# damp

Natural frequencies and damping factors.

### Syntax

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

### Parameters

- sys
A linear dynamical system, in state space, transfer function or zpk representations, in continuous or discrete time.

- P
An array of polynomials.

- R
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 rad/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 ω_{n} (in rad/s) and in `z`

the damping
factors ξ 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
ω_{n} (in rad/s) and in `z`

the
damping factors ξ 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
ω_{n} (in rad/s) and in `z`

the
damping factors ξ 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
ω_{n} (in rad/s) and in `z`

the
damping factors ξ 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 damping ratio for a set of roots:

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

### History

Version | Description |

6.0 | handling zpk representation |

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