#### 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.

#### Stiffness

To get an idea of the rotational stiffness of a cross spring pivot we first take a look at a reference situation in which the leaf springs don’t cross but have a virtual pivot above.

For two leaf springs that both have the same width $w$ and thickness $t$, and for which its pole is located at a distance $a\cdot L$ along the leaf springs the rotational stiffness is given by:

$K_{z}=8\left(1-3a+3a^2\right)\frac{EI}{L}=8\left(1-3a+3a^2\right)\frac{Ewt^3}{12L}$

With $I$ the area moment of inertia of a single leaf spring.

This equation also applies to cross spring pivots in which $a<1$, and where the pole will lie within the mechanism (see examples below).

#### Reference equations

Rotational stiffness (for $a=0.5$):

$K_{ref}=2\frac{EI}{L}=\frac{Ewt^3}{6L}$

Buckle load (radial):

${F_b}_{ref}=8\pi^2\frac{E\cdot I}{L^2}=\frac{\pi^2E\cdot b\cdot t^3}{3L^2}$

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:

$\frac{\rho}{L}=-\frac{\sqrt2}{30}\left(36\left(a-0.5\right)^2-5\right)$

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

For $a=0.5$. Most often assembled from 3 separate plate spring elements with width $\frac{1}{4}w,\frac{1}{2}w,\frac{1}{4}w$. No pure pivot motion, but also parasitic displacement.

ρ ⁄ L | 0.236 |

Angular stiffness * | 0.5 • K_{ref} |

Buckle load * | 0.5 • F_{bref} |

Maximum angle | θ_{ref} |

*A factor 0.5 is included because the total width of the leaf springs is w, while in the reference situation this is 2w.

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

For $a=0.5\pm\frac{\sqrt5}{6}$. Special case classical double symmetric cross spring pivot where $\frac{\rho}{L}=0$ and thus pure pivot behavior.

ρ ⁄ L | 0 |

Angular stiffness * | 2.67 • K_{ref} |

Buckle load * | 0.5 • F_{bref} |

Maximum angle | 1/3.3 • θ_{ref} |

*A factor 0.5 is included because the total width of the leaf springs is w, while in the reference situation this is 2w.

#### ‘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 • K_{ref} |

Buckle load * | F_{bref} |

Maximum angle * | 1/4 • θ_{ref} |

*Not according to reference equations due to coupled leaf springs.