Single Degree of Freedom System Forced Vibration

Transcription

1 Maa Kliah : Dinamia Srr & Penganar Reayasa Kegempaan Kde : TSP 3 SKS : 3 SKS Single Degree f Freedm Sysem Frced Vibrain Pereman - 3

2 TIU : Mahasiswa dapa menelasan enang eri dinamia srr. Mahasiswa dapa memba mdel maemai dari masalah enis yang ada sera mencari slsinya. TIK : Mahasiswa mamp menghing respn srr dengan esiasi harmni dan esiasi peridi

3 Sb P Bahasan : Esiasi Harmni Esiasi Peridi

4 Undamped SDF Harmnic Lading The impressed frce p() acing n he simple scillar in he figre is assmed be harmnic and eqal F sin w, where F is he amplide r maximm vale f he frce and is freqency w is called he exciing freqency r frcing freqency. () () m p() = F sin w f S = f I () = mü F sin w

5 The differenial eqain bained by smming all he frces in he Free Bdy Diagram, is : m F sinw (1) The slin can be expresses as : c p () Cmplemenary Slin c ()= A cs w n + B sin w n (3.a) Pariclar Slin p () = U sin w (3.b)

6 Sbsiing Eq. (3.b) in Eq. (1) gives : Or : mw U U U F mw F F 1 b (4) Which b represens he rai f he applied frced freqency he naral freqency f vibrain f he sysem : w b (5) w n

7 Cmbining Eq. (3.a & b) and (4) wih Eq. () yields : Acs Wih iniial cndiins : F B sinwn 1 b wn sinw (6) cs wn F b F sinwn 1 b 1 b w n sinw (7) Transien Respnse Seady Sae Respnse

8 3 F / Tal Respnse Seady Sae Respnse Transien Respnse n and w, w w F n /

9 Seady sae respnse presen becase f he applied frce, n maer wha he iniial cndiins. Transien respnse depends n he iniial displacemen and velciy. Transien respnse exiss even if In which Eq. (7) specializes F 1 b sinw b sinw I can be seen ha when he frcing freqency is eqal naral freqency (b = 1), he amplide f he min becmes infiniely large. A sysem aced pn by an exernal exciain f freqency cinciding wih he naral freqency is said be a resnance. n (8)

10 If w = w n (b = 1), he slin f Eq. (1) becmes : F w csw sinw 1 n n n (9) F / p

11 Damped SDF Harmnic Lading Inclding viscs damping he differenial eqain gverning he respnse f SDF sysems harmnic lading is : m c F sinw (1) c () f D () = cú () m p() f S () = f I () = mü F sin w

12 The cmplemenary slin f Eq. (9) is : c wn e Acsw B sinw The pariclar slin f Eq. (9) is : p C sinw D D csw D (11) (1) Where : C F 1 b 1 b b (13.a) D F b 1 b b (13.b)

13 The cmplee slin f Eq. (9) is : c wn e Acsw B sinw C sinw D cs D D w (14) Transien Respnse Seady Sae Respnse Respnse f damped sysem harmnic frce wih b =,, =,5, () =, ú() = w n F /

14 The al respnse is shwn by he slid line and he seady sae respnse by he dashed line. The difference beween he w is he ransien respnse, which decays expnenially wih ime a a rae depending n b and. Afer awhile, essenially he frced respnse remains, and called seady sae respnse The larges defrmain pea may ccr befre he sysem has reached seady sae.

15 If w = w n (b = 1), he slin f Eq. (1) becmes : cs sin cs e F n D D n w w w w 1 1 (15)

16 Cnsidering nly he seady sae respnse, Eq. (1) & Eq. (13.a, b), can be rewrien as : Where : U sinw U F 1 b b (16) b an 1 b (17) Rai f he seady sae amplide, U he saic deflecin s (=F /) is nwn as he dynamic magnificain facr, D : U 1 D s (18) 1 b b

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18 Exercise The seel frame in he figre spprs a raing machine ha exers a hriznal frce a he girder level p() = 1 sin 4 g. Assming 5% f criical damping, deermine : (a) he seady-sae amplide f vibrain and (b) he maximm dynamic sress in he clmns. Assme he girder is rigid W = 6,8 ns EI WF 5.15 I = 4,5 cm 4 E = 5. MPa 4,5 m (b)

19 Respnse T Peridic Exciain A peridic fncin can be separaed in is harmnic cmpnens sing Frier Series Where : 1 1 sin b cs a a p w w,...,, T 1 3 p w (19) 1 T T T d sin p T b d cs p T a d p T a w w

20 Respnse peridic frce is given by : ()

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