# Cross spring pivot

## Construction Fundamentals

#### Introduction

Cross spring pivots are an interesting alternative for common flexure pivots in case transverse loads of the pivot are relatively high.

The layout of a cross spring pivot can be chosen to optimize for large angular motion, or for optimal pivot behavior which is unbiased with parasitic displacements.

Orthogonal cross spring pivot
Pivot constructed from two leaf springs which are oriented perpendicular in relation to each other.

Symmetric cross spring pivot
Pivot constructed from two equal leaf springs which are symmetrically located in relation to the pole.

Double symmetric cross spring pivot
Special case of the symmetric cross spring pivot where the pole is exactly in the middle of the leaf spring.

#### Reference equations

Angular stiffness:
$C_{ref}=2\frac{E\cdot I}{L}=\frac{E\cdot b\cdot t^3}{6L}$

${F_b}_{ref}=4\pi^2\frac{E\cdot I}{L^2}=\frac{\pi^2E\cdot b\cdot t^3}{3L^2}$

Maximum angle:
$\theta_{ref}=2\frac{\sigma_{max}\cdot L}{E\cdot t}$

The equivalent rolling radius of an orthogonal cross spring pivot is given by:

$\frac{\rho}{L}=-\frac{\sqrt2}{30}\left(36\lambda^2-5\right)$

For $\frac{\rho}{L}<0$ the virtual rolling surface flips in relation to real mounting surface of the springs (see ‘Haberland’ cross spring)

#### Classical double symmetric cross spring pivot

Most often assembled from 3 plate spring elements with width $\frac{1}{4}b,\frac{1}{2}b,\frac{1}{4}b$. No pure pivot motion, but also parasitic displacement.

 ρ ⁄ L 0.236 Angular stiffness Cref Buckle load Fbref Maximum angle θref

#### ‘Haberland’ cross spring pivot

Ideally suited for monolithic fabrication. Relatively pure pivot behavior but with less angular stroke compared to classical double symmetric cross spring pivot.

 ρ ⁄ L * -0.047 Angular stiffness 4 • Cref Buckle load 2 • Fbref Maximum angle 1/4 • θref

#### Special case classical double symmetric cross spring pivot

Special case classical double symmetric cross spring pivot for $\lambda=\pm\frac{\sqrt5}{6}$ where $\frac{\rho}{L}=0$ and thus pure pivot behavior.

 ρ ⁄ L 0 Angular stiffness 2.67 • Cref Buckle load Fbref Maximum angle 1/3.3 • θref

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