1 Introduction
2 NES Design Process
d[mm] | D[mm] | lf[mm] | dl[mm] | Dl[mm] | l0l[mm] | k0[N/m] | a1[k/m] | st[mm] | a3[N/m3] | pc | |
---|---|---|---|---|---|---|---|---|---|---|---|
Lb | 0.1 | 20 | Course + 2toler | 0.1 | 20 | 20 | 100 | 100 | 0.5Course + toler | 0.2e5 | 0.3 |
Ub | 5 | 60 | 95 | 5 | 60 | 70 | 800 | 800 | 60 | 8e5 | 0.6 |
3 Nonlinear Constraint Analysis
No | Description | Constraint |
---|---|---|
1 | Avoid the occurrence of negative numbers of coils during the calculation | \(k_{0} < a_{1}\) |
2 | Ensure the accuracy of the fitting | \(n_{a} > 8\) |
3 | Maintain spring characteristics | \(n_{{_{l} }} > 4\) |
4 | Strengthen the spring and induce low internal stress. Facilitate the manufacturing process | \(5 < \frac{D}{d} < 20\) |
5 | \(5 < \frac{{D_{l} }}{{d_{l} }} < 20\) | |
6 | Ensure spring free length is smaller than its critical length \(L_{k} \, = \,{{v_{l} l_{0l} } \mathord{\left/ {\vphantom {{v_{l} l_{0l} } {D_{l} }}} \right. \kern-0pt} {D_{l} }}\) to avoid buckling. Sitting factor for linear spring where both ends are connected with hinge joints \(v_{l} \, = \,1\) | \(v_{l} \frac{{l_{0l} }}{{D_{l} }} < 2.6\) |
7 | Sitting factor \(v_{nl} \, = \,0.5\)for nonlinear spring that is embedded | \(v_{nl} \frac{{L_{l} }}{D} < 2.6\) |
8 | Constant helix angle of linear spring | \(\arctan \left( {\frac{{p_{l} }}{{\pi D_{l} }}} \right) < 7.5^\circ\) |
9 | The helix angle is variable for the nonlinear spring, its maximal helix angle is considered | \(\arctan \left( {\frac{{\max \left( {p_{nl} } \right)}}{\pi D}} \right) < 7.5^\circ\) |
10 | Ensure the maximum corrected main stress does not exceed the acceptable maximum stress. Maximum stress: τmax = 8FDcos(α)(1 + 2d/(3D))/(πd3). Stress correction factor k = (w + 0.5)/(w − 0.75). Maximum allowable stress τzul is equal to 0.48Rm. For stainless steel Rm = 1919 − 255.86ln(d) | \(\tau_{zul} > k\tau_{\max }\) |
11 | Reduce the risk of a crash under alternating load. Compressed solid length of spring ln \(\, = \,d_{l} \left( {n_{lin - active} \, + \,1.5} \right)\, + \,Sa\). Sum of minimum spaces between active coils. Sa = nlin−active (0.0015Dl2 /dl + 0.1dl) | \(l_{p} \le l_{0l} - l_{n}\) |
12 | Avoid nonlinear spring 1 or 3 being released central mass (in Fig. 2 right) | \(s_{t} > \frac{Course}{2} + toler\) |
13 | \(l_{f} - s_{t} > \frac{Course}{2} + toler\) | |
14 | Avoid the linear spring 2, 4 separation in Fig. 2 left. Distance between hinges in extreme position \(D_{max}^{2} \, = \,\left( {l_{0l} \, - \,l_{p} \, + \,2l_{c} } \right)^{{2}} \, + \,\left( {l_{f} \, - \,s_{t} } \right)^{{2}}\), free distance \(D_{f} \, = \,l_{0l} \, + \,2l_{c}\) | \(D_{f} - D_{\max } > toler\) |
15 | Error control. Taylor expansion with higher order term \(\rm{\Delta} F\left( u \right)\, = \,0.75 k_{l} \left( {l_{0l} \, + \,2l_{c} } \right)u^{5} /\left( {l_{0l} \, - \,l_{p} \, + \,2l_{c} } \right)^{{5}}\). Maximum force \(F_{max} \, = \,A_{3} \left( {Course/2} \right)^{{3}}\) | \(\Delta F\left( u \right) < 0.05F_{\max }\) |
4 Optimization Result
d[mm] | D[mm] | lf [mm] | dl[mm] | Dl[mm] | l0l[mm] | k0[N/m] | a1[N/m] | st[mm] | a3[N/m3] | pc |
---|---|---|---|---|---|---|---|---|---|---|
1.6 | 25.1 | 90 | 1.71 | 26.9 | 70 | 165.9 | 165.9 | 48.7 | 2.73e5 | 0.3 |
1.5 | 22.2 | 80 | 1.75 | 26.9 | 70 | 185.7 | 185.7 | 40 | 2.58e5 | 0.3 |
1.4 | 20 | 72.7 | 1.87 | 26.9 | 70 | 231.4 | 253.4 | 35 | 2.15e5 | 0.3 |