Tire crown shape optimization problems The grip performance and wear resistance are the tread properties that the tire designer cares about. The two main parameters that determine the shape of the tire crown are the tread width B and the tread arc height H that have a significant effect on these properties. The uniformity of ground pressure distribution can be used to measure grip and wear resistance. Therefore, the uniformity of ground pressure distribution is used as the optimization goal. Reasonable design should consider the stress condition of the shoulder, so the Mises equivalent stress of the shoulder is taken as the constraint condition. Then we can list the mathematical model of tire crown shape optimization: minF(B,H)=∑i∈A2N,stΡjvonm≤1.1Ρ0 vonmj∈K,B≤B≤B,H≤H≤H{.(1) :K is the set of elements at the shoulder; FRYi(B,H) is the normal contact reaction force of the i-th ground node, FRYmean is the average of the normal contact reaction force of all ground nodes, and both are design variables B The function of H and H; Ρ0 vonm is the maximum Mises equivalent stress in the initial design of the shoulder region; B, B, H{, H are the upper and lower bounds of the running surface width and the tread arc height, respectively. The objective function is minimized, even if the square sum of the difference between the normal contact reaction force and the mean value of each node is the smallest, that is, the distribution of the normal contact reaction force is as uniform as possible. Local Response Surface Method In the optimization problem, a global or local response surface is usually constructed based on several different design points in the design space. This method is generally used where it is difficult to give explicit sensitivity, and sometimes it is used in systems that contain random disturbances, but the sensitivity analysis at this time is also unreliable. For the tire shape optimization problem, the positive problem involves multiple nonlinearities, and the optimization problem is again at the shape level. Therefore, it is difficult to guarantee the accuracy of the objective function or the constraint condition approximation by constructing the global response surface. In this paper, the local response surface is constructed and the local response surface is continuously updated during the iterative process of the optimization solution. A series of response surface approximations are obtained for the partial approximation of the objective function and the constrained condition. The objective function is designed from the initial design in the entire optimization process. Optimize the design and gradually use the new design point correction to generate a new response surface, until the optimization convergence. The second-order polynomial is used to construct the response surface: f^(x, Β)=Β0+∑Ni=1Βixi+∑Ni=1,j (2) where Β is the second-order response surface parameter, and N(N+1) accumulated during the iterative process is used. ) 2+N+1 feasible point regression determination. In addition to the second-order response surface described above, the first-order response is constructed simultaneously with the constraints of the original optimization problem to perform local approximation: g^(x,c)=c0+∑Ni=1cixi. (3) where c is the first-order response surface parameter, determined by the cumulative N+1 point regression during the iteration. After the above treatment, the implicit objective function and implicit constraint of the tire crown shape optimization problem are locally approximated by the first-order and second-order response surfaces respectively. The crown shape optimization problem is also replaced by the following approximate problem near the current design point. :minΒ0+∑Ni=1Βixi+∑Ni=1,j Due to the multiple nonlinearity and large scale involved in the local response surface of the adaptive sequence, special processing is required during the construction of the local response surface in order to reduce the limitation in the optimization process. The number of meta-analysis increases the efficiency of the solution. The first-order and second-order response surface adaptive iteration formats as shown are constructed in this paper. First-order and second-order response surface self-adaptive iterative formats At the beginning of the iteration, a first-order response surface is first constructed to locally approximate the objective function and constraints: minΒ0+∑Ni=1Βixi,st Ax ≤ g, xi ≤ xi ≤ x θi. (5) The above equation is a linear programming problem. To solve this problem, we can obtain the optimal direction vector and adopt a small step size to obtain a new design point. Reconstruct the first-order response surface near the new design point and solve the new design point until enough design points are generated to construct the second-order response surface. The advantage of this iterative format is that the multiple design points that construct the first second-order response surface are closer to the negative gradient direction of the objective function, improving the efficiency of the second-order response surface and reducing the number of iterations. When the incremental ray step design point xn+1 satisfies the following equation, it is called a "bad point": f(xn+1)>f(xn). (6) When the iterative design point xn+1 is outside the feasible domain, then it violates the constraint condition and becomes an infeasible point. Ray steps are usually used to deal with violations of constraints, and are usually used only when the constraints are linear inequalities of the design variables. For the case that the constraint condition and the objective function are all nonlinear implicit functions, this paper proposes a modified ray step adjustment scheme for incremental ray steps to uniformly deal with infeasible and dead pixels and pulls the infeasible point back to the feasible region. The dead point keeps the objective function down to suppress the generation of iterative oscillations. The specific formula is: x'n+1=xn+(xn+1-xn)E,(7)E=maxgin+1-gingi-gin,(8a)E=f^(x3n+1)f^(xn )+1,(8b)E=maxgin+1-gingi-gin,f^(x3n+1)f^(xn)+1. (8c) Equation (8a) is the case where the current point is only an infeasible point, Equation (8b) is the case where the current point is a dead point, and Equation (8c) is that the current point is both an infeasible point and a dead point. Where: x′n+1 is a new design point after incremental ray step adjustment, x3n+1 is a dead point or infeasible point, and gin+1 is the ith constraint value at a dead point or an infeasible point. Xn is the feasible design point generated by the last iteration, gin is the ith constraint value of the last feasible design point, gi is the upper limit of the ith constraint, and f^(x3n+1) is the objective function value of the dead point, f^ (xn) is the objective function value of the last feasible point. Application example of tire crown parameter 7 Based on the above method, the crown shape parameters B and H of 26575 radial tire were optimized. The initial design was: H(0)=10.625mm, B(0)=99.33mm.(15) The optimized design obtained through 13 iterations is: H=9.319mm, B=94.024mm. (16) Iteration history The changes in crown ground pressure distribution show that the optimized design has a more uniform ground pressure distribution along the tread width than the original design, with a high ground pressure zone along the tire circumference (Fig. 7c), and the tire The ground pressure in the middle part of the circumferential symmetry line is slightly increased. The cloud map shows that the maximum ground pressure dropped by about 16%, and the original high contact pressure area with obvious design was replaced by a larger and lower value contact pressure area, and the pressure distribution was more even. Comparison of node paths on 3 lines Figure 3 Comparison of normal contact force distributions of 3 node paths before and after optimization 8 Conclusion The first-order and second-order adaptive local response surface methods constructed in this paper are combined with the proposed methods for unfeasible and bad points. The special treatment of incremental ray steps can be satisfactorily optimized after fewer iterations. In particular, incremental ray step methods that deal with infeasible points and reduced oscillations can be applied to other objective functions and optimization conditions where the constraints are implicitly nonlinear. At the same time, the adaptive first-order and second-order response surface methods are also reasonable choices to improve the optimization efficiency and reduce the number of iterations for the optimization design problems with large analysis time for other positive problems. 1150Kg Gearless Elevator Motor 1150Kg 0.75M/S Gearless Motor,Permanent Magnet Synchronous Motor,Gearless Elevator Motor Gie Em Co., Ltd. , http://www.shelevatormotor.com