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# 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 wherez is the damping factor and wnthe 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

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

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