This paper presents a mathematical model to determine the volume or weight of a clutch, which is influenced by the main structural parameters of the spring. The objective is to minimize the clutch's weight, and thus, the design variable is defined as X = ^T = ^T. The objective function for the clutch is expressed as:
**min F(X) = P4d²(PD)(2N) C = P22x1²x2x3C (1)**
Here, *C* represents the specific gravity of the spring material.
The structure of the self-excited overrunning spring clutch is simple, consisting of components such as the drive shaft, key, active housing, spring, seal ring, passive housing, and passive shaft. To ensure proper operation, constraints must be established.
One of the key constraints is the **strength reliability constraint**, based on literature [1]. The mathematical expectation of the compressive stress in the spring wire is given by:
**Ls = 4(D/d) - 14(D/d) - 432T(e²PNf - 1)Pd³ + 8TPd²D (2)**
Where:
- *T* is the torque transmitted by the clutch,
- *f* is the coefficient of friction between the spring and the two shells.
To evaluate the reliability of the spring, the reliability coefficient is calculated using:
**ZR = Lc - nLs / √(Rc² + Rs²) (3)**
Where:
- *Lc* is the load-carrying capacity of the spring wire,
- *Ls* is the compressive stress,
- *n* is the safety factor, typically taken as 1.125.
From literature [2], we have:
**Rc = CkLc, Rs = SkLs**
Where *Ck* and *Sk* are the coefficients of variation for the material and stress, respectively. For carbon spring steel, *Ck = 0.14* and *Sk = 0.08*. Substituting these into equation (3) yields the coupling equation for reliability.
Using this, the spring reliability coefficient can be determined as:
**ZR = 2 (4)**
Based on literature [2], the reliability *RZ* can be derived from *ZR* using the following integral:
**RZ = 1/√(2π) ∫₋∞^ZR e^(-t²/2) dt (5)**
This confirms that the spring’s strength reliability exceeds the allowable level.
Substituting equation (4) into (5), and replacing *d*, *D*, and *N* with the corresponding design variables *x1*, *x2*, and *x3*, we obtain the strength reliability constraint condition:
**g1(x) = 1/√(2π) ∫₋∞^ZR e^(-t²/2) dt > 0.99 (6)**
The integral in equation (6) is a generalized one, difficult to compute manually. It can be efficiently evaluated using numerical integration methods like Simpson’s rule, allowing dynamic observation during optimization.
In addition to the strength constraint, there are **structural parameter constraints**. Based on design experience and structural requirements, the spring must transmit a torque of *T = 452 N·mm* and achieve a spring reliability of at least *RZ ≥ 0.99*.
**Solution:**
Carbon spring steel is selected with a specific gravity of *C = 0.0078 g/mm³*. The yield strength is *Lc = 538 N/mm²*, and the safety factor is *n = 1.125*. The material coefficient of variation is *Ck = 0.14*, and the stress coefficient of variation is *Sk = 0.08*. The two shells are made of 45 steel, with a friction coefficient of *f = 0.1*.
The design variable bounds are set as:
- *d_min = 0.2 mm*, *d_max = 1 mm*
- *D_min = 10 mm*, *D_max = 30 mm*
- *N_min = 5*, *N_max = 10*
After substituting all values into equations (1), (6), and others, and running the optimization program with numerical integration, the optimal dimensions are found to be:
- *d = 0.8 mm*, *D = 25 mm*, *N = 6*, and *RZ = 0.999*.
**Conclusion:**
This study provides an effective and practical design method for the new type of self-excited overrunning spring clutch, making it suitable for widespread application and industrial use.
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