Vib-4wk1Open Access Computer Rules and Regulations V2016Open Access Computer Rules and Regulations V2016Open Access Computer Rules and Regulations V2016Open Access Computer Rules and

8/15/2019 Vib-4wk1Open Access Computer Rules and Regulations V2016Open Access Computer Rules and Regulations V2016Open Access Computer Rules and Regulations V2016Open A…

1/13

MAE 351AE 351:: Wk 4

CHAPTER 3CHAPTER 3

Harmonically ExcitedHarmonically Excited Vibration Vibration

m

ck

xF(t)


2/13

MAE 351AE 351:: Wk 4

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)


3/13

MAE 351AE 351:: Wk 4

Homogeneous, Particular, General Solutions

(Underdamped 1-DOF Vibration)

S3 2 S iti P i i l


4/13

MAE 351AE 351:: Wk 4

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)


5/13MAE 351AE 351:: Wk 4

. . 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’?


6/13MAE 351AE 351:: Wk 4

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


7/13MAE 351AE 351:: Wk 4

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


8/13MAE 351AE 351:: Wk 4

. 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)


9/13MAE 351AE 351:: Wk 4

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


10/13MAE 351AE 351:: Wk 4

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


11/13MAE 351AE 351:: Wk 4

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


12/13MAE 351AE 351:: Wk 4

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


13/13MAE 351AE 351:: Wk 4

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 − ∆

= = − = ∆

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