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 Mapping onto electrical domain (optional) Lowrie et. al 2005
It involves Turning partial differential equations into ordinary differential equations Approximations and more approximations (Ab-)use of available tools
Procedure Partitioning and choice of variables (x,v) Find values for the parameters (m,k,γ) Couple Analyze x
Hvor i Senturia
Partitioning and choice of variables Partitioning and choice of variables (x,v) Find values for the parameters (m,k,γ) Couple Analyze x
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 m, ω Best trial function Approximate eigenmode Approximate eigenfrequency t=0
Rayleigh-Ritz
ω
m x2x2 m= 2
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