1 Overview
The history of urban rail transit using traditional subways, represented by rotary motor drives , is long, dating back over 140 years to the opening of the world's first subway (Metro) in London , England in 1865. Traditional subways primarily rely on the force between wheels and rails to transmit traction (braking) power. Another newer mode of urban rail transit is the linear motor drive system . This technology , primarily developed abroad (Germany, Japan, etc.) since the late 1970s , was only applied to automated production in railway transportation, coal mining, and metallurgy by the mid-20th century. The application of linear motors in railway transportation has attracted particular attention. The linear motors used in urban rail transit are linear synchronous motors . Essentially, the primary (stator) of the linear motor is mounted on the vehicle , while the secondary (rotor) is laid on the track . This technology requires contact rails and converters for traction drive.
In 2003, Guangzhou Metro Line 4 became the first in China to adopt linear motor technology , and the first section began operation in December 2005. In the following years , Guangzhou Metro Lines 5 and 6, as well as the Beijing Airport Express, also adopted this technology. This article mainly analyzes and compares the strength and stability of seamless tracks under the two operating modes.
Comparison of main technical standards for track 2.
2.1 Maximum gradient of the line
The maximum gradient of traditional subway mainlines should not exceed 30‰ , and 35‰ can be used in difficult sections. For linear motor tracks, the maximum gradient in general sections is 50‰ , and 55‰ can be used in difficult sections. The theoretical maximum climbing capacity of linear motors is 100‰ , but the actual application value is around 80‰. In the strength calculation of seamless tracks , attention should be paid to the additional braking force generated under braking conditions.
2.2 Minimum Curve Radius
At a speed of 100 km/h , the minimum curve radius for a traditional subway Type B train on the main line is 500 m , and 400 m under difficult conditions; the minimum curve radius for a linear motor vehicle is 200 m , and 15 m under difficult conditions. The strength and stability of the track structure need further verification under different curve radii.
2.3 Comparison of key vehicle parameters
For linear motor vehicles , the suction force between the rotor and stator should be considered . The linear motor vehicles on Guangzhou Metro Line 4 use Japanese technology , with a suction force of 20kN and a maximum longitudinal thrust of 40kN . This influence should be considered in the track strength calculation process.
3. Strength Calculation of Seamless Track Rails
According to the "Railway Track Strength Detection Method" (TB—2034—88), the rail is considered as a continuous long beam supported on isoelastic continuous point supports for the calculation. The dynamic tensile stress at the bottom of the rail is related to factors such as the track structure stiffness D, speed V, eccentric load coefficient β, and curve horizontal force coefficient f.
During the dynamic operation of linear motor vehicles , the continuity of track structure stiffness is crucial to effectively ensure output power . The strength calculation for linear motor rail support stiffness is 40-50 kN/mm; the stiffness of traditional subway trains is generally less than 30 kN/mm. Because the center of gravity of traditional trains is higher than that of linear motor vehicles , the unbalanced superelevation of traditional subway trains results in approximately 12% more eccentric load compared to linear motor vehicles.
Using the elastically supported continuous long beam method , under the same conditions of a curve radius of 400m and a speed of 100km/h , the tensile stress δgd at the bottom of a traditional subway rail is 107.5MPa , and the dynamic displacement yd is 1.4mm . The tensile stress Md at the bottom of a linear motor rail is 98.9MPa , and the dynamic displacement yd is 1.1mm .
Linear motor vehicles have a lighter axle load and a lower center of gravity , resulting in a greater emergency braking deceleration than traditional subways , but with a smaller overall braking force. Under train operating conditions , the linear motor rails only provide guidance and traction , leading to lower stress in strength calculations and thus extending the rails' service life.
4. Stability Calculation of Seamless Circuit
The main purpose of seamless track stability calculation is to study the track of expanded rail tracks through mechanical models in order to maintain track stability. The expansion of rail tracks is basically divided into three stages: continuous development, gradual expansion, and expansion runway. There are many theories on the stability analysis of seamless tracks in China , among which the "unified seamless track stability calculation formula" and the "wavelength inequality model" are more widely used.
The "Unified Seamless Track Stability Calculation Formula" uses the equivalent track bed resistance Q. It was initially widely applied to 50kg/m rails . Later, Changsha Railway Institute conducted optimization studies on parameters such as the original elastic bending vector foe and plastic vector fop for 60kg/m rails . These parameters were then applied in the design of the seamless track across the Qinhuangdao-Shenyang section. The formula is as follows:
The "wavelength inequality model" employs a power function mode regression to analyze and calculate the lateral resistance equation (Q=Q0-ByZ+Cyn). Using the stationary potential energy theory , a stability calculation formula for seamless tracks is established . The allowable temperature force is related to the rail compression deformation energy τ1, the track frame bending deformation energy τ2, the ballast bed deformation energy τ3, and the fastener deformation energy τ4. While the mathematical derivation of this method is relatively rigorous , the calculation process is quite complex . The formula is as follows:
The VB program was used to program and calculate the results of the two methods. The program results were consistent with the examples in the Railway Engineering Technical Manual "Track" and "Railway Engineering" (Southwest Jiaotong University) under the same conditions. The author mainly focuses on the minimum curve radius standard of traditional and linear motor tracks , using two different stability calculation models . Under the same conditions of 1667 sleepers / km Type III sleeper track bed q= 14.6-357.2y + 784.7y0.75 , the allowable temperature force P of rails with curve radii R=200m and R=500m was calculated .
As can be seen from the two stability calculation formulas , there is no significant difference in the method for calculating the stability of seamless tracks between the two subway models. The difference in the two stability calculation results may be due to factors such as the way the resistance is valued , with the unified formula using a constant resistance method and the safety factor K= 1.25 .
Since linear motors can adapt to smaller curve radii , seamless tracks should be laid as much as possible to ensure track smoothness. As can be seen from the allowable temperature and pressure calculated in Table 2 , the smaller the curve radius, the smaller the allowable temperature and pressure , and the smaller the allowable temperature rise . Therefore, the linear motor track structure should be designed for high-temperature locking as much as possible.
5 Conclusion
As a new urban rail transit mode in China , linear motors require careful consideration of the matching of track structure parameters with vehicle structure , given that the vehicle rotor is installed within the track. By comparing and analyzing the track standards, seamless track strength, and stability calculations of traditional and linear motor metro systems , the curve radius condition for linear motors should be considered a controlling factor for seamless tracks. Due to the lightweight vehicles and small fixed wheelbase of the bogies in linear motor metro systems , appropriately increasing the locking rail temperature can improve track stability. For this new urban rail transit mode , the selection of parameters for the strength and stability calculations of seamless track design still needs to be gradually optimized through practical application.
