Asymmetric fluid-structure dynamics in nanoscale imprint lithography

ase h (for a semi-circular notch,h = t + 2R), and the width b. Twocharacteristic non-dimensional parameters areβ = t and γ = h .2R 2RThe torsional compliance about the z-axis is⎢⎢⎣23 ⎡ 1 ⎤⎧⎡1+β 3 + 2β+ β ⎤=⎡⎢ ⎥⎨⎢ +⎥ 1−γ2 2222EbR⎣2β+ β ⎦⎩⎣ γ γ ( 2β+ β ) ⎦⎢⎣1 2kθMz⎡ ( ) ⎤⎡6 1+ β−13 / 22( 2β+ β )⎥⎢tan⎥⎦⎢⎣⎛⎜⎜⎝2 + β×β1−⎞⎤( γ − β ) ⎟⎥⎟( + − ) ⎥ ⎭ ⎬⎫ 21 β γ⎠⎦( 1+β − ) ⎤ +⎥⎦[B.1]The stiffness along the x-axis is=⎡⎢−⎣γ+2 − γ2β2γβ−1−1kδxFxEb 2 tantan[B.2]( 2 − γ )⎤⎥⎦−1The motion capability of the notched flexure with respect to other directions isnot significant as compared the rotational deflection about the z direction andlinear deflection along the x direction [Paros and Weisbord 1965].106