Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Second Mode Analysis Matrix Iteration Higher Modes Inverse Iteration Matrix Iteration Giacomo Boffi with Shifts Alternative Dipartimento di Ingegneria Strutturale, Politecnico di Milano Procedures May 15, 2012
Outline Matrix Iteration Giacomo Boffi Introduction Introduction Fundamental Mode Analysis Fundamental Mode Analysis Second Mode Analysis Higher Modes Second Mode Analysis Inverse Iteration Matrix Iteration Higher Modes with Shifts Alternative Inverse Iteration Procedures Matrix Iteration with Shifts Alternative Procedures Rayleigh Quotient Rayleigh-Ritz Method Subspace Iteration
Introduction Matrix Iteration Giacomo Boffi Introduction Dynamic analysis of MDOF systems based on modal Fundamental superposition is both simple and efficient Mode Analysis ◮ simple: the modal response can be easily computed, Second Mode Analysis analitically or numerically, with the techniques we have Higher Modes seen for SDOF systems, Inverse Iteration ◮ efficient: in most cases, only the modal responses of a Matrix Iteration with Shifts few lower modes are required to accurately describe the Alternative Procedures structural response.
Introduction Matrix Iteration Giacomo Boffi Introduction Dynamic analysis of MDOF systems based on modal Fundamental superposition is both simple and efficient Mode Analysis ◮ simple: the modal response can be easily computed, Second Mode Analysis analitically or numerically, with the techniques we have Higher Modes seen for SDOF systems, Inverse Iteration ◮ efficient: in most cases, only the modal responses of a Matrix Iteration with Shifts few lower modes are required to accurately describe the Alternative Procedures structural response. As the structural matrices are easily assembled using the FEM , our modal superposition procedure is ready to be applied to structures with tenth, thousands or millions of DOF ’s! except that we can compute the eigenpairs only when the analyzed structure has two, three or maybe four degrees of freedom...
Introduction Matrix Iteration Giacomo Boffi Introduction Dynamic analysis of MDOF systems based on modal Fundamental superposition is both simple and efficient Mode Analysis ◮ simple: the modal response can be easily computed, Second Mode Analysis analitically or numerically, with the techniques we have Higher Modes seen for SDOF systems, Inverse Iteration ◮ efficient: in most cases, only the modal responses of a Matrix Iteration with Shifts few lower modes are required to accurately describe the Alternative Procedures structural response. As the structural matrices are easily assembled using the FEM , our modal superposition procedure is ready to be applied to structures with tenth, thousands or millions of DOF ’s! except that we can compute the eigenpairs only when the analyzed structure has two, three or maybe four degrees of freedom... Enter the various Matrix Iterations procedures!
Equilibrium Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis First, we will see an iterative procedure whose outputs are Second Mode Analysis the first, or fundamental, mode shape vector and the Higher Modes corresponding eigenvalue. Inverse Iteration When an undamped system freely vibrates, the equation of Matrix Iteration with Shifts motion is Alternative Procedures K ψ i = ω 2 i M ψ i .
Equilibrium Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis First, we will see an iterative procedure whose outputs are Second Mode Analysis the first, or fundamental, mode shape vector and the Higher Modes corresponding eigenvalue. Inverse Iteration When an undamped system freely vibrates, the equation of Matrix Iteration with Shifts motion is Alternative Procedures K ψ i = ω 2 i M ψ i . In equilibrium terms, the elastic forces are equal to the inertial forces when the systems oscillates with frequency ω 2 i and mode shape ψ i
Proposal of an iterative procedure Matrix Iteration Giacomo Boffi Introduction Fundamental Our iterative procedure will be based on finding a new Mode Analysis displacement vector x n + 1 such that the elastic forces Second Mode Analysis f S = K x n + 1 are in equilibrium with the inertial forces due Higher Modes to the old displacement vector x n , f I = ω 2 i M x n , Inverse Iteration Matrix Iteration K x n + 1 = ω 2 with Shifts i M x n . Alternative Procedures Premultiplying by the inverse of K and introducing the Dynamic Matrix , D = K − 1 M x n + 1 = ω 2 i K − 1 M x n = ω 2 i D x n .
Proposal of an iterative procedure Matrix Iteration Giacomo Boffi Introduction Fundamental Our iterative procedure will be based on finding a new Mode Analysis displacement vector x n + 1 such that the elastic forces Second Mode Analysis f S = K x n + 1 are in equilibrium with the inertial forces due Higher Modes to the old displacement vector x n , f I = ω 2 i M x n , Inverse Iteration Matrix Iteration K x n + 1 = ω 2 with Shifts i M x n . Alternative Procedures Premultiplying by the inverse of K and introducing the Dynamic Matrix , D = K − 1 M x n + 1 = ω 2 i K − 1 M x n = ω 2 i D x n . In the generative equation above we miss a fundamental part, the square of the free vibration frequency ω 2 i .
The Matrix Iteration Procedure, 1 Matrix Iteration Giacomo Boffi Introduction This problem is solved considering the x n as a sequence of normalized Fundamental vectors and introducing the idea of an unnormalized new displacement Mode Analysis vector, ^ x n + 1 , Second Mode x n + 1 = D x n , ^ Analysis Higher Modes note that we removed the explicit dependency on ω 2 i . Inverse Iteration Matrix Iteration with Shifts Alternative Procedures
The Matrix Iteration Procedure, 1 Matrix Iteration Giacomo Boffi Introduction This problem is solved considering the x n as a sequence of normalized Fundamental vectors and introducing the idea of an unnormalized new displacement Mode Analysis vector, ^ x n + 1 , Second Mode x n + 1 = D x n , ^ Analysis Higher Modes note that we removed the explicit dependency on ω 2 i . The normalized vector is obtained applying to ^ x n + 1 a normalizing Inverse Iteration factor, F n + 1 , Matrix Iteration with Shifts x n + 1 = ^ x n + 1 F n + 1 , Alternative Procedures 1 x n + 1 = ω 2 i D x n = ω 2 F = ω 2 but i ^ x n + 1 , ⇒ i
The Matrix Iteration Procedure, 1 Matrix Iteration Giacomo Boffi Introduction This problem is solved considering the x n as a sequence of normalized Fundamental vectors and introducing the idea of an unnormalized new displacement Mode Analysis vector, ^ x n + 1 , Second Mode ^ x n + 1 = D x n , Analysis Higher Modes note that we removed the explicit dependency on ω 2 i . The normalized vector is obtained applying to ^ x n + 1 a normalizing Inverse Iteration factor, F n + 1 , Matrix Iteration with Shifts x n + 1 = ^ x n + 1 F n + 1 , Alternative Procedures 1 x n + 1 = ω 2 i D x n = ω 2 F = ω 2 but i ^ x n + 1 , ⇒ i If we agree that, near convergence, x n + 1 ≈ x n , substituting in the previous equation we have x n x n + 1 ≈ x n = ω 2 ω 2 i ^ x n + 1 x n + 1 . ⇒ i ≈ ^
The Matrix Iteration Procedure, 1 Matrix Iteration Giacomo Boffi Introduction This problem is solved considering the x n as a sequence of normalized Fundamental vectors and introducing the idea of an unnormalized new displacement Mode Analysis vector, ^ x n + 1 , Second Mode x n + 1 = D x n , ^ Analysis Higher Modes note that we removed the explicit dependency on ω 2 i . The normalized vector is obtained applying to ^ x n + 1 a normalizing Inverse Iteration factor, F n + 1 , Matrix Iteration with Shifts x n + 1 = ^ x n + 1 F n + 1 , Alternative Procedures 1 x n + 1 = ω 2 i D x n = ω 2 F = ω 2 but i ^ x n + 1 , ⇒ i If we agree that, near convergence, x n + 1 ≈ x n , substituting in the previous equation we have x n x n + 1 ≈ x n = ω 2 ω 2 i ^ x n + 1 x n + 1 . ⇒ i ≈ ^ Of course the division of two vectors is not an option, so we want to twist it into something useful.
Normalization Matrix Iteration Giacomo Boffi Introduction First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is Fundamental Mode Analysis x n + 1 , j = ω 2 x n , j / ^ i . Second Mode Analysis Higher Modes Analogously for x n � = ψ i it was demonstrated that Inverse Iteration � x n , j � x n , j � � Matrix Iteration ≤ ω 2 min i ≤ max . with Shifts ^ ^ x n + 1 , j x n + 1 , j j = 1 ,..., N j = 1 ,..., N Alternative Procedures
Normalization Matrix Iteration Giacomo Boffi Introduction First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is Fundamental Mode Analysis x n + 1 , j = ω 2 x n , j / ^ i . Second Mode Analysis Higher Modes Analogously for x n � = ψ i it was demonstrated that Inverse Iteration � x n , j � x n , j � � Matrix Iteration ≤ ω 2 min i ≤ max . with Shifts ^ ^ x n + 1 , j x n + 1 , j j = 1 ,..., N j = 1 ,..., N Alternative Procedures A more rational approach would make reference to a proper vector norm, so using our preferred vector norm we can write x T ^ n + 1 M x n ω 2 i ≈ , x T ^ n + 1 M ^ x n + 1
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