## 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:

$F_d=\gamma\;&space;\dot{x}$

A linear spring would obey:

$F_s=k\;x$

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

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

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):

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

A Fourier transformation yields

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

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

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

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

$F_s(x)=N(x)$

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