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11stopt software introduction 1stopt is a set of mathematical optimization analysis comprehensive tool package developed by Qiwei Hi-Tech Co., Ltd., which can be used in nonlinear regression, curve fitting, nonlinear complex model parameter estimation, and linear and nonlinear programming. The core of its calculation is the general global optimization algorithm. The algorithm overcomes the problem that the iterative method must give the appropriate initial value in the optimization calculation. The user does not need to give the initial value of the parameter, but is given by 1stopt randomly. Through its unique global optimization algorithm, the optimal solution is finally found. . The software interface is simple, the model expression is simple, and the readability is good. It can arbitrarily select a variety of optimization algorithms and freely set the algorithm parameters, the solution accuracy is high, and the solution speed is fast. This paper intends to introduce the application of 1stopt software in solving the optimization problem of mechanical engineering mixed variable through the optimization design of the final transmission gear parameters of the tractor.
2 Tractor final drive gear parameter optimization design The final drive is used for further torque increase and deceleration of the tractor. Since the final drive is heavily stressed and close to the wheel, it must have sufficient load carrying capacity, good support stiffness and a reliable seal. The final transmission structure of the tractor mainly includes externally meshing spur gears and planetary gears. The external drive cylindrical gear final drive has a single-stage and two-stage drive. The authors only optimize the final drive gear parameters of single-stage externally meshed spur gears.
2.1 Objective function and design variables should meet the design requirements and performance constraints, from the aspects of compact structure, weight reduction, material saving and cost reduction, the design goal should be the smallest. Here, in the case of working conditions and material selection, the sum of the optional final drive main and driven gear volumes is the design goal. Since the transmission of the final drive is relatively large (generally i=46), in order to make the structure compact, the drive gear z1 with small number of teeth, the modulus m and the driven gear tooth width b2 are usually selected as independent design variables. In order to ensure the strength of the driving gear and avoid undercutting, the height displacement gear is used, ie the design variable x=[x1,x2,x3]T=[m,z1,b2]T objective function minV(x)=0.25x21x2x3(1.1 I2)
2.2 Constraints require a gear module m of 3 m8 and conform to the standard series.
It is required that the number of teeth of the driving gear z1 is 12z115 and takes an integer.
The driven gear tooth width b2 is required to be 7mb211m; the driving gear tooth width b1=1.1b2.
Drive gear modification coefficient Xz = 0.221z21 (1 tan2) 2.77tan 8.71-0.5z1-1, pressure angle = 20.
The driven gear displacement coefficient Xc=-Xz.
Drive gear addendum thickness Sa1 = da1 4Xztan2z1 inv-inva1, requirements 0.25mSa10.4m, where, a1 = arccos (db1 / da1), db1 = mz1cos, da1 = mz1 2m (1 Xz).
The overlap coefficient a=[z1(tana1-tan) z2(tana2)-tan)]/2, generally requires a 1.11.2.
The drive gear shaft diameter is determined according to the torsional strength condition, ds
17.23T/[](mm), T is the transmitted torque, Nm, [] is the allowable shear stress, MPa.
Gear spoke minimum limit condition: df1-ds-2m 0, where df1=mz1-2m (1.25-Xz).
Tooth surface contact strength condition: 8.8Pt(1 i)imz1b2sin0.5
[H], where Pt=2T/d1(kN), the allowable contact stress [H]=1.21.4GPa.
Root bending strength condition: PtYYsa103/(b2m)[F], where Y is the tooth profile coefficient; Ysa is the root stress correction coefficient; the allowable bending stress [F]=350370MPa.
Y = (5.9286-1.5647lnz1 0.1627ln2z1) 1-18.2705Xz-0.25z1 308.6646Xz-0.25z12Ysa = (1.779-0.146lnz1 0.03495ln2z1) 1 0.70598Xz-0.2lnz1-0.43469Xz-0.2lnz12
2.3 Application example The final drive of a tractor adopts single-stage external meshing spur gear transmission. The drive wheel structure is a coupling gear, and the transmitted torque T=900Nm, transmission ratio i=4.62, []=100MPa, required modulus m = 56, the number of teeth z1 = 1315, using height displacement.
Depending on design requirements, after finishing, the following optimization model minV (x) = 17.628x21x2x3s..tg1 (x) = 0.25- (x2 2Xz 2) 3.1416 1.456Xz2x2 0.014904-tanarccos0.93969x2x2 2Xz 2 arccos0.93969x2x2 2Xz 20g2 (x) = (x2 2Xz 2) 3.1416 1.456Xz2x2 0.014904-tanarccos0.93969x2x2 2Xz 2 arccos0.93969x2x2 2Xz 2-0.40g3 (x) = 1.1-0.1591x2tanarccos0.93969x2x2 2Xz 2-0.364-0.7353x2tanarccos4.3414x24.62x2 2Xz 2-0.3640g4 (x)=4.5x1 40-x1x2-2x1Xz0,g5(x)=704.11/(x1x2x3)-1.30g6(x)=1.8106YYsa/x21x2x3-3500,g7(x)=7x1-x30,g8(x)= X3-11x10Xz=0.25x2 1.01x2 8.71-0.5x2-1
Y = (5.9286-1.5647lnx2 0.1627ln2x2) 1-18.2705Xz-0.25x2 308.6646Xz-0.25x2Ysa = (1.779-0.146lnx2 0.03495ln2x2) 1 0.70598Xz-0.2lnx2-0.43469Xz-0.2lnx22 in the above model input 1stopt The optimal solution obtained by the Levenberg-Marqurdt general global optimization method is x=[5.5,13,52]TminV(x)=4712178.23 displacement coefficient Xz=0.506z2=60b1=58.
3 Conclusion The author uses the tractor's final transmission gear parameters as the optimization object, and uses 1stopt's global optimization algorithm to quickly optimize, which proves that the method has high efficiency and good reliability in solving the mixed variable optimization design problem, and provides a solution to similar mechanical optimization design problems. Reference.
July 09, 2021