set Homework #1 1. Using the results
d 2z M F ( y a) = = 2 dy EI EI for the cantilever shown, (a) Show that the deflection of the cantileverat a distance y from the fixed end is 2
z L a y z F
Fy z= ( y 3a ) 6 EI
(b) Show that, for a cantilever with a rectangular cross-section of width w and thickness t, the spring constant of the cantilever is 3
k=
Ewt 4 L3
(c) For silicon E = 1.69x1011 N/m2, and for silicon nitride E = 2.70x1011 N/m2. What dimensions would you use if you were asked to design a cantilever with a spring constant of 50 pN/nm, out of silicon and silicon nitride, respectively? 2. Using equations (5) (9) in the paper three dimensional displacement of a piezoelectric tube scanner, Rev Sci. Instru, 69, 226 (1998) by Yang and Huang, derive the expression for the lateral displacement
2 2d 31l 2Vx x= Dh
for tube a scanner under the assumption that the tube deformation is small, the wall thickness is much smaller than the radius of the tube, and that V+x and V-x have the same amplitude but opposite signs. Here, d31 is the piezoelectric coefficient, is the length of the tube, D is the diameter of the tube and h is the wall thickness. (a) When the tube bends, a tip, originally placed at the center of the tubes top surface, will sense a change in the surface height. Derive an expression of the...