Reflecting cantilever beam optical modulator

The electrostatic deflection of a cantilever beam has been derived by Petersen [2] and Kovacs [3]. The deflection of a cantilever beam of width w and length I due to a point load F acting on the end of a beam was found to be 

If we consider a normalized point force q x) = F(x)IA acting on a small segment of the cantilever beam wdx at the position x. 

For an electrostatic force on the cantilever with an initial gap d(, the force would be 

The total tip deflection can be found from integrating the incremental deflection along the length of the cantilever beam  

To solve this equation in closed form, Kovacs made a parabolic approximation for the gap d x) as a function of the tip deflection y t)  

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