WrittenHW09-Solutions

WrittenHW09-Solution - Math 216(Section 50 Written Homework#9 – Solutions Fall 2007 1 5.3–6 The mass matrix and stiffness matrix are given by M

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Unformatted text preview: Math 216 (Section 50) Written Homework #9 – Solutions Fall 2007 1 ( 5.3–6 ) The mass matrix and stiffness matrix are given by M = m 1 m 2 = 1 0 0 2 , K =- ( k 1 + k 2 ) k 2 k 2- ( k 2 + k 3 ) =- 6 4 4- 8 Now the equation of motion is M x 00 = K x ⇐⇒ x 00 = A x , where A := M- 1 K . Now, M- 1 = 1 0 1 / 2 and then A = M- 1 K =- 6 4 2- 4 , and thus x 00 =- 6 4 2- 4 x . Let us find the eigenvalues. | A- λI | =- 6- λ 4 2- 4- λ = λ 2 + 10 λ + 16 = ( λ + 2)( λ + 8) = ⇒ λ 1 =- 2 ,λ 2 =- 8 . Thus the corresponding natural frequencies are ω 1 = √ 2 , ω 2 = √ 8 = 2 √ 2 . Let us next find the eigenvectors: • λ 1 =- 2 The eigenvector equation ( A- λI ) v = 0 is- 4 4 2- 2 a b = . Thus an eigenvector is given by v 1 = 1 1 . • λ 1 =- 8 The eigenvector equation ( A- λI ) v = 0 is 2 4 2 4 a b = . Thus an eigenvector is given by v 2 =- 2 1 . 1 Therefore the two natural modes of oscillations are ( a 1 cos ω 1 t + a 2 cos ω 1 t ) v 1 = ( a 1 cos √ 2 t + a 2 cos √ 2 t...
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This homework help was uploaded on 04/02/2008 for the course ENGR 101 taught by Professor Ringenberg during the Fall '07 term at University of Michigan.

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WrittenHW09-Solution - Math 216(Section 50 Written Homework#9 – Solutions Fall 2007 1 5.3–6 The mass matrix and stiffness matrix are given by M

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