Structural Matrices Giacomo Boffi Introductory Remarks Structural Matrices in MDOF Systems Structural Matrices Evaluation of Structural Giacomo Boffi Matrices Choice of Property http://intranet.dica.polimi.it/people/boffi-giacomo Formulation Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 22, 2018
Outline Structural Matrices Giacomo Boffi Introductory Remarks Introductory Structural Matrices Remarks Orthogonality Relationships Structural Matrices Additional Orthogonality Relationships Evaluation of Evaluation of Structural Matrices Structural Matrices Flexibility Matrix Choice of Property Example Formulation Stiffness Matrix Mass Matrix Damping Matrix Geometric Stiffness External Loading Choice of Property Formulation Static Condensation Example
Introductory Remarks Structural Matrices Giacomo Boffi Introductory Remarks Structural Matrices Today we will study the properties of structural matrices, that is the Evaluation of operators that relate the vector of system coordinates x and its time Structural Matrices derivatives ˙ x and ¨ x to the forces acting on the system nodes, f S , f D Choice of Property and f I , respectively. Formulation
Introductory Remarks Structural Matrices Giacomo Boffi Introductory Remarks Structural Matrices Today we will study the properties of structural matrices, that is the Evaluation of operators that relate the vector of system coordinates x and its time Structural Matrices derivatives ˙ x and ¨ x to the forces acting on the system nodes, f S , f D Choice of Property and f I , respectively. Formulation In the end, we will see again the solution of a MDOF problem by superposition, and in general today we will revisit many of the subjects of our previous class.
Introductory Remarks Structural Matrices Orthogonality Relationships Additional Orthogonality Relationships Evaluation of Structural Matrices Choice of Property Formulation
Structural Matrices Structural Matrices Giacomo Boffi We already met the mass and the stiffness matrix, M and K , and tangentially we introduced also the dampig matrix C . Introductory Remarks We have seen that these matrices express the linear relation that holds Structural between the vector of system coordinates x and its time derivatives ˙ x Matrices and ¨ x to the forces acting on the system nodes, f S , f D and f I , elastic, Orthogonality Relationships damping and inertial force vectors. Additional Orthogonality Relationships Evaluation of Structural M ¨ x + C ˙ x + K x = p ( t ) Matrices f I + f D + f S = p ( t ) Choice of Property Formulation Also, we know that M and K are symmetric and definite positive, and that it is possible to uncouple the equation of motion expressing the system coordinates in terms of the eigenvectors , x ( t ) = � q i ψ i , where the q i are the modal coordinates and the eigenvectors ψ i are the non-trivial solutions to the equation of free vibrations, � � K − ω 2 M ψ = 0
Free Vibrations Structural Matrices Giacomo Boffi From the homogeneous, undamped problem Introductory Remarks M ¨ x + K x = 0 Structural Matrices introducing separation of variables Orthogonality Relationships Additional Orthogonality x ( t ) = ψ ( A sin ω t + B cos ω t ) Relationships Evaluation of we wrote the homogeneous linear system Structural Matrices � � K − ω 2 M Choice of ψ = 0 Property Formulation � � � = 0 are whose non-trivial solutions ψ i for ω 2 � K − ω 2 i such that i M the eigenvectors. It was demonstrated that, for each pair of distint eigenvalues ω 2 r and ω 2 s , the corresponding eigenvectors obey the ortogonality condition, ψ T ψ T s K ψ r = δ rs ω 2 s M ψ r = δ rs M r , r M r .
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Starting from the equation of free vibrations (EOFV) Introductory Remarks K ψ s = ω 2 s M ψ s , Structural Matrices r KM − 1 we have Orthogonality pre-multiplying both members by ψ T Relationships Additional Orthogonality Relationships ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s Evaluation of Structural Matrices Choice of Property Formulation
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Starting from the equation of free vibrations (EOFV) Introductory Remarks K ψ s = ω 2 s M ψ s , Structural Matrices r KM − 1 we have Orthogonality pre-multiplying both members by ψ T Relationships Additional Orthogonality Relationships ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 r M r . Evaluation of Structural Matrices Choice of Property Formulation
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Starting from the equation of free vibrations (EOFV) Introductory Remarks K ψ s = ω 2 s M ψ s , Structural Matrices r KM − 1 we have Orthogonality pre-multiplying both members by ψ T Relationships Additional Orthogonality Relationships ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 r M r . Evaluation of Structural Matrices r KM − 1 KM − 1 we Pre-multiplying both members of the EOFV by ψ T Choice of have (compare with our previous result) Property Formulation r KM − 1 KM − 1 K ψ s = ω 2 r KM − 1 K ψ s = ψ T s ψ T
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Starting from the equation of free vibrations (EOFV) Introductory Remarks K ψ s = ω 2 s M ψ s , Structural Matrices r KM − 1 we have Orthogonality pre-multiplying both members by ψ T Relationships Additional Orthogonality Relationships ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 r M r . Evaluation of Structural Matrices r KM − 1 KM − 1 we Pre-multiplying both members of the EOFV by ψ T Choice of have (compare with our previous result) Property Formulation r KM − 1 KM − 1 K ψ s = ω 2 r KM − 1 K ψ s = δ rs ω 6 ψ T s ψ T r M r
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Starting from the equation of free vibrations (EOFV) Introductory Remarks K ψ s = ω 2 s M ψ s , Structural Matrices r KM − 1 we have Orthogonality pre-multiplying both members by ψ T Relationships Additional Orthogonality Relationships ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 r M r . Evaluation of Structural Matrices r KM − 1 KM − 1 we Pre-multiplying both members of the EOFV by ψ T Choice of have (compare with our previous result) Property Formulation r KM − 1 KM − 1 K ψ s = ω 2 r KM − 1 K ψ s = δ rs ω 6 ψ T s ψ T r M r and, generalizing, KM − 1 � b K ψ s = δ rs � b + 1 M r . � � ψ T ω 2 r r
Additional Relationships, 2 Structural Matrices Giacomo Boffi Let’s rearrange the equation of free vibrations Introductory M ψ s = ω − 2 Remarks s K ψ s . Structural r MK − 1 we have Matrices Pre-multiplying both members by ψ T Orthogonality Relationships Additional r MK − 1 M ψ s = ω − 2 Orthogonality ψ T s ψ T r M ψ s Relationships Evaluation of Structural Matrices Choice of Property Formulation
Additional Relationships, 2 Structural Matrices Giacomo Boffi Let’s rearrange the equation of free vibrations Introductory M ψ s = ω − 2 Remarks s K ψ s . Structural r MK − 1 we have Matrices Pre-multiplying both members by ψ T Orthogonality Relationships Additional M s r MK − 1 M ψ s = ω − 2 Orthogonality ψ T s ψ T r M ψ s = δ rs . Relationships ω 2 s Evaluation of Structural Matrices Choice of Property Formulation
Additional Relationships, 2 Structural Matrices Giacomo Boffi Let’s rearrange the equation of free vibrations Introductory M ψ s = ω − 2 Remarks s K ψ s . Structural r MK − 1 we have Matrices Pre-multiplying both members by ψ T Orthogonality Relationships Additional M s r MK − 1 M ψ s = ω − 2 Orthogonality ψ T s ψ T r M ψ s = δ rs . Relationships ω 2 s Evaluation of Structural MK − 1 � 2 we � Matrices Pre-multiplying both members of the EOFV by ψ T r Choice of have Property Formulation MK − 1 � 2 M ψ s = ω − 2 � r MK − 1 M ψ s ψ T s ψ T r
Additional Relationships, 2 Structural Matrices Giacomo Boffi Let’s rearrange the equation of free vibrations Introductory M ψ s = ω − 2 Remarks s K ψ s . Structural r MK − 1 we have Matrices Pre-multiplying both members by ψ T Orthogonality Relationships Additional M s r MK − 1 M ψ s = ω − 2 Orthogonality ψ T s ψ T r M ψ s = δ rs . Relationships ω 2 s Evaluation of Structural MK − 1 � 2 we � Matrices Pre-multiplying both members of the EOFV by ψ T r Choice of have Property Formulation MK − 1 � 2 M ψ s = ω − 2 M s � r MK − 1 M ψ s = δ rs ψ T s ψ T r ω 4 s
Additional Relationships, 2 Structural Matrices Giacomo Boffi Let’s rearrange the equation of free vibrations Introductory M ψ s = ω − 2 Remarks s K ψ s . Structural r MK − 1 we have Matrices Pre-multiplying both members by ψ T Orthogonality Relationships Additional M s r MK − 1 M ψ s = ω − 2 Orthogonality ψ T s ψ T r M ψ s = δ rs . Relationships ω 2 s Evaluation of Structural MK − 1 � 2 we � Matrices Pre-multiplying both members of the EOFV by ψ T r Choice of have Property Formulation MK − 1 � 2 M ψ s = ω − 2 M s � r MK − 1 M ψ s = δ rs ψ T s ψ T r ω 4 s and, generalizing, MK − 1 � b M ψ s = δ rs M s � ψ T r b ω 2 s
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