Truncated sum Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector A truncated sum uses only M < N of the lower frequency modes Expansion Uncoupled x ( t ) ≈ � M<N ψ i q i ( t ) , Equations of i =1 Motion Truncated Sum and, under wide assumptions, gives you a good approximation of the Definition Elastic Forces structural response. Example The importance of truncated sum approximation is twofold: ◮ less computational effort: less eigenpairs to calculate, less equation of motion to integrate etc ◮ in FEM models the higher modes are rough approximations to structural ones (mostly due to uncertainties in mass distribution details) and the truncated sum excludes potentially spurious contributions from the response.
Elastic Forces Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Until now, we showed interest in displacements only, but we are Expansion interested in elastic forces too. We know that elastic forces can be Uncoupled Equations of expressed in terms of displacements and the stiffness matrix: Motion Truncated Sum Definition f S ( t ) = K x ( t ) = Kψ 1 q 1 ( t ) + Kψ 2 q 2 ( t ) + · · · . Elastic Forces Example From the characteristic equation we know that Kψ i = ω 2 i Mψ i substituting in the previous equation f S ( t ) = ω 2 1 Mψ 1 q 1 ( t ) + ω 2 2 Mψ 2 q 2 ( t ) + · · · .
Elastic Forces, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Expansion The high frequency modes contribution to the elastic forces, e.g. Uncoupled Equations of Motion f S ( t ) = ω 2 1 Mψ 1 q 1 ( t ) + · · · + ω 2 20 Mψ 20 q 20 ( t ) + · · · , Truncated Sum Definition Elastic Forces when compared to low frequency mode contributions are more Example important than their contributions to displacement, because of the multiplicative term ω 2 i . From this fact follows that, to estimate internal forces within a given accuracy a greater number of modes must be considered in a truncated sum than the number required to estimate displacements within the same accuracy.
Example: problem statement Truncated Sums, Matrix Iteration Giacomo Boffi m 1 Eigenvector k 1 = 120 MN / m , m 1 = 200 t , Expansion x 1 k 1 k 2 = 240 MN / m , m 2 = 300 t , Uncoupled m 2 Equations of k 3 = 360 MN / m , m 3 = 400 t . Motion x 2 k 2 Truncated Sum m 3 Definition Elastic Forces x 3 Example k 3 1. The above structure is subjected to these initial conditions, � � x T 0 = 5 mm 4 mm 3 mm , � � x T ˙ 0 = 0 9 mm/s 0 . Write the equation of motion using modal superposition. 2. The above structure is subjected to a half-sine impulse, 2 . 5 MN sin π t p T ( t ) = � � 1 2 2 t 1 , with t 1 = 0 . 02 s . Write the equation of motion using modal superposition.
Example: structural matrices Truncated Sums, Matrix Iteration Giacomo Boffi m 1 Eigenvector Expansion x 1 k 1 Uncoupled m 2 Equations of Motion k 1 = 120 MN / m , m 1 = 200 t , x 2 k 2 Truncated Sum k 2 = 240 MN / m , m 2 = 300 t , m 3 Definition Elastic Forces k 3 = 360 MN / m , m 3 = 400 t . Example x 3 k 3 The structural matrices can be written 1 − 1 0 with k = 120 MN = k K , K = k − 1 3 − 2 m , 0 − 2 5 2 0 0 = m M , M = m 0 3 0 with m = 100000 kg. 0 0 4
Example: adimensional eigenvalues Truncated Sums, Matrix Iteration Giacomo Boffi We want the solutions of the characteristic equation, so we start Eigenvector writing that the determinant of the equation must be zero: Expansion � � Uncoupled � � ω 2 � = 0 , � � � K − Ω 2 M � K − k/m M � = Equations of Motion Truncated Sum � rad � 2 Ω 2 . with ω 2 = 1200 Definition s Elastic Forces Example Expanding the determinant � � 1 − 2Ω 2 − 1 0 � � � � 3 − 3Ω 2 − 1 − 2 = 0 � � � � 5 − 4Ω 2 0 − 2 � � we have the following algebraic equation of 3rd order in Ω 2 � � Ω 6 − 11 4 Ω 4 + 15 8 Ω 2 − 1 24 = 0 . 4
Example: table of eigenvalues etc Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Expansion Here are the adimensional roots Ω 2 i , i = 1 , 2 , 3 , the dimensional Uncoupled Equations of i = 1200 rad 2 eigenvalues ω 2 s 2 Ω 2 i and all the derived dimensional Motion quantities: Truncated Sum Definition Elastic Forces Ω 2 Ω 2 Ω 2 1 = 0 . 17573 2 = 0 . 8033 3 = 1 . 7710 Example ω 2 ω 2 ω 2 1 = 210 . 88 2 = 963 . 96 3 = 2125 . 2 ω 1 = 14 . 522 ω 2 = 31 . 048 ω 3 = 46 . 099 f 1 = 2 . 3112 f 2 = 4 . 9414 f 3 = 7 . 3370 T 1 = 0 . 43268 T 3 = 0 . 20237 T 3 = 0 . 1363
Example: eigenvectors and modal matrices Truncated Sums, Matrix Iteration Giacomo Boffi With ψ 1 j = 1 , using the 2nd and 3rd equations, Eigenvector � 3 − 3Ω 2 � � ψ 2 j � � 1 � − 2 Expansion j = 5 − 4Ω 2 − 2 ψ 3 j 0 Uncoupled j Equations of Motion The above equations must be solved for j = 1 , 2 , 3 . The solutions are finally Truncated Sum collected in the eigenmatrix Definition Elastic Forces 1 1 1 Example . Ψ = +0 . 648535272183 − 0 . 606599092464 − 2 . 54193617967 +0 . 301849953585 − 0 . 678977475113 +2 . 43962752148 The Modal Matrices are 362 . 6 0 0 M ⋆ = × 10 3 kg , 0 494 . 7 0 0 0 4519 . 1 76 . 50 0 0 × 10 6 N K ⋆ = 0 477 . 0 0 m 0 0 9603 . 9
Example: initial conditions in modal coordinates Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Expansion Uncoupled Equations of Motion 5 +5 . 9027 Truncated Sum q 0 = ( M ⋆ ) − 1 Ψ T M 4 mm = − 1 . 0968 mm , Definition Elastic Forces 3 +0 . 1941 Example 0 +4 . 8288 mm mm q 0 = ( M ⋆ ) − 1 Ψ T M ˙ 9 = − 3 . 3101 s s 0 − 1 . 5187
Example: structural response Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Expansion These are the displacements, in mm Uncoupled Equations of x 1 = +5 . 91 cos(14 . 5 t + . 06) + 1 . 10 cos(31 . 0 t − 3 . 04) + 0 . 20 cos(46 . 1 t − 0 . 17) Motion x 2 = +3 . 83 cos(14 . 5 t + . 06) − 0 . 67 cos(31 . 0 t − 3 . 04) − 0 . 50 cos(46 . 1 t − 0 . 17) Truncated Sum Definition x 3 = +1 . 78 cos(14 . 5 t + . 06) − 0 . 75 cos(31 . 0 t − 3 . 04) + 0 . 48 cos(46 . 1 t − 0 . 17) Elastic Forces Example
Example: structural response Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Expansion These are the displacements, in mm Uncoupled Equations of x 1 = +5 . 91 cos(14 . 5 t + . 06) + 1 . 10 cos(31 . 0 t − 3 . 04) + 0 . 20 cos(46 . 1 t − 0 . 17) Motion x 2 = +3 . 83 cos(14 . 5 t + . 06) − 0 . 67 cos(31 . 0 t − 3 . 04) − 0 . 50 cos(46 . 1 t − 0 . 17) Truncated Sum Definition x 3 = +1 . 78 cos(14 . 5 t + . 06) − 0 . 75 cos(31 . 0 t − 3 . 04) + 0 . 48 cos(46 . 1 t − 0 . 17) Elastic Forces Example and these the elastic/inertial forces, in kN x 1 = +249 . cos(14 . 5 t + . 06) + 212 . cos(31 . 0 t − 3 . 04) + 084 . cos(46 . 1 t − 0 . 17) x 2 = +243 . cos(14 . 5 t + . 06) − 193 . cos(31 . 0 t − 3 . 04) − 319 . cos(46 . 1 t − 0 . 17) x 3 = +151 . cos(14 . 5 t + . 06) − 288 . cos(31 . 0 t − 3 . 04) + 408 . cos(46 . 1 t − 0 . 17)
Example: structural response Truncated Sums, Matrix Iteration Giacomo Boffi Eigenvector Expansion These are the displacements, in mm Uncoupled Equations of x 1 = +5 . 91 cos(14 . 5 t + . 06) + 1 . 10 cos(31 . 0 t − 3 . 04) + 0 . 20 cos(46 . 1 t − 0 . 17) Motion x 2 = +3 . 83 cos(14 . 5 t + . 06) − 0 . 67 cos(31 . 0 t − 3 . 04) − 0 . 50 cos(46 . 1 t − 0 . 17) Truncated Sum Definition x 3 = +1 . 78 cos(14 . 5 t + . 06) − 0 . 75 cos(31 . 0 t − 3 . 04) + 0 . 48 cos(46 . 1 t − 0 . 17) Elastic Forces Example and these the elastic/inertial forces, in kN x 1 = +249 . cos(14 . 5 t + . 06) + 212 . cos(31 . 0 t − 3 . 04) + 084 . cos(46 . 1 t − 0 . 17) x 2 = +243 . cos(14 . 5 t + . 06) − 193 . cos(31 . 0 t − 3 . 04) − 319 . cos(46 . 1 t − 0 . 17) x 3 = +151 . cos(14 . 5 t + . 06) − 288 . cos(31 . 0 t − 3 . 04) + 408 . cos(46 . 1 t − 0 . 17) As expected, the contributions of the higher modes are more important for the forces, less important for the displacements.
Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Part II Second Mode Analysis Higher Modes Matrix Iteration Procedures Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Dynamic analysis of MDOF systems based on modal superposition is Second Mode Analysis both simple and efficient Higher Modes ◮ simple: the modal response can be easily computed, analitically Inverse Iteration or numerically, with the techniques we have seen for SDOF Matrix Iteration with Shifts systems, Rayleigh ◮ efficient: in most cases, only the modal responses of a few lower Methods modes are required to accurately describe the structural response.
Introduction Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis The structural matrices being easily assembled using the FEM , the Second Mode modal superposition procedure is ready to be applied to structures Analysis Higher Modes with thousands, millions of DOF ’s! Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis The structural matrices being easily assembled using the FEM , the Second Mode modal superposition procedure is ready to be applied to structures Analysis Higher Modes with thousands, millions of DOF ’s! Inverse Iteration But wait, we can know how to compute the eigenpairs only when the Matrix Iteration analyzed structure has very few degrees of freedom... with Shifts Rayleigh Methods
Introduction Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis The structural matrices being easily assembled using the FEM , the Second Mode modal superposition procedure is ready to be applied to structures Analysis Higher Modes with thousands, millions of DOF ’s! Inverse Iteration But wait, we can know how to compute the eigenpairs only when the Matrix Iteration analyzed structure has very few degrees of freedom... with Shifts We will discuss how it is possible to compute the eigenpairs of Rayleigh Methods arbitrarily large dynamic systems using the so called Matrix Iteration procedure (and a number of variations derived from this fundamental idea).
Introduction Fundamental Mode Analysis A Possible Procedure Matrix Iteration Procedure Convergence of Matrix Iteration Procedure Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Equilibrium Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental First, we will see an iterative procedure whose outputs are the first, Mode Analysis Idea or fundamental, mode shape vector and the corresponding Procedure Convergence eigenvalue. Second Mode When an undamped system freely vibrates with a harmonic time Analysis dependency of frequency ω i , the equation of motion, simplifying the Higher Modes time dependency, is Inverse Iteration Matrix Iteration with Shifts K ψ i = ω 2 i M ψ i . Rayleigh Methods 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 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Our iterative procedure will be based on finding a new displacement Fundamental vector x n +1 such that the elastic forces f S = K x i +1 are in Mode Analysis Idea equilibrium with the inertial forces due to the old displacement Procedure vector x n , f I = ω 2 i M x n , that is Convergence Second Mode Analysis K x n +1 = ω 2 i M x n . Higher Modes Inverse Iteration Premultiplying by the inverse of K and introducing the Dynamic Matrix Iteration Matrix , D = K − 1 M with Shifts Rayleigh Methods x n +1 = ω 2 i K − 1 M x n = ω 2 i D x n .
Proposal of an iterative procedure Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Our iterative procedure will be based on finding a new displacement Fundamental vector x n +1 such that the elastic forces f S = K x i +1 are in Mode Analysis Idea equilibrium with the inertial forces due to the old displacement Procedure vector x n , f I = ω 2 i M x n , that is Convergence Second Mode Analysis K x n +1 = ω 2 i M x n . Higher Modes Inverse Iteration Premultiplying by the inverse of K and introducing the Dynamic Matrix Iteration Matrix , D = K − 1 M with Shifts Rayleigh Methods 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 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Idea Procedure This problem is solved considering the x n as a sequence of Convergence normalized vectors and introducing the idea of an unnormalized new Second Mode Analysis displacement vector, ˆ x n +1 , Higher Modes Inverse Iteration x n +1 = D x n , ˆ Matrix Iteration with Shifts note that we removed the explicit dependency on ω 2 i . Rayleigh Methods
The Matrix Iteration Procedure, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The normalized vector is obtained applying to ˆ x n +1 a normalizing Fundamental factor, F n +1 , Mode Analysis x n +1 = ˆ x n +1 Idea , Procedure F n +1 Convergence Second Mode 1 Analysis x n +1 = ω 2 i D x n = ω 2 F = ω 2 but i ˆ x n +1 , ⇒ i Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
The Matrix Iteration Procedure, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The normalized vector is obtained applying to ˆ x n +1 a normalizing Fundamental factor, F n +1 , Mode Analysis x n +1 = ˆ x n +1 Idea , Procedure F n +1 Convergence Second Mode 1 Analysis x n +1 = ω 2 i D x n = ω 2 F = ω 2 but i ˆ x n +1 , ⇒ i Higher Modes Inverse Iteration If we agree that, near convergence, x n +1 ≈ x n , substituting in the Matrix Iteration previous equation we have with Shifts Rayleigh x n Methods x n +1 ≈ x n = ω 2 ω 2 i ˆ ⇒ i ≈ . x n +1 x n +1 ˆ
The Matrix Iteration Procedure, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The normalized vector is obtained applying to ˆ x n +1 a normalizing Fundamental factor, F n +1 , Mode Analysis x n +1 = ˆ x n +1 Idea , Procedure F n +1 Convergence Second Mode 1 Analysis x n +1 = ω 2 i D x n = ω 2 F = ω 2 but i ˆ x n +1 , ⇒ i Higher Modes Inverse Iteration If we agree that, near convergence, x n +1 ≈ x n , substituting in the Matrix Iteration previous equation we have with Shifts Rayleigh x n Methods x n +1 ≈ x n = ω 2 ω 2 i ˆ ⇒ i ≈ . x n +1 x n +1 ˆ Of course the division of two vectors is not an option, so we want to twist it into something useful.
Normalization Truncated Sums, Matrix Iteration Giacomo Boffi First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is Introduction x n +1 ,j = ω 2 x n,j / ˆ i . Fundamental Mode Analysis Idea When x n � = ψ i it is possible to demonstrate that we can bound the Procedure Convergence eigenvalue Second Mode � x n,j � x n,j Analysis � � ≤ ω 2 Higher Modes min i ≤ max . x n +1 ,j ˆ x n +1 ,j ˆ j =1 ,...,N j =1 ,...,N Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Normalization Truncated Sums, Matrix Iteration Giacomo Boffi First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is Introduction x n +1 ,j = ω 2 x n,j / ˆ i . Fundamental Mode Analysis Idea When x n � = ψ i it is possible to demonstrate that we can bound the Procedure Convergence eigenvalue Second Mode � x n,j � x n,j Analysis � � ≤ ω 2 Higher Modes min i ≤ max . x n +1 ,j ˆ x n +1 ,j ˆ j =1 ,...,N j =1 ,...,N Inverse Iteration Matrix Iteration A more rational approach would make reference to a proper vector with Shifts norm, so using our preferred vector norm we can write Rayleigh Methods x T ˆ n +1 M x n ω 2 i ≈ , x T ˆ n +1 M ˆ x n +1
Normalization Truncated Sums, Matrix Iteration Giacomo Boffi First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is Introduction x n +1 ,j = ω 2 x n,j / ˆ i . Fundamental Mode Analysis Idea When x n � = ψ i it is possible to demonstrate that we can bound the Procedure Convergence eigenvalue Second Mode � x n,j � x n,j Analysis � � ≤ ω 2 Higher Modes min i ≤ max . x n +1 ,j ˆ x n +1 ,j ˆ j =1 ,...,N j =1 ,...,N Inverse Iteration Matrix Iteration A more rational approach would make reference to a proper vector with Shifts norm, so using our preferred vector norm we can write Rayleigh Methods x T ˆ n +1 M x n ω 2 i ≈ , x T ˆ n +1 M ˆ x n +1 (if memory helps, this is equivalent to the R 11 approximation, that we introduced studying Rayleigh quotient refinements).
Proof of Convergence, 1 Truncated Sums, Matrix Iteration Giacomo Boffi Until now we postulated that the sequence x n converges to some, Introduction unspecified eigenvector ψ i , now we will demonstrate that the Fundamental sequence converge to the first, or fundamental mode shape, Mode Analysis Idea Procedure n →∞ x n = ψ 1 . lim Convergence Second Mode Analysis 1. Expand x 0 in terms of eigenvectors an modal coordinates: Higher Modes Inverse Iteration x 0 = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · . Matrix Iteration with Shifts 2. The inertial forces, assuming that the system is vibrating Rayleigh Methods according to the fundamental frequency, are f I,n =0 = ω 2 1 M ( ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · ) � � ω 2 ω 2 ω 2 1 + ω 2 1 = M 1 ψ 1 q 1 , 0 2 ψ 2 q 2 , 0 + · · · . ω 2 ω 2 1 2
Proof of Convergence, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental 3. The deflections due to these forces (no hat!, we have multiplied by ω 2 1 ) are Mode Analysis Idea � � 1 ψ 1 q 1 , 0 ω 2 2 ψ 2 q 2 , 0 ω 2 Procedure x n =1 = K − 1 M ω 2 1 + ω 2 1 + · · · , Convergence ω 2 ω 2 1 2 Second Mode Analysis (note that every term has been multiplied and divided by the corresponding eigenvalue ω 2 i ). Higher Modes Inverse Iteration 4. With ω 2 j M ψ j = Kψ j , substituting and simplifying K − 1 K = I , Matrix Iteration � � with Shifts � ω 2 � 1 � ω 2 � 1 � ω 2 � 1 x n =1 = K − 1 1 1 1 Kψ 1 q 1 , 0 + Kψ 2 q 2 , 0 + Kψ 3 q 3 , 0 + · · · Rayleigh ω 2 ω 2 ω 2 Methods 1 2 3 = ψ 1 q 1 , 0 ω 2 + ψ 2 q 2 , 0 ω 2 + ψ 3 q 3 , 0 ω 2 1 1 1 + · · · , ω 2 ω 2 ω 2 1 2 3
Proof of Convergence, 3 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Idea 5. applying again this procedure Procedure Convergence � � � ω 2 � 2 � ω 2 � 2 � ω 2 � 2 Second Mode 1 1 1 x n =2 = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · , Analysis ω 2 ω 2 ω 2 1 2 3 Higher Modes Inverse Iteration 6. applying the procedure n times Matrix Iteration � � ω 2 � n � ω 2 � n � ω 2 � n � with Shifts 1 1 1 x n = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · . Rayleigh ω 2 ω 2 ω 2 1 2 3 Methods
Proof of Convergence, 4 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Going to the limit, Idea n →∞ x n = ψ 1 q 1 , 0 lim Procedure Convergence Second Mode because Analysis � � n ω 2 Higher Modes 1 lim = δ 1 j ω 2 n →∞ Inverse Iteration j Matrix Iteration with Shifts Consequently, | x n | Rayleigh x n | = ω 2 lim Methods 1 | ˆ n →∞
Introduction Fundamental Mode Analysis Second Mode Analysis Purified Vectors Sweeping Matrix Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Purified Vectors Truncated Sums, Matrix Iteration Giacomo Boffi If we know ψ 1 and ω 2 1 from the matrix iteration procedure it is Introduction possible to compute the second eigenpair, following a slightly Fundamental different procedure. Mode Analysis Second Mode Analysis Purified Vectors Sweeping Matrix Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Purified Vectors Truncated Sums, Matrix Iteration Giacomo Boffi If we know ψ 1 and ω 2 1 from the matrix iteration procedure it is Introduction possible to compute the second eigenpair, following a slightly Fundamental different procedure. Mode Analysis Express the initial iterate in terms of the (unknown) eigenvectors, Second Mode Analysis x n =0 = Ψ q n =0 Purified Vectors Sweeping Matrix and premultiply by the (known) ψ T Higher Modes 1 M : Inverse Iteration ψ T 1 M x n =0 = M 1 q 1 ,n =0 Matrix Iteration with Shifts solving for q 1 ,n =0 Rayleigh Methods q 1 ,n =0 = ψ T 1 M x n =0 . M 1
Purified Vectors Truncated Sums, Matrix Iteration Giacomo Boffi If we know ψ 1 and ω 2 1 from the matrix iteration procedure it is Introduction possible to compute the second eigenpair, following a slightly Fundamental different procedure. Mode Analysis Express the initial iterate in terms of the (unknown) eigenvectors, Second Mode Analysis x n =0 = Ψ q n =0 Purified Vectors Sweeping Matrix and premultiply by the (known) ψ T Higher Modes 1 M : Inverse Iteration ψ T 1 M x n =0 = M 1 q 1 ,n =0 Matrix Iteration with Shifts solving for q 1 ,n =0 Rayleigh Methods q 1 ,n =0 = ψ T 1 M x n =0 . M 1 Knowing the amplitude of the 1st modal contribution to x n =0 we can write a purified vector, y n =0 = x n =0 − ψ 1 q 1 ,n =0 .
Convergence (?) Truncated Sums, Matrix Iteration Giacomo Boffi Introduction It is easy to demonstrate that using y n =0 as our starting vector Fundamental Mode Analysis | y n | Second Mode y n | = ω 2 n →∞ y n = ψ 2 q 2 ,n =0 , lim lim 2 . Analysis | ˆ n →∞ Purified Vectors Sweeping Matrix Higher Modes because the initial amplitude of the first mode is null. Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Convergence (?) Truncated Sums, Matrix Iteration Giacomo Boffi Introduction It is easy to demonstrate that using y n =0 as our starting vector Fundamental Mode Analysis | y n | Second Mode y n | = ω 2 n →∞ y n = ψ 2 q 2 ,n =0 , lim lim 2 . Analysis | ˆ n →∞ Purified Vectors Sweeping Matrix Higher Modes because the initial amplitude of the first mode is null. Inverse Iteration Matrix Iteration with Shifts Due to numerical errors in the determination of fundamental Rayleigh mode and in the procedure itself, using a plain matrix iteration Methods the procedure however converges to the 1st eigenvector, so to preserve convergence to the 2nd mode it is necessary that the iterated vector y n is purified at each step n .
Purification Procedure Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The purification procedure is simple, at each step the amplitude of Fundamental the 1st mode is first computed, then removed from the iterated Mode Analysis Second Mode vector y n Analysis q 1 ,n = ψ T 1 My n /M 1 , Purified Vectors Sweeping Matrix � � I − 1 Higher Modes ψ 1 ψ T y n +1 = D ( y n − ψ 1 q 1 ,n ) = D ˆ 1 M y n Inverse Iteration M 1 Matrix Iteration with Shifts Rayleigh Methods
Purification Procedure Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The purification procedure is simple, at each step the amplitude of Fundamental the 1st mode is first computed, then removed from the iterated Mode Analysis Second Mode vector y n Analysis q 1 ,n = ψ T 1 My n /M 1 , Purified Vectors Sweeping Matrix � � I − 1 Higher Modes ψ 1 ψ T y n +1 = D ( y n − ψ 1 q 1 ,n ) = D ˆ 1 M y n Inverse Iteration M 1 Matrix Iteration M 1 ψ 1 ψ T 1 with Shifts Introducing the sweeping matrix S 1 = I − 1 M and the Rayleigh modified dynamic matrix D 2 = DS 1 , we can write Methods y n +1 = DS 1 y n = D 2 y n . ˆ
Purification Procedure Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The purification procedure is simple, at each step the amplitude of Fundamental the 1st mode is first computed, then removed from the iterated Mode Analysis Second Mode vector y n Analysis q 1 ,n = ψ T 1 My n /M 1 , Purified Vectors Sweeping Matrix � � I − 1 Higher Modes ψ 1 ψ T y n +1 = D ( y n − ψ 1 q 1 ,n ) = D ˆ 1 M y n Inverse Iteration M 1 Matrix Iteration M 1 ψ 1 ψ T 1 with Shifts Introducing the sweeping matrix S 1 = I − 1 M and the Rayleigh modified dynamic matrix D 2 = DS 1 , we can write Methods y n +1 = DS 1 y n = D 2 y n . ˆ This is known as matrix iteration with sweeps .
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Third Mode Truncated Sums, Matrix Iteration Giacomo Boffi Using again the idea of purifying the iterated vector, starting with the knowledge Introduction of the first and the second eigenpair, Fundamental Mode Analysis y n +1 = D ( y n − ψ 1 q 1 ,n − ψ 2 q 2 ,n ) ˆ Second Mode Analysis with q n, 1 as before and Higher Modes q 2 ,n = ψ T 2 My n /M 2 , Inverse Iteration substituting in the expression for the purified vector Matrix Iteration with Shifts 1 − 1 � � M 1 ψ 1 ψ T M 2 ψ 2 ψ T y n +1 = D ˆ I − 1 M 2 M Rayleigh Methods � �� � S 1
Third Mode Truncated Sums, Matrix Iteration Giacomo Boffi Using again the idea of purifying the iterated vector, starting with the knowledge Introduction of the first and the second eigenpair, Fundamental Mode Analysis y n +1 = D ( y n − ψ 1 q 1 ,n − ψ 2 q 2 ,n ) ˆ Second Mode Analysis with q n, 1 as before and Higher Modes q 2 ,n = ψ T 2 My n /M 2 , Inverse Iteration substituting in the expression for the purified vector Matrix Iteration with Shifts 1 − 1 � � M 1 ψ 1 ψ T M 2 ψ 2 ψ T y n +1 = D ˆ I − 1 M 2 M Rayleigh Methods � �� � S 1 The conclusion is that the sweeping matrix and the modified dynamic matrix to be used to compute the 3rd eigenvector are 1 M 2 ψ 2 ψ T S 2 = S 1 − 2 M , D 3 = D S 2 .
Generalization to Higher Modes Truncated Sums, Matrix Iteration Giacomo Boffi The results obtained for the third mode are easily generalised. It is easy to verify that the following procedure can be used to compute all the Introduction modes. Fundamental Mode Analysis Define S 0 = I , take i = 1 , Second Mode Analysis 1. compute the modified dynamic matrix to be used for mode i , Higher Modes D i = D S i − i Inverse Iteration Matrix Iteration 2. compute ψ i using the modified dynamic matrix; with Shifts 3. compute the modal mass M i = ψ T M ψ ; Rayleigh Methods 4. compute the sweeping matrix S i that sweeps the contributions of the first i modes from trial vectors, S i = S i − 1 − 1 M i ψ i ψ T i M ; 5. increment i , GOTO 1.
Generalization to Higher Modes Truncated Sums, Matrix Iteration Giacomo Boffi The results obtained for the third mode are easily generalised. It is easy to verify that the following procedure can be used to compute all the Introduction modes. Fundamental Mode Analysis Define S 0 = I , take i = 1 , Second Mode Analysis 1. compute the modified dynamic matrix to be used for mode i , Higher Modes D i = D S i − i Inverse Iteration Matrix Iteration 2. compute ψ i using the modified dynamic matrix; with Shifts 3. compute the modal mass M i = ψ T M ψ ; Rayleigh Methods 4. compute the sweeping matrix S i that sweeps the contributions of the first i modes from trial vectors, S i = S i − 1 − 1 M i ψ i ψ T i M ; 5. increment i , GOTO 1. Well, we finally have a method that can be used to compute all the eigenpairs of our dynamic problems, full circle!
Discussion Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis The method of matrix iteration with sweeping is not used in Second Mode production because Analysis Higher Modes 1. D is a full matrix, even if M and K are banded matrices, and Inverse Iteration the matrix product that is the essential step in every iteration is Matrix Iteration computationally onerous, with Shifts 2. the procedure is however affected by numerical errors, Rayleigh Methods so, after having demonstrated that it is possible to compute all the eigenvectors of a large problem using an iterative procedure it is time to look for different, more efficient iterative procedures.
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration LU Decomposition Back Substitution Matrix Iteration with Shifts Rayleigh Methods
Introduction to Inverse Iteration Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Inverse iteration is based on the fact that the symmetric stiffness Mode Analysis matrix has a banded structure, that is a relatively large triangular Second Mode Analysis portion of the matrix is composed by zeroes. Higher Modes Inverse Iteration LU Decomposition Back Substitution The banded structure is due to the FEM model: in every Matrix Iteration with Shifts equation of equilibrium the only non zero elastic force coef- Rayleigh ficients are due to the degrees of freedom of the few FE’s Methods that contain the degree of freedom for which the equilibrium is written.
Definition of LU decomposition Truncated Sums, Matrix Iteration Giacomo Boffi Every symmetric, banded matrix can be subjected to a so called LU Introduction decomposition, that is, for K we write Fundamental Mode Analysis K = L U Second Mode Analysis Higher Modes where L and U are, respectively, a lower- and an upper-banded Inverse Iteration matrix. LU Decomposition Back Substitution If we denote with b the bandwidth of K , we have Matrix Iteration with Shifts � i < j � � Rayleigh L = l ij with l ij ≡ 0 for Methods j < i − b and � i > j � � U = u ij with u ij ≡ 0 for j > i + b
Twice the equations? Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental In this case, with w n = M x n , the recursion can be written Mode Analysis Second Mode Analysis L U x n +1 = w n Higher Modes Inverse Iteration or as a system of equations, LU Decomposition Back Substitution U x n +1 = z n +1 Matrix Iteration with Shifts L z n +1 = w n Rayleigh Methods Apparently, we have doubled the number of unknowns, but the z j ’s can be easily computed by the procedure of back substitution .
Back Substitution Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Temporarily dropping the n and n + 1 subscripts, we can write Fundamental Mode Analysis Second Mode z 1 = ( w 1 ) /l 11 Analysis Higher Modes z 2 = ( w 2 − l 21 z 1 ) /l 22 Inverse Iteration z 3 = ( w 3 − l 31 z 1 − l 32 z 2 ) /l 33 LU Decomposition Back Substitution · · · Matrix Iteration with Shifts i − 1 � Rayleigh z i = ( w i − l ij z j ) /l ii Methods j = i − b · · ·
Back Substitution Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Temporarily dropping the n and n + 1 subscripts, we can write Fundamental Mode Analysis Second Mode z 1 = ( w 1 ) /l 11 Analysis Higher Modes z 2 = ( w 2 − l 21 z 1 ) /l 22 Inverse Iteration z 3 = ( w 3 − l 31 z 1 − l 32 z 2 ) /l 33 LU Decomposition Back Substitution · · · Matrix Iteration with Shifts i − 1 � Rayleigh z i = ( w i − l ij z j ) /l ii Methods j = i − b · · · The x are then given by U x = z .
Back Substitution Truncated Sums, Matrix Iteration Giacomo Boffi We have computed z by back substitution, we must solve U x = z Introduction but U is upper triangular, so we have Fundamental Mode Analysis Second Mode Analysis x N = ( z N ) /u NN Higher Modes x N − 1 = ( z N − 1 − u N − 1 ,N z N ) /u N − 1 ,N − 1 Inverse Iteration LU Decomposition x N − 2 = ( z N − 2 − u N − 2 ,N z N − u N − 2 ,N − 1 z N − 1 ) /u N − 2 ,N − 2 Back Substitution Matrix Iteration · · · with Shifts j − 1 Rayleigh � Methods x N − j = ( z N − j − u N − j,N − k z N − k ) /u N − j,N − j , k =0
Back Substitution Truncated Sums, Matrix Iteration Giacomo Boffi We have computed z by back substitution, we must solve U x = z Introduction but U is upper triangular, so we have Fundamental Mode Analysis Second Mode Analysis x N = ( z N ) /u NN Higher Modes x N − 1 = ( z N − 1 − u N − 1 ,N z N ) /u N − 1 ,N − 1 Inverse Iteration LU Decomposition x N − 2 = ( z N − 2 − u N − 2 ,N z N − u N − 2 ,N − 1 z N − 1 ) /u N − 2 ,N − 2 Back Substitution Matrix Iteration · · · with Shifts j − 1 Rayleigh � Methods x N − j = ( z N − j − u N − j,N − k z N − k ) /u N − j,N − j , k =0 For moderately large systems, the reduction in operations count given by back substitution with respect to matrix multiplication is so large that the additional cost of the LU decomposition is negligible.
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction to Shifts Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse iteration can be applied to each step of matrix iteration with Inverse Iteration sweeps, or to each step of a different procedure intended to compute Matrix Iteration with Shifts all the eigenpairs, the matrix iteration with shifts . Rayleigh Methods
Matrix Iteration with Shifts, 1 Truncated Sums, Matrix Iteration Giacomo Boffi If we write ω 2 i = µ + λ i , Introduction Fundamental where µ is a shift and λ i is a shifted eigenvalue , the eigenvalue problem Mode Analysis can be formulated as Second Mode K ψ i = ( µ + λ i ) M ψ i Analysis Higher Modes or Inverse Iteration ( K − µ M ) ψ i = λ i M ψ i . Matrix Iteration with Shifts If we introduce a modified stiffness matrix Rayleigh K = K − µ M , Methods we recognize that we have a new problem, that has exactly the same eigenvectors and shifted eigenvalues, K φ i = λ i Mφ i , where λ i = ω 2 φ i = ψ i , i − µ.
Matrix Iteration with Shifts, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The shifted eigenproblem can be solved, e.g., by matrix iteration and the Fundamental procedure will converge to the smallest absolute value shifted eigenvalue and to Mode Analysis the associated eigenvector. After convergence is reached, Second Mode Analysis ω 2 ψ i = φ i , i = λ i + µ. Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Matrix Iteration with Shifts, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The shifted eigenproblem can be solved, e.g., by matrix iteration and the Fundamental procedure will converge to the smallest absolute value shifted eigenvalue and to Mode Analysis the associated eigenvector. After convergence is reached, Second Mode Analysis ω 2 ψ i = φ i , i = λ i + µ. Higher Modes The convergence of the method can be greatly enhanced if the shift µ is updated Inverse Iteration every few steps during the iterative procedure using the current best estimate of Matrix Iteration with Shifts λ i , x n +1 M x n ˆ Rayleigh λ i,n +1 = x n +1 , Methods x n +1 M ˆ ˆ to improve the modified stiffness matrix to be used in the following iterations, K = K − λ i,n +1 M
Matrix Iteration with Shifts, 2 Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The shifted eigenproblem can be solved, e.g., by matrix iteration and the Fundamental procedure will converge to the smallest absolute value shifted eigenvalue and to Mode Analysis the associated eigenvector. After convergence is reached, Second Mode Analysis ω 2 ψ i = φ i , i = λ i + µ. Higher Modes The convergence of the method can be greatly enhanced if the shift µ is updated Inverse Iteration every few steps during the iterative procedure using the current best estimate of Matrix Iteration with Shifts λ i , x n +1 M x n ˆ Rayleigh λ i,n +1 = x n +1 , Methods x n +1 M ˆ ˆ to improve the modified stiffness matrix to be used in the following iterations, K = K − λ i,n +1 M Much thought was spent on the problem of choosing the initial shifts, so that all the eigenvectors can be computed in sequence without missing any of them.
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Rayleigh Quotient for Discrete Systems Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The matrix iteration procedures are usually used in conjunction with methods Fundamental derived from the Rayleigh Quotient method. Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Rayleigh Quotient for Discrete Systems Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The matrix iteration procedures are usually used in conjunction with methods Fundamental derived from the Rayleigh Quotient method. Mode Analysis The Rayleigh Quotient method was introduced using distributed flexibilty Second Mode systems and an assumed shape function, but we have seen also an example Analysis where the Rayleigh Quotient was computed for a discrete system using an Higher Modes assumed shape vector. Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Rayleigh Quotient for Discrete Systems Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The matrix iteration procedures are usually used in conjunction with methods Fundamental derived from the Rayleigh Quotient method. Mode Analysis The Rayleigh Quotient method was introduced using distributed flexibilty Second Mode systems and an assumed shape function, but we have seen also an example Analysis where the Rayleigh Quotient was computed for a discrete system using an Higher Modes assumed shape vector. Inverse Iteration The procedure to be used for discrete systems can be summarized as Matrix Iteration with Shifts x ( t ) = φ Z 0 sin ωt, x ( t ) = ω φ Z 0 cos ωt, ˙ Rayleigh 2 T max = ω 2 φ T M φ , 2 V max = φ T K φ , Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Rayleigh Quotient for Discrete Systems Truncated Sums, Matrix Iteration Giacomo Boffi Introduction The matrix iteration procedures are usually used in conjunction with methods Fundamental derived from the Rayleigh Quotient method. Mode Analysis The Rayleigh Quotient method was introduced using distributed flexibilty Second Mode systems and an assumed shape function, but we have seen also an example Analysis where the Rayleigh Quotient was computed for a discrete system using an Higher Modes assumed shape vector. Inverse Iteration The procedure to be used for discrete systems can be summarized as Matrix Iteration with Shifts x ( t ) = φ Z 0 sin ωt, x ( t ) = ω φ Z 0 cos ωt, ˙ Rayleigh 2 T max = ω 2 φ T M φ , 2 V max = φ T K φ , Methods Rayleigh-Ritz Method Rayleigh-Ritz Example equating the maxima, we have Subspace iteration ω 2 = φ T K φ φ T M φ = k ⋆ m ⋆ , where φ is an assumed shape vector, not an eigenvector.
Ritz Coordinates Truncated Sums, Matrix Iteration Giacomo Boffi Introduction For a N DOF system, an approximation to a displacement vector x Fundamental Mode Analysis can be written in terms of a set of M < N assumed shape, linearly Second Mode independent vectors, Analysis Higher Modes φ i , i = 1 , . . . , M < N Inverse Iteration Matrix Iteration with Shifts and a set of Ritz coordinates z i , i − 1 , . . . , M < N : Rayleigh Methods � x = φ i z i = Φ z . Rayleigh-Ritz Method Rayleigh-Ritz Example i Subspace iteration We say approximation because a linear combination of M < N vectors cannot describe every point in a N -space.
Rayleigh Quotient in Ritz Coordinates Truncated Sums, Matrix Iteration Giacomo Boffi Introduction We can write the Rayleigh quotient as a function of the Ritz Fundamental coordinates, Mode Analysis ω 2 ( z ) = z T Φ T K Φ z z T φ T M φz = k ( z ) Second Mode m ( z ) , Analysis Higher Modes but this is not an explicit function for any modal frequency... Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Rayleigh Quotient in Ritz Coordinates Truncated Sums, Matrix Iteration Giacomo Boffi Introduction We can write the Rayleigh quotient as a function of the Ritz Fundamental coordinates, Mode Analysis ω 2 ( z ) = z T Φ T K Φ z z T φ T M φz = k ( z ) Second Mode m ( z ) , Analysis Higher Modes but this is not an explicit function for any modal frequency... Inverse Iteration Matrix Iteration with Shifts On the other hand, we have seen that frequency estimates are always greater than true frequencies, so our best estimates are the the local Rayleigh Methods minima of ω 2 ( z ) , or the points where all the derivatives of ω 2 ( z ) Rayleigh-Ritz Method Rayleigh-Ritz Example with respect to z i are zero: Subspace iteration m ( z ) ∂k ( z ) − k ( z ) ∂m ( z ) ∂ω 2 ( z ) ∂z i ∂z i = = 0 , for i = 1 , . . . , M < N ∂z j ( m ( z )) 2
Rayleigh Quotient in Ritz Coordinates Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Second Mode Analysis Observing that Higher Modes k ( z ) = ω 2 ( z ) m ( z ) Inverse Iteration we can substitute into and simplify the preceding equation, Matrix Iteration with Shifts ∂k ( z ) − ω 2 ( z ) ∂m ( z ) = 0 , for i = 1 , . . . , M < N Rayleigh ∂z i ∂z i Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Rayleigh Quotient in Ritz Coordinates Truncated Sums, Matrix Iteration Giacomo Boffi With the positions Introduction Φ T K Φ = K Φ T M Φ = M and Fundamental Mode Analysis we have � � Second Mode k ( z ) = z T Kz = k rs z r z s , Analysis r s Higher Modes hence Inverse Iteration �� � Matrix Iteration � ∂k ( z ) � � with Shifts = k is z s + k ri z r . ∂z i Rayleigh s r Methods Due to symmetry, k ri = k ir and consequently Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration � � � ∂k ( z ) � � = 2 k is z s = 2 Kz . ∂z i s Analogously � ∂m ( z ) � = 2 Mz . ∂z i
Reduced Eigenproblem Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Second Mode Analysis Substituting these results in ∂k ( z ) ∂z i − ω 2 ( z ) ∂m ( z ) = 0 we can write a ∂z i Higher Modes new eigenvector problem , in the M DOF Ritz coordinates space, Inverse Iteration with reduced M × M matrices: Matrix Iteration with Shifts K z − ω 2 M z = 0 . Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Modal Superposition? Truncated Sums, Matrix Iteration Giacomo Boffi Introduction After solving the reduced eigenproblem, we have a set of M Fundamental eigenvalues ω 2 i and a corresponding set of M eigenvectors z i . What Mode Analysis is the relation between these results and the eigenpairs of the original Second Mode Analysis problem? Higher Modes The ω 2 i clearly are approximations from above to the real Inverse Iteration eigenvalues, and if we write ψ i = Φ z i we see that, being Matrix Iteration with Shifts T i Φ T M Φ i Mψ j = z T Rayleigh ψ z j = M i δ ij , Methods � �� � Rayleigh-Ritz Method M Rayleigh-Ritz Example Subspace iteration the approximated eigenvectors ψ i are orthogonal with respect to the structural matrices and can be used in ordinary modal superposition techniques.
A Last Question Truncated Sums, Matrix Iteration Giacomo Boffi Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes One last question: how many ω 2 i and ψ i are effective approximations Inverse Iteration to the true eigenpairs? Experience tells that an effective Matrix Iteration with Shifts approximation is to be expected for the first M/ 2 eigenthings. Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
RR Example m x 5 k m x 4 k m x 3 k m x 2 k m k x 1
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k m x 2 k m k x 1
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k The Ritz base vectors and the reduced matrices, m 0 . 2 − 0 . 5 � 0 . 2 � 0 . 2 x 2 ¯ k 0 . 4 − 1 . 0 K = k 0 . 2 2 . 0 m 0 . 6 − 0 . 5 Φ = � 2 . 2 � 0 . 2 ¯ 0 . 8 +0 . 0 M = m k x 1 0 . 2 2 . 5 1 . 0 1 . 0
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k The Ritz base vectors and the reduced matrices, m 0 . 2 − 0 . 5 � 0 . 2 � 0 . 2 x 2 ¯ k 0 . 4 − 1 . 0 K = k 0 . 2 2 . 0 m 0 . 6 − 0 . 5 Φ = � 2 . 2 � 0 . 2 ¯ 0 . 8 +0 . 0 M = m k x 1 0 . 2 2 . 5 1 . 0 1 . 0 � 2 − 22 ρ � � z 1 � � 0 � 2 − 2 ρ Red. eigenproblem ( ρ = ω 2 m/k ): = 2 − 2 ρ 20 − 25 ρ z 2 0
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k The Ritz base vectors and the reduced matrices, m 0 . 2 − 0 . 5 � 0 . 2 � 0 . 2 x 2 ¯ k 0 . 4 − 1 . 0 K = k 0 . 2 2 . 0 m 0 . 6 − 0 . 5 Φ = � 2 . 2 � 0 . 2 ¯ 0 . 8 +0 . 0 M = m k x 1 0 . 2 2 . 5 1 . 0 1 . 0 � 2 − 22 ρ � � z 1 � � 0 � 2 − 2 ρ Red. eigenproblem ( ρ = ω 2 m/k ): = 2 − 2 ρ 20 − 25 ρ z 2 0 The roots are ρ 1 = 0 . 0824 , ρ 2 = 0 . 800 , the frequencies are � � ω 1 = 0 . 287 k/m [ = 0 . 285] , ω 2 = 0 . 850 k/m [ = 0 . 831] , while the k/m normalized exact eigenvalues are [0 . 08101405 , 0 . 69027853] . The first eigenvalue is estimated with good approximation.
Rayleigh-Ritz Example Truncated Sums, Matrix Iteration Giacomo Boffi Introduction � 1 . 329 � 0 . 03170 Fundamental The Ritz coordinates eigenvector matrix is Z = . Mode Analysis − 0 . 1360 1 . 240 Second Mode The RR eigenvector matrix, Φ and the exact one, Ψ : Analysis Higher Modes +0 . 3338 − 0 . 6135 +0 . 3338 − 0 . 8398 Inverse Iteration +0 . 6676 − 1 . 2270 +0 . 6405 − 1 . 0999 Matrix Iteration Φ = +0 . 8654 − 0 . 6008 , Ψ = +0 . 8954 − 0 . 6008 . with Shifts +1 . 0632 +0 . 0254 +1 . 0779 +0 . 3131 Rayleigh +1 . 1932 +1 . 2713 +1 . 1932 +1 . 0108 Methods Rayleigh-Ritz Method The accuracy of the estimates for the 1st mode is very good, on the contrary the Rayleigh-Ritz Example 2nd mode estimates are in the order of a few percents. Subspace iteration
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