CROSS FLEXURE (CROSS SPRING PIVOT)

Construction fundamentals

INTRODUCTION

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

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The layout of a cross flexure 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 \bullet I}{L}=\frac{E \bullet b \bullet t^3}{6L}

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

Buckle load (radial)

F_{b_ref}=4\pi ^2\frac{E \bullet I}{L^2}=\frac {\pi^2E \bullet b \bullet t^3}{3L^2}

\frac{\rho }{L}=-\frac{\sqrt 2}{30}(36\lambda ^2-5)

Maximum angle

\theta _{ref}=2\frac{\sigma_{max} \bullet L}{E \bullet t}

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
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 ⁄ L0.236
Angular stiffnessCref
Buckle loadFbref
Maximum angleθref
‘Haberland’ cross spring pivot
'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 stiffness4 • Cref
Buckle load2 • Fbref
Maximum angle1/4 • θref

*) Not according formula above due to coupled leaf springs

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

Special case classical double symmetric cross spring pivot for \lambda =±\frac{\sqrt 5}{6} where \frac{\rho }{L}=0 and thus pure pivot behavior.

ρ ⁄ L0
Angular stiffness2.67 • Cref
Buckle loadFbref
Maximum angle1/3.3 • θref

Sources:

  • On the design of plate-spring mechanisms – J. van Eijk
  • Elastische geleidingen, een literatuursutdie – M.N. Boneschanscher

This page uses QuickLaTeX to display formulas.