Saturday, April 19, 2008

[NR 1] Nonlinear Kelvin Body (NKB)

Consider a Kelvin body, i.e. a spring in parallel with a dashpot:

The body is externally described by its length x and force F.

Assume that the rest length (at zero force) is zero. The dashpot has the property:

F_d=\gamma\; \dot{x}

A linear spring would obey:


The two forces add up to the total force of the Kelvin body:

F=F_s+F_d=k\;x+\gamma\; \dot{x}

We can solve for the temporal change of x to obtain a diff.eq. describing the evolution of x(t) from its starting value for any given external force F(t):

\dot{x}=\frac{1}{\gamma}\left[ F(t)-kx \right]

A Fourier transformation yields

i\omega \;x(\omega)=\frac{1}{\gamma}\left[ F(\omega)-k \;x(\omega) \right]

which allows to define a linear response function S(w) of the system:

x(\omega)=\left[  \frac{1}{i\omega\gamma+k} \right] \;F(\omega)=S(\omega)\;F(\omega)

Let the spring now have some nonlinear force-length-relation N(x):


The evolution equation then reads:

\dot{x}=\frac{1}{\gamma}\left[ F(t)-N(x) \right]

This nonlinear diff.eq. can also be solved (at least numerically) for any applied F(t). However, it is in general not possible to define a response function. This will have important consequences when we consider more complex systems build from many Kelvin bodies.

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