Numerical Approximation of Dynamic Euler–Bernoulli Beams and a Flexible Satellite
Asti, Kristian (2020)
2020
Teknis-luonnontieteellinen DI-tutkinto-ohjelma - Degree Programme in Science and Engineering, MSc (Tech)
Informaatioteknologian ja viestinnän tiedekunta - Faculty of Information Technology and Communication Sciences
This publication is copyrighted. You may download, display and print it for Your own personal use. Commercial use is prohibited.
Hyväksymispäivämäärä
2020-08-27
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:tuni-202008136475
https://urn.fi/URN:NBN:fi:tuni-202008136475
Tiivistelmä
In this work, we develop a numerical method for simulating Euler–Bernoulli beams. We use this method for simulating a single beam, as well as simulating the solar panels of a flexible satellite. In developing the method, we apply mathematical control theory, Legendre polynomials, Fourier–Legendre series expansion and the spectral Galerkin method.
The main goal of this work is to develop a numerical model that we can use for doing simulations in MATLAB. We obtain a linear system of first-order differential equations, which we can solve with dedicated tools. We included the possibility to apply time-dependent boundary control input in the simulation model developed for a single beam. This allows us to use this model in simulating a flexible satellite. In the approximate model for the flexible satellite, we included the possibility to apply time-dependent control input to the system.
We study the existence of solutions to the dynamic beam equation with homogeneous boundary conditions (i.e. without boundary control input) by applying mathematical control theory and the theory of strongly continuous semigroups in particular. This is not a central part of the work, but we include it as a theoretical foundation for the existence of numerical solutions to the beam equation. As an introduction, we study boundary conditions corresponding to differently mounted beams with static beams first. Boundary conditions corresponding to different beam mountings form an essential part of solving the beam equation and developing the numerical method.
The numerical approximation for beams is based on modal basis functions made of Legendre polynomials and the spectral Galerkin method utilising these polynomials. The Legendre polynomials are computationally well applicable, as they are polynomials with integer coefficients and thus can be represented numerically as coefficient vectors to a high precision. Operations between these polynomials are also computationally simple. The use of Legendre polynomials is further motivated by the fact that they are great for approximating continuously differentiable functions.
The model for a flexible satellite consists of a small, rigid central body and two identical flexible solar panels, which we model as dynamic Euler–Bernoulli beams. The solar panels are mounted symmetrically to the central body and they affect each other via the boundary mountings. We model the satellite numerically by applying the approximate model developed for a single beam to both solar panels, and using a model derived from Newton’s laws for the central body. Using the boundary connections between the components, we combine the three systems into a single system, which we can simulate using MATLAB.
The main goal of this work is to develop a numerical model that we can use for doing simulations in MATLAB. We obtain a linear system of first-order differential equations, which we can solve with dedicated tools. We included the possibility to apply time-dependent boundary control input in the simulation model developed for a single beam. This allows us to use this model in simulating a flexible satellite. In the approximate model for the flexible satellite, we included the possibility to apply time-dependent control input to the system.
We study the existence of solutions to the dynamic beam equation with homogeneous boundary conditions (i.e. without boundary control input) by applying mathematical control theory and the theory of strongly continuous semigroups in particular. This is not a central part of the work, but we include it as a theoretical foundation for the existence of numerical solutions to the beam equation. As an introduction, we study boundary conditions corresponding to differently mounted beams with static beams first. Boundary conditions corresponding to different beam mountings form an essential part of solving the beam equation and developing the numerical method.
The numerical approximation for beams is based on modal basis functions made of Legendre polynomials and the spectral Galerkin method utilising these polynomials. The Legendre polynomials are computationally well applicable, as they are polynomials with integer coefficients and thus can be represented numerically as coefficient vectors to a high precision. Operations between these polynomials are also computationally simple. The use of Legendre polynomials is further motivated by the fact that they are great for approximating continuously differentiable functions.
The model for a flexible satellite consists of a small, rigid central body and two identical flexible solar panels, which we model as dynamic Euler–Bernoulli beams. The solar panels are mounted symmetrically to the central body and they affect each other via the boundary mountings. We model the satellite numerically by applying the approximate model developed for a single beam to both solar panels, and using a model derived from Newton’s laws for the central body. Using the boundary connections between the components, we combine the three systems into a single system, which we can simulate using MATLAB.