FLEXURE OR SPRING SHAPES

Construction fundamentals

INTRODUCTION

Various shapes can be used as a flexure or spring to provide certain stiffness. In this sheet several shapes are described.

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Also the stroke according to the maximum occurring stress in the particular shape is added; maximum stroke at yield stress.

\tau _{0.2}{\approx}0.4{\bullet}\sigma _{0.2}
(JPE Estimate for metals)

Below materials are characterized with: E,\sigma _{0.2},\nu

Spring Shapes
Buckled plate / wire spring

F_{buckling}=\frac{\pi ^2EI}{L^2}=\frac{\pi ^2Ebt^3}{12L^2}
F_{buckling}=\frac{\pi ^2EI}{L^2}=\frac{\pi ^3Ed^4}{64L^2}
C{\approx}0 (Constant force F)
\delta s_{max}= (unknown)

Buckled Plate Wire Spring
Compression / tension spring

C=\frac {F}{\delta s}=\frac{d^4E}{16nD^3(1+\nu )}
\delta s=\frac{2\pi nD^2(1+\nu )}{dE}\tau

Compression Tension Spring
Torsion wire spring

M=Fr
K=\frac {M}{\delta \varphi }=\frac{Ed^4}{64nD}
\delta \varphi =\frac{2\pi nD}{Ed}\sigma

Torsion Wire Spring
Inclined compression spring

C=\frac{F}{\delta s}=\frac{d^4E}{32n(r_1+r_2)(r_1^2+r_2^2)(1+\nu )}
\delta s=\frac{2\pi n(r_1+r_2)(r_1^2+r_2^2)(1+\nu)}{dr_2E}\tau

Inclined Compression Spring
Spiral plate / wire spring

M=Fr
K=\frac {M}{\delta \varphi }=\frac{Ebt^3}{12L}
K=\frac {M}{\delta \varphi }=\frac{\pi Ed^4}{64L}
\delta \varphi =\frac{2L}{Et}\sigma
\delta \varphi =\frac{2L}{Ed}\sigma

Spiral Plate Wire Spring

(L= length of spring)

Ring plate / wire spring*

C=\frac{F}{\delta s}=4.48\frac{Ebt^3}{D^3(1-\nu ^2)}
C=\frac{F}{\delta s}=4.48\frac{Ed^4}{D^3(1-\nu ^2)}
\delta s=6.72\frac{D^2}{2Et}\sigma
\delta s=6.72\frac{D^2}{2Ed}\sigma

Ring Plate Wire Spring
Disc spring (DIN 2092)

r_1=\frac{r_2}{2}
C=\frac {F}{\delta s}=\frac{Et^3}{0.69r_2^2(1-\nu ^2)}
\delta s_{max}=h

Disc Spring DIN 2092*
Plate / wire spring

C=\frac{3EI}{L^3}=\frac{Ebt^3}{4L^3}
C=\frac{3EI}{L^3}=\frac{3\pi Ed^4}{64L^3}
\delta s=\frac{4L^2}{3Et}\sigma
\delta s=\frac{4L^2}{3Ed}\sigma

Plate Wire Spring

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