damp
====

Natural frequencies and damping factors.



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [wn,z] = damp(sys)
    [wn,z] = damp(P [,dt])
    [wn,z] = damp(R [,dt])




Parameters
~~~~~~~~~~

:sys A linear dynamical system (see `syslin`_).
: :P An array of polynomials.
: :P An array of real or complex floating point numbers.
: :dt A non negative scalar, with default value 0.
: :wn vector of floating point numbers in increasing order: the
  natural pulsation in rd/s.
: :z vector of floating point numbers: the damping factors.
:



Description
~~~~~~~~~~~

The denominator second order continuous time transfer function with
complex poles can be written as `s^2+2*z*wn*s+wn^2` where `z` is the
damping factor and `wn`the natural pulsation.

If `sys` is a continuous time system, `[wn,z] = damp(sys)` returns in
`wn` the natural pulsation (in rd/s) and in `z` the damping factors of
the poles of the linear dynamical system `sys`. The `wn` and `z`
arrays are ordered according to the increasing pulsation order.

If `sys` is a discrete time system `[wn,z] = damp(sys)` returns in
`wn` the natural pulsation (in rd/s) and in `z` the damping factors of
the continuous time equivalent poles of `sys`. The `wn` and `z` arrays
are ordered according to the increasing pulsation order.

`[wn,z] = damp(P)` returns in `wn` the natural pulsation (in rd/s) and
in `z` the damping factors of the set of roots of the polynomials
stored in the `P` array. If `dt` is given and non 0, the roots are
first converted to their continuous time equivalents. The `wn` and `z`
arrays are ordered according to the increasing pulsation order.

`[wn,z] = damp(R)` returns in `wn` the natural pulsation (in rd/s) and
in `z` the damping factors of the set of roots stored in the `R`
array. If `dt` is given and non 0, the roots are first converted to
their continuous time equivalents. `wn(i)` and `z(i)` are the the
natural pulsation and damping factor of `R(i)`.



Examples
~~~~~~~~


::

    s=%s;
    num=22801+4406.18*s+382.37*s^2+21.02*s^3+s^4;
    den=22952.25+4117.77*s+490.63*s^2+33.06*s^3+s^4
    h=`syslin`_('c',num/den);
    [wn,z] = damp(h)


The following example illustrates the effect of the damping factor on
the frequency response of a second order system.


::

    s=%s;
    wn=1;
    `clf`_();
    Z=[0.95 0.7 0.5 0.3 0.13 0.0001];
    for k=1:`size`_(Z,'*')
      z=Z(k)
      H=`syslin`_('c',1+5*s+10*s^2,s^2+2*z*wn*s+wn^2);
      `gainplot`_(H,0.01,1)
      p=`gce`_();p=p.children;
      p.foreground=k;
    end
    `title`_("$\frac{1+5 s+10 s^2}{\omega_n^2+2\omega_n\xi s+s^2}, \quad \omega_n=1$")
    `legend`_('$\xi='+`string`_(Z)+'$')
    `plot`_(wn/(2*%pi)*[1 1],[0 70],'r') //natural pulsation


It produces this plot:



Computing the natural pulsations and daping ratio for a set of roots:


::

    [wn,z] = damp((1:5)+%i)




See Also
~~~~~~~~


+ `spec`_
+ `roots`_


.. _roots: roots.html
.. _syslin: syslin.html
.. _spec: spec.html