Structural Matrices Giacomo Boffi Introductory Remarks Structural Matrices Structural Matrices in MDOF Systems Evaluation of Structural Matrices Choice of Property Formulation Giacomo Boffi Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano May 8, 2014
Outline Structural Matrices Giacomo Boffi Introductory Remarks Introductory Remarks Structural Matrices Structural Orthogonality Relationships Matrices Additional Orthogonality Relationships Evaluation of Structural Matrices Evaluation of Structural Matrices Choice of Property Flexibility Matrix Formulation Example 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, Evaluation of Structural that is the operators that relate the vector of system Matrices coordinates x and its time derivatives ˙ x and ¨ x to the forces Choice of Property Formulation acting on the system nodes, f S , f D and f I , respectively.
Introductory Remarks Structural Matrices Giacomo Boffi Introductory Remarks Structural Matrices Today we will study the properties of structural matrices, Evaluation of Structural that is the operators that relate the vector of system Matrices coordinates x and its time derivatives ˙ x and ¨ x to the forces Choice of Property Formulation acting on the system nodes, f S , f D and f I , respectively. 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, but you know
Introductory Remarks Structural Matrices Giacomo Boffi Introductory Remarks Structural Matrices Today we will study the properties of structural matrices, Evaluation of Structural that is the operators that relate the vector of system Matrices coordinates x and its time derivatives ˙ x and ¨ x to the forces Choice of Property Formulation acting on the system nodes, f S , f D and f I , respectively. 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, but you know that a bit of reiteration is really good for developing minds.
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 Structural that holds between the vector of system coordinates x and its Matrices time derivatives ˙ x and ¨ x to the forces acting on the system Orthogonality Relationships Additional Orthogonality nodes, f S , f D and f I , elastic, damping and inertial force vectors. Relationships Evaluation of Structural Matrices M ¨ x + C ˙ x + K x = p ( t ) Choice of Property Formulation f I + f D + f S = p ( t ) 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 Relationships x ( t ) = ψ ( A sin ω t + B cos ω t ) Evaluation of Structural Matrices we wrote the homogeneous linear system Choice of Property Formulation � � K − ω 2 M ψ = 0 whose non-trivial solutions ψ i for ω 2 i such that � � � = 0 are the eigenvectors. � K − ω 2 i M 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 Introductory Remarks From Structural K ψ s = ω 2 s M ψ s Matrices Orthogonality Relationships r KM − 1 we have premultiplying by ψ T Additional Orthogonality Relationships Evaluation of ψ T r KM − 1 K ψ s = ω 2 s ψ T Structural r K ψ s Matrices Choice of Property Formulation
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Introductory Remarks From Structural K ψ s = ω 2 s M ψ s Matrices Orthogonality Relationships r KM − 1 we have premultiplying by ψ T Additional Orthogonality Relationships Evaluation of ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 Structural r M r , Matrices Choice of Property Formulation
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Introductory Remarks From Structural K ψ s = ω 2 s M ψ s Matrices Orthogonality Relationships r KM − 1 we have premultiplying by ψ T Additional Orthogonality Relationships Evaluation of ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 Structural r M r , Matrices Choice of Property premultiplying the first equation by ψ T r KM − 1 KM − 1 Formulation ψ T r KM − 1 KM − 1 K ψ s = ω 2 s ψ T r KM − 1 K ψ s =
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Introductory Remarks From Structural K ψ s = ω 2 s M ψ s Matrices Orthogonality Relationships r KM − 1 we have premultiplying by ψ T Additional Orthogonality Relationships Evaluation of ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 Structural r M r , Matrices Choice of Property premultiplying the first equation by ψ T r KM − 1 KM − 1 Formulation ψ T r KM − 1 KM − 1 K ψ s = ω 2 s ψ T r KM − 1 K ψ s = δ rs ω 6 r M r
Additional Orthogonality Relationships Structural Matrices Giacomo Boffi Introductory Remarks From Structural K ψ s = ω 2 s M ψ s Matrices Orthogonality Relationships r KM − 1 we have premultiplying by ψ T Additional Orthogonality Relationships Evaluation of ψ T r KM − 1 K ψ s = ω 2 s ψ T r K ψ s = δ rs ω 4 Structural r M r , Matrices Choice of Property premultiplying the first equation by ψ T r KM − 1 KM − 1 Formulation ψ T r KM − 1 KM − 1 K ψ s = ω 2 s ψ T r KM − 1 K ψ s = δ rs ω 6 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 From Introductory M ψ s = ω − 2 Remarks s K ψ s Structural r MK − 1 we have premultiplying by ψ T Matrices Orthogonality Relationships Additional Orthogonality Relationships ψ T r MK − 1 M ψ s = ω − 2 s ψ T r M ψ s = Evaluation of Structural Matrices Choice of Property Formulation
Additional Relationships, 2 Structural Matrices Giacomo Boffi From Introductory M ψ s = ω − 2 Remarks s K ψ s Structural r MK − 1 we have premultiplying by ψ T Matrices Orthogonality Relationships Additional Orthogonality Relationships M s ψ T r MK − 1 M ψ s = ω − 2 s ψ T r M ψ s = δ rs Evaluation of ω 2 Structural s Matrices Choice of Property Formulation
Additional Relationships, 2 Structural Matrices Giacomo Boffi From Introductory M ψ s = ω − 2 Remarks s K ψ s Structural r MK − 1 we have premultiplying by ψ T Matrices Orthogonality Relationships Additional Orthogonality Relationships M s ψ T r MK − 1 M ψ s = ω − 2 s ψ T r M ψ s = δ rs Evaluation of ω 2 Structural s Matrices MK − 1 � 2 we have Choice of Property � premultiplying the first eq. by ψ T Formulation r MK − 1 � 2 M ψ s = ω − 2 � ψ T s ψ T r MK − 1 M ψ s = r
Additional Relationships, 2 Structural Matrices Giacomo Boffi From Introductory M ψ s = ω − 2 Remarks s K ψ s Structural r MK − 1 we have premultiplying by ψ T Matrices Orthogonality Relationships Additional Orthogonality Relationships M s ψ T r MK − 1 M ψ s = ω − 2 s ψ T r M ψ s = δ rs Evaluation of ω 2 Structural s Matrices MK − 1 � 2 we have Choice of Property � premultiplying the first eq. by ψ T Formulation r MK − 1 � 2 M ψ s = ω − 2 M s � ψ T s ψ T r MK − 1 M ψ s = δ rs r ω 4 s
Additional Relationships, 2 Structural Matrices Giacomo Boffi From Introductory M ψ s = ω − 2 Remarks s K ψ s Structural r MK − 1 we have premultiplying by ψ T Matrices Orthogonality Relationships Additional Orthogonality Relationships M s ψ T r MK − 1 M ψ s = ω − 2 s ψ T r M ψ s = δ rs Evaluation of ω 2 Structural s Matrices MK − 1 � 2 we have Choice of Property � premultiplying the first eq. by ψ T Formulation r MK − 1 � 2 M ψ s = ω − 2 M s � ψ T s ψ T r MK − 1 M ψ s = δ rs r ω 4 s and, generalizing, M s MK − 1 � b M ψ s = δ rs � ψ T r b ω 2 s
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