Multj Degrees of Freedom Systems MDOF Giacomo Boffj - PowerPoint PPT Presentation

Multj Degrees of Freedom Systems MDOF Giacomo Boffj htup://intranet.dica.polimi.it/people/boffj‐giacomo 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 𝑞 1 (𝑢) 𝑞 2 (𝑢) Propertjes of Structural Matrices An example 𝑛 1 𝑛 2 The 𝑙 1 𝑙 2 𝑙 3 Homogeneous Problem 𝑦 1 𝑦 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 D’Alembert 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 𝑞 1 Operators 𝑙 1 𝑦 1 𝑙 2 (𝑦 1 − 𝑦 2 ) Propertjes of Structural Matrices 𝑛 1 ̈ 𝑦 1 An example The 𝑛 1 ̈ 𝑦 1 + (𝑙 1 + 𝑙 2 )𝑦 1 − 𝑙 2 𝑦 2 = 𝑞 1 (𝑢) Homogeneous Problem Modal Analysis 𝑞 2 Examples 𝑙 2 (𝑦 2 − 𝑦 1 ) 𝑙 3 𝑦 2 𝑛 2 ̈ 𝑦 2 𝑛 2 ̈ 𝑦 2 − 𝑙 2 𝑦 1 + (𝑙 2 + 𝑙 3 )𝑦 2 = 𝑞 2 (𝑢)

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, 𝑦 1 (𝑢) and 𝑦 2 (𝑢) : Operators Propertjes of Structural Matrices An example �𝑛 1 ̈ 𝑦 1 + (𝑙 1 + 𝑙 2 )𝑦 1 − 𝑙 2 𝑦 2 = 𝑞 1 (𝑢), The Homogeneous 𝑛 2 ̈ 𝑦 2 − 𝑙 2 𝑦 1 + (𝑙 2 + 𝑙 3 )𝑦 2 = 𝑞 2 (𝑢). Problem Modal Analysis Examples

The equatjon of motjon of a 2DOF system Multj DoF Systems Giacomo Boffj Introductjon Introducing the loading vector 𝐪 , the vector of inertjal forces 𝐠 𝐽 and the vector of An Example The Equatjon of Motjon elastjc forces 𝐠 𝑇 , Matrices are Linear Operators 𝐪 = �𝑞 1 (𝑢) 𝐠 𝐽 = �𝑔 𝐠 𝑇 = �𝑔 𝐽,1 𝑇,1 Propertjes of 𝑞 2 (𝑢)� , 𝐽,2 � , 𝑇,2 � Structural Matrices 𝑔 𝑔 An example The we can write a vectorial equatjon of equilibrium: Homogeneous Problem Modal Analysis 𝐠 𝐉 + 𝐠 𝐓 = 𝐪(𝑢). Examples

In our example it is 𝐠 𝑇 = �𝑙 1 + 𝑙 2 −𝑙 2 𝑙 2 + 𝑙 3 � 𝐲 = 𝐋 𝐲 −𝑙 2 The stjfgness matrix 𝐋 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 𝐋 is hence a square matrix 𝐋 ndof × ndof 𝐠 𝑇 = 𝐋 𝐲 Multj DoF Systems Giacomo Boffj It is possible to write the linear relatjonship between 𝐠 𝑇 and the vector of Introductjon 𝑈 in terms of a matrix product, introducing the so called An Example displacements 𝐲 = �𝑦 1 𝑦 2 � The Equatjon of Motjon stjfgness matrix 𝐋 . Matrices are Linear Operators Propertjes of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples

The stjfgness matrix 𝐋 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 𝐋 is hence a square matrix 𝐋 ndof × ndof 𝐠 𝑇 = 𝐋 𝐲 Multj DoF Systems Giacomo Boffj It is possible to write the linear relatjonship between 𝐠 𝑇 and the vector of Introductjon 𝑈 in terms of a matrix product, introducing the so called An Example displacements 𝐲 = �𝑦 1 𝑦 2 � The Equatjon of Motjon stjfgness matrix 𝐋 . Matrices are Linear Operators In our example it is Propertjes of Structural Matrices An example 𝐠 𝑇 = �𝑙 1 + 𝑙 2 −𝑙 2 The 𝑙 2 + 𝑙 3 � 𝐲 = 𝐋 𝐲 Homogeneous −𝑙 2 Problem Modal Analysis Examples

𝐠 𝑇 = 𝐋 𝐲 Multj DoF Systems Giacomo Boffj It is possible to write the linear relatjonship between 𝐠 𝑇 and the vector of Introductjon 𝑈 in terms of a matrix product, introducing the so called An Example displacements 𝐲 = �𝑦 1 𝑦 2 � The Equatjon of Motjon stjfgness matrix 𝐋 . Matrices are Linear Operators In our example it is Propertjes of Structural Matrices An example 𝐠 𝑇 = �𝑙 1 + 𝑙 2 −𝑙 2 The 𝑙 2 + 𝑙 3 � 𝐲 = 𝐋 𝐲 Homogeneous −𝑙 2 Problem Modal Analysis The stjfgness matrix 𝐋 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 𝐋 is hence a square matrix 𝐋 ndof × ndof

𝐠 𝐽 = 𝐍 ̈ 𝐲 Multj DoF Systems Giacomo Boffj Introductjon Analogously, introducing the mass matrix 𝐍 that, for our example, is An Example The Equatjon of Motjon Matrices are Linear 𝐍 = �𝑛 1 0 Operators 𝑛 2 � Propertjes of 0 Structural Matrices An example The we can write Homogeneous Problem 𝐠 𝐽 = 𝐍 ̈ 𝐲. Modal Analysis Also the mass matrix 𝐍 is a square matrix, with number of rows and columns equal to Examples the number of DOF ’s.

Of course it is possible to take into consideratjon also the damping forces, taking into account the velocity vector ̇ 𝐲 and introducing a damping matrix 𝐃 too, so that we can eventually write 𝐍 ̈ 𝐲 + 𝐃 ̇ 𝐲 + 𝐋 𝐲 = 𝐪(𝑢). 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 𝐍 ̈ 𝐲 + 𝐋 𝐲 = 𝐪(𝑢). 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 𝐍 ̈ 𝐲 + 𝐋 𝐲 = 𝐪(𝑢). 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 ̇ 𝐲 and introducing a damping matrix 𝐃 too, so that we can Homogeneous Problem eventually write 𝐍 ̈ 𝐲 + 𝐃 ̇ Modal Analysis 𝐲 + 𝐋 𝐲 = 𝐪(𝑢). 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 𝐍 ̈ 𝐲 + 𝐋 𝐲 = 𝐪(𝑢). 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 ̇ 𝐲 and introducing a damping matrix 𝐃 too, so that we can Homogeneous Problem eventually write 𝐍 ̈ 𝐲 + 𝐃 ̇ Modal Analysis 𝐲 + 𝐋 𝐲 = 𝐪(𝑢). Examples But today we are focused on undamped systems...

𝐋 is a positjve defjnite matrix. Propertjes of 𝐋 Multj DoF Systems Giacomo Boffj Introductjon An Example The Equatjon of Motjon Matrices are Linear Operators 𝐋 is symmetrical. Propertjes of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples

Propertjes of 𝐋 Multj DoF Systems Giacomo Boffj Introductjon An Example The Equatjon of Motjon Matrices are Linear Operators 𝐋 is symmetrical. Propertjes of Structural Matrices An example 𝐋 is a positjve defjnite matrix. The Homogeneous Problem Modal Analysis Examples