Problem 4: (30\") An atomic force microscope has a resolution of more than 1000 times the best optical microscope. The basic construction of the atomic microscope is as shown in Fig. 1. Its structure is a cantilever beam AB with a sharp probe BC at its end. The probe contacts the surface, which exerts forces on the tip in the X and Y direction (F, and F,) respectively. Due to these forces the tip deflects in the Y direction. This deflection i1s measured using laser interferometry. iy Lo benen A -"'""'*n-...,,____ AB = 100 gm Figure 1: An atomic force microscope For successful operation of this microscope assuming only vertical forces (i.e. Fx=0). the following criteria needs to be met. 3E5%!2 5 O']. where E is the Young's modulus of polysilicon, & is the tip deflection, b is the thickness of the beam, oy is the yield strength, and / is the length of the beam. The parameter distribution information is shown below. Variable Distribution Mean St. Dev. | E Normal 169 GPa 2GPa | o Normal 5 0.25 b Normal 5 ym 1.5 gm oy Normal 1GPa 0.05GPa | l Deterministic 100 gm | Please (1) formulate a performance function, G, for this requirement, (2) convert it to the standard normal space U, and (3) find out the reliability index, B, using the first order reliability method (FORM) with the following Hasofer-Lind and Rackwitz-Fiessler (HL-RF) iterative MPP search formula. (%) o :[u{k] OVUG(U[\") G(u{\" )] ' VL:G(\" ]' VUG(u{\")| (Note: please start the search in the U-space origin, and only conduct the search for three iterations. Fill the result in the table given below) =71 =3} (>3] S C & 2 Q Iter. u; u; us W B G(u) A. 2u 2 _I_E' =)