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

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.

F14.14
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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.

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