Design methodology and performance analysis of application-oriented flexure hinges Rev. Sci. Instrum. 84, 075005 (2013); A generalized analytical compliance model for transversely symmetric three-segment flexure hinges Rev. Sci. Instrum. 82, 105116 (2011); 10.1063/1.3656075 Tractable model for concave flexure hinges Rev. Sci. Instrum. 82, 015106 (2011); 10.1063/1.3505114 Stiffness characterization of corner-filleted flexure hinges Rev. Sci. Instrum. 75, 4896 (2004);This paper presents closed form equations based on a modification of those originally derived by Paros and Weisbord in 1965, for the mechanical compliance of a simple monolithic flexure hinge of elliptic cross section, the geometry of which is determined by the ratio ⑀ of the major and minor axes. It is shown that these equations converge at ⑀ϭ1 to the Paros and Weisbord equations for a hinge of circular section and at ⑀ ⇒ϱ to the equations predicted from simple beam bending theory for the compliance of a cantilever beam. These equations are then assessed by comparison with results from finite element analysis over a range of geometries typical of many hinge designs. Based on the finite element analysis, stress concentration factors for the elliptical hinge are also presented. As a further verification of these equations, a number of elliptical hinges were manufactured on a CNC milling machine. Experimental data were produced by applying a bending moment using dead weight loading and measuring subsequent angular deflections with a laser interferometer. In general, it was found that predictions for the compliance of elliptical hinges are likely to be within 12% for a range of geometries with the ratio  x ͑ϭt/2a x ͒ between 0.06 and 0.2 and for values of ⑀ between 1 and 10.
The motion of an inertial particle in a viscous streaming flow of Reynolds number order 10 is investigated theoretically and numerically. The streaming flow created by a circular cylinder undergoing rectilinear oscillation with small amplitude is obtained by asymptotic expansion from previous work, and the resulting velocity field is used to integrate the Maxey-Riley equation with the Saffman lift for the motion of an inertial spherical particle immersed in this flow. It is found that inertial particles spiral inward and become trapped inside one of the four streaming cells established by the cylinder oscillation, regardless of the particle size, density and flow Reynolds number. It is shown that the Faxén correction terms divert the particles from the fluid particle trajectories, and once diverted, the Saffman lift force is most responsible for effecting the inward motion and trapping. The speed of this trapping increases with increasing particle size, decreasing particle density, and increasing oscillation Reynolds number. The effects of Reynolds number on the streaming cell topology and the boundaries of particle attraction are also explored. It is found that particles initially outside the streaming cell are repelled by the flow rather than trapped.
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