title: Matrix basis for plane and modal waves in a Timoshenko beam.
abstract: Plane waves and modal waves of the Timoshenko beam model are characterized in closed form by introducing robust matrix basis that behave according to the nature of frequency and wave or modal numbers. These new characterizations are given in terms of a finite number of coupling matrices and closed form generating scalar functions. Through Liouville's technique, these latter are well behaved at critical or static situations. Eigenanalysis is formulated for exponential and modal waves. Modal waves are superposition of four plane waves, but there are plane waves that cannot be modal waves. Reflected and transmitted waves at an interface point are formulated in matrix terms, regardless of having a conservative or a dissipative situation. The matrix representation of modal waves is used in a crack problem for determining the reflected and transmitted matrices. Their euclidean norms are seen to be dominated by certain components at low and high frequencies. The matrix basis technique is also used with a non-local Timoshenko model and with the wave interaction with a boundary. The matrix basis allows to characterize reflected and transmitted waves in spectral and non-spectral form.
authors.affiliation: Instituto de Matemática , Universidade Federal do Rio Grande do Sul , 91.509-900 Porto Alegre, RS, Brazil.
authors.forename: Julio Cesar Ruiz
authors.identifier: 0000-0002-9249-3142
authors.initials: JC
authors.lastname: Claeyssen
authors.affiliation: Universidade Federal do Pampa, Campus Caçapava do Sul , 96.570-000, Caçapava do Sul, RS, Brazil.
authors.forename: Daniela de Rosso
authors.initials: DR
authors.lastname: Tolfo
authors.affiliation: Centro de Engenharias , Universidade Federal de Pelotas , 96.010-020, Pelotas, RS, Brazil.
authors.forename: Leticia
authors.initials: L
authors.lastname: Tonetto
keywords: Timoshenko beam; cracked beam; fundamental response; matrix basis; modal waves; plane waves