Differential parametrization of rail vehicle properties and its impact on hunting

Abstract

In this work, we have developed a serial-chain multibody model of a rail vehicle, using the Decoupled Natural Orthogonal Complement matrices. The governing equations of the system were linearized, and the start of hunting was identified by finding the Smallest Bifurcation Point (SBP) of the trivial solution. Uneven wear creates differences in the properties of similar components in a rail vehicle. It was found that such variations are neglected in the hunting analysis. Hence, we introduce these differences by independently changing 32 parameters. The study found that these variations lead to unique trends. Then, these trends were used to recommend design changes for a rail vehicle, which could be thought of, in order to improve the hunting characteristics. For this purpose, an interactive software tool was created to easily visualize, and manipulate the hunting behavior of a rail vehicle.

Introduction

Hunting is defined as the sustained oscillatory motion of a wheelset, which is caused due to conical wheel profiles. Theoretically, the start of hunting is determined by searching the function-parameter space, by varying the vehicle’s forward velocity (V). The process involves finding the speed at which the trivial solution loses stability, also known as the Smallest Bifurcation Point (SBP).^1,2 SBP (denoted using V _s ) can be found using conventional stability analysis, i.e., identifying the lowest speed for which the real part of the eigenvalues, switches signs from negative to positive. ¹ It is generally regarded that hunting beyond SBP is undesirable. ³ Therefore, the identification of SBP is an important part of railway dynamics. Persistent hunting vibrations occur in two distinct speed intervals. ⁴ Bogie/secondary hunting is when the vibrations are present only in bogie frames, and wheelsets. ⁵ Whereas carbody/primary hunting ⁶ is a parametric excitation, when a vehicle is moving, which is clear from the terms in the damping matrix $\overline{\mathit{C}}$ (See appendix), that contains the forward velocity of the vehicle, as well as the conicity of the wheels.

Hunting of rail vehicles is characterized using parametric analysis, by varying the vehicle properties. Note that, most of the studies^3–10 assume comparable components with same properties. For example, as indicated in Figure 1, all four wheelsets in the carbody system are considered to have the same conicity, i.e., γ^w1 = γ^w2 = γ^w3 = γ^w4. While such an assumption is appropriate for an idealized rail vehicle, realistic operation conditions lead to uneven wear between the components. For instance, an unevenly loaded carbody exerts dissimilar loads on the axles. Extended operations under these conditions creates a difference in the properties of individual components. Notably, the symmetry of a rail vehicle about the YZ plane (See Figure 1) is lost, when these variations exist. It was recognized that such variations have not been studied extensively, although they may assume importance under certain conditions. Therefore, an extensive parametric study was performed, by varying the properties of individual components. Using the results, design improvements on a specific component, to achieve higher vehicle speeds, are suggested. Crucial properties of specific components that adversely effect the hunting behavior were also identified and reported.

Figure 1.

A railway carbody system (Top view).

This paper is organized as follows: The next section describes the carbody model used for hunting analysis. The convention for various symbols is also explained here. Next, we describe important steps of the Decoupled Natural Orthogonal Complement (DeNOC) multibody formulation, which were used to obtain the equations of motion. Linearizations done in the multibody model, and the method adopted to find the SBP is also explained in the same section. Then, application cases used to validate our findings, as well as time response plots are shown. In the next section, we discuss the results of parametric analysis and the corresponding design recommendations. An overview of the interactive user interface for hunting analysis is given in the subsequent section. Finally, the contributions are summarized.

Carbody system

A railway carbody system shown in Figure 1 having two bogies, and four wheelsets is considered here. Its three translations are longitudinal (x), lateral (y), and vertical/bounce (z). Rotations about these axes are roll (ϕ), pitch (θ), and yaw (ψ). Two wheelsets are connected to each bogie frame through primary suspension, whereas secondary suspension links two bogies with a carbody. Each wheelset has two Degrees of Freedom (DOF), which correspond to lateral translation (y ^wi ), and yaw (ψ ^wi ), but the bogies, and carbody have three DOF each, which are their lateral translation (y ^bj , y ^c ), yaw (ψ ^bj , ψ ^c ), and roll (ϕ ^bj , ϕ ^c ). In total, the carbody system has 17 DOF (2 × 4 wheelsets + 3 × 2 bogies + 3 × 1 carbody = 17). The following assumptions were made for the modeling.

1. The carbody system is considered to move at a constant forward velocity (V), about the x axis on a tangent track. Note that, $V=r_0\stackrel{.}{\theta }$ , with r₀ being the nominal rolling radius of wheels.

2. The system is symmetric about the XZ plane (left-right symmetry). For instance, the properties of the left, and right wheels of a wheelset are similar.

3. The wheels are perfectly conical, which means that the generator of a cone is a straight line. Wheelset rolls on knife-edge rails.

The following nomenclatures are followed throughout this paper: Bold symbols denote vectors or matrices. The main character represents the properties, such as γ for wheelset conicity, I for inertia, etc. Superscripts have two characters, in which the first letter (w/b/c/p/s/t) indicates the component type, i.e., w for wheelset, b for bogie, c for carbody, p for primary suspension, s for secondary suspension and t for track, whereas the ensuing number (second character in the superscript) represents the component number, namely i = 1, 2, 3, 4 for four wheelsets, and j = 1, 2 for two bogies. Subscripts are used to denote the axes or other relevant properties. For example, $I_z^{w1}$ is inertia of the first wheelset about z axis. Accents ( ˙ ), and ( ¨ ) denote first, and second time derivatives respectively, whereas ( ¯ ) denotes the linearized matrices, ( ˆ ) is used for zero vectors, and (˜) represents the skew-symmetric form of a 3 - dimensional vector.

Mathematical framework

Equations of motion

A serial-chain modeling approach using the DeNOC matrices ¹¹ was used to derive equations of motion. Each component of the carbody system was modeled as an open-loop serial-chain system having multiple links (one each for every DOF). Figure 2 shows adjacent links in a serial-chain, which are connected through a joint at O _i . The unconstrained Newton-Euler equations for an i ^th link in the serial-chain system are written as

${\mathit{M}}_{\mathit{i}}{\dot{\mathit{t}}}_{\mathit{i}}+{\mathit{W}}_{\mathit{i}}{\mathit{M}}_{\mathit{i}}{\mathit{t}}_{\mathit{i}}={\mathit{w}}_{\mathit{i}}$

(1)

Figure 2.

Adjacent links of a serial-chain multibody system.

Note that, t _i is a 6-dimensional twist vector, which contains the angular velocity ( ω _i ) and linear velocity $\left({\dot{\mathit{c}}}_{\mathit{i}}\right)$ of the center of mass (C _i ): ${\mathit{t}}_{\mathit{i}}={\left[\begin{array}{cc}{\mathit{\omega }}_{\mathit{i}}^{\mathit{T}}& {\dot{\mathit{c}}}_{\mathit{i}}^{\mathit{T}}\end{array}\right]}^{\mathit{T}}$ . w _i is the wrench vector that contains the moments ( n _i ) and forces ( f _i ), such that ${\mathit{w}}_{\mathit{i}}={\left[\begin{array}{cc}{\mathit{n}}_{\mathit{i}}^{\mathit{T}}& {\mathit{f}}_{\mathit{i}}^{\mathit{T}}\end{array}\right]}^{\mathit{T}}$ . M _i and W _i are the 6 × 6 mass and angular velocity matrices, they take up the following form:

${\mathit{M}}_{\mathit{i}}\equiv \left[\begin{array}{cc}{\mathit{I}}_{\mathit{i}}& {\mathit{0}}_{\mathit{3}}\\ {\mathit{0}}_{\mathit{3}}& m_i.{\mathit{1}}_{\mathit{3}}\end{array}\right],{\mathit{W}}_{\mathit{i}}\equiv \left[\begin{array}{cc}{\stackrel{\sim }{\mathit{\omega }}}_{\mathit{i}}& {\mathit{0}}_{\mathit{3}}\\ {\mathit{0}}_{\mathit{3}}& {\mathit{0}}_{\mathit{3}}\end{array}\right],$

(2)

where I _i , 0 ₃ , and 1 ₃ are the 3 × 3 inertia matrix, zero matrices, and identity matrix, respectively. m _i is the mass of the link. The serial-chain architecture of a railway carbody system is shown in Figure 3, where the dotted lines represent the virtual links, i.e., links that have no mass or inertia associated with them, and solid lines denote real links. It has a total of n = 17 links. Newton-Euler equation of this system is written in a compact form, as given below.

$\mathit{M}\dot{\mathit{t}}+\mathit{W}\mathit{M}\mathit{t}=\mathit{w}$

(3)

In the above equation, $\mathit{M}\equiv \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left[\begin{array}{ccc}{\mathit{M}}_{\mathit{1}}& {\mathit{M}}_{\mathit{2}}& \dots {\mathit{M}}_{\mathit{n}}\end{array}\right]$ , $\mathit{W}\equiv \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left[\begin{array}{ccc}{\mathit{W}}_{\mathit{1}}& {\mathit{W}}_{\mathit{2}}& \dots {\mathit{W}}_{\mathit{n}}\end{array}\right]$ , $\mathit{t}\equiv {\left[\begin{array}{ccc}{\mathit{t}}_{\mathit{1}}^{\mathit{T}}& {\mathit{t}}_{\mathit{2}}^{\mathit{T}}& \dots {\mathit{t}}_{\mathit{n}}^{\mathit{T}}\end{array}\right]}^{\mathit{T}}$ and $\mathit{w}\equiv {\left[\begin{array}{ccc}{\mathit{w}}_{\mathit{1}}^{\mathit{T}}& {\mathit{w}}_{\mathit{2}}^{\mathit{T}}& \dots {\mathit{w}}_{\mathit{n}}^{\mathit{T}}\end{array}\right]}^{\mathit{T}}$ . Next, the kinematic constraints for the i ^th joint in the serial-chain, are expressed using the recursive relations for the twist vector (See Figure 2).

${\mathit{t}}_{\mathit{i}}=\left[\begin{array}{c}{\mathit{\omega }}_{\mathit{i}}\\ \stackrel{.}{{\mathit{c}}_{\mathit{i}}}\end{array}\right]=\left[\begin{array}{cc}{\mathit{1}}_{\mathit{3}}& {\mathit{0}}_{\mathit{3}}\\ {\stackrel{\sim }{\mathit{c}}}_{\mathit{i},\mathit{i}-\mathit{1}}& {\mathit{0}}_{\mathit{3}}\end{array}\right]\left[\begin{array}{c}{\mathit{\omega }}_{\mathit{i}-\mathit{1}}\\ {\dot{\mathit{c}}}_{\mathit{i}-\mathit{1}}\end{array}\right]+\left[\begin{array}{c}{\mathit{e}}_{\mathit{i}}\\ {\stackrel{\sim }{\mathit{e}}}_{\mathit{i}}{\mathit{d}}_{\mathit{i}}\end{array}\right]{\dot{q}}_i$

(4)

where ${\dot{q}}_i$ is the independent joint speed, and e _i is the joint axis vector. Recursive relation given in equation (4) is used to express the kinematic constraints of all joints, which populate the generalized 6n - dimensional twist vector ( t ). After simplifications and re-arrangement, ¹¹ t is expressed in the following form

$\mathit{t}=\mathit{N}\dot{\mathit{q}}$

(5)

In the above equation, N is called Natural Orthogonal Complement (NOC) matrix. It is the transformation between Cartesian velocities and generalized velocities $\dot{\mathit{q}}$ , and happens to be the column space of the Constraint Jacobian matrix C _q . Note that, N can be further decomposed ¹² into N _l and N _d , both of which are termed the DeNOC matrices. Pre-multiplying equation (3) with N ^T , results in the constrained dynamics equations of motion of the system, as given below.

$\mathit{I}\ddot{\mathit{q}}+\mathit{h}=\mathit{\tau }$

(6)

where, I ≡ N ^T MN is the 17 × 17 Generalized Inertia Matrix (GIM). The 17-dimensional vector $\mathit{h}\equiv {\mathit{N}}^{\mathit{T}}\left(\mathit{M}\dot{\mathit{N}}+\mathit{W}\mathit{M}\mathit{N}\right)\dot{\mathit{q}}$ is termed as the Vector of Convective Inertia (VCI) terms, containing quadratic velocity components such as Coriolis forces, etc. Finally, the External Forces Vector (EFV), given by τ ≡ N ^T w contains forces, and moments due to (a) wheel-rail contact, (b) axle-load, (c) track stiffness, and (d) suspension components. As shown in Figure 4, at the wheel-rail contact point (C _r ), the normal force ( n _r ) is obtained using Hertz theory, ¹³ whereas the two tangential components ( t _xr and t _yr ), commonly known as creep forces were calculated using Kalker’s linear theory. ¹⁴ During hunting, when the wheelset tries to exceed the flange contact limit (δ = 5 × 10⁻³ m), the track’s lateral stiffness $\left(k_y^t\right)$ exerts a force on the wheel flange, limiting its lateral motion. Recently, the DeNOC-based formulation was used to obtain equations of motion for 2-DOF railway wheelset. ¹⁵ Note that, the DeNOC-based approaches are reported to be highly stable, ¹² and computationally efficient. ¹⁶

Figure 3.

Serial-chain architecture of a railway carbody model.

Figure 4.

Wheel-rail contact pair.

Linearization

The lateral translations (y ^wi , y ^bj , y ^c ), and yaw rotations (ψ ^wi , ψ ^bj , ψ ^c ) of a rail vehicle in motion is often very small. Thus, a set of non-linear ODE’s given in equation (6) can be linearized around the equilibrium point. Briefly, the product of system states, i.e, ψ^w1.y^w1, etc., and higher order terms such as ${{\psi }^c}^2$ , ${y^c}^2$ , etc. were considered trivial. This resulted in the GIM becoming a 17 × 17 diagonal mass matrix $\left(\overline{\mathit{M}}\right)$ . Coefficients of the remaining terms were sorted based on system’s positions ( q ), velocities $\left(\dot{\mathit{q}}\right)$ , resulting in the stiffness, and damping matrices, which are denoted using $\overline{\mathit{K}}$ , and $\overline{\mathit{C}}$ respectively. Equation (6) was then re-written in the following form:

$\overline{\mathit{M}}\ddot{\mathit{q}}+\overline{\mathit{C}}\dot{\mathit{q}}+\overline{\mathit{K}}\mathit{q}={\widehat{\mathbf{0}}}_{\mathbf{17}}$

(7)

Complete expressions for $\overline{\mathit{M}}$ , $\overline{\mathit{C}}$ , $\overline{\mathit{K}}$ , and q are given in the appendix.

Lyapunov’s Indirect method for finding SBP

Lyapunov’s Indirect Method (LIM) ⁷ asserts that, below SBP, all eigenvalues of the linear system shown in equation (7), lie in the open left-half of the complex plane. Beyond SBP however, at-least one of the eigenvalues falls into the open right-half complex plane. Following steps are used to find the SBP:

1. Linear second-order ODE’s given in equation (7) were converted to an equivalent first-order form, as given below.

$\dot{\mathit{p}}=\mathit{D}\mathit{p}$

(8)

where $\mathit{p}\equiv {\left[{\mathit{q}}^{\mathit{T}}\hspace{0.17em}{\dot{\mathit{q}}}^{\mathit{T}}\right]}^{\mathit{T}}$ represents the state vector, and D is the 34 × 34 coefficient matrix, which is given as:

$\mathit{D}=\left[\begin{array}{cc}{\mathit{0}}_{\mathit{17}}& {\mathit{1}}_{\mathit{17}}\\ {\overline{\mathit{M}}}^{-\mathit{1}}\overline{\mathit{K}}& {\overline{\mathit{M}}}^{-}\overline{C}\end{array}\right]$

• 2. At every iteration, real part of the eigenvalues of D were obtained by varying the vehicle’s forward velocity V.

• 3. The stability loss of the trivial solution, i.e. SBP is identified as the speed at which any one of the real part of the eigenvalues become positive.

In this work, LIM was used in conjunction with either (a) dichotomy/bisection method, (b) secant method, (c) monte-carlo method, which were commonly adopted in the literature of railway dynamics. A typical flowchart of the bisection is shown in Figure 5.

Figure 5.

Bisection method for calculating SBP.

Implementation

Validation

In order to verify the correctness of the proposed model and analysis, we compare the results to those given in the literature. They are in close agreement, as evident from Table 1.

Table 1.

SBP in kmph.

System	#DOF	Results
		LiteratureRef.	Obtained
Suspended wheelset	2	168.3 9	166.9
Bogie	7	121.3 5	123.2
Carbody	17	440 6	445.2

Time response

For time integration, an initial perturbation ⁸ of 1 × 10⁻³m was applied to the leading wheelset (w1). When the forward velocity of the vehicle is less than the SBP (V = 425 kmph), as illustrated in Figure 6(a), the magnitude of hunting oscillations decay with time. Persistent hunting can be seen in Figure 6(b), when the vehicle is moving at V = 445.2 kmph, but the amplitude does not exceed the original disturbance. Finally, Figure 6(c) shows lateral translation increasing until the wheelset reaches its flange contact limit (See δ in Figure 4). This happens at speeds higher than the SBP (V = 480 kmph).

Figure 6.

Time response of the leading wheelset’s lateral translation (a) Below SBP, (b) At SBP, (c) Above SBP.

Carbody bifurcation analysis

Main contribution of this work is the parametric studies by varying individual vehicle properties, which gives a clear understanding of their effects on the hunting speeds of a carbody. A total of 32 parameters were examined, as mentioned in Table 2. Unlike many studies which focused hugely on suspension properties, here we have examined a wide range of geometric, mass, and inertia properties. Detailed results are explained in the follow up sections.

Table 2.

Parameter table.

Name	Symbol	Unit	Lower bound	Upper bound	# of variations
Wheelset
Conicity	γ wi	rad	0.025	0.15	4
Nominal rolling radius	$r_0^{wi}$	m	0.35	0.5	4
Mass of wheelset	m wi	kg	2000	2750	4
Pitch inertia of wheelset	$I_y^{wi}$	kg.m2	150	600	4
Yaw inertia of wheelset	$I_z^{wi}$	kg.m2	500	2000	4
Bogie
Mass of bogie	m bj	kg	2000	4500	3
Bogie length	$l_x^{bj}$	m	1.5	4	3
Bogie width	$b_y^p$	m	1.5	4	3
Carbody
Mass of carbody	m c	kg	1 × 105	7 × 105	1
Wheelbase of carbody	$l_x^c$	m	10	20	1
Width of carbody	$l_y^c$	m	1	3	1
				Total	32

Results

Geometric properties of wheelset

From Figure 7(a) it is apparent that, with higher wheelset conicities in the leading wheelsets of both bogies i.e., γ^w1, and γ^w3, are reducing the SBP significantly, in comparison with the effect of second wheelset (γ^w2). On the other hand, after a slight drop, the fourth wheelset (γ^w4) shows an increase in the SBP. This phenomenon could be ascribed to the dominant vibration modes of the leading wheelsets. Due to constant wear, the rolling radius of the wheels changes. Variations of SBP due to change in nominal rolling radii of wheels is shown in Figure 7(b). It is clearly evident that the rolling radius of wheelsets in the trailing bogie i.e., $r_0^{w3}$ , and $r_0^{w4}$ are directly proportional to the SBP. The opposite is true for first $\left(r_0^{w1}\right)$ , and second $\left(r_0^{w2}\right)$ wheelsets, where SBP of begin to drop, despite having an initial rise. Therefore, this graph gives a recommendation that if there is any difference in rolling radius between the wheels, the wheelset with a larger radius, if put on the rear bogie could counteract the hunting oscillations and therefore increase the SBP.

Figure 7.

Geometric properties of wheelset versus SBP (a) Conicity, (b) Nominal rolling radius.

Mass and inertia properties of Wheelset

Figure 8 shows the impact of mass of the wheelset on the SBP. A lower value for SBP is observed when the heavier wheelsets are close to the Center of Gravity (C.G) of the carbody i.e., m^w2, and m^w3. With increased mass away from C.G, the net inertial forces resisting the hunting oscillations are high. Thus, SBP increases with increasing mass of first (m^w1), and fourth (m^w4) wheelset. In the case of pitch inertia of wheelset, the SBP values on the leading bogie i.e., $I_y^{w1}$ , and $I_y^{w2}$ , are directly proportional as seen in Figure 9(a), whereas the ones on the trailing bogie ( $I_y^{w3}$ , and $I_y^{w4}$ ) has marginal effects. Cumulatively, it is clear that increasing yaw inertia reduces the SBP, as evident from Figure 9(b). However, it is worth noting that this effect is low for the first wheelset $\left(I_z^{w1}\right)$ , and keeps increasing until the last wheelset $\left(I_z^{w4}\right)$ .

Figure 8.

Mass of wheelset versus SBP.

Figure 9.

Inertia properties of wheelset versus SBP (a) Pitch inertia of wheelset, (b) Yaw inertia of wheelset.

Mass of bogie and carbody

SBP of the carbody system decreases with increasing mass of the leading bogie (m^b1), whereas the opposite is true for the trailing bogie (m^b2). In general, increasing the mass of both bogies, i.e., m^b1 = m^b2 = m ^b , shown by the solid black line in Figure 10(a), there is a consistent drop in the SBP. However, this effect is marginal when compared with the leading bogie. This is depicted in the same figure using a dashed red line. On the other hand, the mass of carbody has non-linear relationship with the SBP. See Figure 10(b). The SBP is observed to drop until a threshold at m ^c = 2 × 10⁵ kg, at which point it begins to grow steadily with increasing values of carbody mass. Qualitatively, a wagon which is rear-heavy shows greater resistance to hunting phenomenon.

Figure 10.

Mass of bogie and carbody versus SBP (a) bogie mass, (b) carbody mass.

Bogie dimensions

The length or wheelbase of the bogie, denoted as $l_x^{bj}$ , is defined as the distance between the suspension mounts of two wheelsets along x axis (see Figure 1). SBP exhibits a radical trend by increasing the length of front, and rear bogies. A combination of longer leading bogie $\left(l_x^{b1}\right)$ , and shorter trailing bogie $\left(l_x^{b2}\right)$ can reduce the SBP. When the length of both bogies was increased concurrently, the SBP tends to decrease, which is depicted in Figure 11(a) using a solid black line. The width of bogie $\left(l_y^{bj}\right)$ is defined here using the lateral distance between the primary suspension mounts. SBP patterns are similar to those of the length of bogie when the widths are independently increased. However, by simultaneously increasing the width of both bogies, SBP increases. This phenomenon is shown in Figure 11(b). Generally, it can be deduced that a larger rear bogie is favorable for running at high speeds.

Figure 11.

Dimensions of bogie versus SBP (a) Bogie length, (b) Bogie width.

Carbody dimensions

The length $\left(l_x^c\right)$ , and width $\left(l_y^c\right)$ of carbody were also varied, and their effects on the SBP are shown in Figure 12(a) and (b), respectively. Increasing the carbody dimensions essentially elongates the moment arm connecting the secondary suspension mounts. As a consequence, larger dimensions result in higher SBP.

Figure 12.

Dimensions of carbody versus SBP (a) Carbody length, (b) Carbody width.

Design recommendations

It is not necessary that the carbody be essentially made non-symmetric in order to take use of these variances. However, whenever variations exist within the system, they can be directed to achieve higher SBP speeds, which could help to improve the hunting behavior of a rail vehicle. In light of this, these suggestions should be considered. Based on the results, recommendations for improved vehicle speed are summarized below:

1. Worn-wheelsets often have high conicities, which makes them start hunting starting at lower speeds. Thus, frequent re-profiling of especially the leading wheelsets is essential, since they provide help increasing the SBP.

2. Larger wheelsets (that is wheels with a higher rolling radius) are preferable for the rear bogie, whereas small wheels can be chosen for the front bogie.

3. Increased rolling radius results in increased pitch inertia of a wheelset. Hence, a trade-off is required to balance the contrasting effects of rolling radius, and the pitch inertia, see Figures 7(b) and 9(a).

4. Heavier wheelsets should be located well away from the carbody C.G, whereas light wheelsets should be close to the carbody C.G, as seen in Figure 8.

5. A straightforward explanation of wheelset’s yaw inertia is possible using Figure 9(b). Small wheels at the front part of the carbody (i.e., small portrait on the XY plane) enhances resistance to hunting oscillation.

6. Lighter bogies are conducive for improving the vehicle speeds. This is very similar to road vehicle dynamics, ¹⁷ where it is demonstrated that lower unsprung mass results in a better ride quality.

7. Due to enhanced resistance to hunting stimulation, heavier carbodies have higher SBP. However, the associated inertial forces may affect the carbody system differently in case of motion on curved track.

8. In general, a lower length-to-width ratio $\left(l_x^b/l_y^b\right)$ is recommended for bogie dimensions, refer Figure 11(a) and (b).

9. Increased length and width of carbody is suggested to achieve higher vehicle speeds. This too relates to road vehicle dynamics, as large vehicles usually tend to exhibit better ride quality than shorter vehicles. ¹⁷

Railhunt - an interactive tool for hunting analysis

For parametric studies, a Graphical User Interface (GUI) called ‘Railhunt’ (see Figure 13) was developed in Matlab’s app designer^®. In Railhunt, the user can create, modify and store different railway carbody models in XML files. Parameters under study can be independently varied by setting lower bound, upper bound, and sampling intervals. Apart from that, one can also select from a catalog of pre-defined solvers and also be able to change solver settings. The post-processor of Railhunt allows users to superimpose multiple result sets. Using this interactive GUI, SBP trends can be understood easily.

Figure 13.

Railhunt - Hunting analysis tool.

Summary and conclusion

A serial-chain multibody model of a railway carbody system having two bogies, and four wheelsets was developed. The governing equations of motion of this system were obtained using the DeNOC matrices. The start of hunting was identified by finding the Smallest Bifurcation Point (SBP) of the trivial solution, and our results were validated with those given in the literature.^5,6,9 SBP trends were then investigated, by altering the attributes of individual components of a carbody system. Unlike many prior publications, which focused extensively on suspension properties, this paper illustrates the effects of geometric, mass, and inertia properties. Using these trends, it was feasible to systematically comprehend the crucial aspects of rail vehicle design that affect hunting characteristics. Accordingly, design proposals to achieve SBP at the highest speed possible, are given. For this purpose, an interactive tool called Railhunt was developed using Matlab’s app designer^®.