damped Vibration Report

7/21/2019 damped Vibration Report

1/13

Dynamics and Structures ofMachines

Vibration of a SDOF System

Song Ling Leng130198123

SUMMARY

This Report is made of to parts! the "rst being the physi#a$e%periment in hi#h data from free &ibration and for#ed &ibration ofthe beam is #o$$e#ted' The se#ond part of the report #onsists of()meri#a$ St)dies on the sim)$ation of free &ibration response ofSDOF systems )sing *+TL+,' -n the $aboratory! the physi#a$e%periment aims to meas)re the free &ibrating response fo$$oingan imp)$se and the steady state fre.)en#y response of a systeme%#ited by a rotating ba$an#e' There is a #omparison beteen thenat)ra$ fre.)en#y and damping ratio beteen the oi$ and emptydamper ith both free and for#ed &ibrations' /sing *+TL+,! the

re$ationship beteen the steady state and imp)$se response isin&estigated' Time domain #a$#)$ation methods of dis#rete#on&o$)tion integra$ and #entra$ dieren#e method are a$so#ompared )sing *+TL+, sim)$ations' Their performan#e is )dgedby the time taen to #arry o)t #a$#)$ations and the R*S error' +$$data is re#orded in the res)$ts be$o'


2/13

130198123*4 201Vibration of a SDOF System

Lab ResultsPeakNumbe

r

Time, s Period,s

Acceleration, ms!

Lo"Decrem

ent

Dam#in" Ratio

Natural$re%ue

ncy,Rads&

Test &

0 0'535 5'627

1 0'59 0'061 3'982 0'101 0'0239 105'50

2 0'7 0'062 3'307 0'187 0'0296 105'52

3 0'617 0'06 2'692 0'208 0'0327 105'53

5 0'677 0'06 2'38 0'1211 0'0193 105'39

0'737 0'06 1'91 0'219 0'0359 105'55

Test !

0 0'391 3'6621 0'52 0'061 3'06 0'1780 0'0283 105'51

2 0'12 0'06 2'597 0'200 0'0326 105'53

3 0'72 0'06 2'012 0'2160 0'0355 105'53

5 0'632 0'06 1'711 0'1621 0'028 105'51

0'692 0'06 1'309 0'2678 0'0526 105'57

Test '

0 0'62 3'652

1 0'622 0'06 3'03 0'1823 0'0290 105'52

2 0'682 0'06 2'33 0'1808 0'0288 105'52

3 0'752 0'06 2'122 0'1770 0'0282 105'515 0'802 0'06 1'751 0'1979 0'031 105'52

0'861 0'09 1'281 0'3068 0'0588 105'0

$inal

A(era"e

0'1971 0'0315 105'52

S)De(iation

0'052 0'0072 0'03

Table 1: Results of Free Vibration with Oil Damper

1


3/13


PeakNumber

Time, s Period,s

Acceleration, ms!

Lo"Decrement

Dam#in" Ratio

Natural$re%uency,Rads&

Test &

0 0'236 2'331 0'296 0'06 2'329 0'0850 0'0135 103'3

2 0'3 0'09 2'228 0'0553 0'0071 103'35

3 0'51 0'06 1'92 0'1322 0'0210 103'36

5 0'57 0'06 1'857 0'03 0'0088 103'3

0'33 0'08 1'622 0'1299 0'0207 103'36

Test !

0 0' 11'116

1 0'62 0'062 9'71 0'1310 0'0209 103'36

2 0'623 0'061 8'59 0'1316 0'0209 103'36

3 0'68 0'062 7'595 0'1317 0'0210 103'365 0'756 0'061 6'776 0'1007 0'0160 103'36

0'808 0'062 '898 0'1388 0'0221 103'37

Test '

0 0'583 10'566

1 0'56 0'063 9'31 0'116 0'018 103'36

2 0'607 0'061 8'097 0'1501 0'0223 103'37

3 0'669 0'062 7'205 0'1169 0'0186 103'36

5 0'731 0'062 6'351 0'1276 0'0203 103'36

0'79 0'09 '67 0'1118 0'0178 103'36

$inal

A(era"e 0'1128 0'0180 103'36

S)De(iation

0'0297 0'0057 0'01

Table 2: Results of Free Vibration with Empty Damper

S#eed,R#m

$re%uency Acceleration, ms

!

Dis#lacement,mm

*+ Rads&

830 13'82 86'9 0'651 0'0859

882 15'71 92'5 1'307 0'131922 1'37 96'6 2'6 0'285

985 16'5 103'1 8'882 0'836

1032 17'21 108'1 '792 0'597

1083 18'0 113'5 3'82 0'299

1139 18'98 119'3 2'811 0'197

Table 3: Results of Forced Vibration with Oil Damper

2


4/13


S#eed,R#m

$re%uency Acceleration, ms

!

Dis#lacement,mm

*+ Rads

827 13'78 86'6 0'702 0'0936

856 15'1 88'7 0'75 0'0957

885 15'73 92'6 1'38 0'161923 1'39 96'7 2'87 0'307

966 16'1 101'2 11'719 1'1553

975 16'25 102 13'55 1'2932

981 16'36 102'8 13'77 1'2857

989 16'59 103'6 13'3 1'2627

996 16'6 105'3 12'82 1'166

1000 16'67 105'8 9'988 1'086

1002 16'71 10 11'213 1'0171

1051 17'3 109 '886 0'595

1079 17'99 113 5'371 0'3523

1137 18'96 119'1 3'12 0'2222

Table : Results of Forced Vibration with Empty Damper

12 13 15 1 16 17 18 19 20

0

0'2

0'5

0'6

0'8

1

1'2

1'5

Oi$ Damper mpty Damper

$re%uency, *+

Dis#lacement Am#litude, mm

!raph 1: Displacement "mplitude a#ainst Fre$uency

3


5/13


12 13 15 1 16 17 18 19 20

0'5000

0'500

0'000

0'00

0'6000

0'600

0'7000

0'700

0'8000

0'800

0'9000

Response a$f :oer Line

$re%uency, *+

Dis#lacement Am#litude, mm

!raph 2: %alf &ower 'ethod to (nd Dampin# Ratio for Oil Damper

Damping Ratio! ; 17.0015.85

216.4

0.0351

(at)ra$ Fre.)en#y 16.410.03512

16.39 0'01

105'52>0'03

102'98>'1

102'98>'1

Dam#in"Ratio

0'0180>0'0057

0'0315>0'0072

0'030>0'001

0'031>0'0018

Table ): *omparison of Results

From these &a$)es! e #an #a$#)$ate the Damping 4oe?#ient forboth the empty and oi$ dampers' +n a&erage &a$)e #an be obtainedfrom )sing both free and for#ed &ibrations'

For Free Vibration!

stimated mpty Damping 4oe?#ient 0.01824.5103.36

16.74

For For#ed Vibration!


7/13


stimated mpty Damping 4oe?#ient 0.030524.5102.98

28.27

Taing the a&erage!

mpty Damping 4oe?#ient! # 16.74+28.27

2

22.51 (s=1

For Free Vibration!

stimated Oi$ Damping 4oe?#ient 0.031424.5104.42

29.5

For For#ed Vibration!

stimated Oi$ Damping 4oe?#ient 0.035124.5102.98

32.53

Taing the a&erage!

Oi$ Damping 4oe?#ient! # 29.5+32.53

2

31.02 (s=1

+s a res)$t! the Damping 4oe?#ient has in#reased by 8'1(s=1d)eto the oi$'

6


8/13


Matlab Results

!raph : +teady +tate Responses showin# the E,ects of *han#in#'ass and Dampin# on the &hase *han#e

!raph ): Fourier Transform of a Force -mpulse

7


9/13


!raph .: *omparison of +teady +tate Fre$uency Response with theFourier Transform of the Response di/ided by the Fourier Transform

of the Force +i#nal

8


10/13


!raph 0: *omparison between the Eact 'ethod and Discrete

*on/olution -nte#ral

!raph : &eas of !raph 0

9


11/13


10


12/13


Sam#lin"

Rate, s&

/alculation Time,s

RMS -rror

4on&'-nt

4ent'Di

4on&' -nt 4ent' Di

5096 0'21 0'0185 0'118 0'796

1025 0'0339 0'0135 0'119 3'6726 0'017 0'015 0'51 23

Table .: &erformance *omparison between the Discrete *on/olution-nter#ral and *entral Di,erence 'ethod

Discussion-n the :hysi#a$ %periment! damping #a)ses the disp$a#ementamp$it)de to de#rease signi"#ant$y at resonan#e and fre.)en#iesnear resonan#e @Araph 1B' oe&er! disp$a#ement amp$it)des

f)rther aay from the resonan#e fre.)en#y remain )n#hanged'These statements a$so ho$ds tr)e in the sim)$ated res)$ts @Araph 5B'F)rthermore! #hanging the mass i$$ a$so shift the graph either $eftor right! here in#reasing the mass i$$ red)#e the resonan#efre.)en#y &a$)e and &i#e &ersa @Araph 5B' :hysi#a$ properties p$ayan important ro$e in the o)t#ome of the response' For e%amp$e! ifthe $ength of the beam is in#reased! the stiness i$$ red)#e b)t themass i$$ in#rease' +s the nat)ra$ fre.)en#y s.)ared is the stinessdi&ided by the mass! the nat)ra$ fre.)en#y i$$ red)#e if the $engthof the beam is in#reased'

/n#ertainties in the physi#a$ e%periment a$so pro&e to be importanthen disp$aying res)$ts' -n the free &ibration e%periment! thee%periment is repeated three times d)e to the nat)re of thepro#ed)re )sing an imp)$se #reated by h)man for#e' The standardde&iation of the res)$ts is ey to )stifying the a&erage res)$tobtained' +s for the for#ed &ibration e%periment! the damping ratiois $ess than 0'1 and the #a$#)$ations )sing the form)$as pro&ided i$$gi&e an a##)ra#y of C' -t m)st a$so be high$ighted that the #hangein damping #oe?#ient beteen oi$ed and empty dampers in boththe free and for#ed &ibrations dier by a great amo)nt' Thisdieren#e #o)$d ha&e arisen from the dieren#es in h)man for#eshen #arrying o)t the free &ibration e%periment'

Time domain responses disp$ay the &ariations of amp$it)de ithtime' +s a res)$t! Fo)rier Transform #an be )sed to sho the samegraph ith fre.)en#y instead of time' This is #r)#ia$ as responses#an be #omposed of dierent fre.)en#ies at dierent $e&e$s ofe%#itation'

4omparing the to time domain so$&ers! the #entra$ dieren#emethod pro&es to be .)i#er than the #on&o$)tion integra$ method'

oe&er! the speed #omes at the e%pense of a##)ra#y! ith the#on&o$)tion integra$ method pro&iding a more a##)rate res)$t' The

11


13/13


samp$ing rate a$so ae#ts the speed and a##)ra#y of the res)$tspro&ided! ith 1025s=1 being the .)i#est samp$ing rate to )se'oe&er! the higher the samp$ing rate! the more a##)rate theres)$ts i$$ be'

12

Documents

damped Vibration Report