FLEXURE DESIGN: 3 DOF STAGE (X, Y, RZ)

Construction Design & Examples

INTRODUCTION

A flexure design that comprises 3 parallel & tangential folded leaf springs are often used as a z, Rx, Ry fixation. Each leaf spring contributes to the total stiffness of the setup. This sheet describes the stiffness and stroke of the individual and the combination of 3 parallel & tangential folded leaf springs at 120°.

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Individual stiffness

The stiffness varies with the angle θ. If the leaf spring is combined with others the following holds*.

3 Parallel tangential folded leaf springs Individual Stiffness

Angle dependent linear stiffness:
C_{t1}(\theta )=C(1+\frac{3}{5}sin⁡(2\theta ))
With: C=\frac {15}{2}\frac {EI}{L^3}
Linear stiffness perpendicular to the drawing:
C_{z1}=\frac {Etb^3}{(2L)^3+2b^2(2L)(1+\nu )}

* If a folded leaf spring is used individually, the stiffness varies according some boundary conditions, see other Precision Point sheet.

Individual stroke
3 Parallel tangential folded leaf springs Individual Stroke

Stiff direction: \delta _{stiff}=\frac {2}{3\sqrt 2}\bullet d\approx 0.47d
Weak direction: \delta _{weak}=\frac {4}{3\sqrt 2}\bullet d \approx 0.94d
Unidirectional: \delta _{uni} \leq 0.42d
With d=\frac{\sigma L^2}{Et}

Combined leaf springs at 120°
3 Parallel tangential folded leaf springs Combined Leaf Springs 120º - 1
3 Parallel tangential folded leaf springs Combined Leaf Springs 120º - 2
Combined Stiffness

Angle dependent stiffness of the folded leaf springs:
C_{t1}(\theta )=C(1+\frac {3}{5}\sin (2\theta ))
C_{t2}(\theta )=C(1+\frac {3}{5}\sin (2\theta +\frac {2}{3}\pi ))
C_{t3}(\theta )=C(1+\frac {3}{5}\sin (2\theta -\frac {2}{3}\pi ))

Combined radial stiffness:
C_{radial}=C_{t1}(\theta )+C_{t2}(\theta )+C_{t3}(\theta )=3 \bullet C=\frac{45}{2} \frac{EI}{L^3}

Combined linear z stiffness:
C_{axial}=3 \bullet C_{z1}=\frac{3}{2} \frac{Etb^3}{L^3}

Combined rotational stiffness:
K_x=K_y=C_{axial} \bullet r^2=\frac {3}{2}\frac {Etb^3r^2}{L^3}
K_z=(C_{t1}(\theta )+C_{t2}(\theta +120^{\circ} )+C_{t3}(\theta -120^{\circ} )) \bullet r^2= \frac{45}{2}\frac {EIr^2}{L^3}

Combined stroke

\delta _{uni}{\leq}0.42d=0.42\frac{\sigma L^2}{Et}
Note that in the stiff direction of 1 folded leaf spring, the other is not stiff, so
only the unidirectional stroke can be used.

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