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Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications. Northern Illinois University Department of Mathematical Sciences Dissertation Defense Vadim Sokolov Advisor: Biswa N. Datta. October 24, 2008. TexPoint fonts used in EMF.

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Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

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  1. Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications Northern Illinois University Department of Mathematical Sciences Dissertation Defense Vadim Sokolov Advisor: Biswa N. Datta October 24, 2008 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

  2. Matrix Pencil Also called matrix polynomial or - matrix Linear Pencil: Quadratic Pencil: Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  3. Free Vibrations The motion of the system is identified by Natural Frequencies eigenvalues of P() Mode Shapes Eigenvectors of P() Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  4. Coefficient Matrices • M,C,K – symmetric • M - positive definite • C,K – semi-definite They are structured! Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  5. Inverse Eigenvalue Problem Structured Matrices M,C,K which satisfy given properties so that pencil (M,C,K) has prescribed eigendata Given eigendata (eigenvalues, eigenvectors, their multiplicities) CONSTRUCT Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  6. Two Problems This dissertation deals with two Qudaratic Inverse Eigenvalue Problems (QIEP): • Finite Element Model Updating • Affine Quadratic Inverse Eigenvalue Problem Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  7. Model Updating • Vibrating structures: Automobiles, Bridges, Highways, Space and Aircrafts etc, may be modeled by System of Second Order DE (FEM). • Error can be introduced due to: Discretization, geometry, boundary conditions, assumption of linearity, discretization order, values of physical parameters. Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  8. To ensure the validity of the entire FEM: A few small number of eigenvalues and eigenvectors are experimentally measured from a realized practical structure and These measured eigenvalues and eigenvectors are then compared with those of FEM. Frequently, there is a disagreement between these two sets. Model Updating Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  9. Model Updating A vibration engineer needs to update the FEM such that : • Measured eigenvalues and eigenvectors are reproduced by the updated model • The strutural and physical properties of the original model are preserved Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  10. Finite Element Model Updating (QIEP setting) • Given an analytical structured quadratic pencil (Ma,Ca,Ka) • Assume that new measured eigenpairs have been obtained. • Update the pencil (Ma,Ca,Ka) to (Mu,Cu,Ku) of the same structure, such that: • are p eigenpairs of (Mu,Cu,Ku) • Matrices (Mu,Cu,Ku) are as close as possible to (Ma,Ca,Ka) Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  11. What is Available? Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  12. Issues Issues which are not covered in this work: • Model Expansion/Model Reduction • Parameter Selection (to avoid the ill-conditioning) • Vibration testing issues (transducers locations, data acquiring, data scaling) Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  13. Model Updating. Optimization Setting M and C are fixed, Ku is defined as the solution of the following optimization problem Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  14. Existence of Solution Theorem. Given real symmetric matrices M and C, there exists a real symmetric matrix K such that If and only if (diagonal matrix) Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  15. Updating of the measured eigenvectors D is a diagonal matrix W – suitable weighting matrix Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  16. Two-Stage Model Updating Procedure Stage I. Updating of measured eigenvectors Stage II. Updating of stiffness matrix K Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  17. Solution of the Optimization Problems • Stage I: Non convex minimization problem. Has a convex quadratic objective function, and polynomial equality constraints • Stage II: Convex quadratic programming problem with a unique solution • Gradients of functions associated with the problems can be computed in terms of their associated matrices • An augmented Lagrangian method has been used Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  18. Real Valued Representation of Eigenvalues Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  19. Real Valued Representation of Orthogonality Relation Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  20. Real Valued Representation of B Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  21. Stage I with Real-Valued Representation of the Orthogonality Relation Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  22. Closed form solution of Stage II (Friswell, Inman, Pilkey). Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  23. Case Studies • A spring-mass system of 10 dof • A vibrating beam • The mass and stiffness matrices of the system is assumed to be exact; that is, the matrices M and C do not need to be updated • Given: Perturb the Matrix K Compute P eigenpairs Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  24. A mass-spring system of 10 DoF • all rigid bodies have a mass of 1 kg • all springs have stiffness 1 kN/m • to get true model stiffness of the spring between masses 2 and 5 was reduced to to 800 N/m Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  25. System Matrices Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  26. Eigendata Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  27. Results Stage I Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  28. Results Stage II Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  29. Vibrating Beam m = 0.1 kg EI To get the true model, the coefficients k3,k5,k9 are increased by 40%, 50%, 30% respectively … 1 2 3 16 1/16 m Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  30. Governing Equations Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  31. Stage I&II Results Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  32. Affine Inverse Eigenvalue Problem Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  33. Reformulation of the Problem Problem 1 can be reformulated: Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  34. Alternating Projections Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  35. Alternating Projections Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  36. Convergence Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  37. Global Convergence of the Alternating Projections Algorithm Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  38. Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  39. Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  40. Projection onto A Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  41. Alternating Projections Algorithm Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  42. Modification Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  43. Algorithm Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  44. Newton's Method Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  45. Newton’s Step Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  46. Newton’s Algorithm Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  47. Convergence Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  48. Hybrid Method Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  49. Example Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

  50. Numerical Experiment Quadratic Inverse Eigenvalue Problems: Theory, Methods and Applications

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