FLEXURE GUIDING – 2 LEAF SPRINGS IN PARALLEL

Construction Design & Examples

INTRODUCTION

A flexure guiding with 2 leaf springs or flexures in parallel are often use as a (quasi-) linear guidance were play must be eliminated. This sheet allows you to design such a construction, how to reinforce it and how to eliminate the parasitic motion.

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Pro’s & Con’s

Play/backlash free

Well predictable stiffness (Cx)

Parasitic displacements (uz )

Short stroke

(small) Stiffness in direction of movement

Elimination of parasitic displacements

Through a double parallel leaf spring (in series) the parasitic displacement can be eliminated, like:

The drive stiffness (Cx) halves; however the guiding stiffness (Cz) halves as well.

Leaf spring configuration

For machinability, often reinforced leaf springs or 2 elastic hinges in series are used as an alternative per leaf spring. If so use the following guide-lines:

Each flexure with L, b, t then:
L_0=\frac{1}{6}L (matching movement)
t_0=0,9t (matching C_x)
t_{RF}=5t_0 (guideline reinforcement)
L_H=\frac{5}{6}L (matching movement)
h=\frac{1}{2}t
D=2h (elastic hinge guide line)

Stiffness

C_x=2\frac{12EI_z}{L^3}=\frac{2Ebt^3}{L^3}

C_y=2\frac{3EI_x}{L^3}=\frac{Eb^3t}{2L^3}

u_x=0: C_z=2\frac{EA}{L}=\frac{2Ebt}{L}

u_x\not=0: C_z=\frac{2}{\frac{L}{EA}+\frac{{u_x^2}L}{700EI_z}}=\frac{350Eb^3t}{(175b^2+3u_x^2)L}

K_x=2\frac{EI_x}{L}=\frac{Eb^3t}{6L}

K_y=\frac{C_z}{2}(2r)^2=\frac{4Ebtr^2}{L}

K_z=\frac{C_y}{2}(2r)^2=\frac{Eb^3tr^2}{L^3}

Motion

u_x=\frac{L^2\sigma}{3Et}, u_z=\frac{3u_x^2}{5L}

dynamic movements: \sigma_{max}< fatigue stress limit
quasi-static movements: \sigma_{max}< yield stress limit (\sigma_{0,2})

Over constrained design

Essentially, 2 parallel leaf springs are over constrained. This could be overcome if internal elasticity is introduced like low torsion stiffness of the moving body or notching 1 out of 2 leaf springs. Practically, the best way is to machine the fixed world, the leaf spring and the moving body monolithically.

Applying Force Fx

To ensure identical normal force on each leaf spring and thus; a pure linear guidance, the force Fx should be applied at L/2 as depicted below.

Where to apply force to 2 flexures in parallel

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