FLEXURE ENGINEERING FUNDAMENTAL – FOLDED LEAF SPRING

Construction fundamentals

INTRODUCTION

A Folded leaf springs is one of the mostly used flexure engineering fundamentals; it can be used for stiffness in one direction as an alternative for a rod. A major benefit is that the folded leaf spring does not comprise sag when displaced and the range of motion is often larger. A disadvantage in relation to the rod spring is the manufacturability.

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Pro’s & Con’s

No parasitic displacements
Force (grey arrow) and displacement (red arrow) are not Unidirectional. To obtain an identical direction an additional is necessary and thus ‘extra stiffness’ is introduced.

 Equations folded leaf spring with free end

F_x=F{\bullet}\cos (\theta ) , F_y=F{\bullet}\sin (\theta )

I=\frac 1{12}bt^3
\delta s_{x\mathit{at}\theta }=\frac{L_1^2}{\mathit{EI}}\left(\frac 1 3F_xL_1-\frac 1 2F_yL_2\right)

\delta s_{y\mathit{at}\theta }=\frac{L_2}{\mathit{EI}}\left(\frac 1 3F_yL_2^2+F_yL_1L_2-\frac 1 2F_xL_1^2\right)

\delta s_{\mathit{absolute}\mathit{at}\theta }=\sqrt{\delta s_y^2+\delta s_x^2}

\delta s_{\mathit{projected}\mathit{at}\theta }=\delta s_{\mathit{absolute}}{\bullet}\cos}(\theta -\varphi )

C_{\mathit{absolute}\mathit{at}\theta }=\frac F{\delta s_{\mathit{absolute}}} (bidirectional)

C_{\mathit{projected}\mathit{at}\theta }=\frac F{\left|\delta s_{\mathit{projected}}\right|} (unidirectional)

 Moment & Stress

M_{\mathit{vertical}\mathit{beam}}\left(y\right)=F_xL_1-F_yL_2-F_xy
M_{\mathit{horizontal}\mathit{beam}}(x)=F_yL_2-F_yx
\sigma _{\mathit{max}}=\frac{\left|M_{\mathit{max}}\right|{\bullet}\frac 1 2t} I

 Common case:L_1=L_2=L

\delta s_{x\mathit{at}\theta }=\frac{L^3}{\mathit{EI}}\left(\frac 1 3F_x-\frac 1 2F_y\right)
\delta s_{y\mathit{at}\theta }=\frac{L^3}{\mathit{EI}}\left(\frac 5 6F_y-\frac 1 2F_x\right)

 Guided case: \varphi =\theta and L_1=L_2=L

C=\frac{15} 2\frac{\mathit{EI}}{L^3}
C_{\mathit{at}\theta }=C\left(1+\frac 3 5\sin \left(2\theta \right)\right)

Leaf Spring - Folded
 Stiffness of folded leaf spring in stiff direction (all cases)

C_z=\frac {1}{\frac {1}{C_b}+\frac {1}{C_s}}=\frac {Etb^3}{(L_1+L_2)^3+2b^2(L_1+L_2)(1+\nu )}

C_b=\frac {Etb^3}{(L_1+L_2)^3}

(bending)

C_s=\frac {Ebt}{2(1+\nu )(L_1+L_2)}

(shear)

Stiffness graph
Stiffness Graph
Legend
Legend Stiffness Graph

This page uses QuickLaTeX to display formulas.