2.Super Position of Periodic Motions

7/30/2019 2.Super Position of Periodic Motions

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1. Sup erpo sed v ibrat ion s in o ne d im ension

2 . M any superposed v ib rat ions

3 . Com binat ion o f t w o v ib rat ions at r igh tangles

4. Com binat ion o f paralle l and perpend icu larsuperposi t ion

Top ics t o be co ve red


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Su pe rpo sit ion o f vibrat io ns in 1 D

Tw o o r m ore harm on ic v ibrat ion s sup erpo sing

The resul t ant d isplacem ent is sim ple t he sumof t he ind iv idu al d isp lacem ent s

Sup erp osi t io n pr incip le

W hen t he superposit ion p r incip le isappl icable?


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tcosFkxxm 0

0kxxm

tcosFkxxbxm 0

0 kxxbxm Lineardi f ferent ia lequat ions

Depends on ly on t he f i rst pow er o f

var iables and i t s der ivat ives

Sup erp osi t ion pr incip le ho lds on ly for l ineard i f ferent ial equat ion s


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P1

P2

1 t t

2 t

P

OP1 = Ro t at in g vect o rLengt h A1

OP2 = Ro t at in g vect o r

Lengt h A2

OP = Resul t ant vect orLengt h A

)cos(Ax 111 t )cos(Ax 212 t

OX

Y

)Acos(xxx 21 t


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P1

P2

t

O X

1 t

2 t

P

Y

A2A1

A A2PPOPOP 11

Law o f cosines

P)OP(cosAA2AAA121

2

2

2

1

2


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P1

P2

N

K

21 PKP t

Angle bet w een OP1 and P1P

1212 )( tt

12

t

O X

1 t

2 t

P

Y


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P1

OX

1 t

P

12 A

A1

A2

)(180 12

Law o f cosines

)(cosAA2AAA

))((180cosAA2AAA

1221

2

2

2

1

2

1221

2

2

2

1

2


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com plex exponent ia l representa t ion

)(

111 tieAz

)(

22

2 ti

eAz

Resultant 21 zzz

)(i)(i21

)(i

2

)(i

1

112

21

eeAA

eAeAz

t

tt

OP1

OP2

Tw orota t ingvectors


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Sup e rpo se d vibrat ion s of d if fe rent freq ue ncy

tAx 111 cos tAx 222 cos

Dif feren t am pl it udes A1 & A2

Dif ferent f req uen cies - 1 & 2zero phase d i f feren ce

t1

t2A1

A2

P

210 AAOX

Resu ltant d isp lacem ent

OX

O X

Y


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W ill th e com binat ion be per iod ic?

Unless t here is som e sim ple relat ion bet w een1 and2 , t he resul tant d isp lacem ent w i ll bea com pl icated f un ct ion o f t im e, w h ich m ay

alm ost never rep eat i tsel f

2211 TnTnT

Con di t ion for per iod ici ty

W here n 1 and n 2 are int egers


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ttx 2cos)cos(

2211 TnTnT

Con di t ion fo r p er iod ici ty

Is i t per io d ic?

NO! !

Let us analyze

n 2 is no t an int eger


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Is t he resul t ant per iodic?

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

- 2

- 1

0

1

2

Amplitude

T i m e ( s e c )

YES!!

3723

21 nnT sT

nn

1

37;23

3

21

2211 TnTnT Con di t ion for per iod icit y


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If t he v ibrat ion f req uen cies (1 and 2) o f

t he t w o w aves are close to each o t her -W hat happens to t he com bined e f fect ?

Beats are produced

tAx 11 cos

tAx 22 cos

Sl ight ly d i f ferent f req uen cies

t)costA(cosxxx2121


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t

2

2cost

2

2cosA 21212121

t

2

cost

2

2Acosx 2121

t)(cos(t)Ax avmod

is t he m od ulat ion f requency2

21

t)costA(cosxxx 2121

av is t he average fr equency


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1 and 2 are close t o each ot her

Am od

( t )

NOT

2

21vv

Beat /

Am ax

Bea t f requ en cy

21vv

Beat per iod :Time

intervalbe tweentw o beats


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Beat ph eno m eno n is m ain ly used b y m usiciansto t une the ir i nst rum en ts

ht tp : / /www.schoo l - f o r -champions.com/sc ience/sound_beat_f requencies.htm

Hum an ear can d etect beats up t o a f requ encyof 7 per secon d


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Prob lem 2 .3

Tw o vibrat ion s alon g th e sam e l ine aredescr ibed b y t he equ at ion s -

y1 = A cos 10t

y2 = A cos 12 t

Find t he b eat per iod .

Draw a carefu l sketch of t he resul t antd ist ur bance over one b eat per iod


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M any sup erposed v ibrat ions of t he sam eFrequency

tcosAx 01

) tcos(Ax02

])1N([ tcosAx 0N

Resultant x = x1+x2++xN )tcos(A Relevant t o m ul t ip le sou rce int er feren ceeffect s in Op t ics


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)tcos(Ax

Geo m et r ical represen tat ion

O

x

y

P1

P

d

AA0

2

1)(Ntcos

2

sin

2

Nsin

Ax 0

A


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C2

P2

A2

t 2+2

Y

C1

P1

A1

X

t 1+1

Com binat ion o f t w o v ibrat ion at r ight angles

111 tcosAx

222 tcosAy

OA1

-A1

A2

-A2

P


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Point s t o n ot e

Point P is con f ined w i th in t he rectangle

One need t o kno w f requency and ph asere lat ionship for t he t raject or y

If f requen cies are no t com m ensurate Pw i ll never repeat i tsel f


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Sup erp osit ion o f t w o pe rpe nd icular sim pleha rm on ic vibrat ion s having e qu al frequ e ncie s

tcosAx 1

tcosAy 2

is t he p hase d i f ference bet w een t hem o t i o n s


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Case 1 : = 0

tcosAx 1 tcosAy 2

x

A

Ay

1

2

Therefo re t he t raject o ry is a st raigh t linep assin g t h r o ugh t h e o r igin


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Case 2 : = / 2

tAx cos1 tAtAy sin2

cos 22

)sin(;cos21

tAyt

Ax

12

2

2

2

1

2

Ay

Ax

Equ at ion o f el lipse


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Case 3 : =

tAx cos1 tAtAy coscos 22

xA

Ay1

2


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Case 4 : = 3 / 2

tAx cos1 tAtAy

sin2

3cos

22

12

2

2

2

1

2

A

y

A

xEqu at ion o f el lipse


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4

A2

A1

Case 4 : = / 4

1

22

3

2

344

5

6

6

7

7

8

A

B C

D

5 1

1

8


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Sup erp osit ion o f t w o pe rpe nd icular sim pleha rm on ic vibrat ion s having dif feren tfrequencies

tcosAx 11

tcosAy 22

is t he p hase d i f ference bet w een t he

m o t i o n s Lissajou s Figu res

J A Lissajou s (1822 1880) m ade an ext ensivest ud y on sup erposit ion o f m ot ion s


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2 = 2 1 = 0


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2 = 2 1 = 0


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2 = 2 1 = 0


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2 = 2 1 = 0


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2 = 2 1 = 0


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2 = 2 1 = 0


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Lissajou s Figu res

31

2

21

2

0 /2


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The rat io o f t he n um bers o f t angenciesm ade by t he f igure w it h t he ad jacentsides o f t he rect angle gives t he rat io o f

f requenc ies

Except ion : w here t he Lissajous f igu regoes int o t h e exact co rn ers o f t hebo un ding rect angle

Find in g freq u e n cy rat io from Lissajo us figu res


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1:1

2:3

3:4

4:5

= 0 / 4 / 2 3/ 4


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ht tp : / /ngs i r .ne t f i rms.com/eng l ishhtm/Lissajous.htm

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2.Super Position of Periodic Motions

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