Question 14.7: Spiral Torsion Spring: Design for Static Loading For a torsi......

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) $A$ 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 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.

Question: 14.8

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