# Insights into the Effect of WJ-7 Fastener Rubber Pad to Vehicle-Rail-Viaduct Coupled Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of Dynamic Flexibility Method

#### 2.1. Vehicle-Rail-Viaduct Coupling Model via Dynamic Flexibility Method

_{v}means the displacement of the vehicle system, M

_{v}, K

_{v}and C

_{v}indicate the mass, stiffness and damping, respectively. p(ω) represents the vertical rail-wheel force, and ω denotes the excitation circular frequency.

_{w}in the rail, x

_{w}indicates the contacting point for the wth wheel rail, P

_{w}means the wth wheel rail contact force for the wheelset, N

_{w}indicates the amount of the wheelsets; K

_{f}z

_{r}(x

_{n}) represents the fastener’s force when nth fastener acts on the x

_{n}location of the rail. N represents the amount of the fasteners.

_{n}location of the slab, F

_{fn}denotes the fastener’s force at location x

_{n}of the slab. N means the amount of the fasteners. F

_{jm}represents the discrete spring force at x

_{m}in the CA mortar of the slab; M illustrates the amount of the discrete spring dampers in the CA mortar.

_{m}location of the viaduct, F

_{zh}denotes the hth bearing reaction force of the viaduct.

_{fn}, the discrete spring force at CA mortar F

_{jm}, and the bearing reaction force F

_{zh}can be illustrated as

_{f}denotes the complex stiffness of the fastener, K

_{j}means the complex stiffness of the discrete spring damper element in the CA mortar, K

_{z}indicates the complex stiffness of the viaduct bearing.

_{f}means the stiffness of the fastener, η

_{f}denotes the loss factor of fastener, k

_{j}indicates the discrete spring element stiffness of the CA mortar, η

_{j}represents the loss factor of the CA mortar discrete spring element, k

_{z}denotes the stiffness of the viaduct bearing, η

_{z}represents the loss factor of the viaduct bearing stiffness.

#### 2.2. Dynamic Flexibility Calculation for the Wheel

_{j}is applied to the jth wheel-rail contacting point; ${\beta}_{ij}^{V}$ means the displacement at the ith location of the wheel when unit force is applied to the jth contacting point.

#### 2.3. Dynamic Flexibility for the Wheel Rail Interaction

_{H}denotes the contacting stiffness of the wheel-rail.

#### 2.4. The Dynamic Flexibility of the Rail-Viaduct Coupling System

_{r}, E

_{r}, G

_{r}, I

_{r}, A

_{r}, κ, and η

_{r}indicate density, Young’s modulus, shear modulus, sectional inertia moment, section area, shear coefficient, and loss factor of the rail, respectively.

_{n}(x) indicates the nth order mode shape at x location, ω

_{n}represents nth order resonant frequency for the beam, η means the loss factor for the slab, N denotes the order.

_{bn}indicates the nth order mode shape, ω

_{bn}indicates the nth order resonant frequency, NMB means the amount of modes.

_{j}is loaded at jth wheel rail contacting point, ${\beta}_{ij}^{TB}$ means the displacement at the ith location of the rail-viaduct coupling system when unit harmonic force is applied to the jth wheel rail contacting point.

#### 2.5. Harmonic Analysis for the Vehicle-Rail-Viaduct Coupling System

_{wr}can be expressed as

_{wr}into Equations (1) and (7) can determine displacements of the vehicle, the rail, the slab, and the viaduct, illustrated as Z(ω), respectively.

## 3. Experimental Tests

#### 3.1. General information for the tests

#### 3.1.1. Dynamic Mechanical Tests for the Fastener Rubber Pad

_{f}indicates the stiffness of the fastener rubber pad, K

_{f}means the complex stiffness of the fastener rubber pad, A and h represent the section area, and thickness of the fastener rubber pad, respectively.

#### 3.1.2. Testing Configuration

#### 3.2. Model Description

#### 3.2.1. Model Properties

#### 3.2.2. Track Irregularity Spectrum of the Rail

^{−6}m; λ indicates the wavelength (m), V means the velocity of the vehicle.

## 4. Results Analysis

## 5. Comparison of Proposed Method with Conventional Method

#### 5.1. Case I: Three-Span Bridge

#### 5.2. Case II: Multi-Layer Model for Ballastless Track

## 6. Concluding Remarks

- The WJ-7B small resistance fastener rubber pad behaves sensitive at low temperatures, while stable at high temperatures. The WJ-7B small resistance fastener rubber pad has relatively high stiffness at low temperatures; during the transition period from a glassy state to a rubbery state, the stiffness of the WJ-7B small resistance fasteners declines sharply; while at relatively high temperatures, the stiffness of the WJ-7B small resistance fasteners keeps stable, with little change versus the temperature change.
- The temperature dependent stiffness of the WJ-7B small resistance fastener rubber pad has little effect to the vertical vibration responses of the vehicle; as the temperature increases, the dynamic flexibility of the rail-viaduct increases, the amplitudes increases, but the resonant frequencies decrease especially when the frequency is higher than 30 Hz.
- In terms of the temperature dependent stiffness to wheel rail force and the dynamic responses of the wheelsets, they share similar characteristics; i.e., in frequency range below 20 Hz, the temperature dependent stiffness of the WJ-7B small resistance fastener rubber pad has little influence on the wheel-rail contact force and the accelerations of the wheelsets; the peaks of the wheel-rail contact force and the accelerations of the wheelsets decrease as the temperature increases, and the dominant frequencies also decrease as the temperature increases; considering the fact that the temperature dependent stiffness of the WJ-7B fastener rubber pad enables the resonant frequencies of the coupled wheel-rail to decrease as the temperature increases, the peak of the acceleration of the viaduct also decreases, and the dominant frequencies shift leftward.
- In terms of further investigation, the extension of such work to the real engineering tests would be desirable to optimize the proposed methodology. As to engineering application, in the low temperature areas and sudden temperature changing areas, the stiffness of the WJ-7B fastener rubber pad changes sharply, which further affects the vehicle-rail-viaduct coupling analysis. Such an effect has to be considered in certain areas in the engineering projects.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Details of the fastener rubber pad: (

**a**) Fastener rubber pad of one fastener type; (

**b**) Fastener rubber pad of another fastener type.

**Figure 2.**Vertical model of vehicle-track coupling system: (

**a**) high-speed railway (HSR) system; (

**b**) vehicle-rail-viaduct coupled model.

**Figure 6.**Mechanical parameters of WJ-7B small resistance fastener rubber pad: (

**a**) Young’s modulus, (

**b**) stiffness, (

**c**) complex stiffness.

Cases | Temperature (°C) | Stiffness (MN/m) |
---|---|---|

#1 | −40 | 64.71 |

#2 | −20 | 26.66 |

#3 | 0 | 21.00 |

#4 | 20 | 17.94 |

#5 | 40 | 16.65 |

Parameter, Symbol (Unit) | Value |
---|---|

Vehicle’s weight under rated load (kg) | 42,934 |

Bogie weight (kg) | 3300 |

Wheelset weight (kg) | 1780 |

Rotary of inertia for vehicle’s nod (kg m^{2}) | 1.712 × 10^{6} |

Rotary of inertia for bogie’s nod (kg m^{2}) | 1807 |

Vertical stiffness of primary suspension (N m^{−1}) | 1.176 × 10^{6} |

Damping of primary suspension (N s/m) | 1.0 × 10^{4} |

Stiffness of secondary suspension (N m^{−1}) | 2.4 × 10^{5} |

Damping of secondary suspension (N s/m) | 2.0 × 10^{4} |

Length of train (m) | 25 |

Fixed distance of carriage (m) | 17.5 |

Fixed distance between axles (m) | 2.5 |

Component | Symbol, Unit | Value |
---|---|---|

Rail | Stiffness (N/m^{2}) | 2.059 × 10^{11} |

Sectional inertia moment (m^{4}) | 3.217 × 10^{−5} | |

Density (kg/m^{3}) | 7850 | |

Section area (m^{2}) | 7.745 × 10^{−3} | |

Shear modulus (N/m^{2}) | 7.7 × 10^{10} | |

Section coefficient (κ) | 0.45 | |

Loss factor (η_{r}) | 0.01 | |

Fastener | Loss factor | 0.25 |

Distance between fasteners (m) | 0.625 | |

Slab | Stiffness (N/m^{2}) | 3.6 × 10^{10} |

Sectional inertia moment (m^{4}) | 1.7 × 10^{−3} | |

Density (kg/m^{3}) | 2750 | |

Section area (m^{2}) | 0.51 | |

Loss factor | 0.1 | |

CA mortar | Equivalent stiffness (N/m) | 0.78 × 10^{9} |

Loss factor | 0.2 | |

Viaduct | Length (m) | 32 |

Stiffness (N/m^{2}) | 3.45 × 10^{10} | |

Sectional inertia moment (m^{4}) | 5.757 | |

Density (kg/m^{3}) | 2650 | |

Section area (m^{2}) | 4.416 | |

Loss factor | 0.1 | |

Bearing for the viaduct | Stiffness (MN/m) | 6 × 10^{9} |

Distance between bearings (m) | 32 | |

Loss factor | 0.25 |

**Table 4.**The first order and dominant resonant frequencies comparison for five scenarios in the dynamic flexibility amplitude of the rail.

Scenarios | First Order and Dominant Resonant Frequencies | |||||
---|---|---|---|---|---|---|

First Order Resonant Frequency | Dominant Resonant Frequency | |||||

(Hz) | ∆(#-#1) (Hz) | ∆(#1-#) (%) | (Hz) | ∆(#-#1) (Hz) | ∆(#1-#) (%) | |

#1 | 6 | 0 | 0 | 142 | 0 | 0 |

#2 | 5 | 1 | 16.67 | 130 | 12 | 8.45 |

#3 | 5 | 1 | 16.67 | 118 | 24 | 16.90 |

#4 | 5 | 1 | 16.67 | 110 | 32 | 22.53 |

#5 | 5 | 1 | 16.67 | 107 | 35 | 24.65 |

Scenarios | Peak Value and Dominant Frequency | |||
---|---|---|---|---|

Peak Value | Dominant Frequency | |||

(kN) | ∆(#1-#) (%) | (Hz) | ∆(#1-#) (%) | |

#1 | 20.62 | 0 | 74 | 0 |

#2 | 16.79 | 18.57 | 51 | 31.08 |

#3 | 14.41 | 30.12 | 47 | 36.49 |

#4 | 13.50 | 34.53 | 43 | 41.89 |

#5 | 13.35 | 35.26 | 42 | 43.24 |

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## Share and Cite

**MDPI and ACS Style**

Liu, L.; Zuo, Z.; Zhou, Y.; Qin, J.
Insights into the Effect of WJ-7 Fastener Rubber Pad to Vehicle-Rail-Viaduct Coupled Dynamics. *Appl. Sci.* **2020**, *10*, 1889.
https://doi.org/10.3390/app10051889

**AMA Style**

Liu L, Zuo Z, Zhou Y, Qin J.
Insights into the Effect of WJ-7 Fastener Rubber Pad to Vehicle-Rail-Viaduct Coupled Dynamics. *Applied Sciences*. 2020; 10(5):1889.
https://doi.org/10.3390/app10051889

**Chicago/Turabian Style**

Liu, Linya, Zhiyuan Zuo, Yunlai Zhou, and Jialiang Qin.
2020. "Insights into the Effect of WJ-7 Fastener Rubber Pad to Vehicle-Rail-Viaduct Coupled Dynamics" *Applied Sciences* 10, no. 5: 1889.
https://doi.org/10.3390/app10051889