Rosemaira Dalcin Copetti

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.

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.

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.

Matrix Vibration Formulation of Damped Multi-Span Beams

In this work we consider segmented Euler-Bernoulli beams that can have an internal damping of the type Kelvin-Voight and external viscous damping at the discontinuities of the sections. In the literature, the study of this kind of beams has been sufficiently studied with proportional damping only, however the effects of non-proportional damping has been little studied in terms of modal analysis. The obtaining of the modes of segmented beams can be accomplished with a the state space methodology or with the classical Euler construction of responses. Here, we follow a newtonian approach with the use of the impulse response of beams subject both types of damping. The use of the dynamical basis, generated by the fundamental solution of a differential equation of fourth order, allows to formulate the eigenvalue problem and the shapes of the modes in a compact manner. For this, we formulate in a block manner the boundary conditions and intermediate conditions at the beam and values of the fundamental matrix at the ends of the beam and in the points intermediate. We have chosen a basis generated by a fundamental response and it derivatives. The elements of this basis has the same shape with a convenient translation for each segment. This choice reduce computations with the number of constants to be determined to find only the ones that correspond to the first segment. The eigenanalysis will allow to study forced responses of multi-span Euler-Bernoulli beams under classical and non-classical boundary conditions as well as multi-walled carbon nanotubes (MWNT) that are modelled as an assemblage of Euler-Bernoulli beams connected throughout their length by springs subject to van der Waals interaction between any two adjacent nanotubes.Copyright

Nanobeams and AFM Subject to Piezoelectric and Surface Scale Effects

Vibration dynamics of elastic beams that are used in nanotechnology, such as atomic force microscope modeling and carbon nanotubes, are considered in terms of a fundamental response within a matrix framework. The modeling equations with piezoelectric and surface scale effects are written as a matrix differential equation subject to tip-sample general boundary conditions and to compatibility conditions for the case of multispan beams. We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects. Simulations were performed for a two stepped beam with a piezoelectric patch subject to pulse forcing terms. Results with Timoshenko models that include surface effects are presented for micro- and nanoscale. It was observed that the effects are significant just in nanoscale. We also simulate the frequency effects of a double-span beam in which one segment includes rotatory inertia and shear deformation and the other one neglects both phenomena. The proposed analytical methodology can be useful in the design of micro- and nanoresonator structures that involve deformable flexural models for detecting and imaging of physical and biochemical quantities.

Cálculo das Frequências e dos Modos de Vibração de um Sistema de Duas Vigas Acopladas

Neste trabalho e realizado um estudo sobre vibracoes livres e forcadas de um sistema composto por duas vigas Euler-Bernoulli acopladas. As vigas sao paralelas, de mesmo comprimento, simplesmente apoiadas e conectadas por uma camada viscoelastica. O estudo e realizado atraves da analise modal e de uma formulacao matricial em blocos, onde os modos de vibracao do sistema sao escritos em termos da base dinâmica, obtida atraves da solucao dinâmica de uma equacao diferencial de quarta ordem. E considerado o amortecimento de Rayleigh e o sistema e desacoplado usando o teorema dos modos normais. A resposta forcada e escrita em termos da resposta impulso matricial.

FREQUÊNCIAS E AUTOFUNÇÕES DO SISTEMA DE DUAS CORDAS ACOPLADAS

In this work is considered a system formed by two parallel strings, bi-supported, subject to a constant tension and connected by a viscoelastic layer. The transverse displacement of the model is described by two partial differential equations of second order coupled due to the viscoelastic term. The system is solved by the method which uses a block matrix formulation and the dynamic basis for determining the frequencies and the eigenfunctions or mode shapes of the problem.

A base dinâmica em um sistema composto por estruturas acopladas

Neste trabalho, e considerado um sistema composto por duas estruturas acopladas elasticamente. Uma teoria geral em termos de um operador diferencial espacial e da base dinâmica e desenvolvida. As frequencias e os modos de vibracao de dois sistemas, um formado por duas cordas e um formado por duas vigas sao determinados a partir da formulacao matricial em blocos e do uso da base dinâmica, simplificando a solucao da equacao modal de mesma ordem do operador espacial. A resposta forcada e escrita usando a resposta impulso matricial. As frequencias obtidas aparecem aos pares gerando dois tipos de movimentos, um sincrono e outro assincrono, dependendo da frequencia considerada.