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Thus
2
11
11
)
(
1
2
tan
ωτ
ωτξ
φ


=
=
B
A
And,
2
2
2
)
2
(
)
)
(
1
((
ξωυ
+

=
A
A
New
(using
New
A
B
A
=
+
2
2
11
11
Thus,
)
(
)
2
(
)
)
(
1
((
)
(
2
2
2
ω
+
+

=
t
Sin
A
t
Y
proved
8.9)
If
a
second order
system
is
over damped, it
is
more
difficult
to
determine
the
parameters
τ
ξ
experimentally. One
method
for
determining
the
parameters
from
a
step
response
has
been
suggested
by
R.c
Olderboung
and
H.Sartarius
(The
dynamics
of
Automatic
controls,ASME,P7.8,1948),as
described
below.
(a)
Show
that
the
unit
step response
for
the
over damped
case
may
be
written
in
the
form.
2
1
2
1
1
2
1
)
(
r
r
e
r
e
r
t
s
t
r
t
r


=
Where
r
1
and
r
2
are
the
roots
of
0
1
2
2
2
=
+
+
s
s
ξτ
(b)
Show
that
s(t)
has
an
inflection
point
at
)
(
)
/
ln(
1
2
1
2
r
r
r
r
t
i

=
©
Show
that
the
slope
of
the
step
response
at
the
inflection
point
)
(
)
(
1
i
t
t
t
s
dt
s
d
i
=

Where,
i
t
r
t
r
i
e
r
e
r
t
s
2
1
2
1
1
)
(

=

=
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(
1
2
1
2
1
1
r
r
r
r
r
r


=
(d)
Show
that
the
value
of
step
response
at
the
inflection
point
is
)
(
1
)
(
1
2
1
2
1
1
i
i
t
s
r
r
r
r
t
s
+
=
and
that
hence
2
1
1
1
1
)
(
)
(
1
r
r
t
s
t
s
i
i


=

(e)
on
a
typical
sketch
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This note was uploaded on 11/13/2011 for the course COP 4355 taught by Professor Koslov during the Spring '10 term at University of Florida.
 Spring '10
 Koslov

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