Single Degree of Freedom System Forced Vibration

Transcription

1 Mata Kliah : Dinamia Strtr & Pengantar Reayasa Kegempaan Kde : CIV SKS : 3 SKS Single Degree f Freedm System Frced Vibratin Perteman - 3

2 TIU : Mahasiswa dapat menjelasan tentang teri dinamia strtr. Mahasiswa dapat membat mdel matemati dari masalah tenis yang ada serta mencari slsinya. TIK : Mahasiswa mamp menghitng respn strtr dengan esitasi harmni dan esitasi peridi

3 Sb P Bahasan : Esitasi Harmni Esitasi Peridi

4 Undamped SDF Harmnic Lading The impressed frce p(t) acting n the simple scillatr in the figre is assmed t be harmnic and eqal t F sin wt, where F is the amplitde r maximm vale f the frce and its freqency w is called the exciting freqency r frcing freqency. (t) (t) m p(t) = F sin wt f S = f I (t) = mü F sin wt

5 The differential eqatin btained by smming all the frces in the Free Bdy Diagram, is : m F sinwt (1) The sltin can be expresses as : t t t c p () Cmplementary Sltin c (t)= A cs w n t + B sin w n t (3.a) Particlar Sltin p (t) = U sin wt (3.b)

6 Sbstitting Eq. (3.b) int Eq. (1) gives : Or : mw U U U F mw F F 1 b (4) Which b represents the rati f the applied frced freqency t the natral freqency f vibratin f the system : w b (5) w n

7 Cmbining Eq. (3.a & b) and (4) with Eq. () yields : t Acs With initial cnditins : F t B sinwnt 1 b wn sinwt 0 0 (6) t 0 cs t wn 0 F b F sinwnt 1 b 1 b w n sinwt (7) Transient Respnse Steady State Respnse

8 3 t F / ,5 1 1,5,5 3 3, Ttal Respnse Steady State Respnse Transient Respnse 0 n 0 and w 0, 0 w 0 w F n /

9 Steady state respnse present becase f the applied frce, n matter what the initial cnditins. Transient respnse depends n the initial displacement and velcity. Transient respnse exists even if In which Eq. (7) specializes t F 1 b t sinwt b sinw t It can be seen that when the frcing freqency is eqal t natral freqency (b = 1), the amplitde f the mtin becmes infinitely large. A system acted pn by an external excitatin f freqency cinciding with the natral freqency is said t be at resnance. n (8)

10 If w = w n (b = 1), the sltin f Eq. (1) becmes : F t w t csw t sinw t 1 n n n (9) t F / ,5 1 1,5,5 3 3,5 p

11 Assignment 3 If system in the figre have initial cnditin 0 3 cm & 0 0 cm/s And sbjected t harmnic lading p(t) = 5000 sin (wt) Plt a time histry f displacement respnse W = f,5 the tns system, fr t = 0 s ntil t = 5 s, if ξ = 0%, and : b = 0,1 ; 0,5 ; 0,60 ; 0,90; 1,00; 1,5 & 1,75 40x40 cm EI 3 m E= MPa (b)

12 Damped SDF Harmnic Lading Inclding viscs damping the differential eqatin gverning the respnse f SDF systems t harmnic lading is : m c F sinwt (10) c (t) f D (t) = cú (t) m p(t) f S (t) = f I (t) = mü F sin wt

13 The cmplementary sltin f Eq. (9) is : c wnt t e Acsw t B sinw t The particlar sltin f Eq. (9) is : p t C sinwt D D cswt D (11) (1) Where : C F 1 b 1 b b (13.a) D F b 1 b b (13.b)

14 The cmplete sltin f Eq. (9) is : t D cs t C sin t B sin t Acs e t D D t c n w w w w w (14) Transient Respnse Steady State Respnse , F B F A b b b b b b b

15 Respnse f damped system t harmnic frce with b = 0,, = 0,05, (0) = 0, ú(0) = w n F /

16 The ttal respnse is shwn by the slid line and the steady state respnse by the dashed line. The difference between the tw is the transient respnse, which decays expnentially with time at a rate depending n b and. After awhile, essentially the frced respnse remains, and called steady state respnse The largest defrmatin pea may ccr befre the system has reached steady state.

17 If w = w n (b = 1), the sltin f Eq. (10) becmes : t cs t sin t cs e F t n D D t n w w w w 1 1 (15)

18 Cnsidering nly the steady state respnse, Eq. (1) & Eq. (13.a, b), can be rewritten as : Where : t U sinwt U F 1 b b (16) b tan 1 b (17) Rati f the steady state amplitde, U t the static deflectin st (=F /) is nwn as the dynamic magnificatin factr, D : U 1 D st (18) 1 b b

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20 Exercise The steel frame in the figre spprts a rtating machine that exerts a hrizntal frce at the girder level p(t) = 100 sin 4 t g. Assming 5% f critical damping, determine : (a) the steady-state amplitde f vibratin and (b) the maximm dynamic stress in the clmns. Assme the girder is rigid W = 6,8 tns EI WF I = 4,050 cm 4 E = MPa 4,5 m (b)

21 Assignment 4 If system in the figre have initial cnditin 0 3 cm & 0 0 cm/s And sbjected t harmnic lading p(t) = 5000 sin (wt) Plt a time histry f displacement respnse f the system, fr t = 0 s ntil t = 5 s, if : ξ = 5% W =,5 tns EI b = 0,1 ; 0,5 ; 0,60 ; 0,90; 1,00; 1,5 & 1,75 40x40 cm 3 m E= MPa (b)