MIT2_092F09_lec16

MIT2_092F09_lec16 - 2.092/2.093 - Finite Element Analysis...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
2.092/2.093 Finite Element Analysis of Solids Fluids I Fall ‘09 Lecture 16 - Solution of Dynamic Equilibrium Equations, cont’d Prof. K. J. Bathe MIT OpenCourseWare Reading assignment: Sections 9.1-9.3 Recall from our last lecture the general dynamic equilibrium equation and initial conditions: 0 MU ¨ + CU ˙ + KU = R ( t ) ; 0 U , U ˙ (1) This equation can be solved by: Mode superposition Direct integration Mode Superposition i = ω i 2 i (2) The ω i 2 are the eigenvalues and φ i are the eigenvectors for this system. Solve for ω i 2 , φ i : ω 2 ω 2 . . . ω 2 1 2 n 0 ±²³ ±²³ ±²³ for φ 1 for φ 2 for φ n where each φ i refers to a mode shape. Aside: Consider, picking “a” φ , = α ˜ (3) 1 where α is a nonzero scalar. Obviously, K ´ α φ µ = ˜ = R . If φ ˜ is an eigenvector, then the load R 1 obtained using φ ˜ gives us back the vector φ ˜ (now scaled by α ). We also used orthonormality to establish that: φ T i j = δ ij φ T i j = ω i 2 δ ij The de±nition of the Rayleigh quotient is φ T ρ ( φ ) = φ T where φ can be any vector. So, we have ρ ( φ i ) = ω i 2 ω 1 2 ρ ( φ
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/24/2012 for the course MECHANICAL 2.092 taught by Professor Klaus-jürgenbathe during the Fall '09 term at MIT.

Page1 / 5

MIT2_092F09_lec16 - 2.092/2.093 - Finite Element Analysis...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online