the bending moments satisfy
The determination of radial and tangential components of the shearing forces Qr and Qt , as functions of the moments, is derived from the equilibrium of a segment trd6 dr around tangential axis Or' parallel to cdt and around radial axis Offl, respectively (Fig. 3.1).
For the radial shearing force Qr, the resulting components around Ot' are
0, where the third term in Mt is the sum of two components tilted of ±d6/2 from the radial axis Orn. After simplification, the radial shearing force is represented by dMr 1 ( dMrt
The tangential shearing force Qt is derived from the moments around Oa. The resulting components are
and after simplification, the tangential shearing force is
Finally the external load q applied per unit area onto the surface of the elementary segment is in static equilibrium with the shearing forces (Fig. 3.2). After dividing the terms of the equilibrium equation by element area rdQ dr, d (n ï , dQt dr ) + dë
This partial derivative equation linking the shearing forces to external load q is the general relation of variable thickness plates . This equation degenerates into Poisson's equation DV2V2z - q = 0 if D = constant (cf. Sect. 7.2). The VTD t(r, 9) is defined by
Let us consider flexures having the shape of wavefront aberration modes. These modes belong to a circular polynomial series and are represented by rQrde
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