ANALYSIS OF A VISCOELASTIC SPRING-MASS MODEL

1. MAIN QUESTION: When PDE ODE ? ? PDE ODE PDE ODE PDE ODE !!! ANALYSIS OF A VISCOELASTIC SPRING-MASS MODEL Marta Pellicer Sabadí, Joan Solà-Morales Rubió Dpt. Matemàtica Aplicada 1, Universitat Politècnica de Catalunya 1. Abstract • Small internal viscosity ( near 0) The aim of our work is to propose, justify and analyse an alternative PDE model for the classical spring-mass ODE that takes into account possible differences in spring internal deformation as well as its internal viscosity. We can find a finite subset of dominant eigenvalues, but it depends on the value of  ODE model versus PDE model But, in certain situations, we will prove that the PDE model tends to behave as a limit ODE in a certain sense. This is done from a functional analysis point of view based on spectral analysis and dominant eigenvalues. Yes, but the ODE order as well as its coefficients depend on the value of • Large mass at the end ( near 0) We now study the same operator but from a different point of view: we look at it as a perturbation of the limit operator for  =0 2. The model We can prove the existenceof an 0 , depending on and r,such that if  <0 the perturbed operator admits two dominant eigenvalues of the spectrum We consider a system formed by a viscoelastic spring with a rigid mass attached at its end, which is linearly damped by a friction force. • Classical interpretation: the ODE model. • A more realistic point of view: the dimensionless PDE model. Classical continuum mechanics Rheological approach For large times, the solutions of the PDE model can be approximated by a second order ODE’s ones. This limit ODE is NOT the classical one, but its coefficients can be approximately calculated. Balance of momentum with 4. Numerical simulation of the nonhomogeneous problem We can consider an external force f(t)acting on the mass. By looking for the only globally bounded solution it will naturally appear a different notion of limit ODE, in terms of the convergence between transfer functions. We have obtained a wave equation model with strong damping (or continuous Kelvin-Voigt model) and dynamical boundary conditions. 3. Main results Convergence of the coefficients of transfer functions series up to some reasonable order A parametric study of the model is done. Nonhomogeneous limit ODE: • No internal viscosity ( =0) No finite subset of dominant eigenvalues We can simulate and compare these two approaches: Spectrum of the operator 5. References. BUT: all solutions tend to zero when time tends to infinity and we can find some of themthat tend to zero as slow as we wish. 1. M. Grobbelaar-van Dalsen, “On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions”, Appl. Analysis, 53 (1994), 41-54. 2. M. Pellicer, J. Solà-Morales, “Analysis of a viscoelastic spring-mass model”, Preprint 2003. 3. M. Pellicer, “Anàlisi d’un model de suspensió-amortiment”, Thesis (under preparation)