Aim Understand the concept of lumped element modelling

Aim • Understand the concept of lumped element modelling • Understand variational solution in the mechanical domain • Understand Rayleigh Ritz in the mechanical domain

What is lumped element modelling? • Reduction of the number of variables Lowrie et. al 2005 • Mapping onto electrical domain (optional)

It involves • Turning partial differential equations into ordinary differential equations • Approximations and more approximations • (Ab-)use of available tools

Procedure x Partitioning and choice of variables (x, v) Find values for the parameters (m, k, γ) Couple Analyze

Hvor i Senturia

Partitioning and choice of variables x Partitioning and choice of variables (x, v) Find values for the parameters (m, k, γ) Couple Analyze

Parameter extraction • • Solution of partial differential equations Formulas Experiment Guesswork Partitioning and choice of variables (x, v) Find values for the parameters (m, k, γ) Couple Analyze

Løsning av partielle differensialligninger Fokus – på kobling – på dynamikk (egenmoder) Verktøy – FEM/BEM – Analytisk • Eksakt • Tilnærmet

Variational principle Weak form of the differential equation: Minimize U Trial function: Parameters k

Applied to membrane (plate) Pre-stress Bending Elongatio n k

Rayleigh-Ritz - basis Approximate eigenmode Best trial function Approximate eigenfrequency t=0 m, ω

Rayleigh-Ritz

Rayleigh-Ritz ω

Rayleigh-Ritz m= 2 x 2 m

Motstand R

Verktøy for analyse • • Generaliserte impedanser Tilstandsvariable Krets simuleringsverktøy Transferfunksjoner Transfermatriser Laplace transformasjon Fourier transformasjon Konvolusjon

Mekanisk svingesystem x

Elektrisk svingesystem

Bevegelseslikningen

Resonansfrekvens Ved null dempning går transferfunksjonen mot uendelig når: Innfører derfor udempet resonansfrekvens

Q-faktor

Eksperiment Q, f

Systembeskrivelse

Observerbarhet

Two port elements