# An Innovative and Efficient Method for Reverse Design of Wheel-Rail Profiles
## 1 Introduction  
The entire mass of the railway vehicle running on the track is borne by the wheels in contact with the rail and the vehicle is driven and guided by the traction and braking force created by wheel-rail adhesion. For railway vehicles, the proper design of wheel-rail profiles is helpful in achieving ideal vehicle operation, including curve negotiation, prevention of derailment, running stability and safety [1; 2; 3; 4]. The design of wheel-rail profiles has been a topic of concern in many studies and many profile design methods with different targets and measures have been presented. [PERSON] and [PERSON] [5] developed a design program for optimizing single arc wheel profiles. The different design requirements such as the wear characteristics, high critical speed or good curve negotiation could be satisfied by adjusting the radii of the wheel profiles. [PERSON] and [PERSON] [6] introduced a method for designing wheel-rail profiles of steering shaft vehicles. The required rolling radius difference could be satisfied by forming wheel-rail profiles with tangent arcs with different radii and two-point contact could be avoided. [PERSON] and [PERSON] [7] tried the instructive design of freight car wheel profiles based on expansions of rail shapes. [PERSON] [8] designed subway wheel profiles based on the rail profile expansion method that are helpful in improving the compatibilityof wheel-rail profiles, reducing the contact stress and finally reducing wheel-rail wear. [PERSON] and [PERSON] [9] and [PERSON] et al. [10] sought wheel profiles matching with the vehicle suspension parameters directly through dynamic simulation and the genetic algorithm and comprehensively analyzed the constraint conditions of different wheel-rail characteristics such as comfort, lateral track force, derailment coefficient, wear and contact stress. However, this method is very time-consuming. [PERSON] et al. [11; 12] optimized the design of wheel profiles by taking the rolling radius difference as the design target according to multipoint approximation technology based on response surface fitting (MARS) and improved the profile matching and dynamic performance. Reference [13], based on the target functions and optimization method [11; 12], optimized railway wheel profiles considering the wheel-rail rolling contact fatigue and wear etc. [PERSON] et al. [14] proposed a similar method to that used by [PERSON] et al. [11; 12] and they chose a reduced set of generalized coordinates constituted by five variable parameters together with cubic spline curves in a linear programming formulation for the generation of profiles. The objective is to minimize the deviation of the rolling radii difference of the generated profiles from the target one. [PERSON] [15] presented a method which established regardless of the roll angle for designing wheel profiles with target conicity and wide tread wear spreading by adjusting the estimated coefficient \(k_{y}\). [PERSON] and [PERSON] [16], based on the geometric characteristics shapes, developed railway wheel and rail head profiles by iteratively varying the arc radii profile variations. [PERSON] et al. [17; 18] presented a wheel profile optimization method from the wear viewpoint and the method requires the contact point on rail, roll angle, optimization parameter \(k(y)\) and its range Ik. The rolling radius difference meets the design objective by adjusting the parameter \(k(y)\). The greater the resolution of Ik, the smaller the error but the longer the calculation time required and the range of Ik is too small to meet the design requirements and too large to increase the calculation time. [PERSON] et al. [19; 20; 21; 22] presented a method for designing wheel-rail profiles by taking the rolling radius difference or contact angle difference as design targets based on the wheel-rail contact geometric relation equation. Although the above methods each have their advantages, they cannot have high precision and high efficiency simultaneously.