Julio Cesar Ruiz Claeyssen

A direct approach to second-order matrix non-classical vibrating equations

Abstract A direct framework is developed for second-order matrix equations without transforming them into a first-order companion equation. It is done in terms of its matrix impulse response that is directly related to the transfer matrix. We formulate an extension of the Cayley–Hamilton identity, derive the controllability and observability matrices, and discuss Krylovs method in terms of such matrix response. This formulation will allow to further discuss Arnoldi and Lanczos methods as well as time-integration by FFT.

Diagonalization and spectral decomposition of factor block circulant matrices

Abstract We introduce factor circulant matrices: matrices with the structure of circulants, but with the entries below the diagonal multiplied by the same factor. The diagonalization of a circulant matrix and spectral decomposition are conveniently generalized to block matrices with the structure of factor circulants. Differential equations involving factor circulants are considered.

The Integral-Averaging Bifurcation Method and the General One-Delay Equation

Abstract A formula for determining the Hopf direction of bifurcation for periodic solutions of the delay equation x ′( t ) = g ( x ( t ), x ( t − r ), α ) is obtained by applying the integral-averaging method.

Simulation in primitive variables of incompressible flow with pressure Neumann condition

SUMMARY A velocity‐pressure algorithm, in primitive variables and finite differences, is developed for incompressible viscous flow with a Neumann pressure boundary condition. The pressure field is initialized by least-squares and updated from the Poisson equation in a direct weighted manner. Simulations with the cavity problem were made for several Reynolds numbers. The expected displacement of the central vortex was obtained, as well as the development of secondary and tertiary eddies. Copyright

Modal Formulation of Segmented Euler-Bernoulli Beams

We consider the obtention of modes and frequencies of segmented Euler-Bernoulli beams with internal damping and external viscous damping at the discontinuities of the sections. This is done by following a Newtonian approach in terms of a fundamental response of stationary beams subject to both types of damping. The use of a basis generated by the fundamental solution of a differential equation of fourth-order allows to formulate the eigenvalue problem and to write the modes shapes in a compact manner. For this, we consider a block matrix that carries the boundary conditions and intermediate conditions at the beams and values of the fundamental matrix at the ends and intermediate points of the beam. For each segment, the elements of the basis have the same shape since they are chosen as a convenient translation of the elements of the basis for the first segment. Our method avoids the use of the first-order state formulation also to rely on the Euler basis of a differential equation of fourth-order and it allows to envision how conditions will influence a chosen basis.

The impulse response in the symbolic computing of modes for beams and plates

This work seeks to obtain in a systematic manner the symbolic normal modes of beams and plates in terms of the impulse response of an associated fourth-order ordinary differential equation and its derivatives. The problem of finding the modes of a beam subject to general boundary conditions amounts to considering a singular homogeneous matrix system involving the boundary conditions as well as the dynamical basis generated by the impulse response. For a plate, Levys method, that assumes supported conditions on two parallel sides, is modified with the inclusion of the dynamical basis. For several boundary conditions, the initial conditions of the impulse response allows us to reduce computations.

Decomposition of forced responses in vibrating systems

We seek to determine the forced response of evolution systems (discrete, concentrated and distributed) in terms of free and permanent responses. This can be accomplished with the use of the dynamical basis generated by the impulse response. The influence of the transients due to elementary inputs and piecewise linear forcing inputs is observed with the forced response. Finite differences numerical integration schemes, matrix differential systems and a clamped-supported Euler-Bernoulli beam with axial force are considered.

Inversion of Higher-Order Matrix Difference and Differential Equations through Their Dynamical Solutions

Abstract We discuss matrix finite difference and ordinary differential equations in terms of their dynamical solutions which correspond to Green functions for initial-value problems. Explicit formulas, which make no use of Jordan decompositions, are derived by using the Laplace-Stieltjes transform. The situation for inverting matrix polynomials is also considered.

Modal Analysis of a Beam with a Tip Rotor by Using a Fundamental Response

The shape modes of a clamped-free beam model with a tip rotor are determined by using a dynamical basis that is generated by a fundamental spatial free response. This is a non-classical distributed model for the displacements in the transverse directions of the beam which turns out to be coupled through boundary conditions due to rotation. Numerical calculations are performed by using the Ritz-Rayleigh method with several approximating basis.

Thermally driven cavity flow with Neumann condition for the pressure

We develop a velocity-pressure algorithm with a pressure Neumann condition in primitive variables using finite differences, for a 2D thermally driven square cavity flow with the Boussinesq approximation and a fixed Prandtl number. The pressure field is updated in a one-step weighted form. Simulations were made for several Rayleigh numbers and the results are close to those found in the literature.