structural dynamics main cw 423.docx

8/9/2019 Structural Dynamics Main CW 423.docx

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Structural Dynamics(CEGEM071/CEGEG071)

Student: Carmine Russo 14103106Main Coursework Design of a tuned-mass absorber

IntroductionIn general, if te beams CD,

!" and #$ %ere not rigid te

s&stem %ould a'e t%el'e

degrees of freedom (t%o

eac node)* In te +resent

case beams are rigid tus,

te s&stem as nine degrees

of freedom, but b& neglecting

te columns aial

deformations ( E A )

te& become tree D*.*"*

/e coose as lagrangian

coordinates te oriontal

dis+lacements of CD, !" and #$*

Collecting tese 'ariables in te

'ector, %e a'e:

u=

{

u1

u2

u3

}=

{

uCDuEF

uGH

}In order to %rite te euilibriumeuations, %e a'e to 2nd testiness of eac element for eac

dis+lacement*

In tis case, %e can use te direct

metod b& sol'ing te dierential

euation of te elastic beam:

IV=0

' ' '=C1

' '=C1 s+C2

'=1

2C

1s

2+C2 s+C3

=

1

6C

1s

3+1

2C

2s

2+C3 s+C4

/it0 s LAC

C5 Ci'il, !n'ironmental and #eomatic !ngineering

Structural Dynamics 2014 Main Coursework Carmine Russo 14103106page 1 of 34


2/44

/it te boundar& conditions:

(A )=0 ; (A )' =0 ; ( C)

' =0; (C)=u

/e get te constants of integration:

C1=12

LAC3

u ; C2=

6

LAC2

u ; C3=0 ; C

4=0 ;

"inall&:

T( s )=E I '' '=

12E I

LAC3

u M(s )=E I ' '=

6E I

LAC3

(2 sLAC) u

Since teres no lateral load a++lied to te columns, te sear force is constant along

teir lengt*

It is con'enient to indicate %it kte uantit&:

k=12EILAC

3 =7 103[

kNm]

7en, te generic e+ression of te sear force become:

T=k u

/ere 8k re+resent te stiness of te column sub9ected to a oriontal

dis+lacement*

art 1-a) Stiness and mass matri

/e can assemble te stiness matri, columnb& column, sim+l& im+osing one deformation

at time, %ile ;ee+ing te oter one eual to

ero, and 2nding te euilibrium forces*

Displacement u1 :

In tis case, %e a'e te mass m of te

beam CD sub9ected to te dis+lacement u1 .


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(k+k+k+k) u1=4 k u

1

FEF=TECTDF=(k+k) u 1=2 k u1

FGH=0

/e can summarie tese relations as:

{FCDFEFFGH}=( 4 k

2 k0) u1

Displacement u2 :

In tis case, %e consider te mass m of te

beam !" sub9ected to te dis+lacement u2 *

7e total forces for eac mass are:

FCD=TCETDF= (kk) u2=2 k u2

FEF=TCE+TDF+TEG+TFH=4 k u2

FGH=TEGTFH=(kk) u2=2 k u2


{FCDFEFFGH}=(2 k

4 k

2 k) u2

Displacement u3 :

In tis case, %e consider te mass m of te beam #$ sub9ected to te

dis+lacement u3 *

7e total forces for eac mass are:

FCD=0

FEF=TEGTFH=2 k u3

FGH=TEG+TFH=(k+k) u2=2 k u3





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{FCDFEFFGH}=( 0

2 k2 k) u3

7us, te static euilibrium is re+resented b&

te euations:

{FCDFEFFGH}

=[K] u=

[ 4 k 2 k 02 k 4 k 2k

0 2 k 2 k]

{u1u

2

u3}

!e sti"ness matri# ($%ateral Sti"ness

Matri#&):

[K]=[ 4 k 2 k 02 k 4 k 2 k0 2 k 2 k]=2 k[

2 1 01 2 1

0 1 1]=[ 28000 14000 014000 28000 14000

0 14000 14000][ kNm]!e mass matri#:

[M]=[m 0 00 m 00 0 m ]=[

9000 0 0

0 9000 0

0 0 9000] [ kg ]

!e ener'etic approac!

Instead of follo%ing te euilibrium a++roac, %e can 2nd te euation b&calculating te total energ& of te s&stem:

inetic !nergy

T=1

2 mu12

+1

2 mu22

+1

2 mu32

7e mass matri can be found sim+l& b& calculating te >acobian:

[M]=[ 2T

u i uj]=

[

2T

u12

2T

u1

u2

2

T

u1 u

3

2T

u2 u

1

2T

u22

2

T

u2

u3

2T

u3 u1

2T

u3 u2

2

T

u32

]=[m 0 00 m 00 0 m]




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"otential !nergyIn tis eercise te +otential elastic energ& is gi'en b& te oriontaldis+lacements of te beams, terefore b& using te stinesses alread&calculated abo'e, %e can directl& %rite te e+ression of tis energ& %itoutcalculate te integrals* Sim+l& remembering tat te +otential energ& of asingle s+ring is:

Vs!i"g=1

2k # $

2

/e 9ust a'e to sum te +otential energies of eac deformed element:

V= 12

k ui2=

1

2[kACu12+kBDu12+kCE (u1u2 )2+kDF(u1u2 )2+kFH(u2u3 )2+kEG(u2u3 )2 ]=

1

2k[u12+u12+(u1u2 )2+ (u1u2 )2+(u2u3 )2+(u2u3 )2 ]=

1

2 k[2 u1

2

+2 (u1u2 )2

+2 (u2u3 )2

]=

1

2k[4 u12+4 u22+2 u324 u1 u24 u2 u3 ]=k[2 u12+2 u22+u322 u1 u22 u2 u3]

7e stiness matri can be found sim+l& b& calculating te >acobian:

[K]= 2

Vu i uj

=

[

2V

u12

2

V

u1 u2

2

V

u1 u3

2

Vu

2 u

1

2

Vu

2

2 2

Vu

2 u

3

2V

u3

u1

2

V

u3

u2

2

V

u32 ]

=

[ 4 k 2 k 02 k 4 k 2k

0 2 k 2 k]!actl& te same results obtained %it te euilibrium metod?

!e euations o motion:

In case of undam+ed free 'ibrations, te euations of motion are:

[M](u( %))+ [K](u( %))=(0 )


Structural Dynamics 2014 Main Coursework Carmine Russo 14103106page # of 34


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[m 0 00 m 00 0 m](

u1( %)

u2( %)

u3 ( %)

)+[ 4 k 2 k 02 k 4 k 2 k0 2 k 2 k](u

1 (%)

u2 (%)

u3 (%)

)=(000)

art 1-b)


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{1 }={112}

#etting te 'alue:

& 1(K , M ,1 )= {

1

1

2

}

T

[ 2.8 10

7 1.4 107 01.4 107 2.8 107 1.4 107

0 1.4 107 1.4 107

]{1

1

2

}{112}

T

[9 103

0 0

0 9 103

0

0 0 9 103]{112}

=

518.519[ !( )2

s2 ]


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"or te second trial

'ector:& 2(K , M ,2)=

2

{2 }T [M]{2 }

([K]22 [M]){2 }={47.61995.238

79.365}

Clearl&, since te second gradient is closer to ero tan te 2rst one, if te guess of

te mode sa+e is correct (i*e* %e made a coice close to te 2rst eecti'e mode),

{2 } re+resent a better a++roimation of te eecti'e eigen'ector*

7e estimate natural freuenc& is:

&(*=& 2(K , M ,2)=333.333 [ !( )2

s2 ]=18.257 [ !()s]

In general, a good coice for te trial eigen'ector is te 'ector of static

dis+lacements, %it forces +ro+ortional to te %eigt of eac mass ,1+* In tis

eercise te mass are eual, terefore %at +la&s a ;e& role in te 2rst mode is te

geometr& of te s&stem (i*e* te +osition of te masses and te stiness of

columns)*

-n iteratie met!o to minimie aylei'! 2uotient usin' t!e Con3u'ate

Graient -l'orit!m

"rom te +ro+ert& so%n abo'e, te Ra&leig uotient is a 'alue bet%een te

minimum eigen'alue and te maimum eigen'alue of te s&stem:

+1

&(K , M ,! ) +"

7erefore it can be used in min-ma teoremto get eact 'alues of all eigen'alues

anal&ticall& (for eam+le %it a formulation based on 5agrange multi+liers) or,

basing on a 'ariational a++roac, is used in eigen'alue algoritmsto obtain an

a++roimation from an& trial eigen'ector: s+eci2call&, tis is te basis for te

metod called 8Ra&leig uotient iteration*

In +articular, since te +roblem as;s to 2nd te 2rst eigen'alue, %e a'e to sol'e a

+roblem of minimum* 7e basic idea of Ra&leig uotient minimiation is to

construct a seuence {! }!=1,2,3 suc tat:

&(K , M ,!+1 ) &(K , M ,! ); !=0,1,2

7e o+e is tat te seuence {&(K , M ,! ) } con'erges to +1 and b& conseuence

te 'ector seuence {! } to%ards te corres+onding eigen'ector* sing a+erturbation a++roac, de2ning:

{!+1

}= {!}+-!{! }/ere ! re+resent te 8searc% )irection and te +arameter -! is determined

suc tat te Ra&leig uotient of te ne% iterate !+1 becomes minimal:

&(K , M ,!+1 )=mi"-

[&(K , M ,!+-!! ) ]

/e can %rite te Ra&leig uotient of te linear combination {! }+-!{!} of t%o

(linearl& inde+endent) 'ectors {! } and {! } :

1,+*+ Meiro,itc%- !lement of ,i.ration analysis- Mc/raw ill 16- pp 11514


Structural Dynamics 2014 Main Coursework Carmine Russo 14103106page of 34
http://en.wikipedia.org/wiki/Min-max_theoremhttp://en.wikipedia.org/wiki/Eigenvalue_algorithmhttp://en.wikipedia.org/wiki/Rayleigh_quotient_iterationhttp://en.wikipedia.org/wiki/Eigenvalue_algorithmhttp://en.wikipedia.org/wiki/Rayleigh_quotient_iterationhttp://en.wikipedia.org/wiki/Min-max_theorem


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10/44


11/44

2 [( B!+-! C! )(D!+2-!E!+-!2F!)(A!+2 -! B!+-!2 C! )(E !+-!F!)]

(D!+2 -!E!+-!2F! )

2 =0

Sim+lif&ing:

2 [-!2 (C!E!B!F! )+-!(C!D!F!A ! )+ (B !D !A!E ! ) ](-!

2F!+2 -!E !+D! )

2 =0


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Com+ared %it te +re'ious 'alue of natural, te +ercentage error is

&(* 1

1

100=4.012

art 1-c) odal 0 , te

total mecanical energ& of te s&stem (T+V) %ill gro%, %ile {+ }


13/44

{+ }={+1+2+3

}={ 308.0972.419 1035.051 10

3}[ !( )2s2 ]sing te $eig' function in

atlab, te natural freuencies

are:

'"=+=('

1

'2

'3

)=(17.553

49.182

71.069)[!()s]

7e +eriods are:

T=2 8

'"=(

T1

T2

T3

)=(0.3580.1280.088) [ s ]

7e freuencies:

4=1

T=(

41

42

43

)=(2.7947.82711.311) [H9 ]Ei'enectors:Qo% tat %e a'e determined te natural freuencies, %e +roceed to e'aluate te

mode sa+es*/en =1 , 2 or 3 , at least one of te scalar euations described b&

{[K]'i2 [M]}( )=0 is not inde+endent of te oter* /e can retain one of te t%oconditions (for instance te 2rst one), and add a condition on te norm of te 2rst

eigen'ector*

Moe s!ape 1

[

4 k+1

m 2 k 02 k 4 k+

1m 2 k

0 2 k 2 k+1 m

](

11

21

31

)=

(

0

0

0

)/it te condition:112 +212 +312 =1/e a'e:




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{ (4 k+

1m )112 k21=0

2 k11+( 4 k+1 m )212 k31=0

11

2 +21

2 +31

2 =1

{

11=

2k

(4 k+1 m )

21

31=[(4 k+1 m)

24 k2

2 k(4 k+1 m)]21

21=

{ 1

[ 2 k

( 4 k+1 m )]+1+

[(4 k+1m )

2

4 k2

2 k( 4 k+1 m )]2

}0.5

7e 2rst eigen'ector is:

(

11

21

31)=(0.327990.591010.73698)

Moe s!ape

4 k+2 m 2 k 02 k 4 k+

2m 2 k

0 2 k 2 k+2

m(12

22

32)=(

0

0

0)/it te condition:

122 +22

2 +322 =1

/e a'e:

{(4 k+2 m )122 k22=02 k

22+( 2 k+2 m )32=0

122 +22

2 +322 =1

{

12=

2 k

( 4 k+2 m )

22

32= 2 k(2 k+2 m)22

22={

1

[ 2 k( 4 k+2 m ) ]2

+1+[ 2 k(2 k+2 m ) ]2 }

0.5

7e second eigen'ector is:

(

12

22

32

)=

(

0.73698

0.32799

0.59101

) Moe s!ape 84 k+

3m 2 k 0

2 k 4 k+3

m 2 k

0 2 k 2 k+3

m(132333

)=(000)/it te condition:

132 +23

2 +332 =1

/e a'e:




15/44


16/44

02=

2

T [M]2=(122232

)T

[m 0 00 m 00 0 m ](122232

)=[ (12 )2+(22 )2+(32 )2 ] m=16570.498 [ kg ]

03=

3

T [M]3=

(

13

23

33

)

T

[

m 0 0

0 m 0

0 0 m

](

13

23

33

)=[(13 )2+ (23 )2+ (33)2 ]m=16570.498 [ kg ]

Qo% %e can calculate te normal moes(i*e* modes normalied %it te res+ect of

te mass matri)

g ?2 sin (? %)

/it ?=2 8@ * /e can ignore te sign of te ecitation because of it is armonic

and +eriodic*


18/44

/ere . and / can be found considering tat te s&stem is dam+ed at 3 of

te critical dam+ing %en oscillating at its 2rst and tird natural mode freuenc&*

Dia'onaliation 9y aylei'! Dampin'

In order to do tat 2rst, %e ma;e a modal transformation u(%)= [< ] (%) b& using temass normali8e) s%ape matri9, and %e get:

[M] [ < ] ( %)+( .[M]+/ [K])[ < ] ( %)+ [K] [< ] (%)=m ! Fg (? , %)

7en, %e +re-multi+l& b& [ < ]T

and %e get:

[ < ]T [M] [ < ]A( %)+ [ < ]T( .[M]+/ [K])[ < ] A( %)+ [< ]

T [K] [< ] A(%)= [< ]T

m(111)Fg (? , %)

sing te ortogonalit& +ro+ert& of mode sa+e matri, te euation become:

(%)+

[.+/

1

20 0

0 .+/22 0

0 0 .+/3

2] (%)+

[

1

20 0

0 22 0

0 0 3

2] (%)=

m

(

g ?2

sin (? %)

2>g ?2

sin (? %)

3>g ?

2sin (? %))

7e generic armonic res+onse is:

i (%)=>i sin (? %)= {>i 7i ? %}7us, %e a'e:

{ (?2+i 2 i ? i+i

2

)>i7 i ? %}={i>g ?2 7i ? %}




20/44

7is euation must be satis2ed for all 'alues of t, so %e matc te terms contained%itin te brac;ets* 7is leads to:

>i= i>g ?

2

(i2+i 2 i ? i?2)

7e uantit& at te denominator is te d&namic stiness* 7e term i2

is te

circular freuenc& suared, but also re+resents te stiness +er unit of mass of te

eui'alent SD." s&stem= %e can e+ress tis uantit& in a more con'enient form,factoriing it:

>i=i>g(?i)

2

(1!i2+i 2 i !i )=i>g !i

2Di(!i , i )

Di ( !i , i ) is te Com+le "reuenc& Res+onse (te uantit& at te denominator iste d&namic stiness):

Di ( !i , i )= 1

(1!i2+i 2i !i )

; ! i=?

'i

Since Di ( !i , i ) is a com+le number, %e can re+resent in modulus and +ase:

Di ( ! i , i )=|Di(! i , i )|7ii(!i , i)

/ere:

|Di(!i ,i )|= 1

[(1!i2 )2+4 i2 !i2 ]1 /2(M(g"i4i(%i1" F(%1!)

i (

!i,

i )= tan1

(2 i !i

1!i2

)((s7 A"g37 )

"inall&, te stead& state res+onse is te immaginar& +art of te imaginar& number:

( %)={i>g ?2

i2

Di ( !i , i ) 7i ? %}= {i>g?

2

i2 |Di ( !i, i )|7

[ii( !i ,i)]7

i ? %}= {i>g?

2

'i2 |Di ( !i, i)|7

i [ ? % i( !i ,i)]}=i>g!i2|Di(!i ,i )|sin [? %i(!i , i ) ]

( %)=>g(

1 !12

|D1 ( !1 , 1 )|sin [? %

1(!1 ,1)]2!

2

2|D2 ( !2 , 2 )|sin [? %2(!2 ,2 ) ]

3!

3

2|D3 (! 3 , 3 )|sin [? %3(!3 , 3 ) ])!e total solution:


21/44

sing te initial condition %e can 2nd no% te constants Es , E% for eac modal

coordinate A i( %) *

"irst, %e calculate te e+ression of 'elocit&:

i (%)=ii 7i i%[Es cos ()i %)+E%sin ( )i%) ]+)i 7i i%[E s sin ( )i %)+E %cos ()i %) ]+

+i>g ?3

i2 |D

(!

i

, i )|

cos

[? %

i (!

i

, i)]

sing te initial conditions (%=0 )=[ < ]T [M]u( %=0)=0 and (%=0 )=[ < ]

T [M] u( %=0)= 0

{ Es+i>g ?

2

i2 |D (!i , i )|sin [ i ( !i, i )]=0

i iE s+)iE%+i>g?

2

i2

?|D (!i , i )|cos [i(!i ,i ) ]=0

:7 g7%:

{ E s=0i>g?

2

i2 |D (! i , i )|sin [i ( !i ,i ) ]

E%= 1

)i{i>g?2

i2

?|D (! i , i )|cos [i(!i , i ) ]+0+i iEs}If %e consider omogeneous u(%=0)=00=0 and u(%=0)=0 0=0

Es=i>g !i2|D ( !i , i )|sin [i(!i ,i ) ]

E%=i>g?

2

)i i|D ( !i , i)|{!i cos [ i ( !i , i)]+i sin [i(!i , i ) ] }

Conolution inte'ral (Du!amel)

7e solution for eac modal coordinate can also be found b& calculating te

con'olution bet%een te generic res+onse of te underdam+ed s&stem to te single

im+ulse (Dirac delta) at time , and te eecti'e force a++lied at tat instant:

Ai (%)=0

%i>g?

2sin (? )

'i)7i 'i(%)

sin [ 'i)(%)] )

7is e+ression of dis+lacements for eac normal coordinate could a++ear sim+ler

tan te anal&tic solution, but is 9ust more com+act* In fact, te con'olution integral

olds inside a com+lete base of te 'ector s+ace of solutions*

A1 (%)=

1>g?2

'1 )

0

%

sin (? )71 '1(%)sin ['1 )(%)])

2( %)=

2>g?2

2 )

0

%

sin (? )72 2 (% ) sin [2 )( %)])




22/44


23/44

(

;onimensional isplacemento t!e 6rst


24/44

7e mass tat +artici+ates to te 2rst mode is

m(ss1=24680.15 [kg ]

J1*4 of te total mass*

Qo% b& calculating te base total moment, %e can

2nd te eui'alent ele'ation of tis mass:

M(s7="i mi=m(ss744

[ < ] [M](

1

2

3

)[< ] [M] (11

1)=m(ss 744H744

/ereH

744 is a $mo)al e:ecti,eele,ation'+

Its intuiti'e tat b& locating te

absorber on te to+ of te building,

te action (tat is a force) transmitted

b& te absorber to te structure ( and

'ice 'ersa) as a bigger le'el arm, and

tis gi'e to te absorber more

ca+acit& of control of 'ibrations*

art E-b) Sim+li2ed model of te frame*B& using te modal decom+osition, %e can a++roimate te s&stem at te 2rst

mode:




25/44

u(%)= [< ] (%)=(7 and >D are determined from te linear euations:

[K7+KD?

2M7 KD

KD KD?2MD

](>7

>D )=(F

0>g ?

2

0

)

(>7>D )={ (KD?

2MD)F0>g ?

2

(K7+KD?2M7) (KD?

2MD)KD

2

KDF0>g?2

(K7+KD?2M7) (KD?

2MD)KD

2

7e denominators are eual and re+resent te determinant of te s&stem ofcoeGcients* 7is uantit& is identical to te caracteristic euation for teundam+ed s&stem*

7e ob9ecti'e of te absorber is to reduce te 'alue of >7 %en ? matces te

natural freuenc& K7AM7A of te eui'alent s&stem* 7e numerator for >7'anises at ?=KDMD and te denominator is nonero at tat freuenc&* It follo%stat if %e select te u++er s&stem suc tat:




31/44

KD

MD=

K7

M7

7en te mass M7 %ill not oscillates %en ?=K7M7=1 (2rst eigen'alue ofte original s&stem)* 7e corres+onding am+litude for te u++er mass is

>D=

(F

0

>g

?2

KD ), %ic resembles te static deformation of KD resulting from

a com+ressi'e force F0>g?2

* 7e e+lanation is tat because M7 is not

mo'ing, te forces acting on it must euilibrate at e'er& instant, %ic is satis2ed

(%en >7=0 ) if:

KD>D+F0>g?2

sin (? %)=0

7us, %e a'e:

>D=

(F

0>g ?

2

KD

)sin (? %)

Basicall&, if te natural freuenc& of te absorber matces te natural freuenc& ofte 8eui'alent s&stem, and if te armonic motion of te ground as a freuenc&

close to 1 , te s+ring force te 'ibration absorber a++lies to te original s&stem

balances te ecitation force, so te original s&stem does not mo'e*

7e abilit& of te 'ibration absorber to uiet >7 de+ends onl& on te ratio (KDMD )but not on te actual

sie ofM

D and K

D

* "or tat reason, itmigt seem tat teu++er s&stem can beetremel& ligt%eigt*$o%e'er, t%o factorscan ma;e te sie of

MD signi2cant: small

MD leads to small

KD , in %ic case

te am+litude >D %ill

be 'er& large (if te'ibration am+litude ofte absorber is 'er&ecessi'e, ten te s+ring is li;el& to fail)* 7e oter consideration limiting te

smallness of MD and KD +ertains to situations %ere te ecitation freuenc&

migt drift around te nominal 'alue 1 *

If %e coose for instance KD=0.1K7 and MD=0.1M7 , %e can +lot te

am+litudes as function of ? *


32/44

KD

MD=

K7

M7=

1

2

In order to nullif& >7 at te original natural freuenc&, te natural freuencies of

te ne% E degree of freedom s&stem obtained b& sol'ing:

(K7+KD?2M7) (KD?

2MD )KD

2 =0

D , attains a minimum negati'e

'alue in te 'icinit& of 1 and ten increase negati'el&* "urter increase of ?

to%ard te second natural freuenc& '2

causes bot am+litudes to a++roac

in2nit&, but >7 is +ositi'e %ile >D is negati'e* assing te resonance results

in a 180 J +ase sift for bot am+litudes as it did for te fundamental resonance*

Increasing te freuenc& be&ond '2

causes bot am+litudes to decrease in

magnitude because te inertial eect, %ic is +ro+ortional to te suare of tefreuenc&, becomes te dominant eect*

Canging te mass ratio, %e can see o% te ga+ bet%een te t%o freuenciesbecome %ider*




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1 0.73 1=14.995[!()s] 41= 1

2 8=2.386 [H9 ]

21.37 1=20.546 [ !()s] 42= 2

2 8=3.270 [H9 ]

/e can see also tat 42

is also %ell se+arated from te 42 of te original

structure*

MD=0 M71657.05 [ kg ]

KD=0 K7 510531.82 [Nm]

art E-d)


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k=5105318.2[Nm]

-9sor9er:

0=0.1

MD=0 M7 1657.05 [ kg ]

KD=0 K7 510531.82 [Nm]

/e get te follo%ing diagrams:

=or CD= 1

1000


Structural Dynamics 2014 Main Coursework Carmine Russo 14103106page 3# of 34


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It is clear from te diagram tat %en te 'alue of te das+ot constant is too small,te eect of te absorber is detrimental since %e a'e in te 2rst natural freuenc&tat te maimum am+litude of dis+lacements of te frame is greater tan temaimum am+litude of te frame %itout absorber*

=or CD=1000




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5oo;ing at tese diagrams, tere is a clear relation bet%een te 7D dam+ing

ratioand te d&namic am+li2cations*/en tere is a 'er& ig dam+ing ratio

te dam+er constraints te relati'e motions, resulting in te 7D and te

structure %or;ing as one mass (m+MD ) and stiness k

*

.n te oter and, if tere is a 'er& lo% dam+ing ratio large d&namic

am+li2cations %ill occur near te t%o undam+ed resonance freuencies* "or te

small 'alue of te dam+ing, at te 2rst natural freuenc& 1 te s&stem as a

small dis+lacement (not ero), results because tere is a dam+ing force tat is

90 J out of +ase from te inertial resistance and elastic force, %icessentiall& cancel eac oter*


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7e scri+t is a loo+ tat +lot te corres+onding maimum absolute 'alue of te

dis+lacements for eac 'alue of CD * /e can see easil& tat te best 'alue of

te das+ot coeGcient for te absorber can be 2nd 2nd %ere te maimum

dis+lacement is minimum:

7e o+timum 'alue is near CD=12000[N sm] (tat is 0.688C7 )"or tat 'alue of te das+ot constant, te diagram of te stead& state armonic

res+onse and te ste+ res+onse are:




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Broc;Esuggested tat te o+timum-dam+ing ratio could be gi'en b& te em+iric

formula:

E>*!* Broc;, < note on te dam+ed 'ibration absorber* >ournal of


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1 %1= 38 0(1+0 )3It sould be noted tat freuentl& in scienti2c literature te o+timum dam+ing

ratio is also gi'en as:

1 %2= 38 0(1+0 )In our case, %e get:

1 %1

=0.168

1 %2

=0.185

CD 1=2 1%1

0km=9764.2[N sm]CD2=2 1 %

2

0 km=10740.6 [N sm]Bot 'alues are not far from %at %e found %it numeric iteration*

In our case, te 'alue tat %e found is more similar to te a++roimation3:

1%= 12 0(1+0 )CD=21%0km=12402.18 [N sm]In +resence of dam+ing te am+litudes are com+le, tis introduce a +ase

angle in te resulting dis+lacements: te res+onse of te tuned mass is J0 out

of +ase %it te res+onse of te +rimar& mass* 7is dierence in +ase

+roduces te energ& dissi+ation contributed b& te dam+er inertia force*

/en te ecitation freuenc& is lo%, te stiness of te s&stem controls te

res+onse: as conseuence, te dis+lacements are in +ase %it te ecitation*

/en te ecitation a++roaces te natural freuenc&, te res+onse of te

s&stem is controlled to a large etent b& te dam+ing in te s&stem* "inall&, as

? becomes large, te res+onse of te s&stem is largel& controlled b& te mass

(te inertia) of te s&stem*

Coosing te correct 'alue of mass ratio and dam+ing factor, leads to im+ro'edbea'ior (e*g* in te form of am+litude reduction o'er a %ider range of

freuencies): if te t%o natural freuencies of te eui'alent E D." s&stem are

close enoug, te eect of te dam+ing of te absorber etents for all te range

in-bet%een*

3Tren;, S* U $Vgsberg, >* (E00N)* Structural D&namics, Qote 3* 7uned ass Dam+ers* De+artmentof ecanical !ngineering 7ecnical ni'ersit& of Denmar;*




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art 1>9) Con3u'ate Graient -l'orit!m ? Matla9Coe********************* #L ************************ ivilviromet'l ', 3eom'tic %ieeri% ***************** M'i orse4or! *********************************** 'rt 1 ********************************** 6'7lei% 8pproim'tio *********************** o9%'te 3r',iet 8l%oritm ***********************************************************cle'rcle'r 'll

clc,isp(/=======================================/),isp(/ 3M01&3301 /),isp(/


42/44

,isp(/ress eter to se te stiffess m'tri of oit 1'/);M=ipt(/@sert te M'ss m'tri AMC /); M'ss M'tri of te s7stem.ifisempt7(M)==1 ,isp(/M'ss m'tri AMC/) M=AE000 0 0; 0 E000 0; 0 0 E000C; @ A!%Ce,M,=siFe();m=,(1);fi=oes(m1); @iti'liFe te 'l%oritm:ifm G tis is 9st ' prit o scree co,itio,isp(/@iti'l ei%evector:/);fie,v0=*fi;0=M*fi;,isp(/Hirst 6'7lei% Iotiet:/);6=(v0/*fi)&(0/*fi) Hirst 6'7lei% Iotiet3r',6'70=2*(v0-6*0); 'lcl'te te iiti'l %r',iet=orm(3r',6'70); Borm of te iiti'l %r',ietoose ' oFero (', positive) v'le 's co,itio to stop te c7cle:,isp(/BOJ:ress eter to se te ,ef'lt/);,isp(/)-*8**+*(8"2)*D)&(*8">);tri0=("2)->**D+12*8*;tri1=2*(">)-E***D+2*("2)*+2*8*(D"2)-2*8**;I=((tri1+sNrt((tri1"2)-*(tri0)">))&2)"(1&>);)*p+(1&(>*8))*(I+(tri0&I)));sol1=-(&(*8))-


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- X* 7* "eng and D* R* >* .%en, Con9ugate gradient metods for sol'ing te smallesteigen+air of large s&mmetric eigen'alue +roblems, Internat* >* Qumer* etods !ng*,3J (1JJ6), ++* EE0JEEEJ*

structural dynamics main cw 423.docx

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