The development of mathematical models for optimization requires careful selection of design variables, as both an excessive or insufficient number can lead to unsatisfactory results. Too few variables may oversimplify the model, making it difficult to meet real-world production requirements, while too many can complicate the model and hinder obtaining meaningful conclusions. In the case of cylindrical torsion coil spring design, the key design variables are the wire diameter (d), the number of coils (n), and the spring index (C = D2/d). These variables are represented as: X = [x₠x₂ x₃]ᵀ = [d n C]ᵀ.
The objective function is defined based on the goal of minimizing weight, especially since the spring is used in aerospace machinery. The mathematical expression for the objective function is:
f(X) = 0.25P²QD²nd² = 0.25P²Qx₃x₂x₃¹
Constraints in the reliability optimization include performance constraints and structural constraints. Performance constraints involve ensuring the spring's strength meets reliability standards. The maximum bending stress (Rmax) must be less than the allowable bending stress (
b), where b = 1.25 . The curvature coefficient K is calculated as (4C - 1)/(4C - 4), and the section modulus W is Pd³/32. Substituting the design variables into the strength condition gives Rmax = [4x₃ - 1]/[4x₃ - 4] * T / [0.1x₃].
Assuming T follows a normal distribution, Rmax also follows a normal distribution with mean Rmax and standard deviation SR. The reliability coupling equation is expressed as:
LF = (SR - R) / √(R²SR + R²R)
Using material 50CrVA, SR = 25900 N/mm², and RSR = 0.07 * SR = 63 N/mm². Substituting these values leads to the reliability constraint:
gâ‚(X) = LRZ - LF < 0
where LRZ is the required reliability factor.
Geometric constraints include the wire diameter and torsion angle under working conditions. The torsion angle Uj is given by Uj = Wq * n * Mj, and after substituting design variables, the constraint becomes:
gâ‚â‚€(X) = 36b - Uj = 36b - Wq * xâ‚‚ * x₃¹â°.625Rb / [4x₃ - 1 / 4x₃ - 4] < 0
Additionally, a stability constraint is applied:
gâ‚â‚(X) = WqMj - n = Wq * x₃¹â°.625Rb / [4x₃ - 1 / 4x₃ - 4] - xâ‚‚ < 0
An asymmetric fuzzy optimization model is developed, incorporating 11 common constraints and fuzzy constraints. Since d and n are discrete variables, and C is continuous, the MDOP142 method is used for optimization. The optimized result is X = [d n C]áµ€ = [4.15, 6, 4]áµ€, yielding f(X) = 40.09 N. In contrast, the traditional design had Xâ‚€ = [6, 8, 4]áµ€ with f(Xâ‚€) = 126.16 N.
The reliability-based design significantly reduces the wire diameter and number of coils, resulting in a lighter spring. This reduction leads to a 68% improvement in weight, demonstrating substantial economic benefits. By considering uncertainty factors, this model surpasses traditional methods and ensures that the spring meets the high reliability requirement of R = 0.99999 under maximum load. This approach is versatile and applicable to various types of springs.Electric Paint Mixer Single Speed
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