hwk03-solution(1).pdf - CIVE342 Homework 3 Solution...

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CIVE342 Homework 3 Solution Question 1 For a three-storey building 1 2 3 0 0 0 0 0 0 m M m m = ; = 3 2 1 x x x x and [ ] 3 2 1 x x x x T = ; So Since 0 , , 3 2 1 > m m m , the above is always positive. Question 2 Drawing the free body diagram of the floor mass 3 m and writing the Newton’s second law gives ) [ ] [ ] 1 1 1 2 3 2 2 3 3 1 1 1 2 2 3 3 2 3 2 2 2 1 1 2 2 3 3 0 0 0 0 0 0 T m x x M x x x x m x m x x x m x m x m x x m x m x m x = × × = × = + +
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2 ( ) ( ) 1 1 1 2 2 1 1 1 1 1 1 2 1 2 2 1 m x F k x x k x m x k k x k x F = + + + = && && Writing these three equations in matrix form gives ) ( ) ( ) ( t F t x K t x M = + & & & where the stiffness matrix K is given by + + = 3 3 3 3 2 2 2 2 1 0 0 k k k k k k k k k K Performing the matrix multiplication, [ ] ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 2 3 2 2 3 3 2 3 3 3 1 1 1 2 2 2 1 2 2 2 3 3 3 2 3 3

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