Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Generalized Conservative Oscillator: Particular Solutions and Modal Base

In course

ME-332: Mechanical vibrations

Dans ce cours on étudie la dynamique modale des structures mécaniques. Conceptes clés comme Mode Normale, Mass et Raideur effective, et Fréquences Propres sont appris pendant ce cours. ...

Description

This lecture covers the resolution of the free conservative regime using specific solutions and the modal base in the context of a generalized conservative oscillator. Topics include the orthogonality of eigenvectors, normal coordinates, linear combination of solutions, and the reformulation of the free oscillator regime. The instructor explains the resolution by specific solutions, the characteristic equation, proper pulsations, and the writing of eigenmodes in index notations. Additionally, the lecture delves into the normalization of amplitudes, orthogonality of modal vectors, and the response to initial conditions, including the extraction of reference amplitudes and mode phases.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Luis Guillermo Villanueva Torrijo

Guillermo Villanueva is a Tenure Track Assistant Professor at the Ecole Polytechnique Federale de Lausane (EPFL), Switzerland, in the Mechanical Engineering Institute (IGM). Before joining EPFL he was a Marie Curie post-doctoral scholar at DTU (Denmark) and Caltech (California, US); and before a post-doc at EPFL-LMIS1. He received his M.Sc. in Physics in Zaragoza (Spain) and his PhD from the UAB in Barcelona (Spain). Since the start of his PhD (2002), Prof. Villanueva has been active in the fields of NEMS/MEMS for sensing, having expertise from the design and fabrication to the characterization and applicability. He has co-authored more than 75 papers in peer-reviewed journals (h-index of 24 WoK, 32 GoS) and more than 100 contributions to international conferences. He is serving, or has served, on the program committees of IEEE-NEMS, IEEE-Sensors, MNE, IEEE-FCS and Transducers. He is editor of Microelectronic Engineering. He has co-organized MNE2014 and SNC2015; and he is currently co-organizing the short courses at Transducers 2019 and the 16th International Workshop on Nanomechanical Sensors (NMC2019).

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

Generalized eigenvector

In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.

Affine combination

In mathematics, an affine combination of x1, ..., xn is a linear combination such that Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the are elements of K (or for a Euclidean space), and the affine combination is also a point. See for the definition in this case.

Linear independence

In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

Conical combination

Given a finite number of vectors in a real vector space, a conical combination, conical sum, or weighted sum of these vectors is a vector of the form where are non-negative real numbers. The name derives from the fact that a conical sum of vectors defines a cone (possibly in a lower-dimensional subspace). The set of all conical combinations for a given set S is called the conical hull of S and denoted cone(S) or coni(S). That is, By taking k = 0, it follows the zero vector (origin) belongs to all conical hulls (since the summation becomes an empty sum).

Related lectures (46)

Multivariable Control: Eigenvalues and Modes

Explores eigenvalues and modes in multivariable control systems, focusing on their analysis and behavior.

Eigenvalues and Eigenvectors in 3D

Explores eigenvalues and eigenvectors in 3D linear algebra, covering characteristic polynomials, stability under transformations, and real roots.

Method of Undetermined Coefficients

Explores the method of undetermined coefficients for solving non-homogeneous linear differential equations with constant coefficients.

Diagonalization in 3D Linear Algebra

Explores diagonalization in 3D linear algebra, covering conditions for diagonalizability and eigenvectors.

Calcul de valeurs propres