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The weak form of the SDOF and MDOF equation of motion, part I: Theory

Nikolaos Karaliolios, Dimitrios L. Karabalis

TL;DR

This paper develops a weak formulation for the SDOF and extends it to the MDOF case, transforming the initial-value problem into a boundary-value problem that is solved in a Sobolev space setting. It introduces time-weighted inner products, a moving-observer decomposition, and a Duhamel-based representation to guarantee existence, uniqueness, and regularity of solutions under weak forcing $f\in H^{-1}$. The authors provide a general framework for numerical methods by projecting the weak formulation onto finite-dimensional subspaces of $H^1$, analyze convergence via angles between subspaces, and derive error bounds for displacement, velocity, and energy, with clear paths to higher-order and robust numerical schemes. The work lays groundwork for Part II, which will implement Bernstein polynomial bases and establish convergence rates, and Part III, which will present numerical evidence, making the approach practically relevant for structural dynamics and related differential systems.

Abstract

The weak form of the SDOF and MDOF equations of motion are obtained. The original initial conditions problem is transformed into a boundary value problem. The boundary value problem is then solved and transformed back to the initial conditions one. Subsequently, a general method for obtaining numerical methods using an arbitrary number of linearly independent approximating functions is outlined. This is part one of a series of three papers, in the second of which a numerical method is obtained, using Bernstein polynomials of arbitrarily high order. The numerical evidence for the convergence of the method will be presented in the third part paper.

The weak form of the SDOF and MDOF equation of motion, part I: Theory

TL;DR

This paper develops a weak formulation for the SDOF and extends it to the MDOF case, transforming the initial-value problem into a boundary-value problem that is solved in a Sobolev space setting. It introduces time-weighted inner products, a moving-observer decomposition, and a Duhamel-based representation to guarantee existence, uniqueness, and regularity of solutions under weak forcing . The authors provide a general framework for numerical methods by projecting the weak formulation onto finite-dimensional subspaces of , analyze convergence via angles between subspaces, and derive error bounds for displacement, velocity, and energy, with clear paths to higher-order and robust numerical schemes. The work lays groundwork for Part II, which will implement Bernstein polynomial bases and establish convergence rates, and Part III, which will present numerical evidence, making the approach practically relevant for structural dynamics and related differential systems.

Abstract

The weak form of the SDOF and MDOF equations of motion are obtained. The original initial conditions problem is transformed into a boundary value problem. The boundary value problem is then solved and transformed back to the initial conditions one. Subsequently, a general method for obtaining numerical methods using an arbitrary number of linearly independent approximating functions is outlined. This is part one of a series of three papers, in the second of which a numerical method is obtained, using Bernstein polynomials of arbitrarily high order. The numerical evidence for the convergence of the method will be presented in the third part paper.
Paper Structure (19 sections, 24 theorems, 141 equations)

This paper contains 19 sections, 24 theorems, 141 equations.

Table of Contents

  1. The SDOF problem
  2. From initial to boundary conditions
  3. Definitions
  4. The weak formulation of the SDOF problem
  5. Covariance of the Weak form with respect to time reparametrization
  6. Relation with Lagrangian and Hamiltonian mechanics
  7. A conservation law
  8. A dissipation law
  9. Some remarks on the weak formulation
  10. Weak formulation: The MDOF case
  11. From the weak formulation to numerical implementations: the SDOF case
  12. Obtaining a numerical method
  13. The excitation function of the approximate solution
  14. Bounds for the approximation error
  15. Bounds for the error in displacement
  16. ...and 4 more sections

Key Result

Lemma 4.1

Let $f \in H^{-1}$ and $F \in L^{2}$ be a week integral of $e^{c\cdot }f(\cdot )$. Then, where $K>0$ is a constant depending only on $c$ and $\bar{T}$.

Theorems & Definitions (37)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.3
  • Remark
  • proof : Proof of theorem \ref{['weak_form:thm weak form SDOF']}
  • Lemma 4.4
  • proof : Proof of the lemma
  • Remark
  • Proposition 6.1
  • ...and 27 more