest. The imaginary forces are shown in Figures 11, 14, and 16. The second type of simulation results is those representing the differences between solutions for the whole model and the model with particular imaginary force omitted. They are represented with Figures 12, 13, 15, and 17. Results for complete model are drawn with the solid line, while for the model with imaginary torques omitted with the dashed line.
Just two routes appear in the mentioned figures. The route I is as follows: ST (
It is worth noting in Figure 11 that it includes the biggest values of the imaginary torque obtained in the author’s studies. As is seen, it is the roll imaginary torque
In Figure 14, the value of the imaginary torque
In Figure 16, the character of torque
4.2.2. General conclusions from the study
Except the important influences described in Section 4.2.1, some other important influences were found and indicated in Ref. [4]. Therefore, except influences of
Conclusions from the studies, including track section, character of velocity, vehicle elements, direction defining equation, and purpose of calculations, were summarized in table form in Ref. [4]. Here, we present them in Table 1. This table is analogous to that in Ref. [4] in terms of merits. Nevertheless, Table 1 is differently arranged.











Inertia forces of translation  1/2  Force  Yes  ST, CC, TC  Long. ( 
Var.  All  Always, big  
p/r 

Inertia forces of rotation  2/2  Force  No              Yes 
1/4  Torque  Yes  TC  Long. ( vert. ( 
Var. const. (vert.) 
Body  Short TC (long.& vert.), big  
p/r 


Centrifugal forces  3/2  Force  Yes  CC, TC  Lat. ( vert. ( 
Const. var.  All (lat.), body (vert.) 
Always, big small 
p/r 

2/4  Torque  No              Yes  
Gyroscopic forces  4/2  Force  No              Yes 
3/4  Torque  Yes  CC  Long. 
Const.  Wheelset  Small big 
r  No 
Generally, the most important are longitudinal and vertical direction terms related to car body. The influence of eventual term omission is particularly important in TCs. The centrifugal force is important in both the TCs and the CC sections. The influence in TCs increases as compared with CC when the vehicle moves with variable velocity. The stronger the change in velocity (both accelerating and braking), the greater the influence. The last column represents a direct recommendation of terms that should not be neglected.
5. Concluding remarks
The author of this chapter has been involved in technological problems of railways for about 30 years. More precisely, his involvement concerns part of the railway system that are railway vehicles. Within the railway vehicles, many areas of studies can be mentioned as, for example, material issues, production technical issues, braking systems issues, traction systems issues, construction issues, exploitation issues, control issues, safety systems issues, etc. The author is interested in dynamics of vehicle motion with a special regard to the motion in curved sections of the track. It is obvious that the chapter was devoted to these last aspects. The author focused his efforts on giving the idea to the readers how important and powerful tools are methods of modeling and simulation when used for the purposes of the rail vehicle dynamics.
In Sections 1 and 2, the general methods of modeling dynamics in moving coordinate systems, useful for multibody systems of rail vehicle type, were represented. The Newton–Euler and Kane’s equations in the form suitable in AGEM approach were discussed. At the end of Section 2.2, the method profiled for rail vehicle systems was also presented. These methods help build mathematical models of rail vehicle dynamics. In Section 3, example nominal models of rail vehicles were introduced, as used by the author. In that section, the issue of building numerical models corresponding to the mathematical ones was also discussed.
In Section 4, two examples of the author’s simulation studies, with use of the numerical models, were discussed. The first example represented in Section 4.1 concerned the problem of rail vehicle stability in a curved track. Due to the author’s systematic and consequent simulation studies, the method was formulated suitable for rail vehicle nonlinear lateral stability studies in a curved section of the track. The examples of the socalled stability map, being the most important result in the studies, were shown and briefly commented on. The stability maps approach was used by this author to study the influence of different factors on the results of stability analysis in a curved track.
The second example is represented in Section 4.2. It was devoted to the use of the simulation in studying the influence of kinematics accuracy on vehicle dynamics in a curved track at variable velocities. In fact, both the constant and variable velocity cases were of interest. Due to the consequent author’s interest in this issue, it appeared possible, based on the direct results of simulation and equation terms analysis, to resolve the problem in full. The author presents tabular conclusion of the results in Section 4.2.2. The table states precisely which imaginary forces terms, in what conditions, and for which vehicle major elements (bodies) may have an important influence on the simulation results. The last column states that imaginary force of translation, imaginary torque of rotation, and centrifugal force should not be neglected in both theoretical and practical analyses (calculation). The gyroscopic forces should not be neglected in the theoretical issues as well.
Both described results of the simulation studies are original and important contributions to the knowledge on rail vehicle dynamics. These contributions would not be possible without the mathematical, nominal, and numerical models of vehicles build based on the modeling methods discussed in the chapter. Therefore, it is hoped that the readers have obtained an idea of the efficiency and importance of both the modeling and the simulation for the development of contemporary rail vehicle dynamics, and thus the railways in general.
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