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SolSec8.5

# SolSec8.5 - Problems and Solutions Section 8.5(8.44 through...

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Problems and Solutions Section 8.5 (8.44 through 8.49) 8.44 Derive a consistent-mass matrix for the system of Figure 8.9. Compare the natural frequencies of this system with those calculated with the lumped-mass matrix computed in Section 8.5. Solution: Using the vibration toolbox M = ρΑλ .6857 0 0 .7238 The natural frequencies are then ϖ 1 = .8311 1 l E r and w 1 =1.479 1 l E r These are higher than those predicted with the inconsistent mass matrix 8.45 Consider the two beam system of Figure P8.45. Use VTB8_1 to create a two- element, rod/beam element model and compute the first three natural frequencies. Use A = 0.0004 m 2 , I = 1.33 × 10 -8 m 4 , and the properties of aluminum. Assume that nodes 1 and 3 are clamped. Solution: %scipt file for problem 8.45 node=[0 0;1 .5;2 1;1 1.5;0 2]; ncon=[1 2 69e10 .004 1.33e-8 0 2700; 2 3 69e10 .004 1.33e-8 0 2700; 3 4 69e10 .004 1.33e-8 0 2700; 4 5 69e10 .004 1.33e-8 0 2700]; zero=[1 1; 1 2; 1 3; 5 1; 5 2; 5 3]; conm=[]; force=[]; save VTB8_45.con Running this yields that the first three natural frequencies are given as 377.5, 8763.7 and 10951.2 rad/s.

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8.46 Follow the procedure of Problem 8.45 using two elements for each beam.
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