Summary: There are currently several problems in partial differential equations that model vibrations of beams, plates and bridges. Such vibrations may or may not stabilize at any given time, through the characteristics of the studied system or through some dissipative effect introduced into the system, the latter mechanically or naturally. This project aims to study how vibratory effects (thermoelastic, viscoelastic, etc.) act on stability or instability of coupled beam and plate systems. Through recent techniques in functional analysis and equations evolution, the work focuses on developing new methods to detect which components of the problem will lead us to uniform stability (or the best possible), that is, this project aims to establish a mathematical bias to reduce possible unwanted vibrations and avoid collapses in problems involving beams and plates, whether arched or flat.

Keywords: Vibrating beams, oscillation, stabilization, damping.

Financial support: CNPq agency, grant #309929/2022-9.

Registration: PPROPPG UEL - Project number.

Summary: Nowadays, there are several problems in partial differential equations modeling vibrations in beams, plates, and bridges. Such vibrations may (or not) stabilize in some time, depending on the feature of the system or else by means of some dissipative effect introduced into the system, the latter being natural not. This project aims to study how the thermoelastic effects act on the stabilization of coupled beam and plate systems. Through recent techniques in functional analysis and evolution equations, this work focuses on developing new methods to detect which component of the problem shall lead us to a desired stability, reducing as most as possible the undesired vibrations and, therefore, the collapse of the addressed beam and plate problems.

Keywords: thermoelasticity; beams; stability.

Financial support: Fundação Araucária agency, Agreement: 226/2022.

Registration: PROPPG UEL - Project number 13339.

Summary: The main goal of this research project is to develop new mathematical models that represent beams, plates and bridges vibrations. Attempts for models that connect real phenomena with mathematical equations lead us to the following great question: How to understand real physical phenomena like bridge vibrations by means of mathematical models? Such an answer would take humanity to prevent collapse of bridges in case of earthquakes or/and tremors, for instance, once the well-posed mathematical models could give a satisfactory response on how to reduce bridge vibrations to its equilibrium by dissipating the energy therein. There are currently several models in the literature dealing with the subject, but a complete integration with the physical reality is far from a total understanding. The proponent has some articles in the field and intends to develop at least a new model in four years through the research of this project, which will be clarified in its introduction and methodology.

Keywords: Evolution Equations; Stability; Long-time behavior.

Financial support: CNPq agency, grant #301116/2019-9.

Registration: PROPPG UEL - Project number 12367.

Summary: The present research project consists in modeling and solving problems in viscoelastic beams with memory derived from engineering and mathematical physics. In the modeling part, the idea is to establish mathematical models by means of viscoelastic constitutive laws for several models in the field of vibrating beams. In the resolution part, the purpose is to prove mathematical results that ensure the existence of solutions as well as to evaluate their qualitative properties. In this way, the main goal is to develop new techniques of resolution by using theoretical methods existing in the literature in mathematics.

Keywords: Beams; memory; solution; vibrations; stability.

Financial support: Fundação Araucária agency, Agreement: 066/2019.

Registration: PROPPG UEL - Project number 11893.

Summary: The present project of research aims to address in a theoretical and applied way some models in partial differential equations coming from physical-mathematics in what concerns beams and plate equations. The main focus of this project are: work in the modeling of initial-boundary value problems under elastic, viscoelastic and thermoelastic constitutive laws; solubility of the considered problems with respect to existence of solutions, uniqueness and continuous dependence of the initial data; study of the asymptotic behavior of solutions and the long-time behavior of the corresponding dynamical system generated by evolution problems. The topic addressed in this project has been recently studied by several researchers around the world in this field, requires the development of new techniques in a pure and applied mathematic framework and has substantial relevance in different areas of mathematics, physics and engineering.

Keywords: Differential equations; extensible beams; stability.

Financial support: Independent research.

Registration: PROPPG UEL - Project number 10656.

Summary: The present work aims to study new methods on the investigation of solutions for evolution problems related to plate and extensible beams vibrations as well as evaluate its qualitative properties such as regularity and stability.

Keywords: Beam equation; plate equation; Euler-Bernoulli beams.

Financial support: CNPq agency, grant #441414/2014-1.

Registration: PROPPG UEL - Project number 09506.