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.

Pro’s & Con’s

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

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) $

Flexure engineering fundamental: Folded leaf Spring

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)

Stiffness graph

Flexure engineering fundamental: Folded leaf Spring Stiffness

Precision Point sheet download

Please fill in your details to receive the requested Precision Point sheet.