ce-5113 lecture notes 3
TRANSCRIPT
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Dynamics of Structures 7
Undamped Free Vibration
The solution may be written as:
is known as the amplitude of vibration and is called the phase angle
Both amplitude and phase angle depend upon the initial condition and
natural circular frequency of the system
( )
o
ooo
v
uand
vuwhere
tu
=
+=
+=
tan___
sin
2
2
Dynamics of Structures 8
Undamped Free Vibration
It can be shown that:
Thus the vibration repeats itself after and is thenatural periodof
system
The reciprocal of T is call the natural frequency of system;
( )
+=
211 tutu
2=
T
2=f
( ) ( )
( ){ }
( )
+=
+
++=
+
+
+=
+
+=
11
11
11
11
sin2
2sin2
2sin
2
sin
ttu
ttu
ttu
ttu
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Dynamics of Structures 9
Natural Frequency of a Frame
The natural frequency of a 2D frame
with fixed base and rigid beam of
mass m, neglecting mass of
columns, is given by:
3
3
24
24
mL
EI
L
EIk
m
k
=
=
=
Dynamics of Structures 10
Natural Frequency of a Suspended Mass
st
st
st
st
st
stst
gf
gT
g
m
mg
m
k
mgkkmgkW
=
==
=
==
===
2
1
22
/
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Dynamics of Structures 11
Phase Plan Diagram
( )
( )
( ) 22
2
cos
sin
=
+
+=
+=
uu
tu
tu
&
&
Dynamics of Structures 12
Phase Plan Diagram (Initial Displacement)
( ) ( )
tutuu
tutuu
v
u
uv
u
uuu
oo
oo
o
o
oo
sin2
cos
cos2
sin
2tan
00_,0
0
2
2
0
=
+=
=
+=
===
=
+=
==
&
&
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Dynamics of Structures 13
Phase Plan Diagram (Initial Velocity)
( ) ( )
( )
( )tvu
tvu
v
u
vvu
vuu
o
o
o
o
ooo
o
cos
sin
00tan
0_,00
2
2
=
=
===
=
+=
==
&
&
Dynamics of Structures 14
Phase Plan Diagram (Impulse)
( )
( )( ) 1212
111
1
11
2
12
12
1
2
2
1
sin
0sin/
tan__/
tan__
ttforttu
ttfortumIu
uand
mIuu
v
uand
vu
t
ttt
o
ooo
+=
+=+
=
++=
=
+=
&
&
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Dynamics of Structures 15
Phase Plan Diagram (Ground Motion)
( )tuu
tuuu
uu
g
gg
gg
cos12
3sin
2
3_tan
02
2
=
++=
==
=
+=