Flexure engineering fundamental: Folded leaf Spring

Construction Fundamentals

Introduction

Folded leaf springs can be used for stiffness in one direction as an alternative for a rod.

Equations folded leaf spring with free end

$F_x=F\cdot cos\left(\theta\right)$
$F_y=F\cdot sin\left(\theta\right)$

$I=\frac{1}{12}bt^3$

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

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

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

$\delta s_{projected\ at\ \theta}=\delta s_{absolute}\cdot\cos{\left(\theta-\varphi\right)}$

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

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

Moment & Stress

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

Common case: $L_1=\ L_2=L$

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

$\delta\ s_{y\ at\ \theta}=\frac{L^3}{EI}\left(\frac{4}{3}F_y-\frac{1}{2}F_x\right)$

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

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

Stiffness of folded leaf spring in stiff direction (all cases)

$C_z=\frac{1}{\frac{1}{C_b}\ +\ \frac{1}{C_s}}=\frac{Etb^3}{\left(L_1+L_2\right)^3+2b^2(L_1+L_2)(1+\nu)}$
$C_b=\frac{Etb^3}{\left(L_1+L_2\right)^3}$ (bending)
$C_s=\frac{Ebt}{2\left(1+\nu\right)(L_1+L_2)}$ (shear)

Note: this is under the assumption that at $C_z$ the rotations are fixed, which is common in a 3 parallel & tangential folded leaf springs configuration. Torsion of leaf spring 1 is also not taken into consideration.

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