1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
|
<?xml version="1.0" encoding="UTF-8"?>
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="en_US" xml:id="damp">
<refnamediv>
<refname>damp</refname>
<refpurpose>Natural frequencies and damping factors. </refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>
[wn,z] = damp(sys)
[wn,z] = damp(P [,dt])
[wn,z] = damp(R [,dt])
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Parameters</title>
<variablelist>
<varlistentry>
<term>sys</term>
<listitem>
<para>
A linear dynamical system (see <link linkend="syslin">syslin</link>).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>P</term>
<listitem>
<para>
An array of polynomials.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>R</term>
<listitem>
<para>
An array of real or complex floating point numbers.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>dt</term>
<listitem>
<para>
A non negative scalar, with default value 0.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>wn</term>
<listitem>
<para>
vector of floating point numbers in increasing
order: the natural pulsation in rad/s.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>z</term>
<listitem>
<para>
vector of floating point numbers: the damping factors.
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
The denominator second order continuous time transfer function
with complex poles can be written as <literal>s^2 + 2*z*wn*s + wn^2</literal> where <literal>z</literal>
is the damping factor and <literal>wn</literal> the natural pulsation.
</para>
<para>
If <literal>sys</literal> is a continuous time system,
<literal>[wn,z] = damp(sys)</literal> returns in <literal>wn</literal> the natural
pulsation <latex>\omega_n</latex>(in rad/s) and in <literal>z</literal> the damping factors
<latex>\xi</latex> of the poles of the linear dynamical system
<literal>sys</literal>. The <literal>wn</literal> and
<literal>z</literal> arrays are ordered according to the increasing
pulsation order.
</para>
<para>
If <literal>sys</literal> is a discrete time system
<literal>[wn,z] = damp(sys)</literal> returns in
<literal>wn</literal> the natural pulsation
<latex>\omega_n</latex>(in rad/s) and in <literal>z</literal> the
damping factors <latex>\xi</latex> of the continuous time
equivalent poles of <literal>sys</literal>. The
<literal>wn</literal> and <literal>z</literal> arrays are
ordered according to the increasing pulsation order.
</para>
<para>
<literal>[wn,z] = damp(P)</literal> returns in
<literal>wn</literal> the natural pulsation
<latex>\omega_n</latex>(in rad/s) and in <literal>z</literal> the
damping factors <latex>\xi</latex> of the set of roots of the polynomials
stored in the <literal>P</literal> array. If
<literal>dt</literal> is given and non 0, the roots are first
converted to their continuous time equivalents.
The <literal>wn</literal> and <literal>z</literal> arrays are ordered
according to the increasing pulsation order.
</para>
<para>
<literal>[wn,z] = damp(R)</literal> returns in
<literal>wn</literal> the natural pulsation
<latex>\omega_n</latex>(in rad/s) and in <literal>z</literal> the
damping factors <latex>\xi</latex> of the set of roots stored in the
<literal>R</literal> array.
If <literal>dt</literal> is given and non 0, the roots are first
converted to their continuous time equivalents.
<literal>wn(i)</literal> and <literal>z(i)</literal> are the the
natural pulsation and damping factor of <literal>R(i)</literal>.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
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)
]]></programlisting>
<para>
The following example illustrates the effect of the damping factor on
the frequency response of a second order system.
</para>
<programlisting role="example"><![CDATA[
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
]]></programlisting>
<para>
<scilab:image localized="true">
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')
</scilab:image>
</para>
<para>
Computing the natural pulsations and daping ratio for a set of roots:
</para>
<programlisting role="example"><![CDATA[
[wn,z] = damp((1:5)+%i)
]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="spec">spec</link>
</member>
<member>
<link linkend="roots">roots</link>
</member>
</simplelist>
</refsection>
</refentry>
|
