Structural Vibration
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1 Fdameals of Srcral Vibraio Seaer: Prof. FUNG a Chig Dae & ime: Wed Ags 4, :3-5:3 m Vee: CEE Semiar Room D (N-B4C-9B School of Civil ad Eviromeal Egieerig Nayag echological Uiversiy oics i Fdameals of Srcral Vibraio (.5 hrs SDoF Sysems Dyamic Eqilibrim Naral Freq/Period Damig raio Phase lag Dis Res Facor Resose Secrm MDoF Sysems Mode Shaes Modal decomosiio Modal resoses
2 Corse Olie for CV63 Lecre Corse Coe Chaers i ( hors exboo Sigle-Degree-of-Freedom Sysems Eqaios of moio Free vibraio..6.. Sigle-Degree-of-Freedom Sysems Resose o harmoic ad eriodic exciaios Sigle-Degree-of-Freedom g Sysems Resose o arbirary, se ad lse exciaios Mli-Degree-of-Freedom Sysems Eqaios of moio Naral vibraio freqecies ad modes..7 5 Mli-Degree-of-Freedom Sysems Free vibraio resose Mli-Degree-of-Freedom Sysems Forced vibraio resose..7 7 Sysems wih Geeralized Degrees of Freedom Geeralized coordiaes ad heir alicaios 4.3, 7. 3 exboo ad Refereces Mai ex Chora, A. K., Dyamics of Srcres: heory ad Alicaios o Earhqae Egieerig, Preice Hall, 4 rd Ediio,. Refereces Clogh, R. W., ad Pezie, J., Dyamics of Srcres, McGraw-Hill, 993. Meirovich, L. Fdameals of Vibraios, McGraw-Hill,. 4
3 Why Is here A Need o Do Dyamic Aalysis? Saic aalysis Exeral Load Ieral Force Magide of loadig & siffess Dyamic aalysis Exeral Load Ieral Force Magide of loadig & siffess Freqecy characerisics of loadig, ad he dyamic roeries of srcres (mass, siffess, damig 5 Examles of SDOF Sysems Waer a Mass coceraed a oe locaio Sors assmed o be massless Ca be modeled d as a SDOF sysem Pedlm Rod is assmed o be massless Oly allowed o roae abo hige Ca be modelled as a SDOF sysem 6
4 Eqaio of Moio Damig force Resorig force m&& + fd + fs Exeral force Newo s Secod Law of Moio: f S f D m& D Alember s Pricile (Dyamic eqilibrim: f S f D f I wih f Ieria force I m& 7 Sigle-degree-of-freedom (SDoF Sysems yical rereseaio: Mass-srig-damer sysem c m (mass ( ( Assmios: Liear elasic resorig force Liear viscos damig m & + c& + ( Exeral force ( Resorig force d Damig force c d d Ieria force m d & 8
5 Udamed Free Vibraio Eqaio of Moio: m && + c& + ( wih c, ( m & + or && + m Iiial codiios: (, & & ( Exac Solio: ( ( cos + or ( max cos max ( &( ( si ( θ ( & ( + θ a See Page 46 i Chora & ( ( 9 Free Vibraio of a Sysem wiho Damig max max max
6 Periods of Vibraio of Commo Srcres Commo Srcres -sory mome resisig frame -sory mome resisig frame -sory mome resisig frame -sory braced frame -sory braced frame -sory braced frame Graviy dam Ssesio bridge Period.9 sec. sec.5 sec.3 sec.8 sec. sec. sec sec Viscosly Damed Free Vibraio Eqaio of Moio: m && + c& + && c & Divided by m: + + m m c c Le ζ (reasos will be clear laer m zea Hece Noe: c cr && + ζ & + ccr m m criical damig raio m as before
7 ye of Moio hree scearios ζ :damig raio c cr : criical damig coefficie ζ <, i.e. c < c cr der-damed (oscillaig ζ, i.e. c c cr criically damed ζ >, i.e. c > c cr over-damed 3 yical Damig Raio Srcre ζ Welded seel frame. Boled seel frame. Ucraced resressed cocree.5 Ucraced reiforced cocree. Craced reiforced cocree.3535 Gled lywood y shear wall. Nailed lywood shear wall.5 Damaged seel srcre.5 Damaged cocree srcre.75 Srcre wih added damig.5 4
8 Effecs of Damig i Free Vibraio &( + ζ ζ( ( e ( cosd + sid D D ζ ρ max D ζ 5 Decay of Moio Oe way o measre damig is from rae of decay from free vibraio D cos( θ D ζ ( e max D (exacly Sice eas are searaed by D, πζ ( ex( ζ ex D ζ i e ζ + D i + e ζ 6
9 ye of Exciaios Harmoic / Periodic Exciaios Commoly ecoered i egieerig Ubalaced roaig machiery Wave loadig Seady-sae Resoses Basic comoes i more geeral eriodic exciaios Forier series rereseaio More Geeral Exciaios rasie Se/Ram Forces Plses Exciaios Resoses 7 Eqaio of Moio m & + c& + ( Resoace Liear ( ca be relaced by ( + ay free vibraio resoses For examle, c, ( si (-m + si ( ( si Slowly loaded / > Raidly loaded m / ( 8
10 Exciaio si Resose. ( & ( ( ( / 9 si Udamed Resoa Sysems For resoace, ( cos si Derivaio: See Page 7 & 7 i Chora ( Resose grows idefiiely Becomes ifiie afer ifiie draio (, & (
11 Harmoic Vibraio of Viscos Damig Eqaio of Moio Sisoidal force && & m + c + si m Pariclar Solio ( C si + D cos Exciaio freqecy Amlide of force Derivaio: See Page 73 i Chora ( / C [ ( / ] + [ ζ / ] ζ / D [ ( / ] [ / ] + ζ Seady-Sae Solio he ariclar solio ca also be wrie as: ( si ( φ D where C + D φ a C Usig reviosly derived resls for C ad D, max φ a ζ ( / ( / [ ( ] / + ζ ( / [ ]
12 ζ. Saic resose exacly i-hase wih force Dyamic resose has a ime lag, φ/ 3 Geeral Solio Comlemeary solio is he free damed vibraio resose: c ζ ( A cos B si ( e + D D D ζ Comlee solio: ( ( ( e ζ c + Recall: A, B derived by saisfyig iiial codiios ( Acos + Bsi + C si + D cos D D rasie Seady sae Derivaio: See Page 73 i Chora 4
13 Examle /., ζ.5, (, & / ( Observe how he rasie resose decays de o damig, leavig oly he seady sae ar 5 Examle (resoa resose, ζ.5 Wih damig, resose aroach max B ca sill be larger for aoher vale of! s ζ 6
14 Examle 3 7 Sigificace of Seady-sae Solios I cerai roblems, e.g. wave loads o a offshore srcre, he load is assmed o be i lace for a sfficiely log ime, so ha he rasie resose has comleely decayed. he ieres is i he seady-sae solio. 8
15 Maximm Resose ad Phase Agle Seady sae solio is φ a ( ( si ( φ [ ( ] / + [ ζ ( / ] ( / ( / ζ φ is he hase lag ime lag φ/ Called Deformaio Resose Facor (DRF R d i Chora exboo (Page 76 9 DRF DRF ad Phase Resoace For / << Slowly varyig DRF / Dislaceme i-hase wih force Resose domiaed by siffess Phase φ Slowly loaded Raidly For / >> loaded Raidly varyig DRF m Dislaceme ai-hase wih force Resose domiaed by mass 3
16 Whe Forcig freq aral freq Resoace DAF is very large, close o max a / ζ c ( max Resose domiaed by damig / ζ ζ Dislaceme is 9 o of hase wih force whe his is he sceario we wa o avoid! (b o always ossible 3 Periodic Exciaio ( j ( π / + j : ieger i (-, Searae io harmoic comoes sig Forier series ( j j j j j a + a cos( j + b si( j Noe: Arbirary exciaios ca also be rasformed io Forier series wih aroriae echiqe, sch as FF. 3
17 Resose o Arbirary ime- Varyig Forces Eqaio of Moio m && + c& + ( Iiial codiios: (, & ( (: varyig arbirarily wih ime e.g. se forces (wih fiie rise ime, lses ec. Ieresed i he max resose. Max resose Resose/Shoc secrm 33 Simle Examles Se Force Ram or liearly l icreasig force Se force wih fiie rise ime ( ( r r ( / r r ( r r 34
18 Dyamic Resose o Se Forces + e D D ζ ζ ζ si cos ( ζ 35 Dyamic Resose of Ram or Liearly Icreasig Force si r r r si ( ( s ζ.5 r 36
19 Dyamic Resose o Se Force Wih Fiie Rise ime ( ( / r r r r Cosider damed resose: Phase Phase si. Ram hase: ( r r r r ( si si > r. Cosa hase: [ ( ] r 37 Se Force Wih Fiie Rise ime 38
20 Maximm Deformaio R d + si(ππ / π r r / Resose Secrm 39 Sigle Plse Exciaios Examle of lse exciaios: dergrod exlosios Idealized by simle shaes E.g. Force Blas overressre ime 4
21 Resose Secrm R d siπd / d < d / s, / Also called Shoc Secrm for sigle lse 4 Mli-Degree-of-Freedom Sysems ( 4 ( ( 3 ( ( c c c c ( 4 ( 3 ( ( 4
22 Geeral Aroach for Comlex Srcres Elasic resisig forces Same as saic aalysis (i.e. f S Damig forces: sally raher simle f D c& Ieria forces: sally simle f I m & Eqaios of moio m && ( + c & ( + ( ( 43 Arbirary ( NO Simle Harmoic Moio (SHM Freqecy of moio cao be defied ad are o roorioal ( defleced shaes varies wih ime Modal coordiaors 44
23 Whe ( φ ca be SHM wih aroriae iiial codiios roorioal o φ is a aral mode 45 Whe ( φ ca be SHM wih aroriae iiial codiios roorioal o φ is a aral mode 46
24 Naral Freqecy he aral eriod of vibraio he ime reqired for oe cycle of he harmoic moio i oe of hese aral modes. f π f aral cyclic freqecy of vibraio aral circlar freqecy of vibraio A N-DOF sysem has N mber of aral eriods ad N mber of aral modes. 47 How o Fid he aral eriods ad aral modes? EoM: m && ( + c& ( + ( ( Naral freqecy ad mode shae φ ca be obaied by solvig he followig eigevale roblem [ m] φ Differe modes ca be show o be orhogoal wr he m ad marices, i.e. φ mφ φ φ ( r Μ r φ m φ, K φ r φ 48
25 Vibraio Aalysis of MDOF Sysems EoM: m && ( + c& ( + ( ( ime Seig Mehods wih ( ad& ( E.g. Ceral Differece, Newmar s mehod ec. Modal Decomosiio MDoF roblem a mber of SDoF roblems φ q + φ q + L + q φn N q N φ r q r [ φ Lφ ] M r q N Φqq q q Φ φ m φ m m φ 49 Ucoled Eqaios EoM: m && ( + c& ( + ( ( wih ( ad& ( rasformed io N SDoF Sysems, each M q&& ( + C q& ( + Kq( P ( Μ q&& P ( M classical damig φ mφ, K φ φ, P φ ( Cr φr cφ Wih iiial codiios ( + ζ q& ( + q ( K Μ q, φ m ( φ mφ ( q& φ, φ m &( mφ 5
26 Dislaceme Resoses Oce q (,, q N ( are deermied, he resose (,, N ( i( ca be obaied from N ( ( φ q ( ad sbseqely he ieral forces ca also be calclaed if reqired Caio: he exressio cold be very leghy. 5 Modal Coribio I is sefl o defie he coribio of he h mode o ( as ( q ( φ he ieral force de o ( ca be evalaed firs ad he sm for all he modes laer. Frher Imroveme Sice q ( is a scalar fcio, he ieral force de o φ ca be evalaed firs (saic aalysis ad he imes q ( before sm for all he modes laer. 5
27 How o Calclae he Ieral Forces? Direcly from ( or ( or φ Aleraively, he same ieral forces ca be obaied by cosiderig he same srcre sbjeced o he eqivale saic forces ( or ( or φ 53 Eqivale Saic Force m && ( + c& ( + ( ( ( m& ( c& ( f ( ( S 5 ( m 5 m 5& 5( f S5 ( 5 ( 4 ( m 4 5 m 4& 4( f S4 ( 5 4 ( 3 ( m 3 4 m 3& 3( f S3 ( 4 3 ( ( m 3 m && ( ( f S ( 3 ( ( m m & ( ( f S ( ( V( V( 54
28 f ( ( φ q ( mφ q ( Forces : φ or m φ r s P ( M K, ζ q ( r r s ( r q ( Forces: N N φ or mφ N r N s P N ( M N K N, ζ N q N ( r N ( r N s q N ( 55 Reca SDOF Sysems Dyamic Eqilibrim Naral Freq/Period Damig raio Phase lag DAF/DRF Resose Secrm MDOF Sysems Mode Shaes Modal decomosiio Modal resoses 56
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