The development of a mathematical model for the design of a torsion spring involves selecting an appropriate number of design variables based on the optimization objectives. Too few or too many variables can lead to unsatisfactory results. If there are too few, the model becomes overly simplistic, and the design may not meet practical production requirements. On the other hand, having too many variables can make the model excessively complex, leading to difficulties in obtaining meaningful conclusions.
For the design of a cylindrical torsion coil spring, the key design variables typically include the wire diameter (d), the number of coils (n), and the spring index (C = D2/d). These variables are expressed as: X = [x1 x2 x3]^T = [d n C]^T.
The objective function is defined with the goal of minimizing the weight of the spring, which is crucial in aerospace applications. The mathematical expression for the objective function is:
f(X) = 0.25 * P² * Q * D² * n * d² = 0.25 * P² * Q * x3 * x2 * x3¹
Constraints in the reliability-based optimization include both performance and structural constraints. Performance constraints involve ensuring that the spring's strength meets reliability requirements. The maximum bending stress (Rmax) on the spring wire cross-section must be less than the allowable bending stress (). This is expressed as:
Rmax = KT / W <
Where:
- K is the curvature coefficient, given by K = (4C - 1)/(4C - 4)
- W is the section modulus, W = πd³/32
- T is the applied torque
Substituting the design variables into the strength condition yields:
Rmax = [4x3 - 1]/[4x3 - 4] * T / (0.1x3)
Assuming T follows a normal distribution, Rmax also follows a normal distribution. Its mean and standard deviation can be calculated accordingly.
The reliability coupling equation is then used to ensure the spring meets the required reliability level. The formula is:
LF = (SR - R) / √(R² + SR²)
Where:
- SR is the mean material strength
- R is the mean stress
- SR is the standard deviation of the material
For a spring made of 50CrVA steel, SR = 25900 N/mm² and RSR = 63 N/mm². Substituting these values into the equation allows us to compute LF and ensure the reliability constraint g1(X) = LRZ - LF < 0 is satisfied.
Geometric constraints include limits on the wire diameter and the torsion angle under working conditions. For example, the torsion angle Uj is calculated as Uj = Wq * n * Mj, where Wq is the single-turn torsion angle per unit torque. The constraint becomes:
g10(X) = 36b - Uj = 36b - Wq * x2 * x3^10.625Rb / [4x3 - 1 / 4x3 - 4] < 0
Additionally, a stability constraint is considered: g11(X) = Wq * Mj - n = Wq * x3^10.625Rb / [4x3 - 1 / 4x3 - 4] - x2 < 0
An asymmetric fuzzy optimization model has been developed, incorporating 11 common constraints and one fuzzy constraint. Since the wire diameter and number of turns are discrete variables while the spring index is continuous, the MDOP142 method is used for optimization.
After optimization, the best result is found to be X = [d n C]^T = [4.15, 6, 4]^T, with the objective function value f(X) = 40.09 N. In contrast, the traditional design resulted in X0 = [6, 8, 4]^T, with f(X0) = 12616 N.
This optimization significantly reduces the wire diameter and number of coils, resulting in a much lighter spring. The weight reduction is approximately 68%, demonstrating substantial economic benefits. The model accounts for uncertainties in the design process, overcoming traditional limitations and meeting the high reliability requirement of R = 0.99999. This approach is versatile and applicable to various types of springs.
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