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example_III_2_02

# example_III_2_02 - j Guess" j L(2*j-1#\$/2 1 \$/2 1.8751...

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Example 3 - free response of a cantilevered bending beam Find the general form of the free vibration solution of a cantilevered homogeneous bending beam having a flexural rigidity EI, cross-sectional area A and mass/volume ! . x = 0 x=L u(x,t) EI, ! , A From last example, we know that: u x , t ( ) = " x ( ) T t ( ) where T t ( ) = c cos " t + s sin " t " x ( ) = a cosh # x + b sinh # x + c cos # x + d sin # x

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! L/ " -1.5 -1 -0.5 0 0.5 1 1.5 0 1 2 3 4 5 cos ! L - 1/cosh ! L Numerical Results (the first three roots are also presented in Section 9.5 of text):
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Unformatted text preview: j Guess " j L (2*j-1) #\$ /2 1 \$ /2 1.8751 1.5708 2 3 \$ /2 4.6941 4.7124 3 5 \$ /2 7.8548 7.8540 4 7 \$ /2 10.9955 10.9956 5 9 \$ /2 14.1372 14.1372 6 11 \$ /2 17.2788 17.2788 7 13 \$ /2 20.4204 20.4204 8 15 \$ /2 23.5619 23.5619 9 17 \$ /2 26.7035 26.7035 10 18 \$ /2 29.8451 29.8451...
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