In order to ensure sufficient clearance between each pair of friction surfaces when the clutch is fully disengaged, the maximum spring deformation should be calculated as Kmax = Kd + Kb + Zc·S. This calculation is based on the engine's maximum torque (Mmax), the clutch reserve factor (B), and the average friction radius (Rc). The number of friction surfaces (Zc) also plays a role in determining the operating load (Pb) during clutch engagement: Pb = B·Mmax·Lf·Rc·Zc, where Lf represents the coefficient of friction between the friction surfaces.
From equations (1) and (4), six key design variables can be identified: R1, r, H, h, r1, and K1b. This leads to the development of a general multi-objective optimization model, which aims to minimize the objective function f1 under a set of constraints. The mathematical formulation is as follows:
Find: x = [x1]T
Minimize: f1(x)
Subject to: LCj(x) ≤ 0 (for j = 1, 2, ..., I)
Where LCj(x) represents a membership function for fuzzy constraints, and CPT refers to a set of non-fuzzy constraints.
To solve this model, the first step involves fuzzifying each sub-objective function. The problem then reduces to making an optimal fuzzy decision under fuzzy goals and constraints. When using convex, cross, or product fuzzy decision methods, it is necessary to construct a fuzzy maximum or minimum set for each sub-objective. However, the comparative basis used in these constructions typically assumes K=0 on the fuzzy domain, which may lead to sub-optimal results. To address this issue, the paper introduces the concept of ideal point-based fuzzy decision-making.
Establishing the Fuzzy Target Membership Function
Based on the above fuzzy decision framework, the following mathematical model can be developed:
Find: x = [x1]T
Maximize: LF(x) = L - L+ + Ls.t. LCj(x) ∈ C (for j = 1, 2, ..., I)
Here, LF(xC) is a monotonically decreasing function of C, with upper and lower bounds M and m, respectively. These bounds are defined as M = LF(x0) and m = LF(x1). Using these values, a fuzzy target membership function can be established as:
Lf(x) = (LF(x) - m) / (M - m)
As LF(x) approaches M, Lf(x) approaches 1, and as LF(x) approaches m, Lf(x) approaches 0. The membership function is bounded between a lower limit Llf = m/M and an upper limit Luf = 1.0. The solution to the equation C(k) - Lf(x(k)) = 0 provides a specific clear solution to the formula.
For the fuzzy optimization design objective function, four points—A, S, and two others within the ASB range—are considered. The first sub-objective function (b) is defined as the average change in pressing force from each point to point B. The second sub-objective function is derived from the average difference between the separation stroke and the separation force. The design variable is set to zero, simplifying the optimization process while maintaining accuracy.
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