CENTROIDS AND MOMENTS OF INERTIA
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1 CENTODS AND MOMENTS OF NETA
2 CENTODS (AĞLK MEKEZLEİ) A centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.
3 CENTODS (AĞLK MEKEZLEİ) f proper geometrcl odes possess n s of smmetr, te centrod wll le on ts s. f te od possesses two or tree smmetr es, ten te centrod wll e locted t te ntersecton of tese es. f one, two or tree dmensonl odes re defned s nltcl functons, te loctons of ter centrods cn e clculted usng ntegrls.
4 A composte od s one wc s comprsed of te comnton of severl smple odes. n suc odes, te centrod s clculted s follows:
5 Lne tn rod (Çg nce çuuk) Are - flt plte wt constnt tckness (Aln st klınlıklı dü plk) Volume spere or cone (Hcm küre d kon) Composte Composte Composte dl dl l l A A dv dv V V dl dl l l A A dv dv V V dl dl l l A A dv dv V V
6 CENTODS OF SOME GEOMETC SHAPES
7 CENTODS OF SOME GEOMETC SHAPES Tn od n te Spe of Qurter Crcle Çerek Çemer Şeklnde İnce Çuuk dl dl dl dl el r = rdus dl dl dl dl r d el el r cos r sn r cosrd rd r sn r d r r el dl dl r sn rd rd r cos r r
8 Crculr Arc İç Dolu Dre Yı r=rdus r sn G Sold Qurter Crcle İç Dolu Çerek Dre A dd d d =sn =cos ρsn θρdρdθ ρ π - cosθ - - π π ρ sn θdρdθ ρ π π π sn θdθ
9 Sold Hlf Crcle Trngle Üçgen G İç Dolu Yrım Dre d d A cos cos cos d d d d d d w d w? d wd w w d ) ( ) ( w
10 Prol d d w=- Prol 6 d d A 8 d d ( d d wd
11 Are Between Lne nd Curve Br Doğru le Br Eğr Arsınd Kln Aln d d = left w rgt ottom top l (, ) 6 A d d d ld ) ( ) ( ottom top,,,, 6, )d ( d left rgt , )d ( ) ( ) (
12 AEA MOMENT OF NETA ALAN ATALET (EYLEMSİZLİK) MOMENTİ t s often necessr to clculte te moments of unforml dstruted lods out n s lng wtn te plne te re ppled to or perpendculr to ts plne. Generll, te mgntudes of tese forces per unt re (nmed s pressure or stress) re drectl proportonl wt te dstnce of ter lnes of cton from te moment s. Ts w, n elementr force ctng n n elementr re wll e proportonl to, dstnce dfferentl re d df
13 AEA MOMENT OF NETA ALAN ATALET (EYLEMSİZLİK) MOMENTİ dp d, dp df, df d, df d Elementr moment s proportonl to dstnce dfferentl re: dm=d Tus, te totl moment: d df dm d, M d Ts ntegrl s nmed s re moment of nert or second moment of re.
14 AEA MOMENT OF NETA ALAN ATALET (EYLEMSİZLİK) MOMENTİ Moment of nert s not pscl quntt suc s veloct, ccelerton or force, ut t enles ese of clculton; t s functon of te geometr of te re. Snce n Dnmcs tere s no suc concept s te nert of n re, te moment of nert s no pscl menng. But n mecncs, moment of nert s used n te clculton of endng of r, torson of sft nd determnton of te stresses n n cross secton of mcne element or n engneerng structure.
15 ectngulr (Crtesn) nd Polr Are Moments of nert nd Product of nert (Krteen ve Kutupsl Aln Atlet Momentler ve Çrpım Aln Atlet Moment) B defnton, te moments of nert of re wt respect to nd es re d d Te moments of nert of te totl re A wt respect to nd es re Te moment of nert of re A wt respect to s Te moment of nert of re A wt respect to s Tese moments of nert re nmed s ectngulr (Crtesn) moments of nert.
16 Te moment of nert of re wt respect to s or pole O s defnton d or d or J r O Te moment of nert of totl re A wt respect to s or pole O s r Te moment of nert of re A wt respect to s Snce te s s perpendculr to te plne of te re nd cuts te plne t pole O, te moment of nert s nmed polr moment of nert. r Terefore,
17 n certn prolems nvolvng unsmmetrcl cross sectons nd n te clculton of moments of nert out rotted es, n epresson d occurs, wc s te ntegrted form were nd re te coordntes of te element of re. s nmed s te product of nert of te re A wt respect to te es.
18 Propertes of moments of nert: ) Are moments of nert,, re lws postve (+). ) cn e postve (+), negtve (-) or ero wenever eter of te reference es s n s of smmetr, suc s te s n te fgure. ) Te unt for ll re moments of nert s te. power of tt tken for lengt (L ).
19 Propertes of Moments of nert: ) Te smllest vlue of n re moment of nert tt n re cn ve s reled wt respect to n s tt psses from te centrod of ts re. Te re moment of nert of n re ncrees s te re goes furter from ts s. Te re moment of nert wll get smller wen te dstruton of n re gets closer to te s s possle. A A A A A=6 L =. L =8 L = L =75 L
20 dus of Grton Jrson (Atlet Elemslk) Yrıçpı Consder n re A, wc s rectngulr moments of nert nd nd polr moment of nert out O. We now vsule ts re s concentrted nto long nrrow strp of re A dstnce k from te s. B defnton, te moment of nert of te strp out te s wll e te sme s tt of te orgnl re f k A Te dstnce k s clled te rdus of grton of te re out te s.
21 A smlr relton for te s s wrtten consderng te re s concentrted nto nrrow strp prllel to te s s seen n te fgure. Also, f we vsule te re s concentrted nto nrrow rng of rdus k, we m epress te polr moment of nert s. n summr, A k A k A k A k A k A k A k Also snce, k k k
22 Trnsfer of Aes Prlel Eksenler (Stener) Teorem Te moment of nert of n re out noncentrodl s m e esl epressed n terms of te moment of nert out prllel centrodl s. n te fgure, es pss troug te centrod G of te re. Let us now determne te moments of nert of te re out te prllel es. B defnton, te moment of nert of te element out te s s d Epndng to te wole re d O O d O d
23 We see tt te frst ntegrl s defnton, te moment of nert out te centrodl s. Te second ntegrl s ero, snce A O O nd wt te centrod on te term s smpl Ad O O s utomtcll ero s. Te trd. Tus, te epresson for nd te smlr epresson for ecome d O d Ad Ae Te sum of tese two equtons gve Ar nd snce,
24 For product of nert Ade Te prllel es teorems lso old for rd of grton s: k k r were k s te rdus of grton out centrodl s prllel to te s out wc k pples nd r s te perpendculr dstnce etween te two es. For product of nert: k k de
25 Two ponts tt sould e noted n prtculr out te trnsfer of es re: Te two trnsfer es must e prllel to ec oter One of te es must pss troug te centrod of te re f trnsfer s desred etween two prllel es neter of wc psses troug te centrod, t s frst necessr to trnsfer from one s to te prllel centrodl s nd ten to trnsfer from te centrodl s to te second s. So, te prllel s teorem must e used twce.
26 otton of Aes, Prncple Aes of nert, Prncple Moments of nert Eksenlern Döndürülmes, Asl Atlet Eksenler, Asl Atlet Momentler n Mecncs t s often necessr to clculte te moments of nert out rotted es. Te product of nert s useful wen we need to clculte te moment of nert of n re out nclned es. Ts consderton leds drectl to te mportnt prolem of determnng te es out wc te moment of nert s mmum nd mnmum.
27 From te fgure, te moments of nert of te re out te ' nd ' es re: cos sn sn cos Epndng nd susttutng te trgonometrc denttes: cos sn cos cos cos sn cos cos
28 Defnng te reltons n terms of,, gve cos cos sn sn n smlr mnner we wrte te product of nert out te nclnes es s: cos sn sn cos Epndng nd susttutng te trgonometrc denttes: cos cos sn nd sn sn cos, sn cos sn
29 Defnng te reltons for,, gve sn cos Addng, gves Te ngle wc mkes nd eter mmum or mnmum m e determned settng te dervtve of eter or wt respect to equl to ero. Tus, d sn cos d
30 Denotng ts crtcl ngle gves tn Te equton gves two vlues for wc dffer, snce tn = tn ( + ). Consequentl, te two solutons for wll dffer. One vlue defnes te s of mmum moment of nert nd te oter vlue defnes te s of mnmum moment of nert. Tese two rectngulr es re clled prncple es of nert.
31 Wen te crtcl vlue of s susttuted nto te equtons, t s seen tt te product of nert s ero for te prncple es of nert. Susttuton of sn nd cos nto te equtons, gves te epressons for te prncple moments of nert s: m mn
32 Are Moments of nert of Bsc Geometrc Spes
33 Are Moments of nert of Bsc Geometrc Spes ) ectngle d d d d d d Ad d Ad Ae e Ae
34 Are Moments of nert of Bsc Geometrc Spes ) Squre G Snce ==
35 ) Trngle G From smlr trngles n - n d d m n Are Moments of nert of Bsc Geometrc Spes nd d Ad 6 nd Ae 6
36 Are Moments of nert of Bsc Geometrc Spes ) Crcle Due to smmetr π π r π dr r π πrdr r πrdr r o π π π π π Ad 5 5
37 G, O Are Moments of nert of Bsc Geometrc Spes 5) Semcrcle 6) Qurter crcle , 8 Ad 8 6 G
38 Applctons of Are Moment of nert
39 d d w=- * Determne te re moments of nert of te re under te curve wt respect to nd es.?? d d d - - d
40 * Determne te re moments of nert of te re etween curve nd lne wt respect to nd es. d d = left w rgt ottom top l (, ) d - - d - d -,
41 * Determne te product of nerts of te followng res. d d G? dd Ade, dd - Ade - Te product of nert s ero wenever eter of te centrodl es s n s of smmetr. e G d Ade e G d Ade
42 d d - - d d d de d d d d, for ts re wrtng n terms of, G * Determne te product of nert of te trngulr re. d -?? nd Consderng onl te dfferentl rectngulr re For te trngle Ade
43 Products of nert for vrous confgurtons G G 8 7
44 Products of nert for vrous confgurtons =sn =cos d d?? cos sn sn cos sn cos sn sn sn cos d θdθ θdθ θ ρ θ θ θ θρdθdρ θρ ρ ρdθdρ, π π π π - π π π - Ade - - θ π cos 8
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles
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