HW_6-Solutions2007 - 1 EECS 414 Introduction to MEMS Fall...

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Unformatted text preview: 1 EECS 414 Introduction to MEMS Fall 2007 Reading Assignments Class Handouts and Notes on materials, mechanical Structures Homework #7 Solutions Total: 230 Points Handed Out: Friday October 19, 2007 Due: Friday October 26, 2007 1. Given a polysilicon beam with the following properties: Length, L 100 m Width, w 20 m Thickness, t 1 m Youngs Modulus, E 160 GPa Poissons Ratio, v 0.23 (a) The beam is fixed on one end and the other end is free to move (cantilever). A force of 5 N is applied at the very end of the beam (on the free end). Calculate the deflection at the tip of the beam. (b) The beam is fixed at both ends. Derive an equation for the maximum deflection. The deflection is maximum if the applied force, F, (assumed to be constant through the width of the beam) is applied at the very center of the beam. The equation should be in terms of the parameters given above (i.e. F, L, w, etc). (c) Using your result in part (b), find the maximum deflection of a beam fixed at both ends if a force of 100 N is applied at the center of the beam. 30 points Solution: (a) For a point load, deflection is given by: ) 3 ( 6 ) ( 3 2 x L x EI F x y ! = For maximum deflection, x=L and in general, I is: 12 23 2 ) 6 1 )( 6 20 ( 12 1 12 1 3 3 ! = ! ! = = e e e wt I Plugging this into the equation for y(x) and solving gives the maximum deflection as 6.25 um. (b) For a beam fixed at both ends and a point force, the deflection is given by: ) 4 3 ( 48 ) ( 2 x Lx EI Fx x y ! = For maximum deflection, x=L/2 and I is: 3 12 1 wt I = Substituting the above in the equation for deflection gives: 3 3 2 2 3 2 3 max 16 ) 2 3 ( 8 ) 4 4 2 3 ( 48 ) 2 / ( 12 Ewt FL L L Ewt FL L L L Ewt L F y = ! = ! = 2 (c) Plugging F=5e-6, L=100e-6, E=160e9, w=20e-6 and t=1e-6 into the equation for y max and solving gives the maximum deflections as 1.95 uN. 2. Suppose we deposit a thin film on a substrate at a temperature T above room temperature. When the film and the substrate are allowed to return to room temperature, a plane stress will develop in the film. The thermal expansion coefficients of the deposited layer and the substrate are F and S , respectively. The film material has Youngs Modulus E F , and the Poissons ratio is F ! . Derive the formula for the plane stress in the film based on the above variables. Assume that the substrate is much thicker than the film; i.e. assume that its expansion or contraction is not affected by the film. Under what conditions will the stress be tensile, and when will it be compressive? 20 points Solution: If the film was not attached to the substrate, its dimensions in the x and y directions would shrink (in other words, it would experience strain): T F YFilm XFilm F ! = = = " # # # The substrate will also experience strain, but in a different amount....
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HW_6-Solutions2007 - 1 EECS 414 Introduction to MEMS Fall...

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