Suspension parameters for low-damped carbody oscillation of high-speed railway vehicle

Abstract

This study presents an optimization method to design the suspension properties associated with low-damped carbody oscillations for high-speed railway vehicles. In this method, the least damping ratio for the low-frequency modes in the entire service speed range and the critical speed for a worn wheel are proposed for the design objectives. Based on the linearized vehicle model, a genetic algorithm is applied to determine the optimal suspension properties to maximize the least damping ratio while maintaining the critical speed above the desired speed. The optimization results show that the proposed method can enhance lateral ride comfort by eliminating the region where the least damping ratio of the carbody mode decreases excessively and securing a constant for the entire service speed range. The least damping ratio was improved from approximately 5.7% to 15.6% and the critical for the worn wheel increased from approximately 430 to 499 km/h. Parametric studies are conducted to investigate the influence of the tolerances of the suspension properties, and the results provide useful information regarding the manufacture of suspension elements and assembly of the bogie system. The validity of the optimized suspension properties is verified from the simulation results using railway vehicle dynamics software.

Introduction

Low-damped carbody oscillation, which is caused by the coupled mode between sinusoidal motions of the bogies and sway or yaw motion of the carbody,^1–3 sometimes degrades ride comfort even if the instability limit of a railway vehicle is not exceeded. ² It has been recognized that this phenomenon occurs at relatively low vehicle speeds, and disappears at high speeds beyond the resonance zone ³ ; however, in some countries where high-speed railways are widely operated, it is reported that the deterioration of ride quality owing to the carbody oscillation also occurs in high-speed vehicles even when they are traveling above mid-speeds.^3–5

This carbody oscillation can be suppressed by increasing the damping ratio of the coupled mode, and several measures have been introduced based on modified suspension properties.^3–6 However, these suspension properties affect the lateral stability, safety, curving performance, and ride comfort, and some of their performances are conflicting; they should be carefully designed to balance each performance. As high-speed railway vehicles typically operate on large curves, their curving performance is generally not a priority. Thus, it is more important to balance lateral stability and ride quality on a tangent track.

Optimization techniques have been considered effective in compromising conflicting performances in the design process of suspension properties. He and McPhee ⁷ introduced a two-loop method using a genetic algorithm (GA) and sequential quadratic programing (SQP) with multibody dynamic simulation software to optimize the lateral stability of railway vehicles and investigated the cliff phenomenon, in which the critical speed changes dramatically with a small change in suspension stiffness. They also proposed a methodology for multibody systems using a hybrid multidisciplinary optimization method. ⁸ Nishimura et al. ⁹ presented a design strategy for optimizing the vertical stiffness and damping ratio of primary and secondary suspensions, to minimize the objective function consisting of the sum of the cost functions, such as carbody vertical acceleration, suspension travel, and wheel–rail interaction. Martowicz et al. ¹⁰ suggested a response surface method with a GA to enhance the comfort index by optimizing the stiffness of the primary and secondary suspensions. The mass of passengers was assumed to be an uncertain parameter, and the worst scenario of variation in the mass of passengers was considered as the objective function. Johnsson et al. ¹¹ introduced a Pareto optimization method to enhance the safety and comfort of high-speed railway vehicles. The objective functions of safety and comfort were calculated from the response of a train model, constructed using multibody dynamics software, for lateral irregularities modeled by a stationary stochastic process. The sensitivity of the optimal solution was also investigated for train speed, worn wheel profile, load, and friction coefficients. They expanded their research to three objective problems, including the rate of wear of the wheel and rail expressed as the wear number. ¹² They also suggested some adaptive control strategies based on the analysis of Pareto fronts for the variation in vehicle speed. Bideleh Mousavi et al. ¹³ presented a multistep methodology for improving the computational efficiency of Pareto optimization. Five scenarios with different track radii and vehicle speeds were applied to the multibody dynamics software to calculate the objective functions, and the advantages and disadvantages of the asymmetric suspension system were reported. The robustness of Pareto optimized values was also investigated with respect to the uncertainty of suspension properties owing to the manufacturing tolerance of ±25%. ¹⁴ Yao et al. ¹⁵ proposed optimal suspension properties for hunting stability robustness, defined as running speed robustness, suspension parameter robustness, and equivalent conicity robustness. The multi-objective function comprises the damping ratio of the 25 DOF linearized vehicle model at each robustness index. The K-Means clustering method was adopted to categorize a large number of Pareto fronts into six groups. Based on the analysis of the characteristics of each group and the corresponding Pareto sets, matching rules for the key suspension properties were suggested. They also suggested a method for the centralized optimization of key bogie suspension parameters based on vehicle lateral robust stability using damping ratios at vehicle speeds of 250 and 350 km/h. ¹⁶ Low and high equivalent conicities were applied for each vehicle speed.

Most previous studies applied objective functions with respect to a single or several vehicle speeds to increase ride quality and presented significant results. However, this method is not suitable for the problem of low-damped carbody oscillation because the low-damped carbody oscillation occurs at a certain speed range, and the width of the range and the location in the entire vehicle speed range depend on vehicle characteristics such as suspension properties and wheel–rail contact characteristics. For example, assume that the damping ratio of the carbody modes at a certain vehicle speed is selected as the objective function because the damping ratio has the smallest value at this vehicle speed with current suspension properties; however, once the optimization process begins, the vehicle speed at which the damping ratio has a minimum value is continuously changed because the suspension properties are changed at every iteration of the optimization process. Thus, although the damping ratio at that vehicle speed can be enhanced, a low-damped mode may still exist owing to the new minimum damping ratio at another vehicle speed.

The low-damped carbody oscillation can be effectively improved when the damping ratio of the related mode has an appropriate value for the overall vehicle speed. Thus, in this study, the least damping ratio in the service speed range of a railway vehicle, instead of specifying vehicle speeds, is proposed as an objective function for the low-damped carbody oscillation. This method updates the new least damping ratio with each iteration. When the current least damping ratio is no longer the minimum value, a new one is found and improved from the next iteration. Thus, the suspension properties can be optimized to maintain the damping ratios of the carbody modes above an appropriate value for the entire range of service speeds. The lateral stability of a railway vehicle, which usually refers to the bogie hunting stability, often conflicts with the lateral ride quality. When the vehicle speed is higher than the critical speed, bogie hunting occurs, which can directly lead to the rapid deterioration of vehicle safety indicators and the corresponding risk of railway vehicle operation. ³ Thus, it is very important to minimize the influence on the critical speed in optimizing the suspension properties to suppress low-damped carbody oscillation. Because the critical speed typically decreases under high-equivalent conicity conditions such as worn wheel profiles, the critical speed, particularly for a worn wheel condition, was considered as a design constraint or the second objective.

A linearized model was used to calculate the dynamic characteristics of the vehicle, such as damping ratios and damped natural frequencies, because the proposed method requires a slightly larger amount of calculation. In addition, linear computations are useful for concept investigation, parametric studies, and design optimization, although they cannot represent all the characteristics of a railway vehicle.^2,5 The influences of the manufacturing tolerance of the suspension properties on the damping ratio of the carbody mode and the critical speed for the worn wheel condition were investigated through parametric studies. The effectiveness of the optimized suspension properties was verified from simulation results using railway vehicle dynamics software.

Suspension property design

Vehicle model

A 17 degrees-of-freedom (DOF) linearized model was applied to design the suspension properties and estimate their performance. This model was established in a previous study ⁵ to investigate and solve the low-damped carbody oscillation of the M1 car of the HEMU-430X, a high-speed electric-multiple-unit experimental train in Korea. As shown in Figure 1, the vehicle is composed of a carbody and two bogies, and each bogie has two wheelsets. The ends of each wheelset were assembled with journal boxes that were connected to the bogie frame with the primary suspension consisting of rod bushes ( $k_{\mathrm{rx}}$ , $k_{\mathrm{ry}}$ ), coil springs ( $k_{\mathrm{cx}}$ , $k_{\mathrm{cy}}$ , $k_{\mathrm{cz}}$ ), and vertical dampers ( $c_{\mathrm{pz}}$ ). The secondary suspension consisting of air springs ( $k_{\mathrm{sx}}$ , $k_{\mathrm{sy}}$ , $k_{\mathrm{sz}}$ ), lateral dampers ( $c_{\mathrm{sy}}$ , $k_{\mathrm{dy}}$ ), yaw dampers ( $c_{\mathrm{sx}}$ , $k_d$ ), and vertical dampers ( $c_{\mathrm{sz}}$ ) connected the carbody and the bogie frames. For simplification, one wheelset and two corresponding journal boxes were considered as a single body, and the inertia terms of the journal boxes were added to those of the corresponding wheelset. Each component was allowed to move in the lateral, roll, and yaw directions. Assuming that the vehicle is traveling along a straight track at constant speed, the longitudinal motion of each component is negligible. The vertical and pitch motions can also be ignored because they have little effect on the low-damped carbody oscillation and lateral stability. Thus, the longitudinal suspension parameters affect only the yaw motion; the lateral parameters affect the lateral, roll, and yaw motions; and the vertical parameters affect only the roll motion. The yaw damper of the secondary suspension was assumed to follow the Maxwell damper model; however, other dampers were considered typical viscous dampers because the cutting frequencies of the dampers ¹⁷ were sufficiently high compared with the modal frequencies of each component. The wheel–rail contact forces were calculated based on Kalker’s linear theory.

Figure 1.

M1 car of HEMU-430X; schematic model in (a) rear view, (b) top view, and (c) real feature of front bogie.

As the vehicle model was linearized for small perturbations at equilibrium points, high-order terms were neglected, and the wheelset’s roll motion was constrained by the equivalent conicity and lateral motion of the wheelset. ⁵ Thus, the linearized vehicle model has 17 DOFs (four wheelsets with two DOF, two bogie frames with three DOF, and one carbody with three DOF).

Because the state-space equation of the linearized vehicle model has been studied by various researchers,^4–6,11,12 this study briefly introduces its configuration:

$$\stackrel{.}{\mathbf{x}}=\mathbf{A}\mathbf{x},$$

(1)

$$\mathbf{A}=\left[\begin{array}{cc}0\hfill & \mathbf{I}\hfill \\ \multicolumn{1}{c}{\mathbf{-}{\mathbf{M}}^{\mathbf{-}1}\mathbf{K}}& \mathbf{-}{\mathbf{M}}^{\mathbf{-}1}\mathbf{C}\hfill \end{array}\right],$$

(2)

$$\mathbf{x}={[y_{\mathrm{wj}},y_{\mathrm{bi}},y_v,{\psi }_{\mathrm{wj}},{\psi }_{\mathrm{bi}},{\psi }_v,{\varphi }_{\mathrm{bi}},{\varphi }_v]}^T,$$

(3)

for $j=1,2(i=1);j=3,4(i=2)$ , where $\mathbf{A}$ is the system matrix and $\mathbf{x}$ is the state vector. $y_{\mathrm{wj}}$ , $y_{\mathrm{bi}}$ , and $y_v$ are the lateral displacements of the jth wheelset, the ith bogie frame and the carbody, ${\psi }_{\mathrm{wj}}$ , ${\psi }_{\mathrm{bi}}$ , and ${\psi }_v$ are the yaw displacements of the jth wheelset, the ith bogie frame and the carbody, respectively. ${\varphi }_{\mathrm{bi}}$ and ${\varphi }_v$ are the roll displacements of the ith bogie frame and the carbody. $\mathbf{M}$ is a diagonal matrix of the inertia terms, and $\mathbf{C}$ and $\mathbf{K}$ are the assembled matrices for the damping and stiffness coefficients, respectively.

The damped natural frequency ( $f_d$ ) and damping ratio ( ${\zeta }_d$ ) can be determined from the eigenvalue ( $\sigma$ ) of $\mathbf{A}$ :

$$f_d=\mathrm{Im}(\sigma )/2\pi ,$$

(4)

$${\zeta }_d=-\mathrm{Re}(\sigma )/\left|\sigma \right|.$$

(5)

Table 1 lists the wheel–rail contact properties for calculating the creep forces and moments in the linearized vehicle model. New and worn wheel conditions were acquired from the contact geometries between the new and worn XP55 wheels and the new 60E1 rail with 1/20 inclination. The worn wheel had higher equivalent conicity and creep coefficients than the new wheel.

Table 1.

Wheel–rail contact properties in the linearized model.

Property	New wheel	Worn wheel
Equivalent conicity ( $\lambda$ ) (at 3 mm wheel shift)	0.0580	0.3382
$f_{11}$	1.0518 × 107	1.4420 × 107
$f_{12}$	2.5702 × 104	3.2573 × 104
$f_{22}$	92.2002	692.2468
$f_{33}$	1.1694 × 107	1.8106 × 107

Design variables, objective function, and constraint condition

As the longitudinal stiffness of the primary suspension and the series stiffness of the yaw damper mainly affect the magnitude of the yaw directional stiffness of the bogie, which is important for reducing the carbody oscillation, ⁵ these two suspension properties should be included in the design variables. However, to secure the critical speed for the worn wheel, other suspension properties such as the lateral stiffness of the primary suspension, and damping coefficients of the yaw and lateral dampers can also be considered. First, five suspension properties were selected as design variables, as expressed in equations (6) and (7).

$${\mathbf{d}}_1=\{k_{\mathrm{rx}},k_{\mathrm{ry}},k_d,c_{\mathrm{sx}},c_{\mathrm{sy}}\},$$

(6)

$${\mathbf{L}}_{b1}\le {\mathbf{d}}_1\le {\mathbf{U}}_{b1},$$

(7)

where ${\mathbf{L}}_{b1}$ and ${\mathbf{U}}_{b1}$ represent the lower and upper limits of the design variables, respectively. For the five design variables, the boundary conditions were selected within the manufacturable ranges but were somewhat wider. However, as reported in the next section, the critical speed for the worn wheel condition deteriorated significantly when the series stiffness predicted by the damper manufacturer considering the designed damping coefficient was adopted for the lateral damper in the simulation, and its cutting frequency was not sufficiently higher than the lateral modal frequency of the bogie. Thus, to obtain more feasible results, the lateral damper was further modeled as a Maxwell damper, its series stiffness was added to the design variables, and the widths of each boundary condition were adjusted more practically. Equations (8) and (9) present the second set of design variables and boundary conditions, respectively.

$${\mathbf{d}}_2=\{k_{\mathrm{rx}},k_{\mathrm{ry}},k_d,k_{\mathrm{dy}},c_{\mathrm{sx}},c_{\mathrm{sy}}\},$$

(8)

$${\mathbf{L}}_{b2}\le {\mathbf{d}}_2\le {\mathbf{U}}_{b2},$$

(9)

The objective function is defined as the least damping ratio of the carbody mode over the entire service speed range and is expressed as follows:

$${\zeta }_m=\underset{v_1\le v\le v_2}{min}\left(\underset{N}{min}{{{\zeta }_d}_N(v,\mathbf{d})|}_{{f_d}_N\le f_c}\right)$$

(10)

where ${{\zeta }_d}_N(v,\mathbf{d})$ is the damping ratio of the Nth mode at vehicle speed $v$ , with the suspension parameters, d, and the damped natural frequency ${f_d}_N$ , which is lower than $f_c$ . $f_c$ was used to separate the carbody oscillation from bogie hunting, which was determined to be 3 Hz considering the kinematic hunting frequency of the bogie. ⁵ $v_1$ and $v_2$ are the lower and upper bounds of the service speed, respectively.

A constraint condition based on the critical speed was applied to ensure the lateral stability for the worn wheel condition. In this study, the critical speed $v_c$ is defined as the vehicle speed at which any mode with a frequency higher than $f_c$ has a damping ratio of zero. The constraint condition can be expressed as an inequality, $v_c\ge v_d$ with the worn wheel, where $v_d$ denotes the desired critical speed.

Optimization and results

As shown in equations (1)–(5) and (10), because multiple calculation steps are required to obtain the values of the objective and constraint functions, it is likely that the two functions are discontinuous or have several local minima for the design variables. Thus, stochastic optimization techniques are more advantageous than gradient (numerical or analytical)-based techniques for optimizing the proposed objective function. In this study, a GA, which is a widely used stochastic algorithm, was applied. It is known to be highly reliable for global optimization and particularly useful for complicated problems.^7,8 However, because the results in real implementations often depend on the option settings, ¹³ two types of optimization problems were solved as summarized in Table 2 and their validity was verified through comparison with each other. SO and MO in Table 2 denote constrained single-objective and unconstrained multi-objective problems, respectively. Thus, optimization was performed for all four cases.

Table 2.

Optimization problems.

Type	Description	Design variables and boundary conditions $(\times {10}^6)$
		CASE I	CASE II
SO	$\underset{{\mathbf{L}}_b\le \mathbf{d}\le {\mathbf{U}}_b}{max}{\zeta }_m\left(\mathbf{d}\right)$ $\mathrm{s}.\mathrm{t}.v_c\left(\mathbf{d}\right)\ge v_d$ Pop. size: 100 Fun. tol.: $1\times {10}^{-6}$ Stall gen.: 20	$\mathbf{d}={\mathbf{d}}_1$ ${\mathbf{L}}_{\mathbf{b}}={\mathbf{L}}_{\mathbf{b}1},{\mathbf{U}}_{\mathbf{b}}={\mathbf{U}}_{\mathbf{b}1}$ ${\mathbf{L}}_{b1}:\{6,2,4,0.3,0.01,\}$ ${\mathbf{U}}_{b1}:\{35,8,20,3.2,0.06\}$	$\mathbf{d}={\mathbf{d}}_2$ ${\mathbf{L}}_{\mathbf{b}}={\mathbf{L}}_{\mathbf{b}2},{\mathbf{U}}_{\mathbf{b}}={\mathbf{U}}_{\mathbf{b}2}$ ${\mathbf{L}}_{b2}:\{25,3,12,1,0.9,0.01\}$ ${\mathbf{U}}_{b2}:\{33,6,20,4,2.8,0.08\}$
MO	$\underset{{\mathbf{L}}_b\le \mathbf{d}\le {\mathbf{U}}_b}{max}\{{\zeta }_m\left(\mathbf{d}\right),v_c(\mathbf{d})\}$ Pop. size: 200 Fun. tol.: $1\times {10}^{-4}$ Stall gen.: 50

Figure 2 shows the optimization results for CASE I when $v_1$ , $v_2$ , and $v_d$ were 36, 370, and 450 km/h, respectively. To compare the SO and MO results efficiently, the yaw directional stiffness (Yt) and ratio of yaw directional stiffness (Yr) of the secondary and primary suspensions were introduced. Assuming that the cutting frequency of the yaw damper is sufficiently low and that the yaw damper behaves as a stiffness element, Yt and Yr can be expressed as:

$$\begin{array}{c}{\mathrm{Y}}_{\mathrm{t}}=4k_{\mathrm{ry}}{(l_{\mathrm{wb}}-l_{\mathrm{kx}})}^2+4k_{\mathrm{cy}}{l}_{\mathrm{wb}}^2\hfill \\ +4(k_{\mathrm{rx}}+k_{\mathrm{cx}})l_{\mathrm{wa}}^2+2k_d{l}_{\mathrm{bcsxy}}^2+2k_{\mathrm{sx}}{l}_{\mathrm{ba}}^2\hfill \end{array}$$

(11)

$${\mathrm{Y}}_{\mathrm{r}}=\frac{4k_{\mathrm{ry}}{(l_{\mathrm{wb}}-l_{\mathrm{kx}})}^2+4k_{\mathrm{cy}}{l}_{\mathrm{wb}}^2+4(k_{\mathrm{rx}}+k_{\mathrm{cx}})l_{\mathrm{wa}}^2}{2k_d{l}_{\mathrm{bcsxy}}^2+2k_{\mathrm{sx}}{l}_{\mathrm{ba}}^2}.$$

(12)

Figure 2.

Optimization results for five design variables: (a) optimal result of SO and Pareto fronts of MO (least damping ratio: ○ and critical speed for the worn wheel: •) and (b and c) optimal design variables of SO and Pareto set of MO.

As shown in Figure 2(a), the least damping ratio of the carbody mode (○) and critical speed for the worn wheel have a contradictory relationship with the bogie’s yaw directional stiffness, and the results of SO (Δ) lie on the Pareto fronts of MO despite the different GA options of population size, function tolerance, and stall generation. This implies that it is significantly difficult to obtain better optimized results than those of the SO under the given boundary and constraint conditions. The least damping ratio of the carbody mode and critical speed for the worn wheel were estimated to be approximately 19.5% and 460 km/h, respectively. Figure 2(b) presents the results of the design variables of the SO and the Pareto sets of the MO, which are expressed using Yt and Yr. The stiffness of the SO was optimized to place its Yr at the minimum Yr value of the MO, which means that the optimization was performed to reduce and increase the stiffness of the primary and secondary suspensions, respectively. Figure 2(c) shows the ratio of Yt to $c_{\mathrm{sx}}$ according to the change in Yt. The results of the SO show that $c_{\mathrm{sx}}$ moves to the upper bound at a given Yt value, indicating that $c_{\mathrm{sx}}$ should have as high a value as possible. The damping coefficient of the lateral damper $c_{\mathrm{sy}}$ , was designed to be approximately $4.7\times {10}^4$  N s/m.

Figure 3(a) shows the optimized results for CASE II. In the results of SO, the least damping ratio of the carbody mode was approximately 15.6% lower than that of the CASE I, whereas the critical speed for the worn wheel was approximately 499 km/h higher than that of CASE I, despite having the same $v_d$ . As in CASE I, the least damping ratio of the carbody mode and the critical speed for the worn wheel show a conflicting relationship with the yaw directional stiffness of the bogie. However, $k_{\mathrm{rx}}$ , and $k_d$ that moved to their lower bounds in the optimization process caused the SO to lie on the left end of the Pareto fronts of the MO, which implies that better SO results can exist below their lower bounds. As shown in Figure 3(b), the Yr of SO is close to the maximum Yr of MO, because the Yt of SO is located on the left end of the Yt of MO. In Figure 3(c), the ratio between Yt and $c_{\mathrm{sx}}$ of the MO is almost the opposite of the least damping ratio of the carbody mode. It gradually increased along Yt and grew rapidly when Yt was approximately more than $1.97\times {10}^8$ N·m/rad while the least damping ratio of the carbody mode gradually decreased along Yt and reduced rapidly after Yt of approximately $1.97\times {10}^8$  N m/rad. The ratio between Lt and $c_{\mathrm{sy}}$ in Figure 3(d) gradually decreased along Lt and suddenly decreased when Lt was approximately $3.67\times {10}^7$  N/m. Here, Lt, the lateral stiffness of the bogie, is calculated using equation (13).

$${\mathrm{L}}_{\mathrm{t}}=4(k_{\mathrm{cy}}+k_{\mathrm{ry}})+2(k_{\mathrm{sy}}+k_{\mathrm{dy}})$$

(13)

Figure 3.

Optimization results for six design variables: (a) optimal result of SO and Pareto fronts of MO (the least damping ratio: ○ and critical speed for the worn wheel: •) and (b–d) optimal design variables of SO and Pareto set of MO.

Figure 4 shows the least damping ratios and modal frequencies of the M1 car based on the application of the optimized suspension properties. In Figure 4(a), there is no vehicle speed at which the least damping ratio of the carbody mode decreases significantly. In CASE II, the least damping ratio decreased slightly between approximately 160 and 380 km/h, and the modal frequency increased with increasing vehicle speed, which was caused by the remaining coupled mode between the sinusoidal motion of the bogies and the upper sway of the carbody. However, this oscillation mode is expected to be effectively restricted owing to its sufficient damping ratio. CASE I had an almost constant least damping ratio regardless of the vehicle speed because its dominant mode was the lateral motion of the carbody, as shown in the modal frequency in Figure 4(a). The least damping ratio in the bogie mode for the new wheel is shown in Figure 4(b). As the least damping ratio of the bogie mode for each case exceeded 5% even at a vehicle speed of 540 km/h, the critical speed for the new wheel was estimated to be sufficiently secured in all cases. Figure 4(c) shows the least damping ratio of the carbody mode for the worn wheel condition. Over the entire speed range, CASEs I and II exhibited constant damping ratios, similar to those of the new wheel. However, the least damping ratio of the carbody mode for the original suspension remarkably increased and had a value similar to that of CASE I. The modal frequency for the worn wheel implies that the change in the wheel–rail contact parameters changes the dominant mode from the coupled motion between the bogies and the carbody to the lateral motion of the carbody. Figure 4(d) shows the least damping ratio of the bogie mode for the worn wheel. The critical speeds were approximately 430, 460, 499 km/h for the original suspension, CASE I and CASE II, respectively. However, based on the least damping ratio of 5%, the critical speed had the lowest value in CASE I, approximately 270 km/h, followed by CASE II, 355 km/h and the highest value, 375 km/h in the original suspension properties.

Figure 4.

Least damping ratio and modal frequency of M1 car along the vehicle speed: (a) carbody and (b) bogie modes for the new wheel condition, (c) carbody and (d) bogie modes for the worn wheel condition.

Influences of suspension property tolerances

To apply the optimized properties to manufacturing suspension elements, it is important to investigate the influence of manufacturing tolerances on the design results and constraint conditions, particularly when they have contradictory relationships with each other. Generally, in the manufacturing or maintenance process of the bogies, suspension elements with similar values for the same type are assigned to each bogie. For example, the stiffness of rod bushes in one bogie is similar to each other and the characteristics of the yaw dampers in one bogie are similar to each other, but slight differences from those in other bogies are acceptable if they are all within the allowable tolerances. The variation in the least damping ratio according to the changes in Yt, Yr, and Lt indicates that another guide is required to apply the suspension elements to the bogie, although they are produced within their allowable tolerances.

Figure 5 shows the variation in the least damping ratio of the carbody mode for CASE I, and it was assumed that the changes in Yt, Yr, and Lt only affect the parameters of rod bushes, yaw and lateral dampers, and the corresponding parameters can be calculated by combining equations (11)–(13). The result of SO is located in the high damping region close to O.B., in which the critical speed for the worn wheel does not satisfy the desired speed. In Figure 5(a), the grid with solid lines represents the area where the least damping ratio can be changed by ±20% tolerances of $k_{\mathrm{rx}}$ , $k_{\mathrm{ry}}$ , and $k_d$ . Here, $k_{\mathrm{ry}}$ is assumed to change at the same rate as $k_{\mathrm{rx}}$ , as they are usually given by a rod bush. This area shows that the allowable tolerances of rod bushes ( $k_{\mathrm{rx}}$ , $k_{\mathrm{ry}}$ ) and the series stiffness of the yaw damper ( $k_d)$ should be positive, unlikely the commonly used tolerances ¹⁴ for suspension elements of railway vehicles. They should also be managed in conjunction with each other such that Yt and Yr values are within the bounds. For example, if the rod bushes with positive tolerance of 10% are used for the primary suspensions, yaw dampers with a series stiffness of positive tolerance, not nominal value, should be selected. In the maintenance, the Yt and Yr values calculated from the rod bushes and yaw dampers used should be checked to ensure that they are in the bound. When the values are inside the bound and acceptable, the combination of the rod bushes and yaw dampers used to calculate the Yt and Yr can be used again although properties of each of them are changed.

Figure 5.

Variation in the least damping ratio of the carbody mode for CASE I according to: (a) Yt and Yr, and (b) Yt and $c_{\mathrm{sx}}$ . A solid marker ○ denotes the results of SO.

Figure 5(b) shows the influence of the damping coefficient $c_{\mathrm{sx}}$ . The tolerance of $c_{\mathrm{sx}}$ does not have a significant effect on the least damping ratio of the carbody mode and critical speed for the worn wheel condition. Thus, in the maintenance process, the yaw damper does not need to be replaced or repaired unless its damping coefficient is significantly changed or its changed series stiffness can not be compensated by the stiffness of the rod bushes.

The variation in the least damping ratio of the carbody mode for CASE II exhibits characteristics similar to those of CASE I, as shown in Figure 6(a) and (b). However, it is advantageous to control the tolerances more strictly because the carbody oscillation may not be effectively reduced if the least damping ratio of the carbody mode, which is lower than that of CASE I, decreases further owing to the tolerances. As shown in Figure 6(c), the tolerance of the damping coefficient of the lateral damper $c_{\mathrm{sy}}$ , does not affect the critical speed for the worn wheel, but has some effects on the least damping ratio of the carbody. It has a maximum value at the nominal damping coefficient designed by the SO, and gradually decreases as the tolerance increases. The series stiffness of the lateral damper $k_{\mathrm{dy}}$ , it does not influence the least damping ratio of the carbody mode, but reducing $k_{\mathrm{dy}}$ by negative tolerance can significantly decrease the critical speed for the worn wheel. For example, the critical speed for the worn wheel condition decreases to the boundary of O.B. when $k_{\mathrm{dy}}$ is 80% of its designed value. This shows why the critical speed for the worn wheel dramatically decreased when the series stiffness estimated by the damper manufacturer was adopted for the lateral damper in the simulation for CASE I, which implies that the lower limit for the manufacturing tolerance of the series stiffness of the lateral damper should be tighter than the commonly used tolerance. In the maintenance, a lateral damper with a decreased series stiffness should be replaced or repaired, regardless of the damping coefficient.

Figure 6.

Variation in the least damping ratio of the carbody mode for CASE II according to: (a) Yt and Yr, (b) Yt and $c_{\mathrm{sx}}$ , and (c) Lt and $c_{\mathrm{sy}}$ . A solid marker ○ denotes the results of SO.

Simulations

Several simulations were performed using VI-Rail, a commercial software for railway vehicle dynamics, to verify the proposed design. The critical speeds for the worn wheel were evaluated using a vehicle model, and the ride comfort indices for the new wheel were analyzed using a train model consisting of four vehicles. The wheel profiles used in the simulation have the same characteristics as those listed in Table 1. Additionally, the worn wheel has the equivalent conicity that decreases with lateral shift of the wheel, and its nonlinear parameter ¹⁸ is −0.0223.

Figure 7(a) shows the VI-Rail vehicle model, and Figure 7(b) to (e) show the responses of the first wheelset of each vehicle model for a ramp function of the lateral track irregularity. The critical speed for the original suspension was estimated to be 400–420 km/h. As stated earlier, for CASE I, the critical speed decreased remarkably when series stiffness was applied to the lateral damper. However, when only the damping coefficient is adopted for the lateral damper, the critical speed is expected to range between 440 and 460 km/h. In the simulation, a sufficiently high series stiffness yielded almost the same critical speed; however, it was not within the manufacturable range while maintaining the designed damping coefficient. As predicted from the least damping ratio of the bogie, the critical speed for CASE II was higher than that of the other conditions and was in the range of 490–510 km/h. Interestingly, the simulations present critical speeds that are very similar to those of the linearized vehicle model, which is presumed to be due to the declining equivalent conicity for the lateral movement of the wheel. Supercritical Hopf bifurcation often occurs under this wheel–rail contact condition. ¹⁸

Figure 7.

Simulation results for the critical speed under the worn wheel condition: (a) M1 car model built with VI-Rail software, (b) original suspension, (c) CASE I with the series stiffness of the lateral damper, (d) CASE I without the series stiffness of the lateral damper, and (e) CASE II.

The lateral ride comfort was calculated using the lateral accelerations of the carbody measured on the floor in accordance with ISO 2631 when the train model traveled on a straight line of approximately 10 km with ERRI low rail irregularities. Figure 8(a) shows the measurement points of the lateral accelerations, and Figure 8(b) to (d) show the lateral ride comfort indices for each train model. Each shape of the marks, “□,”“⋄,”“Δ,” and “◯” indicates the train speed of 200, 250, 300, and 350 km/h, respectively. In addition, the color of each mark indicates the travel direction of the train model: blue indicates the left (Mc is a leading car), and red indicates the right (Tc is a leading car). The results show that ride comfort is not exactly proportional to the least damping ratio because of the difference between the frequency– and time–domain responses, some nonlinear characteristics applied to the VI-Rail vehicle model, and the difference in the measuring points. However, this was significantly improved owing to optimized suspension properties. The lowest index (best ride comfort) for the original suspension was 0.0523 at p1 of M4 when the train model traveled left at a speed of 200 km/h, and the highest index (worst ride comfort) was 0.2100 at p2 of Tc when the travel direction was left at a train speed of 300 km/h. For CASE I, p2 of Tc showed the lowest index (0.0272) in the travel direction of the right at a speed of 200 km/h, and p2 of M1 had the highest index (0.1070), when the train model traveled left at a speed of 350 km/h. For CASE II, the lowest index, 0.0289, appeared at p2 of Tc when the travel direction was on the right and the train speed was 200 km/h; the highest index was 0.1286 at p2 of M1 when the train model traveled left at a speed of 350 km/h. Thus, CASEs I and II improved the ride comfort by 48% and 45% for the lowest index and 49% and 39% for the highest index, respectively. Comparing the ride comfort of the M1, CASE I improved by 43% and 45% and CASE II improved by 37% and 34%, for the lowest and highest indices, respectively.

Figure 8.

Simulation results for the ride comfort (ISO 2631) in lateral direction: (a) measurement points, (b) original suspension, (c) CASE I, and (d) CASE II. Marks, “□,”“⋄,”“Δ,” and “◯,” represent the train speed of 200, 250, 300, and 350 km/h, respectively. Blue and red colors denote the travel to the left and right direction, respectively.

The simulation results for critical speed and ride comfort indicated that the optimized suspension properties for CASE II were more suitable for actual implementation, although the ride comfort was slightly lower than that of CASE I. In CASE II, all the designed suspension properties were within the manufacturing range.

Conclusions

Low-damped carbody oscillations have been an interesting research area and challenge to be solved for railway vehicle dynamics engineers. In a previous study, the cause of carbody oscillation of the HEMU-430X was investigated, modification of some suspension properties was suggested, and its effectiveness was validated in a field test. ⁵ However, it had a limit that the influence on the lateral stability for high equivalent conicity owing to wheel–rail wear was not sufficiently considered. Hence, in this study, an optimization technique for the suspension properties was proposed to suppress the lateral carbody oscillation while maintaining or enhancing the lateral stability for the worn wheel.

The optimization results indicate that the least damping ratio defined in equation (10) can be an effective design target to suppress the low-damped carbody oscillation by eliminating the region where the damping ratio of the carbody mode decreases excessively and securing a constant least damping ratio for the entire service speed range, as shown in Figure 4(a). The optimization was performed with five design variables (CASE I) and six variables (CASE II). In CASE II, the series stiffness of the lateral damper was added because the lateral damper could not be manufactured with the designed damping coefficient of CASE I unless its series stiffness was lower than the current value. The lower series stiffness caused a deterioration of the critical speed for the worn wheel condition in the simulation. The series stiffness of the lateral damper plays an important role in securing the critical speed, because the lateral motion becomes the dominant mode of the bogie as the vehicle speed increases. However, its effects on the carbody mode, which were not described in this study, were found to be small.

The total stiffness of the bogie (Yt, Lt) and stiffness ratio (Yr) between the primary and secondary suspensions were introduced to efficiently compare the results of the constrained single-objective and unconstrained multi-objective optimizations and investigate the influence of the tolerances. The diagrams of Yt, Yr, and Lt can provide useful information when determining whether the manufactured suspension elements are applicable or when selecting the proper set of suspension elements for each bogie. To ensure the desired critical speed, it is important to construct a set of suspension elements such that Yt and Lt are increased but Yr is maintained or decreased. The change in the lateral and yaw damping coefficients affected the damping ratio of the carbody mode; however, it had a relatively small effect on the critical speed.

In the simulation, the optimized suspension properties exhibited the expected performance for lateral ride comfort and critical speed for the worn wheel. The ride comfort was improved by 48%–49% in CASE I and 39%–45% in CASE II. However, the critical speed for the worn wheel condition was approximately 370 km/h for CASE I. It can be increased to the desired speed by increasing the series stiffness of the lateral damper. However, this value is outside the manufacturable range. In CASE II, the critical speed was approximately 500 km/h.

Finally, the results of the optimization and simulation show that the optimized suspension properties achieved a significant improvement in the ride comfort and lateral stability for the worn wheel condition, and that the designed properties in CASE II can be a good suggestion for high-speed railway vehicles. For actual implementation, the manufacturing tolerances of the rod bushes and the series stiffness of the yaw and lateral dampers should be positive and managed more tightly than their commonly used tolerances in railway vehicle engineering. In a future study, a prototype of a high-speed bogie with optimized suspension properties will be constructed, and its performance will be evaluated using a roller–rig test stand.