# Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Establishment and Solution of the Motion Equation

#### 2.1. Motion Equation of Train System Based on MCM

- (1)
- The interaction is nonexistent among vehicle elements.
- (2)
- Each vehicle element is composed of a rigid car-body, bogies, and wheel sets.
- (3)
- The springs of a train are linearly elastic, while the dampers are linearly viscous.
- (4)
- Each vehicle element is a linear stationary system; namely, the mass, damping, and stiffness matrices of train are constant.
- (5)
- The train moves at a constant speed.
- (6)
- The wheels and rails fit snugly. The relative displacement between the track and the bridge deck, as well as the elastic effect of the track–bridge system, are neglected.
- (7)
- The track irregularity is a zero-mean stationary Gaussian random process.

**M**

_{t},

**C**

_{t}, and

**K**

_{t}are the mass, the damping, and the stiffness matrix, respectively.

**T**

_{s}and

**T**

_{d}are the projector matrices for the spring and the damper, respectively. All of them can be diagonally assembled from

**M**

_{ve},

**C**

_{ve},

**K**

_{ve},

**T**

_{sve}, and

**T**

_{dve}of the vehicle elements, as given in Equation (2), where

**u**

_{t}is the vibratory displacement vector containing the DOFs of the car body and the bogies in Table 1, while

**u**

_{w}is the vibratory displacement vector containing the DOFs of the wheel-sets in Table 1.

**K**

_{ve}in Equation (3) contains the auto-coupling stiffness matrices of the car body and bogies, namely

**K**

_{cc}and

**K**

_{bb}expressed in Equations (5) and (6), as well as the cross-coupling stiffness matrix

**K**

_{bc}(η), which represents the car–bogie interaction and differentiates the front and rear bogies using η = 1 and −1, respectively.

**M**

_{ve}is diagonally assembled using the mass of different components and their moments of inertia around different axes.

**T**

_{sve}in Equation (8) is a stiffness projector matrix multiplied by the displacement vector

**u**

_{w}of wheel–track contact points to form the force vector of the train. The expression of

**C**

_{ve}and

**T**

_{sve}, respectively similar to Equations (3) and (8), can be obtained simply by replacing the spring factor k with the corresponding damping factor c.

**u**

_{w}. Therefore,

**u**

_{w}can be expressed in Equation (9) with the delay vector

**t**

_{d}, which reflects the phase difference among the wheel sets in Equation (10). In addition, the symbol “∘” in Equation (9) represents the Hadamard product, also known as the element-wise product between two matrices.

_{v}is the total number of vehicle elements;

**S**

_{ir}is the given vertical track irregularity PSD matrix in Equation (12) composed of the alignment, vertical, and cross-level PSDs; n

_{k}is an angular wave number sample; dn is the bandwidth; and θ

_{k}is a random phase angle obeying the uniform distribution U(0,2π). The integer order of k ranges from 1 to N.

#### 2.2. Motion Equation of Train System Based on the PEM

_{a}, the vertical S

_{v}, and the cross-level S

_{c}. This is because the PEM does not allow the linear superposition of different pseudo inputs defined by mutually independent PSDs, and only their PSDs obey the linear superposition principle. Meanwhile, the pseudo velocity $\dot{\tilde{U}}\mathrm{t}$ and acceleration $\ddot{\tilde{U}}\mathrm{t}$ vibratory response vectors can be expressed as Equations (14) and (15). The pseudo displacement of wheel sets can be defined in Equation (16) by converting the angular wave number domain to the angular frequency domain; it contains three types of pseudo track irregularity motions, similar to the definition of Equation (13):

## 3. Sperling Index Based on Vibratory Acceleration History and PSD

#### 3.1. Zonal Distribution Deduction of Sperling Index Based on Vibratory Acceleration History

**A**

_{k}(unit: m/s

^{2}) represents the amplitude vector of the vibration acceleration at the frequency point f

_{k}(unit: Hz) after Fourier transform,

**F**(f

_{k}) represents the frequency correction coefficient vector whose vertical and horizontal elements are sectionally expressed in Equations (23) and (24), and the number of frequency N is strongly related to the interval limit of frequency and the measurement duration. In accordance with the code GB/T5599-2019, the standard measurement duration is 5 s, the reciprocal of which is the bandwidth frequency df of the fast Fourier transform.

#### 3.2. Zonal Distribution Deduction of Sperling Index Based on Vibratory Acceleration PSD

## 4. Case Study

#### 4.1. Parameters of Track Irregularity and Train

_{c}and n

_{s}with units rad/m) and roughness constants (A

_{a}and A

_{v}with units m/rad). The expressions of the PSDs for the alignment, vertical, and cross-level PSDs are given in Equation (34) through (36) and are shown in Figure 3. The PSDs were divided into Grades 1–6, and the corresponding parameters from Equations (34)–(36) are listed in Table 2.

#### 4.2. Demonstration of the Methodological Correctness

_{j}represents a particular random time sample of the family of random variables

**V**. For the stationary Gaussian random process with a zero mean, the variance can be expressed as Equation (38). By combining Equation (25) with the time lag τ = 0, the autocorrelation of the signal is equal to its variance.

- In Figure 4a,c, the expected value μ and the limits of the 3σ normal distribution of the vibration response of the Car-Body center using MCM–NβM and PEM-FRF, respectively, are in substantial agreement. The response patterns are also well within the bounds of the 3σ principle for normal distributions. Compared to MCM, which required multiple solutions, the PEM was much more accurate and efficient.
- In Figure 4b,d, the preliminary probability density curve calculated using MCM-NβM agrees with the theoretical normal distribution calculated using PEM-FRF.
- In general, the train vibration acceleration under track irregularities obeyed the stationary zero-mean normal distribution, the statistical properties of which can be described by the 3σ-principle, and which is consistent with the original assumption regarding the track irregularities as a stationary zero-mean Gaussian random process. Therefore, it was demonstrated that the methodology was correct and suitable for the analysis outlined in the following sections.

#### 4.3. Analysis of the Car-Body Center Comfort in Different Vehicle Elements

- The variations in the theoretical Sperling index of the car-body center according to the PEM between the different vehicle elements were insignificant for both the horizontal and vertical components. The theoretical Sperling index for the horizontal component was obviously higher than that for the vertical component;
- In all sequences of vehicle elements, the difference between the upper and lower provisional range limits of the Sperling index sampling distribution according to MCM did not exceed 0.12 for both the horizontal and vertical components;
- In general, the discomfort was strongly related to the horizontal vibration acceleration of the car body. In addition, the Sperling index was a stationary indicator of comfort that was independent of the order of vehicle elements when the train was subjected to track irregularities. In this way, the comfort of the center of the car body could be characterized by simply selecting a vehicle element to analyze.

#### 4.4. Analysis of the Zonal Distribution Characteristics of Train Comfort

- The mean network of the zonal distribution of the Sperling index calculated using MCM–NβM was almost identical to the theoretical network calculated using PEM-FRF for both the horizontal and vertical components;
- In Figure 6a, the zonal distribution of the horizontal Sperling index is symmetric with respect to the pitch and roll axis of the car-body center. It has the cylindrical shape of the letter “V” and reaches the highest line at the rear and front edges of the car-body, respectively, while it reaches the lowest line on the pitch axis of the car-body;
- In Figure 6b, the zonal distribution of the vertical Sperling index is a symmetrically curved surface with respect to the roll axis of the Car-Body center. It reaches the highest points at the four vertices of the X−Y plane, where the center of the car body is located, while the lowest point is slightly in front of the center of the car body in the direction of travel;
- Compared to Figure 5 in Section 4.3, the Sperling index of the center of the car body is smaller for both the horizontal and vertical components for Grade-6 track irregularities than for Grade-4 track irregularities;
- In general, a small track irregularity had a negative effect on train comfort, and the zonal distribution of train comfort was not strictly symmetrical with respect to the center of the car body. The most comfortable area for the vertical component was near the front of the center of the car body, while the most comfortable area for the horizontal component was on the pitching axis of the center of the car body. The realistic evaluation of train comfort could be roughly characterized by the average value of the Sperling index during train operation, while the theoretical design of train comfort could be accurately determined by the PEM.

#### 4.5. Analysis of Influence of the Quality of Track Irregularity on the Zonal Distribution Characteristics of Train Comfort

- The symmetrical characteristic of the zonal distribution of the car-body Sperling index of the car body for the horizontal and vertical components was identical to that in Section 4.4;
- With the deterioration in the quality of the track irregularities, the values of the Sperling index for the entire network increased significantly for both the horizontal and vertical components;
- In general, train comfort deteriorated with the deterioration of the quality of track irregularities, so regular track maintenance is of great importance.

#### 4.6. Analysis of Influence of the Train Speed on the Zonal Distribution Characteristics of Train Comfort

- The symmetrical characteristic of the zonal distribution of the car-body Sperling index of the car body was identical to that shown in Section 4.4 for both the horizontal and vertical components;
- As the train accelerated, the values of the car-body Sperling index for the entire network increased significantly for both the horizontal and vertical components;
- In general, the train comfort deteriorated when the train traveled too fast. This was because the amplitude of the vibration velocity and the acceleration of the track irregularity contained the linear and quadratic terms of the train speed V, respectively. When the train accelerated, the amplitudes increased rapidly, which increased the input excitation and led to a significant increase in the vibration response of the car body. Therefore, appropriate control of a train’s speed can help to improve passenger comfort.

#### 4.7. Analysis of Influence of the Car-Body Mass on the Zonal Distribution Characteristics of Train Comfort

- The symmetrical characteristics of the zonal distribution of the car body’s Sperling index for the horizontal and vertical components were identical to those shown in Section 4.4;
- With the additional mass of the car body, the values of the Sperling index for the entire mesh decreased significantly for both the horizontal and vertical components;
- In general, the comfort of the train deteriorated with the loss in the car-body mass. Therefore, it is important to reasonably distribute the number of passengers during transfer and optimize the original mass design of the car body.

#### 4.8. Analysis of Influence of the Damage of Secondary Suspension System on the Zonal Distribution Characteristics of Train Comfort

_{fk}and d

_{fc}, respectively. Therefore, the corresponding stiffness submatrix of the vehicle element including Equations (3) and (5)–(7) was updated in Equations (41)–(44).

_{fk}with the damping coefficient c and the damper damaged factor d

_{fc}. The top row of the submatrix in the following equations means that damage was present.

_{fk}= 1 and d

_{fc}= 1) and the damaged secondary suspension system (d

_{fk}= 10 and d

_{fc}= 0.1) on the zonal distribution of the car body’s Sperling index, as shown in Figure 10. In Figure 10, it can be seen that:

- In Figure 10a, the symmetrical characteristic of the network of the horizontal Sperling index after damage to a vertical spring damper in the secondary suspension system was still consistent with the initial condition, but the overall magnitude of the horizontal Sperling index was much higher than the initial condition. This was because the damaged vertical spring damper affected the lateral car-body sway by influencing the car-body roll in conjunction with the vertical car-body sway;
- It can be seen in Figure 10b that after the damage to the vertical spring damper in the secondary suspension system, the network of the vertical Sperling index could not maintain the uniformly curved surface as in the initial state, and reached the highest values when the damaged spring damper was located and gradually decreased toward the areas where the other healthy vertical spring dampers were located. The overall magnitude of the vertical Sperling index far exceeded that of the initial condition;
- In general, the damage to the secondary suspension system led to an overall deterioration in the train comfort for both the horizontal and vertical components, even if only one local vertical spring damper in the secondary suspension system was damaged. The comfort of the train suffered more on the side where the damage occurred. The train comfort for the horizontal and vertical components were not independent of each other. Therefore, it is important to consider both components comprehensively when optimizing the train comfort.

## 5. Conclusions

- Compared with the Monte Carlo method and depending on a large amount of samples, the pseudo-excitation Method, based on the linearity of the power spectrum density of the car-body vibratory acceleration, was more efficient to derive the accurate zonal distribution of the Sperling index.
- The realistic evaluation of train comfort can be roughly characterized by the average value of the Sperling index during train operation, while the theoretical design of train comfort can be accurately determined using the pseudo–excitation method.
- The vibration acceleration of the train during track irregularities is a stationary Gaussian random process with a zero mean value, the statistical properties of which can be described by the 3σ–principle.
- The Sperling index is a stationary indicator of comfort that is independent of the order of the vehicle elements when the train is subjected to a track irregularity. In this way, train comfort can be characterized by simply selecting a vehicle element to analyze.
- The zonal distribution of train comfort is not strictly symmetrical with respect to the center of the car body. The most comfortable area for the vertical component was located near the front of the center of the car body, while the most comfortable area for the horizontal component was located on the axis of the tilt axis of the center of the car-body center.
- The comfort of the train deteriorated with a loss in mass of the Car-Body and with irregularities in the tracks, while reasonable control of the train speed, regular maintenance of the tracks, and reasonable distribution of the number of passengers when changing trains improved the comfort of the train.
- The overall comfort of the train deteriorated even if only one local vertical spring damper in the secondary suspension system was damaged. It suffered more on the side where the damage was present. The comfort of the train was not independent of the horizontal and vertical components. To optimize the comfort of the train, it is therefore important to consider both components comprehensively.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Linear zonal relation model of car-body motion between the center and a certain point. (

**a**) front view of car-body motion of the vertical levitation and the pitch; (

**b**) lateral view of car-body motion of the vertical levitation and the roll; (

**c**) top view of car-body motion of the lateral sway and the yaw.

**Figure 3.**Illustrations of track irregularities: (

**a**) vertical profile; (

**b**) alignment; (

**c**) cross-level.

**Figure 4.**Car-Body center vertical and horizontal vibratory acceleration responses: (

**a**) history samples distribution for the horizontal; (

**b**) possibility density curve for the horizontal; (

**c**) history samples distribution for the vertical; (

**d**) possibility density curve for the horizontal.

**Figure 5.**Car-body center Sperling index as the function of the order of vehicle element using MCM and PEM: (

**a**) horizontal Sperling index; (

**b**) vertical Sperling index.

**Figure 6.**Zonal distribution of car-body Sperling index using MCM and PEM: (

**a**) horizontal Sperling index; (

**b**) vertical Sperling index.

**Figure 7.**Zonal distribution of car-body Sperling index as influenced by the quality of the track irregularities using PEM: (

**a**) horizontal Sperling index; (

**b**) vertical Sperling index.

**Figure 8.**Zonal distribution of car-body Sperling index influenced by the train speed according to PEM: (

**a**) horizontal Sperling index; (

**b**) vertical Sperling index.

**Figure 9.**Zonal distribution of car-body Sperling index influenced by the addition of the car-body mass using PEM: (

**a**) horizontal Sperling index; (

**b**) vertical Sperling index.

**Figure 10.**Zonal distribution of Car-Body Sperling index influenced by the damage of a vertical spring damper in secondary suspension system using PEM: (

**a**) horizontal Sperling index; (

**b**) vertical Sperling index.

Vehicle Element i | DOF |
---|---|

Car body | 5 DOFs: y_{ci}, z_{ci}, θ_{ci}, φ_{ci}, ψ_{ci} ^{1} |

Bogies | 10 DOFs: y_{fbi}, z_{fbi}, θ_{fbi}, φ_{fbi}, ψ_{fbi,} y_{rbi}, z_{rbi}, θ_{rbi}, φ_{rbi}, ψ_{rbi} |

Wheel sets | 12 DOFs: y_{w1i}, z_{w1i}, θ_{w1i}, y_{w2i}, z_{w2i}, θ_{w2i}, y_{w3i}, z_{w3i}, θ_{w3i}, y_{w4i}, z_{w4i}, θ_{w4i} |

^{1}y is the lateral sway, z is the vertical levitation, θ is the roll, φ is the pitch, and ψ is the yaw. Subscript “f” and “r” respectively denote the front and the rear bogie, while the numbers represent the orders of wheel sets. Subscript i denotes the number of a certain vehicle element and equals 1, 2, 3 … 8. Back-and-forth along the x-axis was neglected because the train ran at a constant speed according to the 7th assumption. Subscripts “c”, “b”, and “w”, respectively refer to “car body”, “bogies”, and “wheel sets”.

Items | Grade-1 | Grade-2 | Grade-3 | Grade-4 | Grade-5 | Grade-6 |
---|---|---|---|---|---|---|

Alignment roughness constant A_{a} | 1.211 × 10^{−4} | 1.018 × 10^{−4} | 0.682 × 10^{−4} | 0.538 × 10^{−4} | 0.210 × 10^{−4} | 0.034 × 10^{−4} |

Vertical roughness constant A_{v} | 3.363 × 10^{−4} | 1.211 × 10^{−4} | 0.413 × 10^{−4} | 0.303 × 10^{−4} | 0.076 × 10^{−4} | 0.034 × 10^{−4} |

Cutoff angular wave number n_{c} | 0.6046 | 0.9308 | 0.8520 | 1.1312 | 0.8209 | 0.4380 |

Cutoff angular wave number n_{s} | 0.8245 | 0.8245 | 0.8245 | 0.8245 | 0.8245 | 0.8245 |

Items | Values |
---|---|

Half of longitudinal distance of wheel sets d_{1} | 1.25 m |

Half of longitudinal distance of bogies d_{2} | 8.75 m |

Half of lateral distance of springs in 1st suspension system b_{1} | 1.00 m |

Half of lateral distance of springs in 2nd suspension system b_{2} | 1.00 m |

Vertical distance, car body center to 2nd suspension system h_{1} | 0.80 m |

Vertical distance, 2nd suspension system to bogie center h_{2} | 0.20 m |

Vertical distance, bogie center to 1st suspension system h_{3} | 0.10 m |

Mass of wheel set m_{w} | 2000 kg |

Mass of bogie m_{b} | 3000 kg |

Moment of inertia of bogie in longitudinal direction I_{bx} | 3000 kg·m^{2} |

Moment of inertia of bogie in lateral direction I_{by} | 3000 kg·m^{2} |

Moment of inertia of bogie in vertical direction I_{bz} | 3000 kg·m^{2} |

Mass of car body m_{b} | 40 t·m^{2} |

Moment of inertia of car body in longitudinal direction I_{cx} | 100 t·m^{2} |

Moment of inertia of car body in lateral direction I_{cy} | 2000 t·m^{2} |

Moment of inertia of car body, about vertical direction I_{cz} | 2000 t·m^{2} |

Longitudinal damping of 1st suspension system/bogie side c_{x1} | 1 kN·s/m |

Lateral damping of 1st suspension system/bogie side c_{y1} | 1 kN·s/m |

Vertical damping of 1st suspension system/bogie side c_{z1} | 20 kN·s/m |

Longitudinal damping of 2nd suspension system/car-body side c_{x2} | 60 kN·s/m |

Lateral damping of 2nd suspension system/car-body side c_{y2} | 60 kN·s/m |

Vertical damping of 2nd suspension system/car-body side c_{z2} | 10 kN·s/m |

Longitudinal stiffness of 1st suspension system/bogie side k_{x1} | 1000 kN/m |

Lateral stiffness of 1st suspension system/bogie side k_{y1} | 1000 kN/m |

Vertical stiffness of 1st suspension system/bogie side k_{z1} | 1000 kN/m |

Longitudinal stiffness of 2nd suspension system/car-body side k_{x2} | 200 kN/m |

Lateral stiffness of 2nd suspension system/car-body side k_{y2} | 200 kN/m |

Vertical stiffness of 2nd suspension system/car-body side k_{z2} | 200 kN/m |

Lower limit of angular wave number n_{min} | π/V rad/m |

Upper limit of angular wave number n_{max} | 80π/V rad/m |

Sampling rate of angular wave number dn | 2π/5V rad/m |

Lower limit of time t_{min} | 0 s |

Upper limit of time t_{max} | 10 s |

Time sampling rate dt | 1/80 s |

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## Share and Cite

**MDPI and ACS Style**

Wu, Z.; Zhang, N.; Yao, J.; Poliakov, V.
Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method. *Appl. Sci.* **2022**, *12*, 7442.
https://doi.org/10.3390/app12157442

**AMA Style**

Wu Z, Zhang N, Yao J, Poliakov V.
Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method. *Applied Sciences*. 2022; 12(15):7442.
https://doi.org/10.3390/app12157442

**Chicago/Turabian Style**

Wu, Zhaozhi, Nan Zhang, Jinbao Yao, and Vladimir Poliakov.
2022. "Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method" *Applied Sciences* 12, no. 15: 7442.
https://doi.org/10.3390/app12157442