**Spiral Torsion Spring: Design for Static Loading**

For a torsional window-shade spring (Figure 14.14), determine the maximum operating moment and corresponding angular deflection.

**Design Decisions:** We select a music wire of E =207 GPa; d =1.625 mm, D =25 mm, and N_a =350. A safety factor of 1.5 is used.

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By Equation (14.12) and Table 14.2,

S_{u s}=A d^b (14.12)

S_u A d^b=2060\left(1.625^{-0.163}\right)=1903 MPa

From Equation (7.5b) and Table 14.3,

S_{y s}=0.577 S_y (7.5b)

S_y=\frac{S_{y s}}{0.577}=0.4 \frac{1903}{0.577}=1319 MPa

Applying Equation (14.36) with C =25/1.625=15.38,

\begin{gathered} K_i=\frac{4 C^2-C-1}{4 C(C-1)} \\ K_o=\frac{4 C^2+C-1}{4 C(C+1)} \end{gathered} (14.36)

K_i=\frac{4(15.38)^2-15.38-1}{4(15.38)(15.38-1)}=1.051

Through the use of Equation (14.39), we have

\sigma_i=\frac{32 P a}{\pi d^3} K_i (round wire) (14.39)

\begin{aligned} M & =P a=\frac{\pi d^3 S_y / n}{32 K_i}=\frac{\pi(1.625)^3(1319 / 1.5)}{32(1.051)} \\ & =352.5 N \cdot mm \end{aligned}

The geometric properties of the spring are L_{ w }=\pi D N_a=\pi(25)(350)=27,489 \quad mm \quad \text { and } \quad I=\pi (1.625)^4 / 64=0.342 mm ^4 . Equation (14.41) results in

\theta_{ rev }=\frac{1}{2 \pi} \theta_{ rad }=\frac{1}{2 \pi} \frac{M L_w}{E I} (14.41)

\theta_{ rad }=\frac{M L_w}{E I}=\frac{352.5(27,489)}{\left(207 \times 10^3\right)(0.342)}=136.9 rad

**Comment:** The maximum moment winds the spring 136.9 / 2 \pi=21.8 turns.

TABLE 14.2 Coefficients and Exponents for Equation (14.12) |
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A |
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Material |
ASTM No. |
b |
MPa |
ksi |

Hard-drawn wire | A227 | −0.201 | 1510 | 237 |

Music wire | A228 | −0.163 | 2060 | 186 |

Oil-tempered wire | A229 | −0 193 | 1610 | 146 |

Chrome-vanadium wire | A232 | −0.155 | 1790 | 173 |

Chrome-silicon wire | A401 | −0 091 | 1960 | 218 |

Source: Associated Spring-Barnes Group, Design Handbook, Associated Spring-Barnes Group, Bristol, CN, 1987. |

TABLE 14.3 Approximate Strength Ratios of Some Common Spring Materials |
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Material |
S_{y s} / S_u |
S_{e s}^{\prime} / S_u |

Hard-drawn wire | 0.42 | 0.21 |

Music wire | 0.40 | 0.23 |

Oil-tempered wire | 0.45 | 0.22 |

Chrome-vanadium wire | 0.52 | 0.20 |

Chrome-silicon wire | 0.52 | 0.20 |

Source: Associated Spring-Barnes Group, Design Handbook, Associated Spring-Barnes Group, Bristol, CN, 1987. | ||

Notes: S_{y s} , yield strength in shear; S_u, ultimate strength in tension; S_{e s}^{\prime} , endurance limit (or strength) in shear. |

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