Advances on differential equations with uncertainties and their applications to probabilistic mechanics engineering

• Autores: Elena López Navarro
• Directores de la Tesis: María Dolores Roselló Ferragud (dir. tes.) , Juan Carlos Cortés López (dir. tes.)
• Lectura: En la Universitat Politècnica de València ( España ) en 2024
• Idioma: inglés
• Tribunal Calificador de la Tesis: Juan Angel Aledo Sánchez (presid.) , Cristina Santamaría Navarro (secret.) , Ioan-Lucian Popa (voc.)
• Enlaces
  • Tesis en acceso abierto en: RiuNet
• Resumen

  • Differential equations in Engineering are fundamental tools for modelling and analysing dynamical systems. Differential equations allow engineers to describe how physical quantities change over time and/or space, such as vibratory systems, mechanical structures, etc. However, many real-world systems are influenced by uncertainties. For instance, measurement errors, incomplete understanding of complex physical phenomena, random fluctuations like electronic circuit noise, and unpredictable material properties variations are aleatoric factors. Understanding both deterministic and random/stochastic differential equations is, therefore, vital for developing robust and reliable engineering solutions in a random world.

    This thesis presents a comprehensive probabilistic analysis of three mechanical engineering problems: vibratory systems (oscillators), mechanical structures (deflection of beams), and a foundational mechanical problem modelled by a random fractional differential equation. Throughout our work, we have applied different mathematical techniques to better understand these system's behavior under random excitations. A significant focus has been on accurately approximating not only the main statistical moments but also the probability density function of the model's response (solution) of the models studied throughout this dissertation. Providing a complete probabilistic description of such types of mechanical models is a topic that has attracted the interest of mathematicians and engineers during the last decades.

    In the first place, we will study two nonlinear random oscillators where the restoring term depends on the position, in the first case, and on the position and velocity, in the second one. The nonlinear term is affected by a small perturbative parameter. As in both cases, we cannot obtain the solution explicitly, we will use the stochastic perturbation method to construct approximations of the stochastic solution and its first statistical moments. This, in combination with the principle of maximum entropy, will result in obtaining reliable approximations of the stationary probability density function of the response. Second, we will study two models describing the deflection of a random static cantilever beam. We distinguish two different scenarios with respect to the type of stochastic processes modelling the distribution of the load spanned the beam and assuming randomness for some model parameters such as the Young's modulus or the flexural rigidity parameter. We then conveniently adapt different stochastic techniques to calculate exactly or approximately the probability density function of the deflection of the cantilever.

    Finally, we will revisit a simple model recently proposed to study a class of random oscillators formulated via the Caputo fractional derivative. We will construct approximations of the probability density function of the stochastic response, taking advantage of the random variable transformation method. We rigorously prove the convergence of these approximations under mild conditions of the model's parameters. This approach can inspire the study of more complex oscillators formulated via fractional differential equations.