INTRODUCTION. F v. v v v v. M α M=

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1 INTROUTION Newtn s laws and aims devised in 600 s. The cannt be prved arithmeticall. N eperimental evidence up till nw has been bserved t vilate them. These are three laws: Newtn s irst Law: bd at rest tends t sta at rest, and a bd in mtin tends t sta in unifrm mtin (directin & magnitude f the velcit nt changing) unless a net nn-er frce act n it. Newtn s first law is epressed as mathematicall; v = 0 M = 0 Velcities d nt necessaril have t be er. vectr cnstant means that bth its magnitude and directin are cnstant. If the directin f the velcit f translating bd des nt change it mves n a straight line. If angular velcit is cnstant, whatever its magnitude, sense f rtatin i.e. clckwise r cunterclckwise des nt change. Newtn s Secnd Law: When a net, nn-er frce acts n a bd, the bd accelerates in prprtin t and in directin f the acting frce. v v v v a = m a v v v v M α M= I α Where m is an inertial (related with bd n eistence f) prpert f the mving bd. It is called mass and defined as the amunt f matter in a bd. efinitin f the mass in terms f dnamics is resistance f a bd against translatin I is als inertial prpert f the mving bd. It is called mass mment f inertia and defined as the resistance f a bd against rtatin. It is functin f a mass & dimensins f the bd and is a relative indicatin f the distance f the mass particles frm the ais f indicatin.

2 Newtn s secnd law describes the mtin f particles, but particles are hpthetical, nnrealistic. Realistic bdies have finite dimensins. uler s law describes the mtin f rigid bdies f finite dimensins, which can bth rtate and translate, these are mre realistic. Newtn s Third Law: ver frce has a reactin; equal in magnitude, cllinear, and ppsite in directin t the riginal frce. In determinatin f the dnamic behavir f a bd r sstem f bdies, we use Newtn- uler law. Mass and mass mment f inertia can be easil measured, et in mst cases the are cnstant. There are tw variables; frce and acceleratin. S, tw tpes f prblem can cme up: i. rward namics Prblem: cceleratins are given, frces are required. Slutin is simpl multipling mass with acceleratin. This is slved with algebraic equatin and hence eas. ii. Inverse namics Prblem: rces are given, acceleratins are asked. T find the acceleratin, we pre multipl bth sides f the mtin equatin b the inverse f mass. In a simple equatin; inverse f mass is simpl, the reciprcal f mass and calculatin is a simple divisin. Hwever, in a multi degree f freedm sstem, we have n equatin fr each freedm. S, we have a set f equatins, mass epressins becme a square matri. Then, t find the acceleratins we have t find the inverse f it and pre-multipl bth sides f the equatins with it. t the end f dnamic analsis, we get acceleratins, which are still nt t much meaningful. We generall need velcities and psitins. cceleratins must be integrated twice t btain psitins. Integratin is a difficult peratin, ften impssible. S, we generall d numerical integratin. Inverse dnamics prblem is difficult. STTI OR NLYSIS Subject matter f M 0 is t appl Newtn s laws t multi-bd, multi degree f freedm mechanical sstems (mechanisms) t understand their mechanical behavir. We start with the simple prblem where the bdies acceleratins are er. In mst machines, acceleratins are negligible small. Then all the frces and mments will add t er. This

3 is the static cnditin. ver sub-cmpnent f a static sstem is als static. S, If a mechanism is static, each f its links are als static. That means we can eamine each link b ne and appl the law f statics t them. ig prblem f the whle mechanism is nw brken int several simpler prblems. ach bd with all the acting frces is called a free bd. Pictrial representatin f a free bd is called the free bd diagram. Mechanisms are under the effect f tw different frces: i. ternal frces are generated b effects eternal t the mechanism, like actuatin frces frm a mtr r actuatr. rces eerted b the mechanism n the eternal machiner r material ding functinal and useful task. Reactins f the eternal frces are utside ur sstem bundar. ii. nstraint frces cnstraint means limitatin f freedm, in ur case, limitatin f free mtin. These frces are applied nt each link t prevent their free mtin. ree mtin in plane is f three degrees. Translatin in and directins and rtatin abut -ais. nstraint frces are applied thrugh the jints and the act in prper directin and at prper magnitude t limit the mtin t the ne epected frm the mechanism. Reactins f cnstraint frces are inside the sstem bundar. T T

4 rces are vectr and have magnitudes and directins. v v v v = i + j + k frce generates a mment (r trque) abut a pint which is nt in its line f actin mment is the turning effect f a frce. Pint f applicatin v M = v v r Mment directin can be r line f applicatin fund b right hand rule. İt is perpendicular t the plane frmed b r v and v.

5 JOINT TYPS N ONSTRINT ORS nstraint frces are eerted b the jints and depend n the shape and cnstructin f the jints. jint cannt transfer frce and mtin in directin f its egrees f reedm. In this sense, freedm is a free mtin. i. Revlute jint: It has ne degree f freedm, which is in rtatin abut -ais. It cannt transfer frce r trque r angular mtin abut -ais, but it transmits frces and trques and related mtins in all the ther remaining directins. These are; rces in, and directins, Trques abut and -ais. M M M M In case f tw dimensins, a revlute jint transmits frces in and directins. ii. Prismatic jint: It has ne degree f freedm, which is a translatin alng ais. It cannt transmit a frce in ais, but it can transmit frces in and directins and mments,, and ais.

6 M M M M M M In case f tw dimensins, a prismatic jint in a planar mechanism can transmit a frce nl in directin, i.e., perpendicular t the sliding ais. iii. lindrical Jint: It has tw degree f freedm; a translatin alng and rtatin abut the same ais, called the clindrical ais. S, n trque transmissin abut -ais, n frce transmissin alng -ais is pssible. In all ther directins, there can be frces and mments transmitted, which are frces in and -ais and mments abut and -ais. M M M M This jint cannt be used in a planar mechanism.

7 iv. Screw jint: There are tw apparent mtins, which are translatin alng and rtatin abut -ais, are nt independent frm each ther. Therefre degree f freedm is nl ne. This is a rtatin abut -ais. It cannt take mments abut -ais. It can transmits frces in,, - ais and mments abut and -ais. M M M M This jint is nt used in planar mechanism. v. Planar pair: It has three degree f freedm, translatin alng and directins and a rtatin abut -ais. There are n frces and a trque transmissin in these directins. In all ther directins there can be frce and mment transmitted, which are frce in -ais and mments and -ais.

8 M M M M M M This jint is nt used in planar mechanisms. vi. Spherical jint: It has three rtatinal degrees f freedm, s it cannt transmit an mments. It can transmit frces in all directins. MTHO O SOLVING STTI OR PROLMS ) Separate the mechanism int its links, cnsidering each a free bd with all the acting eternal and cnstraint frces n it. ) ppl the rules f statics each free bd which are v v = 0 and M = 0 Slutin f vectr equatins can be b arithmetical and r graphical.

9 Graphical pprach: We draw straight lines t represent vectrs which are in prper directins and lengths prprtinal t the magnitudes f the vectrs and in an articulated manner as depicted in igure. Vectrs frm a clsed plgn called a vectr lp rithmetical apprach: The simplest arithmetical apprach is t separate vectr equatin int cmpnents. θ θ θ csθ + csθ + csθ = sinθ + sinθ + sinθ = 0 0 These tw cmpnent equatins are nt n lnger vectr equatins. The are scalar and can be simultanusl slved t find ma tw f the fllwing;,,, θ, θ, and θ. similar analsis is requierd fr the trques. Trques are calculated with respect t a certain pint. The selectin f the reference pint is immaterial. If the mments acting n an bject are in static equilibrium abut a pint, the are in static equilibrium abut an ther pint als.

10 Simple cases: i. If there are nl tw frces acting n a bd and n mments, it is called a tw frce member. T satisf sum f frces eqaual t er, the tw frces shuld be equal and ppsite. v v v = 0 ; + = 0 v v = T satisf sum f trque is equal t er distance between the frces must be er. This means that the frces are cllinear. ii. Three member is a cmpnent n which nl three frces acts and n mments. T satisf v = 0, the vectrs must frm a clsed plgn and cplanar. T satisf M v = 0, the lines f applicatin f all the three frces intersects at ne single pint. This pint is called the pint f cncurrenc.

11 ample: n eternal frce f 0 N is acting hrintall n the rcker link, 0 mm frm the pint. ind the amunt f trque t be applied t the crank t keep the mechanism in static equilibrium. a a θ b a a a a a a 0 N 0 0 fφ a = 80 mm a = 0 mm a = 70 mm a = 50 mm Slutin: irst step is a psitin analsis t find the angles f the crank and the cupler links. The simplest wa is t draw the mechanism t scale and measure the required angles b a prtractr directl frm the figure. Or we can take a purel arithmetical apprach. n implicit relatin between the psitin variables θ and φ, reudenstein s equatin can be written in the frm: where tan φ φ + tan + = 0 = csθ ( K = sinθ ) + K ) K = csθ ( + K + K + K a K = a a K = a

12 K a = + a a a a + a Substituting the link lengths a, a, a, a and angle θ int abve equatins: K K K 80 0 = = = = = *0 *50 =.66 = cs60(.60) =.0 = sin 60 =.7 = cs 60( +.60) =.0 φ φ.0 * tan.7* tan +.0 = 0 Slving this quadratic equatin fr φ iels: φ = and φ =.5 These tw angles refer t tw different cnfiguratins f the fur-bar mechanism as depicted in figure. a a a θ a Ø Ø igure

13 Nw, t find the angle β: X Y = 0*cs60 = 5,00 mm = 0*sin 60 = 5,98 mm X Y = * cs89,86 = 80,75mm = 50sin 89,86 = 9,90 mm X Y X X = 80,75 5,00 = 65,75 mm = 9,90 5,98 =,9 mm β =,9 tan = 65,75 0,06 Slutin f static frce prblems can be b arithmetical r graphical. rithmetical methd: a) Separate the mechanism int free bdies f links, b) Put all the acting and interacting frces, c) Then, appl the law f statics fr each free bd.

14 0,06 0 N T 89,86 60 Static equatins fr link ; = 0 ; + 0 = 0 + = 0 () = 0 ; + = 0 = () M = 0;0*0*sin89,86 *50 *sin 89,86 + *50*cs89,856 = 0 () 9,99 * 0, * = 99,99 Static equatins fr link ; 0 ; = 0 = = Y 0 ; = 0 = = M = 0; * 70 *sin 0,06 * 70 *cs 0,06 = 0,0* 60,75* = 0 Static equatins fr link ; 0 ; = 0 = =

15 0 ; = 0 = = M = 0 ; *0 *cs60,00 + *0 *sin 60, 00 = T 5,00 * 5,98 * = T We have 9 equatins t slve simultaneusl;,,,,,,,, and T. rm equatin and 6 : + 9,99* (,08) * (,0* 99,99 = =, 7 N 6,6 then, using the equatin 6 fr :,98 = = 5, 99 N,0 rm equatins, 5, 8 : = = = =, 7 N rm equatins,, 7 : 0, * 60,75* 6,6 * = = = 0 = 5, 99 N and using the last equatin 9: = 99,99 = 0) = 99,99 T = 5,00 * 5,98* = 5,00 *,7 5,98*5,99 = 0,07 N mm NS.. Graphical methd:. raw the mechanism in scale,. Measure the unknwn quantities directl frm the scaled drawing,. Separate the mechanism int free bdies f links (scaled drawing),. State whether the link is tw frce - three frce member and then put all the acting and interacting frces, 5. ppl the law f statics fr each free bd. igure is given in : scale. The unknwn quantities φ and β can be directl measured frm this figure b a prtractr. The becme: φ 89,5 and β = 0 Link has frces & trque.

16 Link is a tw frce member, frces at the jints & are equal but ppsite directin. Link is a three frce member, three frces frm a clsed vectr plgn and intersect at ne pint. 0 N d T 0 N O & are measured directl frm the scaled frce plgn. 0 N stands fr 50 mm 0 *.5 stands fr.5 mm = =. 5 N 50 0 *.5 stands fr.5 mm = = 6. 5 N 50 T = d = 9.5* N * =

17 ample: In the figure a Stephensn mechanism is shwn with apprpriate dimensins. W trque f N-m is acting eternall n crank G. alculate the trque required n crank t keep the mechanism in static equilibrium. 5 H 90 N-m 6 60 G = G = cm. = = cm. = = 7 cm. H = H = cm. G = 5 cm ˆ =60 ˆ H =90 Slutin: Link 5 is tw frce member, frces at and are equal magnitude and ppsite directin. Link 6 is tw frce and ne trque member. Link is tw frce member, frces at and are equal magnitude and ppsite directin. Link is three frce member. Line f applicatin, directin and magnitude f the frce at is knwn. Line f applicatin f the frce at is als knwn. Intersecting these alread knwn tw lines, cncurrent pint can be easil fund. Line f applicatin f frce at shuld pass frm and this cncurrenc pint. irectins and magnitudes f the frce at and frce at can be fund frm the vectr lp scaled graph. Link is the tw frce and ne mment member.

18 5 H 90 N-m 6 G d=.5 mm T O = TG = = 0 N and = G G 0,05 = rm frce plgn =, 5 N and = 9 N,5 T = d * = *9 = 0,85 N. m W NS 000

19 ample: Ont link 6 f the mechanism given, a 00 N vertical frce acting. alculate the amunt f the trque required n the crank t keep the mechanism in static equilibrium, using the graphical apprach. θ N = cm. = cm. = 6 cm. = 5 cm. = 8 cm θ =50 Slutin: Link 6 is three frce member. rm the scaled vectr plgn N and N = N becme: = 05 N Link 5 is tw frce member, frces at and are equal magnitude and ppsite directin. = = 05 N Link is tw frce member. Link is three frce member. Line f applicatin, directin and magnitude f the frce at is knwn. Line f applicatin f the frce at is als knwn. Intersecting these alread knwn tw lines, cncurrent pint can be easil fund. Line f applicatin f frce at shuld pass frm and this cncurrenc pint. irectins and magnitudes f the frce at and frce at can be fund frm the vectr lp scaled graph.

20 rm frce plgn = 05 N and = N Link is the tw frce and ne mment member. 6 T = d * = * =,7 N. m W NS 000 T d=6 mm 00 N N 6 =05 N =0 N =05 N 00 N = N 5 N= N

21 ample: Ont pint f the Peucellier s inversr mechanism shwn in the figure, a frce f 00 N is acting verticall dwnward. alculate the amunt f trque required n link t keep the mechanism in static equilibrium N = 0 cm. = 0 cm. = cm. = cm. = cm = cm. = cm. = cm θ =50 θ Slutin: Link,, 5, 6, 7 and 8 are tw frce member. Jint pin at,,, and are three frce member. nsider jint pin at, which is gd starting pint fr static frce analsis. 00 N =5 N rm vectr lp diagram and becme: 00 N =09 N = 5 N 86 = 09 N 7,5

22 Net, cnsider jint pin at that is three frce member. =09 N = =09 N =87 N rm vectr lp diagram and becme: = 87 N 6 = N 86 Net, cnsider jint pin at that is three frce member. =5 N rm vectr lp diagram and becme: =9 N =7 N = 9 N = 7 N 7,5 =5 N

23 Nw, draw the free bd diagrams f the links d T rm the free bd diagram f Link, trque T becmes: T = d NS 0 d * = *66 * 06 = 7,8 N. m W * d

24 ample: n eternal trque f 00 N-m ia acting n the crank f the mechanism shwn in the figure in W directin. alculate the magnitude and directin f the eternal trque required n crank t keep the sstem in static equilibrium. 5 T 00 N.m 6 0 = = 7 cm. = = 8 cm. = 6 cm. = cm. = 5 cm θ =0 Slutin: Link 6 is tw frce and ne trque member. Link 5,, and are tw frce member. Jint pin at is three frce member. In this eample we are beginning with link 5. Since link 5 is tw frce member, this requires that 65 and 5 have equal magnitudes and ppsite directins. We can nw select the link 6 frm which the frce analsis can be started. Then, the frce f 56 can be calculated as: v v 56 T 00 = = = 695 N d v v v = = = Link and are tw frce member, s line f applicatins f frces acting n these links are knwn. Then, we can anale the jint pin at that is three frce member.

25 T d N-m 6 8 Link and are tw frce member, s line f applicatins f frces acting n these links are knwn. Then, we can anale the jint pin at that is three frce member. 5 =00 N 5=695 N =660 N rm the vectr lp unknwn frces becme: v = 660 N,5 v = 00 N 8 rm the free bd diagram f Link, trque T becmes: 0 T v = d * = *660 = 9,8 N. m W NS 000

26 ample: Ont the slider at f the mechanism shwn in the figure, a frce f 00 N is applied. alculate the amunt f the trque required n link. ssume that mechanism is n the hrintal plane and there is n frictin between the mating surfaces. reebd diagram scale: 0 cm stands fr m. 00 N 00 N 6 =09. N = N 5 d T T = d * = 0.0*09. =.09N. m W

27 ample: ind the magnitude and directin f the mment must be applied t link t drive the linkage 00 N against 00 N frce at the mid pint f the link. nd als determine the frces acting n the bearings at,, and. Use graphical apprach. = cm, =7 cm, =5cm, // Slutin:Link is tw frce member Link is three frce member. reebd diagram f these links are shwn belw. T=? 00 N T=? Line f applicatins intersect at infinit rm the freebd diagram f Link, since the frces intersects at = c = 00 / = 50 N and = = = = 50 N rm the free bd diagram f Link, trque T becmes: T v = d * = 0 * 50 = 0 NS ± :

28 ample: In the figure Rbert s fur-bar apprimate straight line mechanism is shwn in scale. 00 Nm W trque is applied nt crank. mplete the freebd diagram shwing all frces acting nt the links. alculate the trque required n crank t keep the mechanism at the given state. Use graphical methd. = = = = = 5 cm = 0 cm 00 Nm Link is tw frce member = d = d 00 Nm = rm the freebd diagram th link; = d = =. 8 N 0. 0 rm the freebd diagram th link; T = d T =. 8* Nm W NS. =

29 ample: ind the magnitude and directin f the mment must be applied t link t drive the linkage against a 00 N frce at the pint f the (6cm, cm) 00 N link as shwn. Use graphical apprach. 60 ==.5 cm, = cm, =5.7cm Slutin: Link is three frce member, Link is tw frce member, 00 N T d=.6 mm 60 =7.8 N =5.8 N 00 N. 6 T = d * = * 7. 8 =. 8 Nm W NSWR 000

30 ample: ind the magnitude and directin f the mment must be applied t link t drive the linkage 00 N against 00 N frce at the mid pint f the link. Use graphical apprach. = cm, =7 cm, =5cm, // Slutin: Link is tw frce member Link is three frce member. T=? reebd diagram f these links are shwn belw. 00 N T=? d

31 rm the vectr lp unknwn frces becme: v = 58. N 5 v = 58. N 90 rm the free bd diagram f Link, trque T becmes: 5. 5 T v = d * = * 58. =. 07 N.m W NS N =58, N =58, N ample: igure shws the si-bar linkage used t btain a duble beat-up n a lm. Ont pint H f the mechanism a hrintal frce f 00 N is acting rightwards. alculate the eternal mtr trque n crank t keep the mechanism in static equilibrium. = cm, =.5 cm, =.5 cm =7 cm, H=0 cm, = cm =6 cm, =0deg. 00 N 5 5 cm 6 5 cm Link 6 is three frce member Link 5 is tw frce member Link is tw frce member Link is tw frce member Jint at is three frce member.

32 = 00 N = 5 = 6 00 N 00 N 6 =65N =8N rm the frce plgn f the 6 th link, = 8 N

33 =8N =0N =85N rm jint frce plgn = 85 N d T. 6 T = d * = 85 =. Nm 000 W

34 - (0%) ind the magnitude and directin f the mment must be applied t link t drive the linkage against a 00 N frce at the pint f the link as shwn. Use graphical apprach. Mechanism is given is scale. 00 N (6cm, cm) 0 τ = * = * 6. 9 Nm W NSWR = Link is tw frce + ne trque member. Link is tw frce member, Link is tw frce and ne trque member. t pint there are tw frces, the resultant f these frce (line f applicatin f this frce shuld be parallel t line f applicatin f frce) can be fund easl b drawing scaled vectr plgn, as shwn belw. Then; = = = N resul tan t 6 τ 00 N = τ 00 N O 0 resultant=6 N

35 - (0%) mplete the freebd diagrams f the links given belw. ind the magnitude and directin f the mment must be applied t link t drive the linkage against a 00 N frce at the pint f the link as shwn. Use graphical apprach. Mechanism is given in scale. 6 5 T N =. N 00 N 00 N N T = * = *. =. 95 Nm, W