Multj Degrees of Freedom Systems MDOF Giacomo Boffj htup://intranet.dica.polimi.it/people/boffjβgiacomo Dipartjmento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 27, 2020
Outline Multj DoF Introductory Remarks Systems An Example Giacomo Boffj The Equatjon of Motjon is a System of Linear Difgerentjal Equatjons Introductjon Matrices are Linear Operators The Propertjes of Structural Matrices Homogeneous Problem An example Modal Analysis The Homogeneous Problem Examples The Homogeneous Equatjon of Motjon Eigenvalues and Eigenvectors Eigenvectors are Orthogonal Modal Analysis Eigenvectors are a base EoM in Modal Coordinates Initjal Conditjons Examples 2 DOF System
Sectjon 1 Multj DoF Systems Giacomo Boffj Introductory Remarks Introductjon An Example The Equatjon of Motjon Matrices are Linear Introductory Remarks Operators Propertjes of An Example Structural Matrices An example The Equatjon of Motjon is a System of Linear Difgerentjal Equatjons The Homogeneous Matrices are Linear Operators Problem Propertjes of Structural Matrices Modal Analysis An example Examples The Homogeneous Problem Modal Analysis Examples
Introductory Remarks Multj DoF Systems Giacomo Boffj Introductjon An Example Consider an undamped system with two masses and two degrees of freedom. The Equatjon of Motjon Matrices are Linear Operators π 1 (π’) π 2 (π’) Propertjes of Structural Matrices An example π 1 π 2 The π 1 π 2 π 3 Homogeneous Problem π¦ 1 π¦ 2 Modal Analysis Examples
Introductory Remarks Multj DoF Systems We can separate the two masses, single out the spring forces and, using the Giacomo Boffj DβAlembert Principle, the inertjal forces and, fjnally. write an equatjon of dynamic Introductjon equilibrium for each mass. An Example The Equatjon of Motjon Matrices are Linear π 1 Operators π 1 π¦ 1 π 2 (π¦ 1 β π¦ 2 ) Propertjes of Structural Matrices π 1 Μ π¦ 1 An example The π 1 Μ π¦ 1 + (π 1 + π 2 )π¦ 1 β π 2 π¦ 2 = π 1 (π’) Homogeneous Problem Modal Analysis π 2 Examples π 2 (π¦ 2 β π¦ 1 ) π 3 π¦ 2 π 2 Μ π¦ 2 π 2 Μ π¦ 2 β π 2 π¦ 1 + (π 2 + π 3 )π¦ 2 = π 2 (π’)
The equatjon of motjon of a 2DOF system Multj DoF Systems Giacomo Boffj Introductjon An Example The Equatjon of With some litule rearrangement we have a system of two linear difgerentjal equatjons Motjon Matrices are Linear in two variables, π¦ 1 (π’) and π¦ 2 (π’) : Operators Propertjes of Structural Matrices An example οΏ½π 1 Μ π¦ 1 + (π 1 + π 2 )π¦ 1 β π 2 π¦ 2 = π 1 (π’), The Homogeneous π 2 Μ π¦ 2 β π 2 π¦ 1 + (π 2 + π 3 )π¦ 2 = π 2 (π’). Problem Modal Analysis Examples
The equatjon of motjon of a 2DOF system Multj DoF Systems Giacomo Boffj Introductjon Introducing the loading vector πͺ , the vector of inertjal forces π π½ and the vector of An Example The Equatjon of Motjon elastjc forces π π , Matrices are Linear Operators πͺ = οΏ½π 1 (π’) π π½ = οΏ½π π π = οΏ½π π½,1 π,1 Propertjes of π 2 (π’)οΏ½ , π½,2 οΏ½ , π,2 οΏ½ Structural Matrices π π An example The we can write a vectorial equatjon of equilibrium: Homogeneous Problem Modal Analysis π π + π π = πͺ(π’). Examples
In our example it is π π = οΏ½π 1 + π 2 βπ 2 π 2 + π 3 οΏ½ π² = π π² βπ 2 The stjfgness matrix π has a number of rows equal to the number of elastjc forces, i.e., one force for each DOF and a number of columns equal to the number of the DOF . The stjfgness matrix π is hence a square matrix π ndof Γ ndof π π = π π² Multj DoF Systems Giacomo Boffj It is possible to write the linear relatjonship between π π and the vector of Introductjon π in terms of a matrix product, introducing the so called An Example displacements π² = οΏ½π¦ 1 π¦ 2 οΏ½ The Equatjon of Motjon stjfgness matrix π . Matrices are Linear Operators Propertjes of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples
The stjfgness matrix π has a number of rows equal to the number of elastjc forces, i.e., one force for each DOF and a number of columns equal to the number of the DOF . The stjfgness matrix π is hence a square matrix π ndof Γ ndof π π = π π² Multj DoF Systems Giacomo Boffj It is possible to write the linear relatjonship between π π and the vector of Introductjon π in terms of a matrix product, introducing the so called An Example displacements π² = οΏ½π¦ 1 π¦ 2 οΏ½ The Equatjon of Motjon stjfgness matrix π . Matrices are Linear Operators In our example it is Propertjes of Structural Matrices An example π π = οΏ½π 1 + π 2 βπ 2 The π 2 + π 3 οΏ½ π² = π π² Homogeneous βπ 2 Problem Modal Analysis Examples
π π = π π² Multj DoF Systems Giacomo Boffj It is possible to write the linear relatjonship between π π and the vector of Introductjon π in terms of a matrix product, introducing the so called An Example displacements π² = οΏ½π¦ 1 π¦ 2 οΏ½ The Equatjon of Motjon stjfgness matrix π . Matrices are Linear Operators In our example it is Propertjes of Structural Matrices An example π π = οΏ½π 1 + π 2 βπ 2 The π 2 + π 3 οΏ½ π² = π π² Homogeneous βπ 2 Problem Modal Analysis The stjfgness matrix π has a number of rows equal to the number of elastjc forces, i.e., Examples one force for each DOF and a number of columns equal to the number of the DOF . The stjfgness matrix π is hence a square matrix π ndof Γ ndof
π π½ = π Μ π² Multj DoF Systems Giacomo Boffj Introductjon Analogously, introducing the mass matrix π that, for our example, is An Example The Equatjon of Motjon Matrices are Linear π = οΏ½π 1 0 Operators π 2 οΏ½ Propertjes of 0 Structural Matrices An example The we can write Homogeneous Problem π π½ = π Μ π². Modal Analysis Also the mass matrix π is a square matrix, with number of rows and columns equal to Examples the number of DOF βs.
Of course it is possible to take into consideratjon also the damping forces, taking into account the velocity vector Μ π² and introducing a damping matrix π too, so that we can eventually write π Μ π² + π Μ π² + π π² = πͺ(π’). But today we are focused on undamped systems... Matrix Equatjon Multj DoF Systems Giacomo Boffj Introductjon Finally it is possible to write the equatjon of motjon in matrix format: An Example The Equatjon of Motjon Matrices are Linear π Μ π² + π π² = πͺ(π’). Operators Propertjes of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples
But today we are focused on undamped systems... Matrix Equatjon Multj DoF Systems Giacomo Boffj Introductjon Finally it is possible to write the equatjon of motjon in matrix format: An Example The Equatjon of Motjon Matrices are Linear π Μ π² + π π² = πͺ(π’). Operators Propertjes of Structural Matrices An example Of course it is possible to take into consideratjon also the damping forces, taking into The account the velocity vector Μ π² and introducing a damping matrix π too, so that we can Homogeneous Problem eventually write π Μ π² + π Μ Modal Analysis π² + π π² = πͺ(π’). Examples
Matrix Equatjon Multj DoF Systems Giacomo Boffj Introductjon Finally it is possible to write the equatjon of motjon in matrix format: An Example The Equatjon of Motjon Matrices are Linear π Μ π² + π π² = πͺ(π’). Operators Propertjes of Structural Matrices An example Of course it is possible to take into consideratjon also the damping forces, taking into The account the velocity vector Μ π² and introducing a damping matrix π too, so that we can Homogeneous Problem eventually write π Μ π² + π Μ Modal Analysis π² + π π² = πͺ(π’). Examples But today we are focused on undamped systems...
π is a positjve defjnite matrix. Propertjes of π Multj DoF Systems Giacomo Boffj Introductjon An Example The Equatjon of Motjon Matrices are Linear Operators π is symmetrical. Propertjes of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples
Propertjes of π Multj DoF Systems Giacomo Boffj Introductjon An Example The Equatjon of Motjon Matrices are Linear Operators π is symmetrical. Propertjes of Structural Matrices An example π is a positjve defjnite matrix. The Homogeneous Problem Modal Analysis Examples
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