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2nd Order Example

# 2nd Order Example - = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − − − =...

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Transverse Vibrations of Beams We now wish to develop a time dependent model of the dynamic motion of beams using simple Euler-Bernoulli theory. This simplified model neglects the effects of rotary inertia and shear on beam elements, and thus simply adds a vertical translation inertia term to the previous static governing equation. The dynamic governing equation is thus given by Going through the usual variational – Ritz Galerkin construction and employing the hermite interpolation functions leads to the finite element model x w ( x,t ) q ( x,t )

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Unformatted text preview: = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − − − = ⋅ ⋅ ⋅ ⋅ ⋅ − − ⋅ − ρ = = + e e e e e e e e e e e e e e e e e e e e e Q Q Q Q Q h h h h h h h h EI K h h h h h h h h A M Q w K w M 4 3 2 1 2 2 2 3 2 2 2 } { 2 3 6 3 2 3 6 3 6 2 ] [ 4 22 156 3 13 4 13 54 22 156 420 ] [ where , } { } ]{ [ } ]{ [ ) , ( 2 2 2 2 2 2 t x q x w EI x t w A = ∂ ∂ ∂ ∂ + ∂ ∂ ρ...
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