the sample stage in terms of the length l width w and thickness t of the illustrated bending element 42 see fig4 b the self resonance is given by technical publication tp 237 from morgan electro ceramics f r 0 162 ue89e ue89e t l 2 ue89e y 11 e u03c1 where y 11 e is the bending young 39 s modulus applied in short circuit conditions and u03c1 is the material density for an unloaded bending element i e the sample stage mass m 0 the geometry of the bending element leads to f r 0 162 ue89e t 3 l 3 ue89e y 11 e ue89e w m 1 when the mass of the stage is added m can be replaced by u03b1m where m is the mass of the stage and u03b1 is a factor on the order of unity that reflects the difference between the mass distribution in this simple model and the real mass distribution cast in terms of a bending spring constant for the system f r 0 162 ue89e k u03b1 ue89e ue89e m ue89e ue89e where ue89e ue89e k w ue89e ue89e t 3 l 3 ue89e y 11 e 2 for pzt 5b bimorph materials technical publication tp 237 from morgan electro ceramics y 11 e is 6 u00d7 10 10 n m 2 as an example.