FLEXURE DESIGN: 3DOF STAGE z, Rx, Ry

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 z, Rx and Ry. This sheet provides formulas for stiffness and displacement of the main body as function of the individual stiffness’s and displacements of the struts.

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Strut stiffness

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

Strut stiffness
Triangular body in struts Complete
XYZ coordinate system
UVW coordinate system

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_I}
then the following stiffness’s can be derived for the mechanism:

C_x=C_y=\frac {3}{2}\centerdot c_L
C_z=3\centerdot c_T

K_{Rx}=K_{Ry}=\frac {3}{2}\centerdot c_z\centerdot R^2
K_{Rz}=3\centerdot c_L\centerdot R^2

C_u=C_v=\frac {3}{2}\centerdot c_L
C_z=3\centerdot c_T

K_{Ru}=K_{Rv}=\frac {3}{2}\centerdot c_z\centerdot R^2
K_{Rw}=3\centerdot c_L\centerdot R^2

Mechanism displacement
Displacements based on movement input (s_1,s_2,s_3{\ll}R)

\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}\sqrt 3&-\frac{1}{3}\sqrt 3\\\frac {R}{3}&\frac {R}{3}&\frac {R}{3}\end{matrix}\right]\centerdot \left[\begin{matrix}s_1\\s_2\\s_3\end{matrix}\right]

\left[\begin{matrix}u\\v\\R_w\end{matrix}\right]=\left[\begin{matrix}\cos (\alpha )&-\sin (\alpha )& 0\\sin (\alpha )&\cos (\alpha )& 0\\0& 0& 1\end{matrix}\right]\centerdot \left[\begin{matrix}x\\y\\R_z\end{matrix}\right]=

\left[\begin{matrix}\cos (\alpha )&-\sin (\alpha )&0\\\sin (\alpha )&\cos (\alpha )&0\\0&0&1\end{matrix}\right]\centerdot \left[\begin{matrix}\frac {2}{3}& -\frac{1}{3}& -\frac{1}{3}\\0&\frac {1}{3}\sqrt 3& -\frac{1}{3}\sqrt 3\\\frac {R}{3}&\frac {R}{3}&\frac {R}{3}\end{matrix}\right]\centerdot \left[\begin{matrix}s_1\\s_2\\s_3\end{matrix}\right]=

\left[\begin{matrix}\frac {2}{3}\centerdot \cos (\alpha )&-\frac{1}{3}\centerdot (\cos (\alpha )-\sqrt 3\centerdot \sin (\alpha ))&-\frac{1}{3}\centerdot (\cos (\alpha )+\sqrt 3\centerdot \sin (\alpha ))\\\frac {2}{3}\centerdot \sin (\alpha )&-\frac{1}{3}\centerdot (\sin (\alpha )+\sqrt 3\centerdot \cos (\alpha ))&-\frac{1}{3}\centerdot (\sin (\alpha ) -\sqrt 3\centerdot \cos (\alpha ))\\\frac {R}{3}&\frac {R}{3}&\frac {R}{3}\end{matrix}\right]\centerdot \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]\centerdot \left[\begin{matrix}\frac 2 3&\frac{-1} 3&\frac{-1} 3\\0&\frac 1 3\sqrt 3&\frac{-1} 3\sqrt 3\\\frac R 3&\frac R 3&\frac R 3\end{matrix}\right]\right]^{-1}\centerdot \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}1&0&\frac 1 R\\-\frac{1}{2}&\frac {1}{2}\sqrt 3&\frac {1}{R}\\-\frac{1}{2}&-\frac{1}{2}\sqrt 3&\frac {1}{R}\end{matrix}\right]\centerdot \left[\begin{matrix}x\\y\\R_z\end{matrix}\right]

\left[\begin{matrix}s_1\\s_2\\s_3\end{matrix}\right]=

\left[\begin{matrix}\cos (\alpha )&\sin (\alpha )&\frac {1}{R}\\\frac{1}{2}\centerdot (-\sqrt 3\centerdot \sin (\alpha )-\cos (\alpha ))&\frac{1}{2}\centerdot (\sqrt 3 \cos (\alpha )-\sin (\alpha ))&\frac {1}{R}\\\frac{1}{2}\centerdot (\sqrt 3\centerdot \sin (\alpha )-\cos (\alpha ))&\frac{1}{2}\centerdot (-\sqrt 3\centerdot \cos (\alpha )-\sin (\alpha ))&\frac{1}{R}\end{matrix}\right]\centerdot \left[\begin{matrix}u\\v\\R_w\end{matrix}\right]

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