example_IV_4_01

example_IV_4_01 - Example IV.4.1 Find the response of the...

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Example IV.4.1 Find the response of the building shown below for x 0 ( ) = ˙ x 0 ( ) = 0 and the forcing f t ( ) = f 0 g t ( ) acting on the fourth floor. Recall from our earlier work that the system has the following natural frequencies and (normalized) modal vectors are: ω 1 = 0.4912 k/m ; ω 2 = 1.4142 k/m ω 3 = 2.1667 k/m ; ω 4 = 2.6579 k/m ˜ P [ ] = φ (1) , (2) , (3) , (4 ) [ ] = 1 m 0.2280 0.5774 0.6565 0.4285 0.4285 0.5774 0.2280 0.6565 0.5774 0 0.5774 0.5774 0.6565 0.5774 0.4285 0.2280 g(t)
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The EOM’s for this model of the building are given by: M [ ] ˙ ˙ x + K [ ] x = f t ( ) where: f t ( ) = 0 0 f t ( ) 0 The uncoupled EOM’s become: ˙ ˙ q j + ω j 2 q j = ˆ f j t ( ) where x t ( ) = ˜ φ j ( ) q j t ( ) j = 1 4 and the modal forcings are given by: ˆ f j t ( ) = ˜ j ( ) T f t ( ) Explicitly we have: ˆ f 1 t ( ) = 0.5774 f 0 m g t ( ) ˆ f 2 t ( ) = 0 ˆ f 3 t ( ) = 0.5774 f 0 m g t ( ) ˆ f 4 t ( ) = 0.5774 f 0 m g t ( ) Solving the modal EOM’s using the convolution integral gives: q j t ( ) = 1 j ˆ f j τ ( ) sin j t ( ) d 0 t Explicitly these responses are:
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q 1 t ( ) = 0.5774 f 0 ω 1 g τ ( ) sin 1 t ( ) d 0 t = g ( ) h 1 t ( ) d 0 t q 2 t ( ) = 0 q 3 t ( ) = 0.5774
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example_IV_4_01 - Example IV.4.1 Find the response of the...

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