Multi Degrees of Freedom Systems Remarks The MDOFs Homogeneous - PowerPoint PPT Presentation

Generalized SDOF’s Giacomo Boffi Introductory Multi Degrees of Freedom Systems Remarks The MDOF’s Homogeneous Problem Modal Analysis Examples Giacomo Boffi http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 28, 2017

Outline Generalized SDOF’s Giacomo Boffi Introductory Remarks Introductory An Example Remarks The Equation of Motion, a System of Linear Differential Equations The Matrices are Linear Operators Homogeneous Problem Properties of Structural Matrices Modal Analysis An example Examples The Homogeneous Problem The Homogeneous Equation of Motion Eigenvalues and Eigenvectors Eigenvectors are Orthogonal Modal Analysis Eigenvectors are a base EoM in Modal Coordinates Initial Conditions Examples 2 DOF System

Introductory Remarks Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example The Equation of Motion Matrices are Consider an undamped system with two masses and two degrees of Linear Operators Properties of freedom. Structural Matrices p 1 ( t ) p 2 ( t ) An example The Homogeneous m 1 m 2 Problem k 1 k 2 k 3 Modal Analysis Examples x 1 x 2

Introductory Remarks Generalized SDOF’s Giacomo Boffi Introductory We can separate the two masses, single out the spring forces and, Remarks using the D’Alembert Principle, the inertial forces and, finally. write An Example The Equation of Motion an equation of dynamic equilibrium for each mass. Matrices are Linear Operators Properties of Structural p 1 Matrices k 2 ( x 1 − x 2 ) An example k 1 x 1 m 1 ¨ x 1 The Homogeneous Problem m 1 ¨ x 1 + ( k 1 + k 2 ) x 1 − k 2 x 2 = p 1 ( t ) Modal Analysis Examples p 2 k 2 ( x 2 − x 1 ) k 3 x 2 m 2 ¨ x 2 x 2 − k 2 x 1 + ( k 2 + k 3 ) x 2 = p 2 ( t ) m 2 ¨

The equation of motion of a 2DOF system Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example The Equation of Motion Matrices are Linear Operators With some little rearrangement we have a system of two linear Properties of Structural differential equations in two variables, x 1 ( t ) and x 2 ( t ) : Matrices An example The � Homogeneous x 1 + ( k 1 + k 2 ) x 1 − k 2 x 2 = p 1 ( t ) , m 1 ¨ Problem m 2 ¨ x 2 − k 2 x 1 + ( k 2 + k 3 ) x 2 = p 2 ( t ) . Modal Analysis Examples

The equation of motion of a 2DOF system Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example The Equation of Introducing the loading vector p , the vector of inertial forces f I and Motion Matrices are the vector of elastic forces f S , Linear Operators Properties of Structural Matrices � p 1 ( t ) � � f I ,1 � � f S ,1 � An example p = , f I = , f S = The p 2 ( t ) f I ,2 f S ,2 Homogeneous Problem we can write a vectorial equation of equilibrium: Modal Analysis Examples f I + f S = p ( t ) .

f S = K x Generalized SDOF’s Giacomo Boffi Introductory Remarks It is possible to write the linear relationship between f S and the An Example � T in terms of a matrix product, The Equation of � vector of displacements x = x 1 x 2 Motion Matrices are Linear Operators introducing the so called stiffness matrix K . Properties of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples

f S = K x Generalized SDOF’s Giacomo Boffi Introductory Remarks It is possible to write the linear relationship between f S and the An Example � T in terms of a matrix product, The Equation of � vector of displacements x = x 1 x 2 Motion Matrices are Linear Operators introducing the so called stiffness matrix K . Properties of Structural In our example it is Matrices An example The � k 1 + k 2 � − k 2 Homogeneous f S = x = K x Problem − k 2 k 2 + k 3 Modal Analysis Examples

f S = K x Generalized SDOF’s Giacomo Boffi Introductory Remarks It is possible to write the linear relationship between f S and the An Example � T in terms of a matrix product, The Equation of � vector of displacements x = x 1 x 2 Motion Matrices are Linear Operators introducing the so called stiffness matrix K . Properties of Structural In our example it is Matrices An example The � k 1 + k 2 � − k 2 Homogeneous f S = x = K x Problem − k 2 k 2 + k 3 Modal Analysis The stiffness matrix K has a number of rows equal to the number of Examples elastic forces, i.e., one force for each DOF and a number of columns equal to the number of the DOF . The stiffness matrix K is hence a square matrix K ndof × ndof

f I = M ¨ x Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example The Equation of Analogously, introducing the mass matrix M that, for our example, is Motion Matrices are Linear Operators Properties of � m 1 � 0 Structural Matrices M = An example 0 m 2 The Homogeneous we can write Problem Modal Analysis f I = M ¨ x . Examples Also the mass matrix M is a square matrix, with number of rows and columns equal to the number of DOF ’s.

Matrix Equation Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example Finally it is possible to write the equation of motion in matrix format: The Equation of Motion Matrices are Linear Operators x + K x = p ( t ) . M ¨ Properties of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples

Matrix Equation Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example Finally it is possible to write the equation of motion in matrix format: The Equation of Motion Matrices are Linear Operators x + K x = p ( t ) . M ¨ Properties of Structural Matrices An example Of course it is possible to take into consideration also the The Homogeneous damping forces, taking into account the velocity vector ˙ x and Problem introducing a damping matrix C too, so that we can eventually Modal Analysis write Examples M ¨ x + C ˙ x + K x = p ( t ) .

Matrix Equation Generalized SDOF’s Giacomo Boffi Introductory Remarks An Example Finally it is possible to write the equation of motion in matrix format: The Equation of Motion Matrices are Linear Operators x + K x = p ( t ) . M ¨ Properties of Structural Matrices An example Of course it is possible to take into consideration also the The Homogeneous damping forces, taking into account the velocity vector ˙ x and Problem introducing a damping matrix C too, so that we can eventually Modal Analysis write Examples M ¨ x + C ˙ x + K x = p ( t ) . But today we are focused on undamped systems...

Properties of K Generalized SDOF’s Giacomo Boffi ◮ K is symmetrical. Introductory The elastic force exerted on mass i due to an unit displacement Remarks An Example of mass j , f S , i = k ij is equal to the force k ji exerted on mass j The Equation of Motion due to an unit diplacement of mass i , in virtue of Betti’s Matrices are Linear Operators Properties of theorem (also known as Maxwell-Betti reciprocal work theorem). Structural Matrices An example The Homogeneous Problem Modal Analysis Examples

Properties of K Generalized SDOF’s Giacomo Boffi ◮ K is symmetrical. Introductory The elastic force exerted on mass i due to an unit displacement Remarks An Example of mass j , f S , i = k ij is equal to the force k ji exerted on mass j The Equation of Motion due to an unit diplacement of mass i , in virtue of Betti’s Matrices are Linear Operators Properties of theorem (also known as Maxwell-Betti reciprocal work theorem). Structural Matrices ◮ K is a positive definite matrix. An example The The strain energy V for a discrete system is Homogeneous Problem V = 1 2 x T f S , Modal Analysis Examples and expressing f S in terms of K and x we have V = 1 2 x T K x , and because the strain energy is positive for x � = 0 it follows that K is definite positive.

Properties of M Generalized SDOF’s Giacomo Boffi Introductory Remarks Restricting our discussion to systems whose degrees of freedom are An Example The Equation of the displacements of a set of discrete masses, we have that the mass Motion Matrices are matrix is a diagonal matrix, with all its diagonal elements greater Linear Operators Properties of than zero. Such a matrix is symmetrical and definite positive. Structural Matrices An example Both the mass and the stiffness matrix are symmetrical and definite The positive. Homogeneous Problem Modal Analysis Examples

Properties of M Generalized SDOF’s Giacomo Boffi Introductory Remarks Restricting our discussion to systems whose degrees of freedom are An Example The Equation of the displacements of a set of discrete masses, we have that the mass Motion Matrices are matrix is a diagonal matrix, with all its diagonal elements greater Linear Operators Properties of than zero. Such a matrix is symmetrical and definite positive. Structural Matrices An example Both the mass and the stiffness matrix are symmetrical and definite The positive. Homogeneous Problem Modal Analysis Examples Note that the kinetic energy for a discrete system can be written T = 1 x T M ˙ 2 ˙ x .

Generalisation of previous results Generalized SDOF’s Giacomo Boffi Introductory Remarks The findings in the previous two slides can be generalised to the An Example The Equation of Motion structural matrices of generic structural systems, with two main Matrices are Linear Operators exceptions. Properties of Structural Matrices An example The Homogeneous Problem Modal Analysis Examples