hw_4_solu

# hw_4_solu - ME 125NT Intro to Nanotechnology Due Solutions...

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ME 125NT Intro to Nanotechnology Due: 4/29/2008 1 Solutions Set 4 Problem 5.7 in book A silicon cantilever beam is 300 μm long, 100 μm wide and 6 μm thick. Silicon’s modulus of elasticity is 110 GPa; its density is 2330 kg/m 3 . (a) Determine the spring constant, k , of this beam. (b) Determine the natural frequency in radians per second. (c) Express your answer from part (b) In hertz. ANSWER A) ( )( )( ) ( ) = × × × × = = - - - 3 6 3 6 9 9 3 3 m 10 300 4 m 10 6 m 10 100 Pa 10 110 4 L wt E k M beam 0.022 N/m B) The mass of the beam is: ( )( )( )( ) kg 10 2 4 m 10 6 m 10 100 m 10 300 kg/m 330 2 10 6 6 6 3 - - - - × = × × × = = . V m ρ Natural frequency, ω n , using the effective mass of the beam, m eff =0.24 m , is: ( ) = × = = - kg 10 2 4 24 0 N/m 022 0 24 0 10 . . . m . k n ω 1.48 × 10 4 rad/s C) = = π 2 n n f 2400 Hz Problem 5.13 in book A cantilever beam has length, l=100 μm; width, w=40 μm ; thickness, t=5 μm ; modulus of elasticity, E M =100 GPa; density, ρ= 2000 kg/m 3 ; and damping coefficient , b=10 - 7 N* s/m . (a) What is the beam’s damped natural frequency, f (Hz)? (Do not forget, for a cantilever beam m eff =0.24 m in the equation for damped natural frequency.) (b) What is its quality factor? (c) A periodically varying driving force with F 0 =10 nN acts on the beam. Plot the amplitude, A, of the beam’s oscillation as a function of the drive frequency, F drive . Include F d on the graph so that the resonance peak is visable. (d) On the same graph, plot the amplitude if the damping coefficient quadruples. Label this curve “extra damping.” (e) On the same graph, plot the amplitude if, instead, the effective mass of the beam increases by 500 femtograms. Label this curve “extra mass.” (f) Finally, on the same graph, plot the amplitude if, instead, the spring constant increases by 5 N/m. ANSWER

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ME 125NT Intro to Nanotechnology Due: 4/29/2008 2 A) ( ) ρ π 96 0 2 24 0 4 2 1 24 0 2 1 2 3 3 . E l t lwt . l wt E m . k f M M n = = = B) ( ) M M E lc wt c lwt l wt E c km Q 2 3 3 2 4 = = = C) ( ) ( ) = × × × = = - - 3 9 2 6 6 2 kg/m 2000 96 0 Pa 10 100 m 10 100 2 m 10 5 96 0 2 . . E l t f M n 574,300 Hz D, E, F, G) 0.E+00 5.E-09 1.E-08 2.E-08 2.E-08 3.E-08 3.E-08 550000 555000 560000 565000 570000 575000 580000 585000 590000 595000 600000 drive frequency, f d [Hz] amplitude, A [m] part (e) extra damping part (f) extra mass
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hw_4_solu - ME 125NT Intro to Nanotechnology Due Solutions...

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