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CHAPTER 3CHAPTER 3
Harmonically ExcitedHarmonically Excited Vibration Vibration
m
ck
xF(t)
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3.1. General3.1. General
• System subject to external dynamic forces
– Forcing function, Exciting function
• Harmonic excitation Harmonic response
Nonperiodic excitation Transient response
• Harmonic excitations
• Governing equation
(nonhomogeneous equation)
General sol’n: x (t ) = x h(t ) + x p(t )
where x h(t ) = homogeneous sol’n of
x p(t ) = particular sol’n
• x h(t ) : exponentially decaying (initial transient) vibration
x p(t ) : steady-state vibration
( )
( )0
0
0
( )
( ) cos
( ) sin( )
i t F t F e
F t F t
F t F t
ω φ
ω φ
ω φ
+=
= +
= +
( )mx cx kx F t + + =
0mx cx kx+ + =
m
ck
xF(t)
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Homogeneous, Particular, General Solutions
(Underdamped 1-DOF Vibration)
S3 2 S iti P i i l
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3.2. Superposition Principle3.2. Superposition Principle
x(t) = output (response); F(t) = input (excitation)
Introduce a linear differential operator
Then, G = ‘Black-box’ of the 2nd order system
( )mx cx kx F t + + = 2
2 ( )
d x dxm c kx F t
dt dt + + =
2
2
d d G m c k
dt dt
≡ + +
[ ]( ) ( )G x t F t =
or
Consider two excitations and responses:
F 1(t )=G [ x 1(t )], F 2 (t )=G [ x 2 (t )]
Next, consider
F 3(t )=c 1F 1(t ) + c 2 F 2 (t ) : linear combination of F 1(t ) & F 2 (t )c 1,c 2 : known constants
If x 3(t ) = c 1 x 1(t ) +c 2 x 2 (t ), the system is “linear ”: otherwise “nonlinea
G [ x 3(t )] = G [c 1 x 1(t ) + c 2 x 2 (t )] = c 1G [ x 1(t )] + c 2 G [ x 2 (t )]= c 1F 1(t ) + c 2 F 2 (t) = F 3(t ) ; principle of superpositio
F (t)
G
x (t)
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. . esponse o an. . esponse o an n ampen ampe ys emys emunder Harmonic Forceunder Harmonic Force
For
F (t ) : harmonic x p(t ): harmonic
with ω with ω
Assume x p
(t ) = X cos ω
t, then
cosomx kx F t ω + =
1 2
0,
( ) cos sinh n n
n
mx kx
x t C t C t k
m
ω ω
ω
+ =
= +
=
2
0( cos cos ) cos X m t k t F t ω ω ω ω − + =
0
2
F X
k mω
=
−
m
k
xF(t)
; Amplitude of particular
solution
Q: What is the physical meaning of ‘particular solution’?
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x(t) = total solution
=
Then,
01 2cos sin cosn n F C t C t t
k mω ω ω
ω 2+ +
−
1 2
0 0
0 01 0 22
Two I.C.'s of (0)& (0) ,Let (0) & (0)
;n
x x C C x x x x
F x
C x C k mω ω
⇒= =
= − =−
0 00 2
0
2
( ) cos sin
cos
n n
n
F x x t x t t k m
F
t k m
ω ω ω ω
ω ω
= − + −
+
−
Homogeneous
solutionRecall: free vib.
Recall:
In free vib. case:
01 0 2;
n
xC x C ω = =
General Solution
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From 0
02
2 2
0
n
and
1 11
1
where static deflection,
= frequency ratio ,
Dynamic ampl.
Static ampl.
magnification factor
or amplification factor
or amplification ratio
st
st
n
st
st
F X F k
k m
X r
F
k
r
X
δ ω
δ ω
ω
δ
ω ω
δ
= =−
⇒ = ≡−
−
= =
≡
=
=
Look! X infinity as r 1
“RESONANCE”
a ac e s csa ac e s cs
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. arac er s cs. arac er s cs
③
: 0 < ω /ωn < 1
:ω /ωn > 1
③
:ω /ωn = 1
F (t ) & x p(t ) are in phase
out-of phase ( r > 1 ) x p(t ) = -X cosωt
( 0 < r < 1
(r=1)
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③
(r=1)
X ; Resonance ω =ωn
1 2 2
00 2
2
2
00
( ) cos sin cos1 ( )
1cos sin (cos cos )
1 ( )cos cos ( sin ) 1
lim lim sin (L'Hospital's rul1 ( ) 2
2
( ) cos sin
n n
st
n nn
n n st n
n n
nn n
n
n
sn n
n
x t C t C t t
x x t t t t
t t t t t t
x x t x t t
ω ω ω ω
δ
ω ω ω ω ω
ω ω δ ω ω
ω ω ω ω ω ω
ω ω ω ω ω
ω δ
ω ω ω
→ →
= + +− /
= + + −− /
− −= =
− / −
⇒ = + +
sin
2t n
n
t t
ω ω
B T t l RB T t l R
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B. Total ResponseB. Total Response
2
2 2 1 21 2
1
( ) cos( ) cos1 ( )
where ; tan
st n
n
x t A t t
C A C C
C
δ ω φ ω
ω ω
φ −
= − ±
− /
= + =
+ for ω/ωn < 1
- for ω/ωn > 1
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C. BeatingC. Beating
Linear superposition of individual vibrations:
1 1 1 2 2 2exp[ ( )] exp[ ( )] x A i t A i t ω φ ω φ = + + +
Nonperiodic (aperiodic) oscillations in general
When ω2 = ω1 + ω, then1 2 1 1( ) ( )
1 2[ ]i i t i t i t x A e A e e Aeφ φ ω ω ω φ +∆ += − ≡
2 2 1/ 2
1 2 1 2[ 2 cos( )] A A A A t φ φ ω 1 2= + + − − ∆
1 1 1 2 2
1 1 2 2
sin sin( )
tan s cos( )
A t
co A t
φ φ ω
φ φ φ ω
− + + ∆
= + + ∆
where
; Approximately simple harmonic vibratio
with slowly varying ampl. A and phase φ
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Consider a special case of A1 = A2 and φ1 = φ2 = 0
Then,
1/ 21 1
1
1 2
(2 2cos ) 0 2
sintan
1 cos
Beat frequency b
A t A A
t
t
f f f f
ω
ω φ
ω
ω
−
= + ∆ ≤ ≤∆ = + ∆
≡ = ∆ = − = ∆
(Forcing freq., ω) ≈ (Natural freq., ωn (system))
0 0
0
2
0
2
0,
( / )( ) (cos cos )
( / )2sin sin
2
n
n
n n
n
When x x
F m x t t t
F mt t
ω ω ω ω
ω ω ω ω
ω ω
2
2
= =
= −−
+ −= ⋅
− 2
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As ωn - ω = ω ≡ 2 ( : very small),
ω +ωn ≈ 2ω & ωn2 - ω 2 = (ωn + ω)(ωn - ω ) ≈ 4 ω
Then,0 /( ) sin sin
2
F m x t t t ε ω
εω
≈
variable amplitude2 2 2
period of beating
2
b
n
b n
π π π τ
ε ω ω ω
ω ε ω ω ω
≡ = = =2 − ∆
= = − = ∆