In order to ensure proper clearance between each pair of friction surfaces after 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's reserve factor B, and the average friction radius Rc. The number of friction surfaces, Zc, also plays a key role in determining the operating load Pb when the clutch is engaged. This can be expressed as: Pb = B·Mmax·Lf·Rc·Zc, where Lf represents the coefficient of friction between the friction surfaces.
From equations (1) and (4), six design variables can be identified: R1, r, H, h, r1, and K1b. Based on these, a general multi-objective optimization model can be formulated as follows: Find x = [x1]T, minimize f1, subject to LCj(x) ≤ 0 for j = 1, 2, ..., I, where LCj(x) represents the membership function for fuzzy constraints, and CPT refers to a set of non-fuzzy constraints.
To solve this model, the first step is to fuzzify each sub-objective function. Then, the problem reduces to making an optimal fuzzy decision under fuzzy targets and conditions. If convex, cross, or product fuzzy decisions are used, the fuzzy maximum or minimum sets of sub-objectives must be constructed. However, during this process, the comparison basis is typically set at K = 0 within the fuzzy domain. As a result, the fuzzy optimization value of individual sub-objectives may end up being lower than that of single-objective fuzzy optimization, which is not logically consistent. To address this issue, this paper introduces the concept of an ideal point-based fuzzy decision.
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 (j = 1, 2, ..., I). Here, LF(xC) is a monotonic decreasing function of C, with upper and lower bounds M and m, respectively: M = LF(x0), m = LF(x1). Using this, the fuzzy target membership function can be defined as Lf(x) = (LF(x) - m)/(M - m). When LF(x) approaches M, Lf(x) approaches 1; when it approaches m, Lf(x) approaches 0. With bounds set as Llf = m/N and Luf = 1.0, the goal is to find Lf(x) such that C(k) - Lf(x(k)) = 0, leading to a specific clear solution.
Establishing the Fuzzy Optimization Design Objective Function
To construct the objective function, four points including the endpoint A and the bump S in 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 determined by the average difference between the separation stroke and the separation force. The design variable is then set to zero, allowing for a more streamlined optimization process.
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