Class Notes 6:
Second Order Differential Equation –
Non Homogeneous – Force Vibration
MAE 82 – Engineering Mathematics
Tacoma Narrow (WA)
Video - Galloping Gertie - The Collapse of the Tacoma Narrows Bridge
Dynamic Balance
Video - The vibrations resulting from the rotor destructs this Chinook helicopter
Force Vibrations – Introduction
vxtx
dxtxtfkxxcxm
0
0
)0(
)0()(
I.C.
mkmw
c
m
kw
n
n2
2
2;
m
tfxwxwx nn
)(2 2
motionDamped
motionUndamped
motionForced
motionFree
0
0
0)(
0)(
)(
c
c
tf
tf
tf
)(tf
impulse
step
ramp
sin
Forced Vibration Tests Aircraft
Video - CSeries Ground Vibration Test (GVT)
Force Vibrations – Introduction –
Frequency Response – Bode Plot
Force Vibrations – Introduction
Frequency Analysis – Fourier Series
f(t)T)f(t
functionPeriodic
T
mtb
T
mtaatf
m
m
m
m
2sin
2cos)(
11
0
Force Vibrations – Effect of Gravity
)(
)(
)()(
0
0
tfkxxcxm
tfmgxckxkxxm
xmtfxcxxkmgF
0
Force Vibrations – Classification
Damped Oscillation – Undamped Oscillation
wtm
Fxwxwx nn cos2 2
1.3
1.2
10.1
0
oscillatorDamped
n
n
n
ww
ww
ww
.3
.2
.1
0
oscillatorUndamped
Force Vibrations
Undamped Oscillation – Case 1
wtm
FxwxwtFkxxm
ww
n
n
coscos
)1Case(0oscillatorUndamped
2
h
c
x
xpx
systemwn : inputw :
wtm
FwtBwtAwwtBwwtAw
x
n
p
cos))sin()cos(()sin()cos(222
atingDifferenti
x x
onlycosoffunctionaisitsincebemustB 0B
pc xxx
)sin()cos()sin()cos( 21 wtBwtAtwctwcx nn
• Using the undermined coefficient method
• Solve for A
• Resulted in
Force Vibrations
Undamped Oscillation – Case 1
wtm
FwtwwA n coscos)( 22
)( 22wwm
FA
n
wtwwm
Ftwctwctx
n
nn cos)(
)sin()cos()(2221
• Given the solution
• Using the Initial Conditions
• Resulted in
Force Vibrations
Undamped Oscillation – Case 1
n
n
nnnn
n
n
nn
w
xc
xwwwwm
Fwwcwwctx
wwm
Fxc
xwwwm
Fwcwctx
02
02221
2201
02221
)0sin()(
)0cos()0sin()0(
)(
)0cos()(
)0sin()0cos()0(
1 0 1
0 1 0
0
0
)0(
)0(..
xx
xxCI
wtwwm
Ftwctwctx
n
nn cos)(
)sin()cos()(2221
Force Vibrations
Undamped Oscillation – Case 1
twwtr
tww
xtwxtx
wtr
tww
xtw
rxtx
wtrmw
Ftw
w
xtw
rmw
Fxtx
w
wr
k
F
wtwwm
Ftw
w
xtw
wwm
Fxtx
nn
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
coscos1
sincos)(
cos1
sincos1
)(
cos)1(
sincos)1(
)(
cos)(
sincos)(
)(
2
00
2
0
20
22
0
220
22
0
220
ratiofrequency
ndifflectiostatic
Free response(Depends on I.C.) Force response(Depends on the input)
• Examining the factor of the force Response
• Static Deflection Factor
• Magnitude Factor
Force Vibrations
Undamped Oscillation – Case 1
twwtr
tww
xtwxtx nn
n
n coscos1
sincos)(2
00
)(frequencynaturaltheand)(frequency
forcingthebetweenratiotheoffunctionaisIt .attenuatedor
amplifiedeitherisdeflectionstaticthewhichbyamountThe
nww
M
Fk
F
22
1
1
1
1
nw
w-
rM
Free response(Depends on I.C.) Force response(Depends on the input)
Force Vibrations
Undamped Oscillation – Case 1
2
1
1
nw
wM
Force Vibrations
Undamped Oscillation – Case 1
)cos(cos)(
)cos(cos1
)(
00)0(,0)0((*)
2
twwttx
twwtr
tx
w
wwwxxFor
n
n
n
n
and
2
1
1
nw
wM
Force Vibrations
Undamped Oscillation – Case 1
Force Vibrations
Undamped Oscillation – Case 1
)cos(cos)(
01
1
and0)0(,0)0((**)
2
twwttx
rM
w
wwwxxFor
n
n
n
2
1
1
nw
wM
Force Vibrations
Undamped Oscillation – Case 1
Force Vibrations
Undamped Oscillation – Case 3
• Undamped Oscillator (case 3)
• The solution is similar to the case when
• Same as
0 n
tm
Fxx n cos
n ,)0(
0)0(
xx
x
IC
n
ttm
Ft
xtxtx
tm
Ft
xt
m
Fxtx
n
n
n
n
n
n
n
n
n
n
coscossincos)(
cossincos)(
22
00
22
0
220
n
Force Vibrations
Undamped Oscillation – Case 3
Force Vibrations
Undamped Oscillation – Case 2 (Resonance)
• Undamped Oscillator
0 n
tm
Fxx nn cos2
pc xxx
tBtAttCtCx nn
x
nn
c
sincossincos 1
21
n n
n
n
2
1
22 0
Force Vibrations
Undamped Oscillation – Case 2 (Resonance)
tBtAttCtCx nn
x
nn
c
sincossincos 21
n n
0
0
)0(
)0(
xx
xx
IC
pn
x
nn
x
n
n
n tt
tx
txtx
sin2
sincos)( 00
IC Forced
Force Vibrations
Undamped Oscillation – Case 2 (Resonance)
• For
)sin(2
)( tttx nn
0)0(
0)0( x
x
IC
Glass
Video - Wine glass resonance in slow motion
Force Vibrations – Damped Oscillations
taxxx
m
F
nn cos2 2
tataxp sincos 21
pc xxx
21
21
21
,
21
,, ,C :
d
2121
2121
sincos
sin
aa
AC
dn
t
t
tt
tata
jtAe
etCC
eCeC
x
c
n
I.C. on Based
real
Force Vibrations – Damped Oscillations
• Substitute into the differential equation
Coefficient of
Coefficient of
px
tatata
tatatata
x
n
x
n
x
cossincos
cossin2sincos
21
2
212
2
1
2
02
2
2
2
12
2
1
2
21
2
aaa
aaaa
nn
nn
tcos
tsin
02
2
2
1
22
22 a
a
a
nn
nn
2222 2 nn
Force Vibrations – Damped Oscillations
• Convert the solution to the form of a single trigonometric function
with a phase shift
• where
aa
aa nn 2
; 2
22
1
tt
ax nn
nn
m
F
p
sin2cos2
22
2222
tBtAx
ttCtCx
p
p
sincos
sinsincoscoscos
A
BBAC
CB
CA1-22 tan , ,
sin
cos
Force Vibrations – Damped Oscillations
22222
2
22
222
22
nn
n
m
FB
nn
n
m
FA
22
21tan
22
222
1
22
2222
22
222
22
2222
22
22
2222
222
22
n
n
nnm
F
nn
nn
m
F
nn
n
nn
n
m
F
BAC
Force Vibrations – Damped Oscillations
• Factor out of the denominator
tm
Fx
nn
p cos
2
1
2222
tm
Ftx
nn
n
p cos
21
1)(
22
2
22
k
F
mm
k
F
m
F
n
2
ttx
M
nn
p cos
21
1)(
22
2
2
2
n
Force Vibrations – Damped Oscillations
2
2
1
22
2
2
1
2
tan
21
1
n
n
nn
M
Single DOF - Resonance
Video - SDOF Resonance Vibration Test