Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems

Transcription

1 Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008

2 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude, lne of acton, and sense. Epemental evdence shows that the combned effect of two foces ma be epesented b a sngle esultant foce. The esultant s equvalent to the dagonal of a paallelogam whch contans the two foces n adacent legs. oce s a vecto quantt. Tata cgaw-hll Companes, 008

3 Resultant of Seveal Concuent oces Concuent foces: set of foces whch all pass though the same pont. set of concuent foces appled to a patcle ma be eplaced b a sngle esultant foce whch s the vecto sum of the appled foces. Vecto foce components: two o moe foce vectos whch, togethe, have the same effect as a sngle foce vecto. Tata cgaw-hll Companes, 008

4 Rectangula Components of a oce: Unt Vectos a esolve a foce vecto nto pependcula components so that the esultng paallelogam s a ectangle. and ae efeed to as ectangula vecto components and Defne pependcula unt vectos and paallel to the and aes. whch ae Vecto components ma be epessed as poducts of the unt vectos wth the scala magntudes of the vecto components. and ae efeed to as the scala components of Tata cgaw-hll Companes, 008

5 Tata cgaw-hll Companes, 008 ddton of oces b Summng Components S R Wsh to fnd the esultant of 3 o moe concuent foces, ( ) ( ) S S S S R R Resolve each foce nto ectangula components S R The scala components of the esultant ae equal to the sum of the coespondng scala components of the gven foces. S R R R R R R 1 tan θ To fnd the esultant magntude and decton,

6 Sample oblem.3 SOLUTION: Resolve each foce nto ectangula components. Detemne the components of the esultant b addng the coespondng foce components. ou foces act on bolt as shown. Detemne the esultant of the foce on the bolt. Calculate the magntude and decton of the esultant. Tata cgaw-hll Companes, 008

7 Sample oblem.3 SOLUTION: Resolve each foce nto ectangula components. foce mag comp comp R R Detemne the components of the esultant b addng the coespondng foce components. Calculate the magntude and decton. R R 199.6N 14.3N tan α α N Tata cgaw-hll Companes, 008

8 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. Vecto poduct of two vectos and s defned as the vecto V whch satsfes the followng condtons: 1.Lne of acton of V s pependcula to plane contanng and..agntude of V s, V snθ 3.Decton of V s obtaned fom the ghthand ule. Vecto poducts: 1.ae not commutatve, ( ).ae dstbutve, 3.ae not assocatve, ( 1 ) 1 ( ) S ( S) Tata cgaw-hll Companes, 008

9 Tata cgaw-hll Companes, 008 Vecto oducts: Rectangula Components Vecto poducts of Catesan unt vectos, " Vecto poducts n tems of ectangula coodnates ( ) ( ) V ( ) ( ) ( )

10 oment of a oce bout a ont foce vecto s defned b ts magntude and decton. Its effect on the gd bod also depends on t pont of applcaton. The moment of about O s defned as O The moment vecto O s pependcula to the plane contanng O and the foce. agntude of O measues the tendenc of the foce to cause otaton of the bod about an as along O. O snθ d R R? The sense of the moment ma be detemned b the ght-hand ule. n foce that has the same magntude and decton as, s equvalent f t also has the same lne of acton and theefoe, poduces the same moment. Tata cgaw-hll Companes, 008

11 oment of a oce bout a ont Two-dmensonal stuctues have length and beadth but neglgble depth and ae subected to foces contaned n the plane of the stuctue. The plane of the stuctue contans the pont O and the foce. O, the moment of the foce about O s pependcula to the plane. If the foce tends to otate the stuctue clocwse, the sense of the moment vecto s out of the plane of the stuctue and the magntude of the moment s postve. If the foce tends to otate the stuctue counteclocwse, the sense of the moment vecto s nto the plane of the stuctue and the magntude of the moment s negatve. Tata cgaw-hll Companes, 008

12 Vagnon s Theoem The moment about a gve 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. " " " " " ( ) 1 1 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. " " Tata cgaw-hll Companes, 008

13 Eamples of Toque n cton Tata cgaw-hll Companes, 008

14 Tata cgaw-hll Companes, 008 Rectangula Components of the oment of a oce The moment of about B, B B / ( ) ( ) ( ) B B B B B / ( ) ( ) ( ) B B B B

15 Tata cgaw-hll Companes, 008 Rectangula Components of the oment of a oce o two-dmensonal stuctues, ( ) Z O O ( ) ( ) [ ] ( ) ( ) B B Z B B B B

16 Sample oblem N vetcal foce s appled to the end of a leve whch s attached to a shaft at O. 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 40-N vetcal foce to poduce the same moment, e) whethe an of the foces fom b, c, and d s equvalent to the ognal foce. Tata cgaw-hll Companes, 008

17 Sample oblem 3.1 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 ( 4cm) cos60 ( 100 N)( 1 cm) 1 cm O 100 N cm Tata cgaw-hll Companes, 008

18 Sample oblem 3.1 c) Hoontal foce at that poduces the same moment, d O 100 N cm ( 4 cm) d sn 60 ( 0.8 cm) 100 N cm 0.8 cm 0.8 cm 57.7 N Tata cgaw-hll Companes, 008

19 Sample oblem 3.1 c) The smallest foce to poduce the same moment occus when the pependcula dstance s a mamum o when s pependcula to O. O 100 N cm d ( 4 cm) 100 N cm 4 cm 50 N Tata cgaw-hll Companes, 008

20 Sample oblem 3.1 d) To detemne the pont of applcaton of a 40 lb foce to poduce the same moment, O 100 N cm d OBcos60 d ( 40 N) d 100 N cm 40 N 5 cm 5 cm OB 10 cm Tata cgaw-hll Companes, 008

21 Sample oblem 3.1 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. Tata cgaw-hll Companes, 008

22 Sample oblem 3.4 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 B and b a we CD. Knowng that the tenson n the we s 00 N, detemne the moment about of the foce eeted b the we at C. Tata cgaw-hll Companes, 008

23 Sample oblem 3.4 SOLUTION: C " C C λ ( 00 N) ( 0.3 m) ( 0.08 m) C C ( ) ( 0.3 m ) ( 0.4 m ) ( 0.3 m 00 N ) 10 N 96 N ( ) ( ) ( 18 N) D D 0.5 m ( 7.68 N m) ( 8.8 N m) ( 8.8 N m) Tata cgaw-hll Companes, 008

24 Tata cgaw-hll Companes, 008 Scala oduct of Two Vectos The scala poduct o dot poduct between two vectos and s defned as ( ) esult scala cosθ Scala poducts: - ae commutatve, - ae dstbutve, - ae not assocatve, ( ) 1 1 ( ) undefned S Scala poducts wth Catesan unt components, " ( ) ( )

25 Scala oduct of Two Vectos: pplcatons ngle between two vectos: cosθ cosθ oecton of a vecto on a gven as: OL cosθ poecton of cosθ cosθ OL OL cos θ cosθ along OL o an as defned b a unt vecto: λ cosθ Tata cgaw-hll Companes, 008

26 Tata cgaw-hll Companes, 008 ed Tple oduct of Thee Vectos ed tple poduct of thee vectos, ( ) esult scala S The s med tple poducts fomed fom S,, and have equal magntudes but not the same sgn, ( ) ( ) ( ) ( ) ( ) ( ) S S S S S S ( ) ( ) ( ) ( ) S S S S S S S Evaluatng the med tple poduct,

27 Some basc facts n Eucldean geomet Two paallel lnes fom a unque plane Thee non-colnea ponts fom a unque plane lne and a non-colnea pont fom a unque plane Two non-paallel but ntesectng lnes fom unque plane

28 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, λ λ OL O ( ) oments of about the coodnate aes, Tata cgaw-hll Companes, 008

29 Tata cgaw-hll Companes, 008 oment of a oce bout a Gven s oment of a foce about an abta as, ( ) B B B B BL λ λ The esult s ndependent of the pont B along the gven as.

30 Sample oblem 3.5 cube s acted on b a foce as shown. Detemne the moment of a) about b) about the edge B and c) about the dagonal G of the cube. d) Detemne the pependcula dstance between G and C. Tata cgaw-hll Companes, 008

31 Tata cgaw-hll Companes, 008 Sample oblem 3.5 oment of about, ( ) ( ) ( ) ( ) ( ) a a a a / / / ( )( ) a / oment of about B, ( )( ) a B / B a /

32 Tata cgaw-hll Companes, 008 Sample oblem 3.5 oment of about the dagonal G, ( ) ( ) ( ) ( ) ( ) a a a a a a a G G G G λ λ 6 a G

33 Sample oblem 3.5 ependcula dstance between G and C, λ 0 1 ( ) ( ) ( 0 1 1) Theefoe, s pependcula to G. 3 6 a G 6 d d a 6 Ths s staghtfowad cuentl snce Tata cgaw-hll Companes, 008

34 Geneal ule to get dstance between lnes

35 oment of a Couple Two foces and - havng the same magntude, paallel lnes of acton, and opposte sense ae sad to fom a couple. oment of the couple, B ( B ) snθ d ( ) 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. Tata cgaw-hll Companes, 008

36 oment of a Couple Two couples wll have equal moments f 1d1 d 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. Tata cgaw-hll Companes, 008

37 Tata cgaw-hll Companes, 008 ddton of Couples Consde two ntesectng planes 1 and wth each contanng a couple n plane n plane Resultants of the vectos also fom a couple ( ) 1 R B Vagnon s theoem 1 1 Sum of two couples s also a couple that s equal to the vecto sum of the two couples

38 Couples Can Be Repesented b Vectos 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. Tata cgaw-hll Companes, 008

39 Resoluton of a oce Into a oce at O and a Couple 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 foce-couple sstem. Tata cgaw-hll Companes, 008

40 Resoluton of a oce Into a oce at O and a Couple ovng fom to a dffeent pont O eques the addton of a dffeent couple vecto O O ʹ ' The moments of about O and O ae elated, O' ' ( s) s s O ovng the foce-couple sstem fom O to O eques the addton of the moment of the foce at O about O. Tata cgaw-hll Companes, 008

41 Sstem of oces: Reducton to a oce and Couple sstem of foces ma be eplaced b a collecton of foce-couple 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 R O ( ) The foce-couple sstem at O ma be moved to O wth the addton of the moment of R about O, R R s R O' O Two sstems of foces ae equvalent f the can be educed to the same foce-couple sstem. Tata cgaw-hll Companes, 008

42 uthe Reducton of a Sstem of oces 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 foce-couple sstem fo a sstem of foces wll be mutuall pependcula f: 1) the foces ae concuent, ) the foces ae coplana, o 3) the foces ae paallel. Tata cgaw-hll Companes, 008

43 uthe Reducton of a Sstem of oces Sstem of coplana foces s educed to a R foce-couple 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 In tems of ectangula coodnates, R R R O R O Smplest Resultant Tata cgaw-hll Companes, 008