Triangular body supported by tangential struts

Construction Fundamentals

Introduction

In many precision engineering solutions, flexure design with a frame configuration is used, where an equi-triangular body is supported by three tangentially oriented struts, to provide motion in $x$, $y$ and $R_z$. This sheet provides formulas for stiffness and displacement of the main body as function of the individual stiffness’s and displacements of the struts.

Strut stiffness

Assume each individual strut has a longitudinal $(c_L)$ and an out-of-plane Z-direction stiffness $(c_T)$ like depicted in the following sketch:

Strut stiffness
Triangular body in struts Complete

Mechanism stiffness

Assuming the typical case where ${c_1}_L=\ {c_2}_L=\ {c_3}_L=c_L$ and ${c_1}_T=\ {c_2}_T=\ {c_3}_T=c_T$, then the following stiffness’s can be derived for the mechanism:

XYZ coordinate system

$C_x=C_y=\frac{3}{2}\cdot c_L$
$C_z=3{\cdot c}_T$

$K_{Rx}=K_{Ry}={\frac{3}{2}}{\cdot c}_z\cdot R^2$
$K_{Rz}=3{\cdot c}_L\cdot R^2$

UVW coordinate system

$C_u=C_v={\frac{3}{2}}{\cdot c}_L$
$C_z=3{\cdot c}_T$

$K_{Ru}=K_{Rv}={\frac{3}{2}}{\cdot c}_z\cdot R^2$
$K_{Rw}=3{\cdot c}_L\cdot R^2$

Mechanism displacement

Displacements based on movement input $(s_1,s_2,s_3\ll\ R)$

XYZ coordinate system

$\left[\begin{matrix}x\\y\\R_z\\\end{matrix}\right]=\left[\begin{matrix}{\frac{2}{3}}&-{\frac{1}{3}}&-{\frac{1}{3}}\\0&{\frac{1}{3}}\sqrt3&-{\frac{1}{3}}\sqrt3\\{\frac{R}{3}}&{\frac{R}{3}}&{\frac{R}{3}}\\\end{matrix}\right]\cdot\left[\begin{matrix}s_1\\s_2\\s_3\\\end{matrix}\right]$

$\left[\begin{matrix}s_1\\s_2\\s_3\\\end{matrix}\right]=\left[\begin{matrix}1&0&{\frac{1}{R}}\\-{\frac{1}{2}}&{\frac{1}{2}}\sqrt3&{\frac{1}{R}}\\-{\frac{1}{2}}&-{\frac{1}{2}}\sqrt3&{\frac{1}{R}}\\\end{matrix}\right]\cdot\left[\begin{matrix}x\\y\\R_z\\\end{matrix}\right]$

UVW coordinate system

$\left[\begin{matrix}u\\v\\R_w\\\end{matrix}\right]=\left[\begin{matrix}\cos{\left(\alpha\right)}&-\sin{\left(\alpha\right)}&0\\\sin{\left(\alpha\right)}&\cos{\left(\alpha\right)}&0\\0&0&1\\\end{matrix}\right]\cdot\left[\begin{matrix}x\\y\\R_z\\\end{matrix}\right]=\left[\begin{matrix}\cos{\left(\alpha\right)}&-\sin{\left(\alpha\right)}&0\\\sin{\left(\alpha\right)}&\cos{\left(\alpha\right)}&0\\0&0&1\\\end{matrix}\right]\cdot\left[\begin{matrix}{\frac{2}{3}}&-{\frac{1}{3}}&-{\frac{1}{3}}\\0&{\frac{1}{3}}\sqrt3&-{\frac{1}{3}}\sqrt3\\{\frac{R}{3}}&{\frac{R}{3}}&{\frac{R}{3}}\\\end{matrix}\right]\cdot\left[\begin{matrix}s_1\\s_2\\s_3\\\end{matrix}\right]$

$=\left[\begin{matrix}{\frac{2}{3}}\cdot\cos{\left(\alpha\right)}&-{\frac{1}{3}}\cdot{\left(\cos{\left(\alpha\right)}-\sqrt3\cdot\sin(\alpha)\right)}&-{\frac{1}{3}}\cdot{\left(\cos{\left(\alpha\right)}+\sqrt3\cdot\sin(\alpha)\right)}\\{\frac{2}{3}}\cdot\sin{\left(\alpha\right)}&-{\frac{1}{3}}\cdot{\left(\sin{\left(\alpha\right)}+\sqrt3\cdot\cos{(\alpha)}\right)}&-{\frac{1}{3}}\cdot{\left(\sin{\left(\alpha\right)}-\sqrt3\cdot\cos{(\alpha)}\right)}\\{\frac{R}{3}}&{\frac{R}{3}}&{\frac{R}{3}}\\\end{matrix}\right]\cdot\left[\begin{matrix}s_1\\s_2\\s_3\\\end{matrix}\right]$

$\left[\begin{matrix}s_1\\s_2\\s_3\\\end{matrix}\right]=\left[\left[\begin{matrix}\cos{\left(\alpha\right)}&-\sin{\left(\alpha\right)}&0\\\sin{\left(\alpha\right)}&\cos{\left(\alpha\right)}&0\\0&0&1\\\end{matrix}\right]\cdot\left[\begin{matrix}{\frac{2}{3}}&-{\frac{1}{3}}&-{\frac{1}{3}}\\0&{\frac{1}{3}}\sqrt3&-{\frac{1}{3}}\sqrt3\\{\frac{R}{3}}&{\frac{R}{3}}&{\frac{R}{3}}\\\end{matrix}\right]\right]^{-1}\cdot\left[\begin{matrix}u\\v\\R_w\\\end{matrix}\right]$

$\left[\begin{matrix}s_1\\s_2\\s_3\\\end{matrix}\right]=\left[\begin{matrix}\cos{\left(\alpha\right)}&\sin(\alpha)&{\frac{1}{R}}\\{\frac{1}{2}}\cdot{\left(-\sqrt3\cdot\sin{\left(\alpha\right)}-\cos{(\alpha)}\right)}&{\frac{1}{2}}\cdot{\left(\sqrt3\cdot\cos{\left(\alpha\right)}-\sin{(\alpha)}\right)}&{\frac{1}{R}}\\{\frac{1}{2}}\cdot{\left(\sqrt3\cdot\sin{\left(\alpha\right)}-\cos{(\alpha)}\right)}&{\frac{1}{2}}\cdot{\left(-\sqrt3\cdot\cos{\left(\alpha\right)}-\sin{(\alpha)}\right)}&{\frac{1}{R}}\\\end{matrix}\right]\cdot\left[\begin{matrix}u\\v\\R_w\\\end{matrix}\right]$

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