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MEM423Su07-08_Ex2_soln_pdf

# MEM423Su07-08_Ex2_soln_pdf - MEM 423 Mechanics of...

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MEM 423: Mechanics of Vibrations ______________________________________________________________________________ Mid-term Examination I take-home Team project relevance to a-k criteria: {a, b, e, g, k} Summer 2007-08 due: 08/16/08 11:59 PM submit electronic copies(pdf or word) to Halim, with cc to me ______________________________________________________________________________

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Consider a bridge represented as a simply supported uniform beam shown below. The effect of an operating equipment at the middle of the bridge is represented by a concentrated sinusoidal force F of magnitude F o = 1000 N, and frequency o = 2 rad/sec . Its material and geometric properties are (Note : these are team specific): Cross Sectional Area,A (m 2 ) Length,L (m) Inertia, I (m 4 ) Modulus of Elasticity,E (N/m 2 ) Density, (kg/m 3 ) A- Team 5 30 20 1.0 10 6 1.0 10 4 B- Team 7.5 30 30 1.0 10 6 1.0 10 4 C- Team 5 30 20 2.0 10 6 2.0 10 4 D- Team 7.5 30 30 2.0 10 6 2.0 10 4 E- Team 2.5 30 10 2.0 10 6 2.0 10 4 F- Team 2.5 30 10 1.0 10 6 1.0 10 4 G- Team 5 30 20 3.0 10 6 3.0 10 4 1. Using the lumped parameter approach a. determine the natural frequencies (as many as appropriate) b. determine and plot the modal vectors c. determine the nodes, if any d. plot the response of the mid-point with zero initial conditions. for each of the following cases (be careful in locating the masses): i) 1-mass approximation ii) 2-mass approximation iii) 3-mass approximation iv) 4-mass approximation v) 5-mass approximation
2. Compare the results you get as the # of masses increases in your approximation, and make observations. Can you anticipate the “actual” mode shapes as the # of masses increases to a large number? 3. Submit a professional written report including the code you have used. 2 2 2 2 2 ( ) ( );0 ......... ( 1) 6 ( ) ( ) ( 2 ); ......... ( 2) 6 fbx w x l b x x a E EIl fa l x w x a x lx a x l E EIl  

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solution Setup the system in the form: Mq Kq Eu  where q are the DOFs (x 1 , x 2 , ..etc; as shown in class for the wing example), and u = F o
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