#### Introduction

2 leaf springs in parallel are often use as a (quasi-) linear guidance were play must be eliminated.

#### Pro’s & con’s

- Play/backlash free
- Well predictable stiffness ($C_x$)

- (small) Stiffness in direction of movement

- Parasitic displacements ($u_z$)
- Short stroke

#### Elimination of parasitic displacements

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

The drive stiffness ($C_x$) halves; however the guiding stiffness ($C_z$) 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:

Leaf spring with L,b,t then:

$L_0=\frac{1}{6}L$

$t_0=0.9t$

$t_{RF}=5t_0$

$L_H=\frac{5}{6}L$

$h=\frac{1}{2}t$

$D=2h$

(matching movement)

(matching $C_x$)

(guideline for reinforcement)

(matching movement)

(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} $

$C_z=2\frac{EA}{L}=\frac{2Ebt}{L}$ only if $u_x=0$

$C_z=\frac{2}{\frac{L}{EA}+\frac{u_x^2L}{700EI_z}}=\frac{350Eb^3t}{\left(175b^2+3u_x^2\right)L}$ for $u_x\neq0$

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

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

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

#### Motion

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

dynamic movements: $\sigma_{max}<$ fatigue stress limit

static deformation: $\sigma_{max}<$ yield stress limit ($\sigma_{0.2}$)

#### Overconstrained 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 $F_x$ should be applied at $L/2$ as depicted below.