3 1 Rgd odes: Equvalent Sstems of oces 기계공학부최해진 School of echancal Engneeng
Contents 3 2 Intoducton Etenal and Intenal oces ncple of Tansmssblt: Equvalent oces Vecto oducts of Two Vectos oment of a oce bout a ont Vagon s Theoem Rectangula Components of the oment of a oce Sample oblem 3.1 Scala oduct of Two Vectos Scala oduct of Two Vectos: pplcatons ed Tple oduct of Thee Vectos oment of a oce bout a Gven s Sample oblem 3.5 oment of a Couple ddton of Couples Couples Can e Repesented Vectos Resoluton of a oce Into a oce at O and a Couple Sample oblem 3.6 Sstem of oces: Reducton to a oce and a Couple uthe Reducton of a Sstem of oces Sample oblem 3.8 Sample oblem 3.10 School of echancal Engneeng
3.1 Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton of the foces must be consdeed. 3 3 ost bodes n elementa mechancs ae assumed to be gd,.e., the actual defomatons ae small and do not affect the condtons of equlbum o moton of the bod. Cuent chapte descbes the effect of foces eeted on a gd bod and how to eplace a gven sstem of foces wth a smple equvalent sstem. moment of a foce about a pont moment of a foce about an as moment due to a couple n sstem of foces actng on a gd bod can be eplaced b an equvalent sstem consstng of one foce actng at a gven pont and one couple. School of echancal Engneeng
ncple of Tansmssblt: Equvalent oces 3 4 ncple of Tansmssblt Condtons of equlbum o moton ae not affected b tansmttng a foce along ts lne of acton. NOTE: and ae equvalent foces. ovng the pont of applcaton of the foce to the ea bumpe does not affect the moton o the othe foces actng on the tuc. ncple of tansmssblt ma not alwas appl n detemnng ntenal foces and defomatons. 내력, 변형을결정할때는주의! (a) : 인장상태, (d) : 압축상태 School of echancal Engneeng
Vecto oduct of Two Vectos Concept of the moment of a foce about a pont s moe easl undestood though applcatons of the vecto poduct o coss poduct. 3 5 Vecto poduct of two vectos and Q s defned as the vecto V whch satsfes the followng condtons: 1. Lne of acton of V s pependcula to plane contanng and Q. 2. agntude of V s V Q snq 3. Decton of V s obtaned fom the ghthand ule. Vecto poducts: ae not commutatve( 비가역성, 교환법칙성립안됨 ), ae dstbutve( 분배법칙성립 ), ae not assocatve( 비결합법칙 ), Q ( Q) ( Q1 Q2 ) Q1 Q2 ( Q) S ¹ ( Q S) School of echancal Engneeng
Vecto oducts: Rectangula Components Vecto poducts of Catesan unt vectos, 0 0 0 v Vecto poducts n tems of ectangula coodnates ( ) ( ) Q Q Q V 3 6 School of echancal Engneeng ( ) ( ) Q Q Q Q Q Q Q Q Q V ) ( ) ( ) ( Q Q Q
oment of a oce bout a ont 힘의모멘트 (moment) o 토크 (toque) 3 7 정의 강체에가해지는힘은그물체를병진운동을하게할수도있으며회전운동을하게할수도있다. 물체를회전시키는힘의영향즉한점에대한힘의모멘트 (moment of a foce about a pont) 를소개한다 축에대한힘의모멘트 (moment of a foce about an as) 그림과같이임의의힘 와 의작용선상에있지않은임의의점 O를생각해보자 O 를중심으로하는힘 의모멘트는 O 기하학적해석 모멘트의크기에대한스칼라값계산 o snq o 수직거리 d 는모멘트팔 (moment am) 이라고부르며, 모멘트크기는 snq d 크기 : 0 d School of echancal Engneeng
oment of a oce bout a ont 3 8 O 의크기는힘의크기와수직거리 d 에의존하므로이힘 ( 미끄럼벡터 ) 은모멘트의변화없이 작용선을따라어디든지옮겨놓을수있다. School of echancal Engneeng
oment of a oce bout a ont Q d sldng vecto a 3 9 lne of acton O sn α d d (N m) d : moment am ( 모멘트 팔) School of echancal Engneeng
oment of a oce bout a ont 3 10 School of echancal Engneeng
Vagnon s Theoem 3 11 The moment about a gven pont O of the esultant of seveal concuent foces s equal to the sum of the moments of the vaous moments about the same pont O. ( L) L 1 2 1 2 Æ 임의의점에대한힘의모멘트는그점에대한힘의분력의모멘트의합과같다. : Vagnon 이론 Vagon s Theoem maes t possble to eplace the dect detemnaton of the moment of a foce b the moments of two o moe component foces of. School of echancal Engneeng
Vagnon s Theoem 3 12 Æ 임의의점에대한힘의모멘트는그점에대한힘의분력의모멘트의합과같다. : Vagnon 이론 그림 (a) 와같이힘 R 에의한모멘트를생각해보자. o R R Q o ( Q) Q 그림 (b) 와같이 d 대신에 p 나 q 를쉽게구할경우에힘 R 대신에힘, Q 을사용하는것이편리함. o R d p q Q School of echancal Engneeng
Rectangula Components of the oment of a oce The moment of about O, O, 3 13 School of echancal Engneeng ( ) ( ) ( ) O
Rectangula Components of the oment of a oce The moment of about, / ( ) ( ) ( ) 기준 점을 : / 3 14 School of echancal Engneeng ( ) ( ) ( )
Rectangula Components of the oment of a oce o twodmensonal stuctues, ( ) Z O O 3 15 School of echancal Engneeng ( ) ( ) ( ) ( ) Z O O ] [
Sample oblem 3.1 3 16 100N vetcal foce s appled to the end of a leve whch s attached to a shaft at O. 24 m Detemne: a) moment about O, b) hoontal foce at whch ceates the same moment, c) smallest foce at whch poduces the same moment, d) locaton fo a 240N vetcal foce to poduce the same moment, e) whethe an of the foces fom b, c, and d s equvalent to the ognal foce. School of echancal Engneeng
Sample oblem 3.1 3 17 a) oment about O s equal to the poduct of the foce and the pependcula dstance between the lne of acton of the foce and O. Snce the foce tends to otate the leve clocwse, the moment vecto s nto the plane of the pape. O d O d ( 24m) cos60 ( 100 N)( 12 m) 12 m O 1200 N m School of echancal Engneeng
Sample oblem 3.1 3 18 c) Hoontal foce at that poduces the same moment, 24 m d O 1200 N m ( 24 m) d sn 60 ( 20.8 m) 1200 N m 20.8 m 20.8 m 57.7 N School of echancal Engneeng
Sample oblem 3.1 3 19 24 m c) The smallest foce to poduce the same moment occus when the pependcula dstance s a mamum o when s pependcula to O. O d 1200 N m ( 24 m) 1200 N m 24 m 50 N School of echancal Engneeng
Sample oblem 3.1 3 20 d) To detemne the pont of applcaton of a 240 N foce to poduce the same moment, O d ( ) 1200 N m 240 N 1200 N m d 5 m 240 N O cos60 5 m d O 10 m School of echancal Engneeng
Sample oblem 3.1 3 21 e) lthough each of the foces n pats b), c), and d) poduces the same moment as the 100 N foce, none ae of the same magntude and sense, o on the same lne of acton. None of the foces s equvalent to the 100 N foce. School of echancal Engneeng
Sample oblem 3.4 3 22 SOLUTION: The moment of the foce eeted b the we s obtaned b evaluatng the vecto poduct, C The ectangula plate s suppoted b the bacets at and and b a we CD. Knowng that the tenson n the we s 200 N, detemne the moment about of the foce eeted b the we at C. School of echancal Engneeng
Sample oblem 3.4 SOLUTION: C C C C D l ( 200 N) C D 0.3 m 200 N v ( ) D (0, 0.24, 0.08), C (0.3, 0, 0.4) (0.3 0.24 0.32 ) C / D 0.5 C / D ( 0.3 m) ( 0.08 m) ( ) ( 0.24 m ) ( 0.32 m)) ( 120 N) ( ) ( 128 N) 96 N 0.3 0 0.08 120 96 128 0.5 m 3 23 ( 7.68 N m) ( 28.8 N m) ( 28.8 N m) School of echancal Engneeng
Scala oduct of Two Vectos 3 24 The scala poduct o dot poduct between two vectos and Q s defned as Q Q cosq ( scala esult) Scala poducts: ae commutatve, ( 교환법칙성립 ), Q Q ae dstbutve, ( 분배법칙성립 ), Q Q2 Q ae not assocatve, ( 결합법칙무의미 ), Q S ( 1 ) 1 Q2 ( ) undefned Scala poducts wth Catesan unt components, Q Q Q Q ( ) ( ) v 1 1 1 0 0 Q Q Q 2 2 2 Q 2 0 School of echancal Engneeng
Scala oduct of Two Vectos: pplcatons ngle between two vectos: Q Q cosq Q Q cosq Q Q Q Q Q 3 25 oecton of a vecto on a gven as: 주어진축에벡터의투영 OL cosq poecton of along OL Q Q cos q Q cosq OL Q o an as defned b a unt vecto: 축의단위벡터에대한벡터의투영 l cos q cosq cosq OL cos q cosq ( ) ( ) cosq School of echancal Engneeng
ed Tple oduct of Thee Vectos ed tple poduct of thee vectos, ( ) esult scala Q S The s med tple poducts fomed fom S,, and Q have equal magntudes but not the same sgn, ( ) ( ) ( ) 스칼라삼중곱 (scala tple poduct) 은두벡터의외적을다른세번째벡터와내적을한것이다. 이는, Q, S 를변으로하는평행육면체의부피와같다. 3 26 School of echancal Engneeng ( ) ( ) ( ) ( ) ( ) ( ) S Q Q S Q S S Q S Q Q S ( ) ( ) ( ) ( ) Q Q Q S S S Q Q S Q Q S Q Q S Q S Evaluatng the med tple poduct, ( ) 부피평행육면체의 : Q S
oment of a oce bout a Gven s oment O of a foce appled at the pont about a pont O, O Scala moment OL about an as OL s the poecton of the moment vecto O onto the as, 주어진축 OL에대한힘 의모멘트크기 OL : 축 OL 에관한모멘트 O 의투영 OC 이다. OL l l O ( ) oments of about the coodnate aes, OL l o λ ( ) l l l l l l 3 27 l,,,, : 힘 가강체를, 및 축방향으로작용하는힘의크기 : 힘 가강체를, 및 축대하여회전시키려는모멘트크기 School of echancal Engneeng
oment of a oce bout a Gven s 3 28 oment of a foce about an abta as, : 임의의주어진축에힘의모멘트 L l l ( ) The esult s ndependent of the pont along the gven as. L / / / C 점을기준으로구하여도동일한결과로유도됨. l l l School of echancal Engneeng
Sample oblem 3.5 3 29 cube s acted on b a foce as shown. Detemne the moment of a) about b) about the edge and c) about the dagonal G of the cube. d) Detemne the pependcula dstance between G and C. School of echancal Engneeng
Sample oblem 3.5 oment of about? ) ( 2 ) 2 2 ( ) ( 2 2 a a a a a 3 30 School of echancal Engneeng 2 2 2 ( )( ) a 2 / oment of about? ( ) ( ) a 2 / 2 a
Sample oblem 3.5 oment of about the dagonal G? ( ) ( ) ( ) ( ) 1 2 3 1 3 a a a a a a G G G G l l 3 31 School of echancal Engneeng ( ) ( ) ( ) 1 1 1 6 2 3 a G 6 a G 6 / 2 / 2 / 0 0 3 1/ 3 1/ 3 1/ ) ( / / / a a a G l l l 별해 ), (0, 0), 0,, ( a a a G ( / ) G l l
Sample oblem 3.5 3 32 l 방향 힘 방향 ependcula dstance d between G and C? 1 l 0 1 1 2 3 6 0 ( ) ( ) ( ) Theefoe, s pependcula to G. a G 6 d d a 6 School of echancal Engneeng
oment of a Couple Two foces and havng the same magntude, paallel lnes of acton, and opposte sense ae sad to fom a couple. : 크기가같고작용선이평행하며방향이반대인두힘은우력 (couple) 을구성함. 합력은제로이지만회전시키려는모멘트가존재함. 3 33 oment of the couple, o ( ) o snq d ( ) : 우력의모멘트 The moment vecto of the couple s ndependent of the choce of the ogn of the coodnate aes,.e., t s a fee vecto that can be appled at an pont wth the same effect. School of echancal Engneeng
oment of a Couple 3 34 Two couples wll have equal moments f 1d1 2d 2 the two couples le n paallel planes, and the two couples have the same sense o the tendenc to cause otaton n the same decton. School of echancal Engneeng
oment of a Couple 우력 ( couple) the moment poduced b two foces equal opposte non_colnea 3 35 단일힘으로합할수없으며, 그효과는전적으로회전만을일으킴 o ( a d ) a d 우력에의한모멘트는선택한모멘트중심 a 에무관하다. o ( o ) ( ) School of echancal Engneeng
oment of a Couple 3 36 우력에의한모멘트는선택한모멘트중심 a 에무관하므로우력모멘트 를자유벡터라고할수있으며방향은우력이작용하는평면에수직이고, 오른손법칙에준한다. Equvalent Couples ( 등가우력 ) d 가일정하면우력에의한모멘트는변함이없다. 임의의어느평면에힘들이작용하여도우력모멘트는동일한자유벡터를갖는다. School of echancal Engneeng
ddton of Couples ( 우력의합성 ) Consde two ntesectng planes 1 and 2 wth each contanng a couple 1 1 n plane 1 n plane 2 2 2 3 37 Resultants of the vectos also fom a couple R ( 1 2 ) Vagon s theoem 1 2 1 2 Sum of two couples s also a couple that s equal to the vecto sum of the two couples School of echancal Engneeng
Couples Can e Repesented b Vectos 3 38 couple can be epesented b a vecto wth magntude and decton equal to the moment of the couple. Couple vectos obe the law of addton of vectos. Couple vectos ae fee vectos,.e., the pont of applcaton s not sgnfcant. Couple vectos ma be esolved nto component vectos. School of echancal Engneeng
Resoluton of a oce Into a oce at O and a Couple 3 39 oce vecto can not be smpl moved to O wthout modfng ts acton on the bod. ttachng equal and opposte foce vectos at O poduces no net effect on the bod. The thee foces ma be eplaced b an equvalent foce vecto and couple vecto,.e, a focecouple sstem. School of echancal Engneeng
Resoluton of a oce Into a oce at O and a Couple 3 40 ovng fom to a dffeent pont O' eques the addton of a dffeent couple vecto O 0 The moments of about O and O' ae elated, O s O ( s ) s School of echancal Engneeng
Sample oblem 3.6 3 41 SOLUTION: ttach equal and opposte 90 N foces n the decton at, theeb poducng 3 couples fo whch the moment components ae easl computed. Detemne the components of the sngle couple equvalent to the couples shown. ltenatvel, compute the sum of the moments of the fou foces about an abta sngle pont. The pont D s a good choce as onl two of the foces wll poduce noneo moment contbutons. School of echancal Engneeng
Sample oblem 3.6 3 42 ttach equal and opposte 90 N foces n the decton at The thee couples ma be epesented b thee couple vectos, ( 135 N)( 450 mm) 60.75 N ( 90 N )( 300 mm ) 27.0 N m ( 90 N)( 225 mm) 20.25 N m m ( 27.0 N m) ( 60.75 N m) ( 60.75 N m) ( 27.0 N m) ( 20.25 N m) ( 20.25 N m) School of echancal Engneeng
Sample oblem 3.6 3 43 별해 임의의점 D 에서의모멘트계산하여결정 ltenatvel, compute the sum of the moments of the fou foces about D. Onl the foces at C and E contbute to the moment about D. D ( 450 mm ) ( 135 N)) ( 225 mm) ( 300 mm) [ ] ( 90 N) 두힘 90N, 135N 는 D 점에서모멘트제로 ( 60.75 N m) ( 27.0 N m) ( 20.25 N m) School of echancal Engneeng
Sstem of oces : Reducton to a oce and Couple 3 44 sstem of foces ma be eplaced b a collecton of focecouple sstems actng a gven pont O The foce and couple vectos ma be combned nto a esultant foce vecto and a esultant couple vecto, R å( ) R å O å o The focecouple sstem at O ma be moved to O' wth the addton of the moment of R about O', R R O O s R Two sstems of foces ae equvalent f the can be educed to the same focecouple sstem. School of echancal Engneeng
uthe Reducton of a Sstem of oces 3 45 주어진힘계가단일힘으로변환될수있는조건 : If the esultant foce and couple at O ae mutuall pependcula, the can be eplaced b a sngle foce actng along a new lne of acton. The esultant focecouple sstem fo a sstem of foces wll be mutuall pependcula f: 1) the foces ae concuent, ( 동일점에작용하는힘들 ) 2) the foces ae coplana, ( 동일평면상의힘들 ) 3) the foces ae paallel. ( 평행한힘들 ) 1) the foces ae concuent, ( 동일점에작용하는경우 ) 합우력이제로이고, 힘 우력계는합력벡터만남아서합력만으로변환되어진다. (2 장참조 ) R R å O å o å ( ) 0 School of echancal Engneeng
uthe Reducton of a Sstem of oces 3 46 2) the foces ae coplana, ( 동일평면상의힘들인경우 ) Sstem of coplana foces s educed to a R focecouple sstem R and O that s mutuall pependcula. Sstem can be educed to a sngle foce b movng the lne of acton of untl ts moment about O becomes R R O In tems of ectangula coodnates, R R R O School of echancal Engneeng
Sample oblem 3.8 SOLUTION: a) Compute the esultant foce fo the foces shown and the esultant couple fo the moments of the foces about. 3 47 o the beam, educe the sstem of foces shown to (a) an equvalent focecouple sstem at, (b) an equvalent foce couple sstem at, and (c) a sngle foce o esultant. Note: Snce the suppot eactons ae not ncluded, the gven sstem wll not mantan the beam n equlbum. b) nd an equvalent focecouple sstem at based on the foce couple sstem at. c) Detemne the pont of applcaton fo the esultant foce such that ts moment about s equal to the esultant couple at. School of echancal Engneeng
Sample oblem 3.8 3 48 SOLUTION: a) Compute the esultant foce and the esultant couple at. R å 150 N 600 N 100 N 250 N R 600 N R ( ) ( ) ( ) ( ) å ( ) ( ) ( 1.6 ) ( 600 ) ( 2.8 ) ( 100 ) ( 4.8 ) ( 250 ) R ( 1880 N m) School of echancal Engneeng
Sample oblem 3.8 3 49 b) nd an equvalent focecouple sstem at based on the focecouple sstem at. The foce s unchanged b the movement of the focecouple sstem fom to. R 600 N ( ) The couple at s equal to the moment about of the focecouple sstem found at. R R / R ( 1880 N m) ( 4.8 m) ( 600 N) ( 1880 N m) ( 2880 N m) R ( 1000 N m) c) 단일힘또는합력 : 합력은 R과같고, 그작용점은 R의 점에대한모멘트가다음같다. R R Þ R ( 600 N), 3.13 m ( ) ( 600 ) ( 1880 ) Þ 3.13m School of echancal Engneeng
Sample oblem 3.10 3 50 SOLUTION: Detemne the elatve poston vectos fo the ponts of applcaton of the cable foces wth espect to. Resolve the foces nto ectangula components. Thee cables ae attached to the bacet as shown. Replace the foces wth an equvalent focecouple sstem at. Compute the equvalent foce, R å Compute the equvalent couple, v R å ( ) School of echancal Engneeng
Sample oblem 3.10 3 51 SOLUTION: Detemne the elatve poston vectos wth espect to. 0.075 0.050 ( m) C 0.075 0.050 ( m) 0.100 0.100 D ( m) Resolve the foces nto ectangula components. ( 700 N) l E 75 150 50 l E 175 0.429 0.857 0.289 300 600 200 C D ( N) ( 1000 N)( cos 45 cos 45 ) 707 707 ( 1200 N)( cos60 cos30 ) 600 1039 ( N) ( N) School of echancal Engneeng
Sample oblem 3.10 3 52 Compute the equvalent foce, R å ( 300 707 600) ( 600 1039) ( 200 707) R 1607 439 507 ( N) Compute the equvalent couple, v R å ( ) 0.075 0 0.050 C D c D 300 0.075 707 0.100 600 600 0 0 200 0.050 707 0.100 1039 0 0 30 45 17.68 163.9 R 30 17.68 118.9 School of echancal Engneeng
actce oblems 3 53 q 3.72, 3.80, 3.96, 3.106, 3.119, 3.126, 3.147 School of echancal Engneeng