Topic 1 SDOF Free Vibration 1
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Transcript of Topic 1 SDOF Free Vibration 1
S
S
S
S
Single‐Degre
Free undam
Spring‐mass
Static deflec
Force balanc
Circular freq
Solution to t
Constants A
ee‐Of‐Freed
mped vibratio
s system: M
ction:
ce: Apply Ne
quency n in
the 2nd orde
A and B are e
dom (SDOF)
on
Mass m; stiffn
ewton’s seco
n radians/sec
r linear diffe
evaluated fro
system
ness k
ond law
cond:
erential equa
om initial con
;
ation (of mo
nditions 0
otion)
0 and 0 .
. The resultin
:
ng response
1
is:
S
t
S
w
Since natura
the natural
Natural freq
Since
we get
al period of v
period of vib
quency fn in c
vibration is
bration in s
cycles/secon
s related to c
seconds/cyc
nd, is given b
circular freq
cle, is given b
by
quency throu
by
ugh
2
S
V
A
S
Free dampe
Spring‐mass
Viscous dam
Apply Newto
For free vibr
Solution:
Upon substi
Characterist
Roots of the
ed vibration:
s‐damper sys
mping force:
on’s second
ration, F(t) =
tution into t
tic equation:
e characteris
:
stem:
law of moti
= 0.
the different
:
stic equation
Mass m; st
on, for force
tial equation
n:
tiffness k; da
e balance:
n,
amping cons
stant c
3
T
w
T
The general
Constants A
Critically‐da
If (c/2m)2 is
become equ
with A and B
Typical resp
solution usi
A and B are e
mped system
equal to k/m
ual to each o
B given by in
onse:
ing superpos
evaluated fro
m:
m, the roots
other (doubl
nitial conditio
sition:
om initial con
e root); s1 =
ons.
nditions 0
s2 = ‐n. The
0 and 0 .
e general so
. The resultin
olution is
ng response
4
is:
From (c/2m
Critical Dam
Damping Ra
For critically
Under‐damp
For this case
become com
Letting
m)2 is equal t
mping Ratio c
atio ζ is defin
y‐damped sy
ped system:
e, ζ < 1; that
mplex numb
o k/m, we g
cc.
ned as
ystem, ζ = 1.
is, (c/2m)2 i
ers. Since
et
s less than k
k/m. The rooots
5
T
w
T
The general
Re‐writing t
with initial c
Frequency o
Typical resp
solution to
he response
conditions us
of damped v
onse:
response is:
e equation as
sed to deter
ibration:
:
s
rmine C1 andd C2.
6
T
Overdampe
For this case
become rea
The motion
d system:
e, ζ > 1; that
l and distinc
is aperiodic
is, (c/2m)2 i
ct. The gener
and non‐os
s more than
ral solution i
cillatory. Typ
n k/m. The ro
is given by
pical respon
oots
se is shown
below.
7
V
Measureme
Ratio of two
Variation of
ent of dampi
o successive
as a funct
ng in a syste
vibration am
tion of damp
em:
mplitudes is
ping:
called Logar
rithmic decre
ement, .
8
SShown below
Calculate
w is that
,
,
9
10