The fundamental principles and assumptions of "Vehicle-Orbital Vibrancy Dynamics" have provided formulas for calculating P and ar, but these calculations require the use of electronic computers and cannot theoretically reflect the full impact of P and ar. From a maintenance and repair perspective, if the factors that influence P and ar can be eliminated, the formation of screw hole cracks can be significantly reduced. Therefore, it is necessary to make several key assumptions regarding the orbital system:
1. The track is assumed to be rigidly supported, with the wheels acting as "excitation points."
2. The track is only subjected to vertical impact forces (since P and ar are primarily influenced by vertical impacts). In this case, the wheel acts as the impact load, and the rail is considered the impacted member. This assumption includes:
- The impacted member obeys Hooke’s law under impact.
- The mass of the impacted component is negligible.
- No energy losses are considered.
When these two conditions are met, P and ar can be simplified using the energy method and the principle of energy conservation to calculate the impact force and the resulting vibration acceleration of the rail. When a wheel strikes the rail joint, the rail experiences a vertical impact force PI, while the sleeper exerts a reaction force Frs. The vibration equation for the rail is given by: mr * ar = PI - Frs, where mr is the rail mass. The calculation of Frs follows the same principles as P. The mass of the wheel, including its self-weight and load, is denoted as am, and Kp represents the stiffness of the underlay cushion, fasteners, and sleepers.
According to "Vehicle-Axis Coordination Dynamics," the joints in ordinary tracks are the primary source of disturbance in the wheel-rail system. When a wheel passes over a joint, the sudden change in the wheel's center causes an immediate vertical downward impact on the track. As the wheel leaves the joint, the impact velocity disappears. The calculation of the joint's impact velocity uses the formula outlined in "Vehicle-Operational Occurrence Dynamics."
Several factors affect P and ar, as shown in equations (1) and (2):
(1) P and ar are closely related to axle weight, speed, and disturbances such as rail joints, misaligned welds, low joints, and peeling blocks. These factors directly increase the likelihood of screw hole cracks in rails. This correlation has been confirmed through statistical analysis.
(2) P and ar increase linearly with vehicle speed, which aligns with findings from "Vehicle-Track Accidental Dynamics."
(3) Disturbances in the track, such as rail joints and structural imperfections, are the root causes of P and ar. The magnitude of these disturbances directly influences the level of P and ar. Without disturbances, there would be no shocks or vibrations, making disturbance control essential for managing P and ar.
(4) The mass of the rail slightly increases with its weight, while ar decreases slightly. This is consistent with previous studies. Statistical data shows that P60 rails have more screw hole cracks than P50 rails.
(5) The stiffness of the track system (K) greatly affects P. A higher K leads to a smaller 6j and a larger impact force. K is influenced by the stiffness of the pad, track bed, and subgrade. Temperature changes also affect K; for example, the track is stiffer in winter, leading to higher P values and a greater risk of screw hole cracks during colder months.
(6) The stiffness of the under-slab pad (KP) has a significant effect on ar. If the track system becomes loose, damaged, or fails, KP may decrease, increasing ar and intensifying rail vibrations. This can lead to fastener loosening, detachment, or breakage, further exacerbating rail vibrations and increasing the likelihood of screw hole cracks.
(7) The volume of traffic and the size of the wheel radius have a major impact on P and ar. Higher traffic volumes and smaller wheel radii (such as those found in freight vehicles) result in larger P and ar values, increasing the probability of rail screw hole cracks. For instance, the Beijing-Kowloon Railway has more screw hole cracks on the downline due to higher traffic volume, even though its speed is relatively low. This highlights the importance of traffic volume in determining crack formation.
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