String Dynamics 5&6 2015

7/21/2019 String Dynamics 5&6 2015

1/35

NANYANG TECHNOLOGICAL

UNIVERSITY, SINGAPORE

Tuesday, November 17, 2015

SCHOOL OF MECHANICAL AND

AEROSPACE ENGINEERING

Uniform String Dynami!L"t#r" $ % & '()$

E* + A #niform !tring a! a -"ngt l=1m, a ."n!ity

=0.01kg/m/ T" !tring i! fi0". at 1ot "n.! 2it initia-

t"n!ion !"t at T=100N/ T" !tring i! 3-#4". at t" mi..-"a! !o2n 2it h=0.01man. i! r"-"a!". at t=0/ D"t"rmin"

5i6 t" fir!t ) nat#ra- fr"7#"ni"! of t" !tring8 5ii6 t"

orr"!3on.ing fir!t ) mo." !a3"! of t" !tring8 5iii6 t"

!#1!"7#"nt 9i1ration of t" !tring/


2/35






Uniform String Dynami!L"t#r" $ % & :()$

From 2at i! gi9"n, t" initia- .i!3-a"m"nt an 1"

."!ri1". a!,

( )


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Uniform String Dynami!L"t#r" $ % & )()$

5ii6 T" mo." !a3"! ar" gi9"n a!,

( ) ( ),,sin 21ilxixi =

( ) ( )xx1 sinSo,

( ) ( )x2x2 sin

( ) ( )x3x3 sin

5iii6 Sin" g(x)=0, 2" a9"

( ) dxl

xjxg

Tj

2E

l

0j

=

sin ( ),,21j=0=


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Uniform String Dynami!L"t#r" $ % &


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Uniform String Dynami!L"t#r" $ % & $()$

G"n"ra-i;". E7#ation! of Motion + For". Vi1ration

On" fr"" 9i1ration mo."! 5i, i(x)6 ar" ."t"rmin"., it i!!traigtfor2ar. to ."t"rmin" t" r"!3on!" of a !y!t"mto any "0t"rna- -oa.ing/ E03r"!! .i!3-a"m"nt in t"rm!of 4no2n mo." !a3"! an. g"n"ra-i;". oor.inat"!8

( ) ( ) ( )txtx ii1i

==, 5'6

T" 4in"ti "n"rgy of t" !y!t"m 1"om"!,

2l

0 t

dx2

1

!

=

5:6

( ) ( )

2

ii1i

l

0 txdx2

1

=

=

( ) ( ) dxtx2

2

ii1i

l

0

=

=


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Uniform String Dynami!L"t#r" $ % & &()$

U!ing t" mat"matia- i."ntity,

ji1j1i

2

i1i

aaa =

=

==

To 3ro9" #!ing !im3-" "0am3-",

2

i

3

1i

a

=

j

3

1j3j

3

1j2j

3

1j1 aaaaaa === ++=

( )2321 aaa ++= 31322123

22

21 aa2aa2aa2aaa +++++=

( ) ( ) ( )321332123211 aaaaaaaaaaaa ++++++++=

ji

3

1j

3

1i

aa==

=


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Uniform String Dynami!L"t#r" $ % & =()$

T"n 5:6 an 1" r">2ritt"n a!,

5)6

( ) ( ) ( ) ( )dxtxtx2

! jjii1j1i

l0

=

==

( ) ( ) ( ) ( ) dxxxtt2

ji

l

0ji

1j1i

=

=

=

( )t4

l 2i

1i

==

Simi-ar-y, t" 3ot"ntia- "n"rgy of t" !y!t"m 1"om"!,

dxx

T

2

1"

2l

0

= ( ) ( ) dxtxT2

12

ii1i

l

0

=

=


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Uniform String Dynami!L"t#r" $ % & @()$

T" g"n"ra-i;". for" i(t)orr"!3on.ing "a

g"n"ra-i;". oor.inat" i(t)( ) ( ) ( ) dxxtxFt i

l

0i ,=

No2, 2" a9",

5$6

( )t4

l" 2i

2i

1i

== 5


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Uniform String Dynami!L"t#r" $ % & ''()$

E7#ation 5&6 an 1" !o-9". for i(t)!in" a-- t"

o"ffii"nt! an. t" RHS ar" 4no2n/ On" i(t) ar"!o-9"., t" for". r"!3on!" of t" !y!t"m an 1"

o1tain".,

( ) ( ) ( )txtxii1i

==,

2"r" i(x)a! 1""n "!ta1-i!". from t" fr"" 9i1ration

ana-y!i! of t" !y!t"m/


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Uniform String Dynami!L"t#r" $ % & ':()$

E* + A #niform !tring of -"ngt l, t"n!ion Tan. ."n!ity

i! !#1"t". to a #niform-y .i!tri1#t". -oa. of !im3-"armoni am3-it#." 5in tim"6 a! !o2n 5t" magnit#."

i! F0an. fr"7#"ny i! 6/ D"t"rmin" t" for".

9i1ration r"!3on!" of t" !tring/


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Uniform String Dynami!L"t#r" $ % & ')()$

( ),,21iTli

i =

From fr"" 9i1ration ana-y!i!, 2" a9"

( ) ( ),,sin 21i

l

xixi =

From 5$6, t" g"n"ra-i;". for" 1"om"!

( ) ( ) ( ) dxxtxFt il

0i ,= dx

l

xitF0

l

0

= sinsin

( )

( )

642i

531i

0

ti

lF2 0

,,

,,sin

==

=

SCHOO OF MECHANICA AND


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Uniform String Dynami!L"t#r" $ % & ')6

T" 3arti#-ar !o-#tion i! t" for". 9i1ration, tat i!

( )t2

l'i

( )t2

l'i

2i

+ ( ),,31i=t

i

lF2 0

sin= 5E'>


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Uniform String Dynami!L"t#r" $ % & '$()$

( )t'i ( )t'i2i + ( ),,31i=t

i

F4 0

sin= 5E'>$6

A!!#m" t" !o-#tion i!,

( ) t&t 'i'i sin=

Fr"7#"ny i! t" !am" t" foring fr"7#"ny , 1#t2at i! &'i

( ) t&t 'i'i sin= ( ) t&t 2'i'i sin=

S#1!tit#t" t"!" into 5E'>$6, t"n

t& 2i'i sin+ ( ),,31i=ti

F4 0

sin=t& 2

'i sin

( ) = 22i'i& i

F4 0

)(

422

0

=

i

'i

i

F&

NANYANG TECHNOLOGICALSCHOOL OF MECHANICAL AND


16/35






Uniform String Dynami!L"t#r" $ % & '&()$

T" g"n"ra- !o-#tion of 5E'>'6 i! t"r"for",

( ) t

i

F4t%t&t

22

i

0iiiii

sin

)(

cossin

++= ( ),, 31i=

T" for". 9i1ration r"!3on!" 1"om"!,

( ) ( ) ( )txtx ii31i

=

=,,

,

++

=

=t

i

F4t%t&

l

xi22

i

0iiii

31i

sin)(

cossinsin,,

( ) ti

F4t

22

i

0'i

sin

)( = 5E'>&6



17/35






Uniform String Dynami!L"t#r" $ % & '=()$

T" initia- on.ition ar",

( ) ( ) 00x00x == ,and,

H"n",

i31i

%l

xi

=

=

sin

,,

( ) ( ) ( )0x0x ii31i

==

,,

, 0=

M#-ti3-y 1ot !i."! 1y sin(jx/l)an. int"grat" o9"r [0,l],

0dxl

xj

l

xi%

l

0i31i

=

=

sinsin

,,

02

l%

i

= ( )531i0%i

,,

==U3.at" t" !o-#tion,

( )

+

=

=

ti

F4t&

l

xitx

22

i

0ii

31i

sin)(

sinsin,,,



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Uniform String Dynami!L"t#r" $ % & '?()$

A33-y t" !"on. initia- on.ition,

( )0t

22i

0iii

31i

ti

F4t&lxi0x

=

=

+

=

cos)(

cossin,,,

+

=

= )(

sin

,,

22

i

0ii

31i i

F4&

l

xi

0=

0dxl

xj

l

xi

i

F4&

l

022i

0

ii31i =

+

=

sinsin

)(,,

02

l

i

F4&

22i

0ii =

+

)(

)( 22ii

0i

i

F4&

=



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20/35






Uniform String Dynami!L"t#r" $ % & :()$

Tr"atm"nt of Con"ntrat". For"!

A on"ntrat". for" an 1" 2ritt"n,


0i ,=

( ) ( ) ( )(( xxtFtxF = , ( )

=

(

(( xx0

xx1xx2"r"

( ) ( ) ( ) dxxxxtF i((l

0 =

( ) ( ) ( )(i(i xtFt =



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Uniform String Dynami!L"t#r" $ % & :'()$


i! !#1"t". to a on"ntrat". -oa. of am3-it#." F0a!

!o2n/ D"t"rmin" t" 9i1ration of t" !tring gi9"n initia-

9"-oity ;"ro an. initia- .i!3-a"m"nt a!

( )

= lx4

h0x

sin,



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Uniform String Dynami!L"t#r" $ % & ::()$

T" g"n"ra-i;". for" 1"om"!


0i ,= ( ) dxlxixxF (0l0 = sin

( ) ( )

( )))*i

+ddi

0

1F 21i

0

=

/)(

=

l

xiF (0

sin

=

2

iF0

sin

( )t2

li

( )t2

li

2i

+ i= ( ),,21i= 5&6

R"a--,

( ) ( ) ( ) ( )

( )

=+

*i0

+ddi1Fl

2

tt21i

0i

2ii

/)(

T"n,



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Uniform String Dynami!L"t#r" $ % & :)()$

T" orr"!3on.ing g"n"ra- !o-#tion! ar",

( ) ( )))*it%t&t iiiii cossin +=

( ) ( )+ddi,t%t&t iiiiii ++= cossin

( ) 21i0i2i 1

l

F2,

/)( =

( ) ( )+ddi1Ti

lF2,

21i

220

i/)(=

So, t" !o-#tion an 1" 2ritt"n,

( ) ( ) ( )txtx ii1i

==,

( )t%t&l

xiiiii

42i

cossinsin,,

+

+

=

( )

++=

= 21i

220iiii

31i

1TilF2t%t&

lxi /)(

,,

cossinsin



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Uniform String Dynami!L"t#r" $ % & :


25/35






Uniform String Dynami!L"t#r" $ % & :$()$

H"n", !o-#tion an 1" #3.at". to 1"om",

t%l

xiii

42i

cossin,,

+

=

( ) ( )

+

=

= 21i22

0ii

31i

1Ti

lF2t%l

xitx /)(

,,

cossin,

Con!i."r t" initia- .i!3-a"m"nt,

( )

= l

4h0x

sin,

+

= l

xi%i

42i

sin

,,

( )

+

=

l

xi1

Ti

lF2%

21i

220

i31i

sin

/)(

,,

=

l

xh

4sin



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Uniform String Dynami!L"t#r" $ % & :&()$


dxl

xj

l

xi%

l

0i42i

+

=

sinsin

,,

( ) dxl

xj

l

xi1

Ti

lF2%

l

0

21i

220

i31i

+

=

sinsin

/)(

,,

( )

===

==

==

+

)(

),(

)(/)(

4ij

2

lh

2

l%

4i))*ij02

l%

+ddij0

2

l1

Ti

lF2%

i

i

21i

22

0i

+

= l

xi%i

42i

sin

,,

( )

+

=

l

xi1

Ti

lF2%

21i

220

i31i

sin

/)(

,,

=

l

xh

4sin

dxl

xj

l

x4h

l

0

=

sinsin



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Uniform String Dynami!L"t#r" $ % & :=()$

( )

==

=

=

+

)(

),(

)(/)(

4i2

lh

2

l%

4i))*i02

l%

+ddi02

l1

Ti

lF2%

i

i

21i

220

i

So-9ing t" a1o9", 2" a9"

( ) )(/)( +ddi1Ti

lF2%

21i

220

i=

h%4=

);( 4i))*i0%i =

H"n" t" fina- !o-#tion 1"om"!,



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Uniform String Dynami!L"t#r" $ % & :?()$

tl

x4h 4

cossin

+

( ) ( )

( )t1l

xi

i

1

T

lF2tx i

2

21i

31i2

0

cossin,/)(

,,

=

=

2"r",

T

l

ii=



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Uniform String Dynami!L"t#r" $ % & :@()$


i! !#1"t". to a on"ntrat". -oa.! of am3-it#." F0

an. a .i!tri1#t". -oa., a-- of fr"7#"ny , a! !o2n/

D"t"rmin" t" 9i1ration of t" !tring gi9"n tat t"

initia- .i!3-a"m"nt an. initia- 9"-oity ar" ;"ro!/



30/35





Uniform String Dynami!L"t#r" $ % & )()$

T" g"n"ra-i;". for" 1"om"!


0i ,= dxlxi

2

lxtF0

l

0 =

sinsin

dxl

xit

l

x31-0

l

0

+

sinsinsin

t2

iF0

sinsin

= ( ) ( ) t3i

2

l-i1

i

l- 00

sincos

++

( ) ( ) t3i2

l-i1

i

l-

2

iF 000

sincossin

++

=

( ) ( ) ( ) ( ) t3i2

l-i1

i

l-

2

iF

l

2tt 000i

2ii

sincossin

++

=+



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Uniform String Dynami!L"t#r" $ % & )'()$

To !im3-ify, -"t

( ) ( ) ( )

++

3i2l-

i1i

l-

2

i

Fl

2

iF 00

0

cossin

T"n,

( ) ( ) ( ) tiFtt i2ii sin=+

T" orr"!3on.ing g"n"ra- !o-#tion 1"om"!,

( ) ( )

tiF

t%t&t22

i

iiiii

sincossin

++=

So, t" !o-#tion an 1" 2ritt"n,

( ) ( ) ( )txtx ii1i

=

=,

( )

++

=

=

tiF

t%t&l

xi22

i

iiii1i

sincossinsin



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Uniform String Dynami!L"t#r" $ % & ):()$

Con!i."r t" initia- .i!3-a"m"nt,

( ) 00x =, ( ),,21i0%i ==

5Pr"9io#!-y .i!#!!".6

0%l

xii

1i =

=

sin

Con!i."r t" initia- 9"-oity,

( ) ( )

+=

=tiFt%t&

lxitx 22

i

iiiiii1i

cossincossin,

( ) 00x =, ( )

0iF

&l

xi22

i

ii1i

=

+

=

sin


( )0dx

l

xj

l

xiiF&

l

022i

ii1i

=

+

=

sinsin



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Uniform String Dynami!L"t#r" $ % & ))()$

( )( ),,21i0

2

liF&

22i

ii ==

+

( )( ),,

)(21i

iF&

22ii

i =

=

( ) ( )

=

=ttl

xiiF

tx ii

22i1i

sinsinsin,

( )

( )

( ) ( ) ( )

( )32

3

4

3,4

1

0

2

000

02/10

=+

+

=

il

F--

i+ddii

-

))*i

l

FiF

i

C-o!" "0amination of F(i)r"9"a-! tat,



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Uniform String Dynami!L"t#r" $ % & )


35/35

UNIVERSITY, SINGAPOREAEROSPACE ENGINEERING

SUMMARY

'/ For". 9i1ration of !tring8

:/ D"ri9ation of g"n"ra-i;". for"8

)/ D"ri9ation of g"n"ra-i;". "7#ation! of

motion8

String Dynamics 5&6 2015

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Transcript of String Dynamics 5&6 2015