Class Notes 6:

Second Order Differential Equation –

Non Homogeneous – Force Vibration

MAE 82 – Engineering Mathematics

Tacoma Narrow (WA)

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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