Subject: KINEMATICS OF MACHINES Topic: VELOCITY AND ACCELERATION Session I

Transcription

1 Subject: KINEMTIS OF MHINES Tpic: VELOITY ND ELERTION Sessin I Intrductin Kinematics deals with study f relative mtin between the varius parts f the machines. Kinematics des nt invlve study f frces. Thus mtin leads study f displacement, velcity and acceleratin f a part f the machine. Study f Mtins f varius parts f a machine is imprtant fr determining their velcities and acceleratins at different mments. s dynamic frces are a functin f acceleratin and acceleratin is a functin f velcities, study f velcity and acceleratin will be useful in the design f mechanism f a machine. The mechanism will be represented by a line diagram which is knwn as cnfiguratin diagram. The analysis can be carried ut bth by graphical methd as well as analytical methd. Sme imprtant Definitins Displacement: ll particles f a bdy mve in parallel planes and travel by same distance is knwn, linear displacement and is dented by x. bdy rtating abut a fired pint in such a way that all particular mve in circular path angular displacement and is dented by. Velcity: Rate f change f displacement is velcity. Velcity can be linear velcity f angular velcity. dx Linear velcity is Rate f change f linear displacement= V = dt d ngular velcity is Rate f change f angular displacement = = dt Relatin between linear velcity and angular velcity. x = r dx d = r dt dt V = r d = dt cceleratin: Rate f change f velcity 1

2 dv d x f = Linear cceleratin (Rate f change f linear velcity) dt dt d d Thirdly = ngular cceleratin (Rate f change f angular velcity) dt dt We als have, bslute velcity: Velcity f a pint with respect t a fixed pint (zer velcity pint). O V a = x r V a = x O Ex: Va is abslute velcity. Relative velcity: Velcity f a pint with respect t anther pint x 3 4 O O 4 Ex: V ba Velcity f pint with respect t Nte: apital letters are used fr cnfiguratin diagram. Small letters are used fr velcity vectr diagram. This is abslute velcity Velcity f pint with respect t O fixed pint, zer velcity pint. 3

3 V ba = r V ab V ba = r V ab Equal in magnitude but ppsite in directin. O 4 V b bslute velcity is velcity f with respect t O 4 (fixed pint, zer velcity pint) b V ba O, O 4 V b V ab a Velcity vectr diagram Vectr a O = V a = bslute velcity Vectr ab = V ab ba = V a Relative velcity V ab is equal magnitude with V ba but is appsite in directin. Vectr O 4 b = V b abslute velcity. T illustrate the difference between abslute velcity and relative velcity. Let, us cnsider a simple situatin. link mving in a vertical plane such that the link is inclined at 30 t the hrizntal with pint is mving hrizntally at 4 m/s and pint mving vertically upwards. Find velcity f. 3

4 V a = 4 m/s ab bslute velcity Hrizntal directin (knwn in magnitude and directrs) V b =? ab bslute velcity Vertical directin (knwn in directrs nly) 30 4 m/s O V a a V ab V b V ba Velcity f with respect t is equal in magnitude t velcity f with respect t but ppsite in directin. Relative Velcity Equatin y Rigid bdy y a O R O 4 x x Fig. 1 Pint O is fixed and End is a pint n rigid bdy. Rtatin f a rigid link abut a fixed centre. nsider rigid link rtating abut a fixed centre O, as shwn in figure. The distance between O and is R and O makes and angle with x-axis next link x = R cs, y = R sin. Differentiating x with respect t time gives velcity. 4

5 d x dt dθ R sin θ dt = - R sin dy R csθ dt dθ Similarly, dt = - R cs Let, d dt x V x dθ = dt d y dt V y = angular velcity f O x V = - Rω sin y V = - Rω cs Ttal velcity f pint is given by R θ V = V = Rω R sin cs Relative Velcity Equatin f Tw Pints n a Rigid link Rigid bdy y R sin y x R cs x x Fig. Pints and are lcated n rigid bdy 5

6 Frm Fig. x = x + R cs y = y + R sin we get, Differentiating x and y with respect t time d x V dt x d dt x d x R dt dθ R sin θ dt sin θ V x R sin d V dt d dt y y y dθ Similarly, dt R csθ d y y R csθ V R csθ dt V = Similarly, V = = = ( = ( x V x V y V = Ttal velcity f pint y V = Ttal velcity f pint x V (Rω sin ) x V x V y V Rω cs y V ) (Rω sin + R ωcs ) y V ) V Similarly, ( R ωsin + Rω cs ) = Rω V = V Rω = V V V = V V Velcity analysis f any mechanism can be carried ut by varius methds. 1. y graphical methd. y relative velcity methd 3. y instantaneus methd 6

7 y Graphical Methd The fllwing pints are t be cnsidered while slving prblems by this methd. 1. Draw the cnfiguratin design t a suitable scale.. Lcate all fixed pint in a mechanism as a cmmn pint in velcity diagram. 3. hse a suitable scale fr the vectr diagram velcity. 4. The velcity vectr f each rtating link is r t the link. 5. Velcity f each link in mechanism has bth magnitude and directin. Start frm a pint whse magnitude and directin is knwn. 6. The pints f the velcity diagram are indicated by small letters. T explain the methd let us take a few specific examples. 1. Fur ar Mechanism: In a fur bar chain D link D is fixed and in 15 cm lng. The crank is 4 cm lng rtates at 180 rpm (cw) while link D rtates abut D is 8 cm lng = D and D = 60. Find angular velcity f link D. 15 cm 8 cm 60 ω 15 cm nfiguratin Diagram D Velcity vectr diagram πx10 V b = r = ba x = x4 60 = 50.4 cm/sec hse a suitable scale 1 cm = 0 m/s = ab 7

8 r t D c Vcb a, d r t r t b V cb = bc V c = dc = 38 cm/sec = V cd We knw that V =ω R V cd = D x D V cd 38 ω cd = rad/sec (cw) D 8. Slider rank Mechanism: In a crank and sltted lever mechanism crank rtates f 300 rpm in a cunter clckwise directin. Find (i) (ii) ngular velcity f cnnecting rd and Velcity f slider. 60 mm 150 mm 45 nfiguratin diagram Step 1: Determine the magnitude and velcity f pint with respect t 0, π x 300 V = O1 x O = x = 600 mm/sec Step : hse a suitable scale t draw velcity vectr diagram. 8

9 a V a r t r t O b O lng sides Velcity vectr diagram V ab = ab =1300mm/sec V ba 1300 ba = rad/sec V b = b velcity f slider Nte: Velcity f slider is alng the line f sliding. 3. Shaper Mechanism: In a crank and sltted lever mechanisms crank O rtates at rad/sec in W directin. Determine the velcity f slider. 6 D 5 Scale 1 cm = x. m O ω 3 4 O 1 nfiguratin diagram 9

10 Scale 1 cm = x. m/s c V b a V O = V V O1 V D d O 1 O Velcity vectr diagram V a = x O O1b O1c O O 1 T lcate pint O1 O 1c O1b O1 1 T Determine Velcity f Rubbing Tw links f a mechanism having turning pint will be cnnected by pins. When the links are mtin they rub against pin surface. The velcity f rubbing f pins depends n the angular velcity f links relative t each ther as well as directin. Fr example: In a fur bar mechanism we have pins at pints,, and D. V ra = ab x ratis f pin (r pa ) + sign is used ab is W and W bc is W i.e. when angular velcities are in ppsite directins use + sign when angular velcities are in sme directins use - ve sign. V rb = ( ab + bc ) radius r pb V r = ( bc + cd ) radius r pc V rd = cd r pd 10

11 Prblems n velcity by velcity vectr methd (Graphical slutins) Prblem 1: In a fur bar mechanism, the dimensins f the links are as given belw: = 50 mm, = 66 mm D = 56 mm and D = 100 mm t a given instant when rad/sec in W directin. D 60 the angular velcity f link is 10.5 Determine, i) Velcity f pint ii) iii) Velcity f pint E n link when E = 40 mm The angular velcity f link and D iv) The velcity f an ffset pint F n link, if F = 45 mm, F = 30 mm and F is read clckwise. v) The velcity f an ffset pint G n link D, if G = 4 mm, DG = 44 mm and DG is read clckwise. vi) The velcity f rubbing f pins,, and D. The rati f the pins are 30 mm, 40 mm, 5 mm and 35 mm respectively. Slutin: Step -1: nstruct the cnfiguratin diagram selecting a suitable scale. Scale: 1 cm = 0 mm G F 60 D Step : Given the angular velcity f link and its directin f rtatin determine velcity f pint with respect t ( is fixed hence, it is zer velcity pint). V ba = x = 10.5 x 0.05 = 0.55 m/s 11

12 Step 3: T draw velcity vectr diagram chse a suitable scale, say 1 cm = 0. m/s. First lcate zer velcity pints. Draw a line r t link in the directin f rtatin f link (W) equal t 0.55 m/s. b V ba = 0.55 m/s e, g f V ed a, d Frm b draw a line r t and frm d. Draw d line r t D t interest at. V cb is given vectr bc V bc = 0.44 m/s V cd is given vectr dc V cd = 0.39 m/s Step 4: T determine velcity f pint E (bslute velcity) n link, first lcate the psitin f pint E n velcity vectr diagram. This can be dne by taking crrespnding ratis f lengths f links t vectr distance i.e. be E bc E be = x Vcb = x 0.44 = 0.4 m/s Jin e n velcity vectr diagram t zer velcity pints a, d / vectr de = V e = m/s. Step 5: T determine angular velcity f links and D, we knw V bc and V cd. Similarly, V bc =ω x V bc 0.44 ω = 6.6 r / s. ( cw) V cd = ω D x D ω D = V cd D r / s (W) Step 6: T determine velcity f an ffset pint F Draw a line r t F frm n velcity vectr diagram. 1

13 Draw a line r t F frm b n velcity vectr diagram t intersect the previusly drawn line at f. Frm the pint f t zer velcity pint a, d and measure vectr fa t get V f = m/s. Step 7: T determine velcity f an ffset pint. Draw a line r t G frm n velcity vectr diagram. Draw a line r t DG frm d n velcity vectr diagram t intersect previusly drawn line at g. Measure vectr dg t get velcity f pint G. V g = dg m / s Step 8: T determine rubbing velcity at pins Rubbing velcity at pin will be V pa = ab x r f pin V pa = 10.5 x 0.03 = m/s Rubbing velcity at pin will be V pb = ( ab + cb ) x r pb f pint at. [ ab W and cb W] V pb = ( ) x 0.04 = m/s. Rubbing velcity at pint will be = 6.96 x = 0.44 m/s Prblem : In a slider crank mechanism the crank is 00 mm lng and rtates at 40 rad/sec in a W directin. The length f the cnnecting rd is 800 mm. When the crank turns thrugh 60 frm Inner-dead centre. Determine, i) The velcity f the slider ii) iii) iv) Velcity f pint E lcated at a distance f 00 mm n the cnnecting rd extended. The psitin and velcity f pint F n the cnnecting rd having the least abslute velcity. The angular velcity f cnnecting rd. 13

14 v) The velcity f rubbing f pins f crank shaft, crank and crss head having pins diameters 80,60 and 100 mm respectively. Slutin: Step 1: Draw the cnfiguratin diagram by selecting a suitable scale. E F 45 O G V a = W a x O V a = 40 x 0. V a = 8 m/s Step : hse a suitable scale fr velcity vectr diagram and draw the velcity vectr diagram. Mark zer velcity pint, g. Draw a r t link O equal t 8 m/s e a Scale: 1 cm = m/s b f, g Frm a draw a line r t and frm, g draw a hrizntal line (representing the line f mtin f slider ) t intersect the previusly drawn line at b. ab give V ba =4.8 m/sec Step 3: T mark pint e since E is n the extensin f link drawn be = E x ab mark the pint e n extensin f vectr ba. Jin e t, g. ge will give velcity f pint E. V e = ge =8.4 m/sec 14

15 Step 4: T mark pint F n link such that this has least velcity (abslute). Draw a line r t ab passing thrugh, g t cut the vectr ab at f. Frm f t, g. gf will have the least abslute velcity. T mark the psitin f F n link. Find F by using the relatin. fb F ab F fb ab x =00mm Step 5: T determine the angular velcity f cnnecting rd. We knw that V ab = ab x V ab = ab = 6 rad/sec Step 6: T determine velcity f rubbing f pins. V pcrankshaft = a x radius f crankshaft pin = 8 x 0.08 = 0.64 m/s V Pcrank pin = ( ab + a ) r crank pin = (6 +8)0.06 =0.84 m/sec V P crss head = ab x r crss head = 6 x 0.1 = 0.6 m/sec 15

16 Prblem 3: quick return mechanism f crank and sltted lever type shaping machine is shwn in Fig. the dimensins f varius links are as fllws. O 1 O = 800 mm, O 1 = 300 mm, O D = 1300 mm and DR = 400 mm The crank O 1 makes an angle f 45 with the vertical and relates at 40 rpm in the W directin. Find: i) Velcity f the Ram R, velcity f cutting tl, and ii) ngular velcity f link O D. Slutin: Step 1: Draw the cnfiguratin diagram. R Tl R 00 D n rank, O, n O D O 1 45 D O 1 O O Step : Determine velcity f pint. V b = O1 x O 1 O1 = N O1 60 x V b = 4.18 x 0.3 = 1.54 m/sec 4.18rad / sec 16

17 Step 3: Draw velcity vectr diagram. hse a suitable scale 1 cm = 0.3 m/sec d b c r O 1 O Draw O 1 b r t link O 1 equal t 1.54 m/s. Frm b draw a line alng the line f O and frm O 1 O draw a line r t O. This intersects at c bc will measure velcity f sliding f slider and O will measure the velcity f n link O. Since pint D is n the extensin f link O measure O d such that O d = OD O. O d will give velcity f pint D. O Frm d draw a line r t link DR and frm O 1 O. Draw a line alng the line f strke f Ram R (hrizntal), These tw lines will intersect at pint r O r will give the velcity f Ram R. T determine the angular velcity f link O D determine V d = O d. We knw that V d = OD x O D. O d Od O D r/s 17

18 Prblem 4: Figure belw shws a tggle mechanisms in which the crank O rtates at 10 rpm. Find the velcity and acceleratin f the slider D. Slutin: ll the dimensins in mm D nfiguratin Diagram Step 1: Draw the cnfiguratin diagram chsing a suitable scal. Step : Determine velcity f pint with respect t O. V a = O x O V a = x m / s Step 3: Draw the velcity vectr diagram. hse a suitable scale Mark zer velcity pints O,q Draw vectr a r t link O and magnitude = 5.04 m/s. a b O,q D Velcity vectr diagram 18

19 Frm a draw a line r t and frm q draw a line r t Q t intersect at b. ab V and qb. ba V bq Draw a line r t D frm b frm q draw a line alng the slide t intersect at d. dq V d (slider velcity) Prblem 5: whitwrth quick return mechanism shwn in figure has the fllwing dimensins f the links. The crank rtates at an angular velcity f.5 r/s at the mment when crank makes an angle f 45 with vertical. alculate OP (crank) = 40 mm a) the velcity f the Ram S O = 150 mm b) the velcity f slider P n the sltted level R = 165 mm c) the angular velcity f the link RS. RS = 430 mm Slutin: Step 1: T draw cnfiguratin diagram t a suitable scale. R S O 45 P n slider Q n nfiguratin Diagram 19

20 Step : T determine the abslute velcity f pint P. V P = OP x OP V a = x x m / s Step 3: Draw the velcity vectr diagram by chsing a suitable scale. P 0.6 m q S O, a, g r Velcity vectr diagram Draw p r link OP = 0.6 m. Frm O, a, g draw a line r t P/Q and frm P draw a line alng P t intersect previusly draw, line at q. Pq = Velcity f sliding. aq = Velcity f Q with respect t. V qa = aq = ngular velcity f link RS = RS sr SR rad/sec 0

21 Prblem 6: tggle mechanism is shwn in figure alng with the diagrams f the links in mm. find the velcities f the pints and and the angular velcities f links, Q and. The crank rtates at 50 rpm in the clckwise directin. Q 100 O = 30 = 80 Q = 100 = rpm 140 ll dimensins are in mm O Slutin Step 1: Draw the cnfiguratin diagram t a suitable scale. Step : alculate the magnitude f velcity f with respect t O. V a = O x O x 50 V a = x m / s m / s 60 b a O, q c Vectr velcity diagram 1

22 Step 3: Draw the velcity vectr diagram by chsing a suitable scale. Draw Oa r t link O = 0.15 m/s Frm a draw a link r t and frm O, q draw a link r t Q t intersect at b. ab and qb V b 0.13m / s V ba ab = ab qb 0.74r / s (ccw) bq 1.3 r / s (ccw a ) Frm b draw a line r t e and frm O, q these tw lines intersect at. O V m / s b V b bc 1.33 r / s (ccw) Prblem 7: The mechanism f a stne crusher has the dimensins as shwn in figure in mm. If crank rtates at 10 rpm W. Find the velcity f pint K when crank O is inclined at 30 t the hrizntal. What will be the trque required at the crank t vercme a hrizntal frce f 40 kn at K hz h 100 M K D 00 nfiguratin diagram Slutin: Step 1: Draw the cnfiguratin diagram t a suitable scale.

23 Step : Given speed f crank O determine velcity f with respect t. x 10 V a = O x O = x m / s 60 d V k (hz), q, m k b c a Velcity vectr diagram Step 3: Draw the velcity vectr diagram by selecting a suitable scale. Draw Oa r t link O = 1.6 m/s Frm a draw a link r t and frm q draw a link r t Q t intersect at b. Frm b draw a line r t and frm a, draw a line r t t intersect at c. Frm c draw a line r t D and frm m draw a line r t MD t intersect at d. Frm d draw a line r t KD and frm m draw a line r t KM t x intersect the previusly drawn line at k. Since we have t determine the trque required at O t vercme a hrizntal frce f 40 kn at K. Draw a the hrizntal line frm, q, m and c line r t this line frm k. T I T O P P V = R T = F x P F = r T O T O = F k V k hrizntal T O = T O = k V k F O hz X = N-m 3

24 Prblem 8: In the mechanism shwn in figure link O = 30 mm, = 680 mm and OQ = 650 mm. Determine, i) The angular velcity f the cylinder ii) iii) The sliding velcity f the plunger The abslute velcity f the plunger When the crank O rtates at 0 rad/sec clckwise. Slutin: Step 1: Draw the cnfiguratin diagram. 60 n R (pint n R belw Q) O R Step : Draw the velcity vectr diagram Determine velcity f pint with respect t O. V a = O x O = 0 x 0.3 = 6.4 m/s Select a suitable scale t draw the velcity vectr diagram. Mark the zer velcity pint. Draw vectr a r t link O equal t 6.4 m/s. a O, q c b Frm a draw a line r t and frm, q, draw a line perpendicular t. T mark pint c n ab We knw that ab ac 4

25 ab x ac = Mark pint c n ab and jint this t zer velcity pint. ngular velcity f cylinder will be. V ab = ab = 5.61 rad/sec (c) Studying velcity f player will be qb = 4.1 m/s O bslute velcity f plunger = = 4. m/s qc Prblem 9: In a swiveling jint mechanism shwn in figure link is the driving crank which rtates at 300 rpm clckwise. The length f the varius links are: Determine, i) The velcity f slider blck S ii) iii) Slutin: The angular velcity f link EF The velcity f link EF in the swivel blck. Step 1: Draw the cnfiguratin diagram. = 650 mm = 100 mm = 800 mm D = 50 mm E = F EF = 400 mm OF = 40 mm FS = 400 mm 400 F O P G S E D 5

26 Step : Determine the velcity f pint with respect t. V b = x V b = x x 0.1 = 3.14 m/s Step 3: Draw the velcity vectr diagram chsing a suitable scale. Mark zer velcity pint a, d,, g. f b P S a, d,, g Velcity vectr diagram c Frm a draw a line r t and equal t 3.14 m/s. Frm b draw a line r t D t intersect at. Mark a pint e n vectr bc such that E be bc x Frm e draw a line r t PE and frm a,d draw a line alng PE t intersect at P. ef Extend the vectr ep t ef such that ef x EF EP Frm f draw a line r t Sf and frm zer velcity pint draw a line alng the slider S t intersect the previusly drawn line at S. Velcity f slider gs.6m / s. ngular Velcity f link EF. Velcity f link F in the swivel blck = OP 1.85 m / s. Prblem 10: Figure shws tw wheels and 4 which rlls n a fixed link 1. The angular unifrm velcity f wheel is is 10 rd/sec. Determine the angular velcity f links 3 and 4, and als the relative velcity f pint D with respect t pint E. 6

27 50 mm mm 40 mm D 4 G F Slutin: Step 1: Draw the cnfiguratin diagram. Step : Given = 10 rad/sec. alculate velcity f with respect t G. V b = x G V b = 10 x 43 = 430 mm/sec. Step 3: Draw the velcity vectr diagram by chsing a suitable scale. 30 E D G 50 mm Redrawn cnfiguratin diagram F 7

28 Velcity vectr diagram c b e d g, f Draw gb = 0.43 m/s r t G. Frm b draw a line r t and frm f draw a line r t F t intersect at. Frm b draw a line r t E and frm g, f draw a line r t GE t intersect at e. Frm c draw a line r t D and frm f draw a line r t FD t intersect at d. Prblem 11: Fr the mechanism shwn in figure link rtates at cnstant angular velcity f 1 rad/sec cnstruct the velcity plygn and determine. i) Velcity f pint D. ii) ngular velcity f link D. iii) Velcity f slider. Slutin: Step 1: Draw cnfiguratin diagram. O D O = 50.8 mm = 10 mm D = 10 mm DO 6 = 10 mm = 03 mm 10 mm O 8

29 Step : Determine velcity f with respect t O. V b = x O V b = 1 x 50.8 = 50.8 mm/sec. Step 3: Draw the velcity vectr diagram, lcate zer velcity pints O O 6. V d d U db a b O O 6 Frm O, O 6 draw a line r t O in the directin f rtatin equal t 50.8 mm/sec. Frm a draw a line r t c and frm O, O 6 draw a line alng the line f stcks f c t intersect the previusly drawn line at c. ab Mark pint b n vectr ac such that ab x Frm b draw a line r t D and frm O, O 6 draw a line r t O 6 D t intersect at d. Step 4: V d = O 6 d = 3 mm/sec bd bd = = D V c = O = DDITIONL PROLEMS FOR PRTIE Prblem 1: In a slider crank mechanism shwn in ffset by a perpendicular distance f 50 mm frm the centre. and are 750 mm and 00 mm lng respectively crank is rtating e at a unifrm speed f 00 rpm. Draw the velcity vectr diagram and determine velcity f slider and angular velcity f link. 9

30 50 mm Prblem : Fr the mechanism shwn in figure determine the velcities at pints, E and F and the angular velcities f links,, DE and EF. 100 rpm E ll dimensins are in mm 150 F 50 D The crank p f a crank and sltted lever mechanism shwn in figure rtates at 100 rpm in the W directin. Varius lengths f the links are OP = 90 mm, O = 300 mm, R = 480 mm and RS = 330 mm. The slider mves alng an axis perpendicular t r O and in 10 mm frm O. Determine the velcity f the slider when OP is 135 and als mentin the maximum velcity f slider. O 45 D 30

31 Prblem 4: Find the velcity f link 4 f the sctch yke mechanism shwn in figure. The angular speed f link is 00 rad/sec W, link O P = 40 mm. P 4 3 Q n link 4 45 Prblem 5: In the mechanism shwn in figure link rtates unifrmly in directin at 40 rpm. Determine the linear velcity f and angular velcity f EF. E = 160 mm = 160 mm 45 D = 100 mm D = 00 mm 100 mm F EF = 00 mm E = 40 mm 31

32 II Methd Instantaneus Methd T explain instantaneus centre let us cnsider a plane bdy P having a nnlinear mtin relative t anther bdy q cnsider tw pints and n bdy P having velcities as V a and V b respectively in the directin shwn. I P V a V b q Fig. 1 If a line is drawn r t V a, at the bdy can be imagined t rtate abut sme pint n the line. Thirdly, centre f rtatin f the bdy als lies n a line r t the directin f V b at. If the intersectin f the tw lines is at I, the bdy P will be rtating abut I at that instant. The pint I is knwn as the instantaneus centre f rtatin fr the bdy P. The psitin f instantaneus centre changes with the mtin f the bdy. I V a P V b q Fig. In case f the r lines drawn frm and meet utside the bdy P as shwn in Fig. V a V b I at Fig. 3 If the directin f V a and V b are parallel t the r at and met at. This is the case when the bdy has linear mtin. 3

33 Number f Instantaneus enters The number f instantaneus centers in a mechanism depends upn number f links. If N is the number f instantaneus centers and n is the number f links. n N = n 1 Types f Instantaneus enters There are three types f instantaneus centers namely fixed, permanent and neither fixed nr permanent. Example: Fur bar mechanism. n = 4. n N = n 1 4 = I 13 3 I 34 I 3 4 I 4 I 1 1 I 14 Fixed instantaneus center I 1, I 14 Permanent instantaneus center I 3, I 34 Neither fixed nr permanent instantaneus center I 13, I 4 rnld Kennedy therem f three centers: Statement: If three bdies have mtin relative t each ther, their instantaneus centers shuld lie in a straight line. 33

34 Prf: 1 I 1 V 3 V I 13 I 3 3 nsider a three link mechanism with link 1 being fixed link rtating abut I 1 and link 3 rtating abut I 13. Hence, I 1 and I 13 are the instantaneus centers fr link and link 3. Let us assume that instantaneus center f link and 3 be at pint i.e. I 3. Pint is a cincident pint n link and link 3. nsidering n link, velcity f with respect t I 1 will be a vectr V r t link I 1. Similarly fr pint n link 3, velcity f with respect t I 13 will be r t I 13. It is seen that velcity vectr f V and V 3 are in different directins which is impssible. Hence, the instantaneus center f the tw links cannt be at the assumed psitin. It can be seen that when I 3 lies n the line jining I 1 and I 13 the V and V 3 will be same in magnitude and directin. Hence, fr the three links t be in relative mtin all the three centers shuld lie in a same straight line. Hence, the prf. Steps t lcate instantaneus centers: Step 1: Draw the cnfiguratin diagram. Step : Identify the number f instantaneus centers by using the relatin N = n 1 n. Step 3: Identify the instantaneus centers by circle diagram. Step 4: Lcate all the instantaneus centers by making use f Kennedy s therem. T illustrate the prcedure let us cnsider an example. 34

35 slider crank mechanism has lengths f crank and cnnecting rd equal t 00 mm and 00 mm respectively lcate all the instantaneus centers f the mechanism fr the psitin f the crank when it has turned thrugh 30 frm IO. ls find velcity f slider and angular velcity f cnnecting rd if crank rtates at 40 rad/sec. Step 1: Draw cnfiguratin diagram t a suitable scale. Step : Determine the number f links in the mechanism and find number f instantaneus centers. N = n 1 n 4 n = 4 links N = 4 1 = 6 I 13 I 4 3 I I O 1 1 I 1 I 14 t I 14 t Step 3: Identify instantaneus centers. Suit it is a 4-bar link the resulting figure will be a square. 1 I 1 I 4 I 41 I 3 I 13 4 I 34 3 OR I 1 I 3 I 34 I 13 I 4 I 14 Lcate fixed and permanent instantaneus centers. T lcate neither fixed nr permanent instantaneus centers use Kennedy s three centers therem. 35

36 Step 4: Velcity f different pints. V a = I 1 = 40 x 0. = 8 m/s als V a = x 13 3 = V I a 13 V b = 3 x I 13 = Velcity f slider. Prblem : fur bar mechanisms has links = 300 mm, = D = 360 mm and D = 600 mm. ngle D 60. rank rtates in directin at a speed f 100 rpm. Lcate all the instantaneus centers and determine the angular velcity f link. Slutin: Step 1: Draw the cnfiguratin diagram t a suitable scale. Step : Find the number f Instantaneus centers N = n 1 n 4 = 4 1 = 6 Step 3: Identify the I s by circular methd r bk keeping methd. 1 I 1 I I 14 I 3 I 13 4 I 34 3 OR I 1 I 3 I 34 I 13 I 4 I 14 Step 4: Lcate all the visible I s and lcate ther I s by Kennedy s therem. 36

37 I 13 3 I 34 I 3 4 I 4 I 1 I 14 1 D V b = x I 1 = x x 0.3 m / sec ls V b = 3 x I 13 3 = V I b 13 rad / sec Fr a mechanism in figure crank O rtates at 100 rpm clckwise using I.. methd determine the linear velcities f pints,, D and angular velcities f links, and D. O = 0 cm = 150 cm = 60 cm D = 50 cm E = 40 cm OE = 135 cm O 4 E 5 D 6 10 mm V a = O x O V a = x x 0..1 m / s n = 6 links n n 1 N = 15 37

38 I I I I 13 I 45 1 I 3 3 I I 56 I 1 I 34 I 15 Link 3 I 13 3 V a = 3 I 13 3 = Va I 13.5 rad / sec V b = 3 x I 13 =.675 m/s 38

39 Link 4 I 14 4 ls V b = 4 x I 14 4 = V I b rad / sec V = 4 x I 14 = 1.73 m/s Link 5 5 D I 15 V = 5 x I 15 5 = V I rad / sec V d = 5 x DI 15 = 0.86 m/s nswers V b =.675 m/s V = 1.73 m/s V d = 0.86 m/s ab =.5 rad/sec bc = 6.37 rad/sec cd = 1.7 rad/sec Find, In the tggle mechanism shwn in figure the slider D is cnstrained t mve in a hrizntal path the crank O is rtating in W directin at a speed f 180 rpm the dimensins f varius links are as fllws: O = 180 mm = 360 mm i) Velcity f slider ii) = 40 mm D = 540 mm ngular velcity f links, and D. 39

40 45 O n = 6 links n n 1 N = 15 D I I 1 I 46 I 13 4 I I 15 I 3 O I 45 3 I 35 I 34 I 5 6 I 4 I 56 40

41 V a = x I 1 = 3.4 m/s Link 3 3 I 13 V a = 3 x I 13 3 = V I a V b = 3 x I 13 rad / sec Link 4 4 I 14 V b = 4 x I 14 4 = V I b rad / sec Link 5 I 15 5 D V b = 5 x I 15 5 = V I b rad / sec V d = 5 x DI 15 = m/s nswers V d = m/s ab =.44 rad/sec bc = rad/sec cd = 4.37 rad/sec 41

42 Figure shws a six link mechanism. What will be the velcity f cutting tl D and the angular velcities f links and D if crank rtates at 10 rad/sec. Q ll dimensins are in mm O D I 13 I I 14 4 I 34 I 46 3 I 3 I 4 I 45 I 6 I O 5 I 1 I 56 6 I I 15 4

43 V a = x I 1 = 10 x V a = x I 1 = 0.15 m/s Link 3 I 13 3 V a = 3 x I 13 3 = V I a 13 V b = 3 x I 13 Link 4 Q I 14 4 V b = 4 x I 14 4 = V I b V = 4 x I 14 rad / sec Link 5 5 D I 15 V = 5 x I 15 5 = V I rad / sec V d = 5 x DI 15 = 1.66 m/s nswers V d = 1.66 m/s bc = 4.5 rad/sec cd = 1.98 rad/sec 43

44 whitwrth quick return mechanism shwn in figure has a fixed link O and crank OP having length 00 mm and 350 mm respectively. Other lengths are R = 00 mm and RS = 40 mm. Find the velcity f the rtatin using I methd when crank makes an angle f 10 with fixed link and rtates at 10 rad/sec. R 5 S P O 3 Lcate the I s n = 6 links n n 1 N =

45 I 15 I I 45 I I 14 I I 1 I 3 I 34 3 I 4 V P = x OP = m/s cceleratin nalysis Rate f change f velcity is acceleratin. change in velcity requires any ne f the fllwing cnditins t be fulfilled: hange in magnitude nly hange in directin nly hange in bth magnitude and directin When the velcity f a particle changes in magnitude and directin it has tw cmpnent f acceleratin. 1. Radial r centripetal acceleratin f c = r cceleratin is parallel t the link and acting twards centre. 45

46 V a r δθ Va csδθ δθ O 1 f t a a f c a O Va Va sinδθ f a a 1 Va = (ω + δ t) r Velcity f parallel t O = 0 Velcity f parallel t O = Va sin δ θ Therefre change in velcity = Va sin δ θ 0 entripetal acceleratin = f c = t r sin t as δt tends t Zer sin δ θ tends t δ θ r rt t f c = ωr (dθ/ dt) =ω r. Tnagential cceleratin: Va = (ω + δ t) r Velcity f perpendicular t O = Va Velcity f perpendicular t O = Va cs δ θ Therefre change in velcity = Va cs δ θ Va Tnagnetial acceleratin = f t = t r cs r t as δt tends t Zer cs δ θ tends t 1 ut V = ωr r ω = V/r Hence, f c =ω r = V /r 46

47 r rt t r f t = r Example: f ab = cts parallel t and acts frm t. f r ab f ab f t ab f t = acts r t link. f = f r + f t Prblem 1: Fur bar mechanism. Fr a 4-bar mechanism shwn in figure draw velcity and acceleratin diagram. ll dimensins are in mm = 10.5 rad/sec D 47

48 Slutin: Step 1: Draw cnfiguratin diagram t a scale. Step : Draw velcity vectr diagram t a scale. V b = x V b = 10.5 x 0.05 V b = 0.55 m/s V c a 1 d V bc b Step 3: Prepare a table as shwn belw: Sl. N. Link Magnitude Directin Sense 1. f c = r f c = (10.5) /0.55 f c = 5.51 m/s. f c = r f c = 1.75 f t = r 3. D f c = Dr f c =.75 f t =? Parallel t Parallel t r t Parallel t D r t D D Step 4: Draw the acceleratin diagram. 11 el t D a 1 d 1 11 el t D t c 1 f bc c 1 b 1 11 el t b 1 11 el t 48

49 hse a suitable scale t draw acceleratin diagram. Mark the zer acceleratin pint a 1 d 1. Link has nly centripetal acceleratin. Therefre, draw a line parallel t and tward frm a 1 d 1 equal t 5.51 m/s i.e. pint b 1. Frm b 1 draw a vectr parallel t pints twards equal t 1.75 m/s (b 1 1). Frm b 1 1 draw a line r t. The magnitude is nt knwn. Frm a 1 d 1 draw a vectr parallel t D and pinting twards D equal t.7 m/s i.e. pint c 1. Frm c 1 1 draw a line r t D t intersect the line drawn r t at c 1, d 1 c 1 = f D and b 1 c 1 = f bc. T determine angular acceleratin. = f t bc 1 c1b rad / sec (W) t 1 fcd c1c1 D = 79.11rad / sec(w) D D Prblem : Fr the cnfiguratin f slider crank mechanism shwn in figure belw. alculate i) cceleratin f slider. ii) cceleratin f pint E. iii) ngular acceleratin f link. If crank O rtates at 0 rad/sec W. Slutin: E 450 ll dimensins are mm G 49

50 Step 1: Draw cnfiguratin diagram. Step : Find velcity f with respect t O. V a = O x O V a = 0 x 0.48 V a = 9.6 m/s Step 4: Draw velcity vectr diagram. e a 5.5 b 9.7 O 1 g Step 4: Sl. N. Link Magnitude Directin Sense 1. O f c ao = Or = 19 Parallel t O O. f c ab = abr = 17. f t ab Parallel t r t 3. Slider Parallel t Slider Step 5: Draw the acceleratin diagram chsing a suitable scale. f b f ab 1 g 1 f t ab b f c ab 17 a 1 e 1 e e1 50

51 Mark 1 g 1 (zer acceleratin pint) Draw 1 g 1 = acceleratin f O twards O. Frm a 1 draw a 1 b 1 1 = 17. m/s twards frm b 1 1 draw a line r t. Frm 1 g 1 draw a line alng the slider t intersect previusly drawn line at b 1, a b 1 1 f ab g 1b 1 = f b = 7 m/s. Extend a b = a e such that a1b1 1R E Jin e 1 t 1 g 1, g e = f 1 1 e = 36 m/s. t f 1 ab ab = nswers: f b = 7 m/sec f e = 36 m/sec ab = 104 rad/sec b1b = 104 rad/sec (W). Prblem 3: In a tggle mechanism shwn in figure the crank O rtates at 10 rpm W increasing at the rate f 60 rad/s. Velcity f slider D and angular velcity f link D. cceleratin f slider D and angular acceleratin f link D D D Q G Step 1 Draw the cnfiguratin diagram t a scale. 51

52 Step Find V a = O x O V a = x 0. = 4.4 m/s Step 3: Draw the velcity vectr diagram. a b d 1,q,g Step 4: Sl. N O Q Link Magnitude m/s Directin Sense f c ao = r = 96.8 f t ao = r = 1 f c ab = r = 5.93 f t ab = r = f c bq = r = 38.3 f t bq = r = Parallel t O r t O Parallel t r t Parallel t Q r t Q O Q 4. D f c bd = r = 0 r t D 5. Slider D f t bd = r = r t D Parallel t slider mtin Step 5: Draw the acceleratin diagram chsing a suitable scale. Mark zer acceleratin pint. 5

53 f bd d 1 f d O 1 q 1 g 1 q 1 1 fab b 1 1 f c O d 1 1 b 1 a 1 f t O a 1 1 Draw 1 a 1 1 = f c O and a 1 1a = f t O r t O frm 1 a 1 = f a c Frm a 1 draw a1b1 f ab, frm b 1 1 draw a line r t. 1 Frm 1 q 1 g 1 draw 1q 1 = f c bq and frm q 1 1 draw a line a line r t Q t intersect the previusly drawn line at b 1 q1b1 f a b bq 1 1 = f ab Frm b 1 draw a line parallel t D = f c bd such that b d = fc bd. Frm d 1 1 draw a line r t D, frm 1 q 1 g 1 draw a line alng slider D t meet the previusly drawn line at. g = 16.4 m/sec. 1d1 f d b = 5.46 m/sec. D = 1d1 f bd f bd D rad / sec nswers: V d =.54 m/s bd = 6.3 rad/s F d = 16.4 m/s bd = 109. rad/s 53

54 rilis cceleratin: It has been seen that the acceleratin f a bdy may have tw cmpnents. entripetal acceleratin and Tangential acceleratin. Hwever, in same cases there will be a third cmpnent called as crilis acceleratin t illustrate this let us take an example f crank and sltted lever mechanisms. P P 1 Q 1 n link 3 d 3 1 n link d ssume link having cnstant angular velcity, in its mtins frm OP t OP 1 in a small interval f time t. During this time slider 3 mves utwards frm psitin t. ssume this mtin als t have cnstant velcity V /. nsider the mtin f slider frm t in 3 stages. 1. t 1 due t rtatin f link.. 1 t 1 due t utward velcity f slider V /. O 3. 1 t due t acceleratin r t link this cmpnent in the crilis cmpnent f acceleratin. We have rc 1 = rc Q rc Q 1 = rc Q rc 1 rc 1 = OQ d - O d = 1 1 d = V / dt 54

55 The tangential cmpnent f velcity is r t the link and is given by V t = r. In this case has been assumed cnstant and the slider is mving n the link with cnstant velcity. Therefre, tangential velcity f any pint n the slider 3 will result in unifrm increase in tangential velcity. The equatin V t = r remain same but r increases unifrmly i.e. there is a cnstant acceleratin r t rd. Displacement 1 = ½ at = ½ f (dt) ½ f (dt) = V / dt f cr / = V / crilis acceleratin The directin f crilis cmpnent is the directin f relative velcity vectr fr the tw cincident pints rtated at 90 in the directin f angular velcity f rtatin f the link. Figure belw shws the directin f crilis acceleratin in different situatin. f cr (a) Rtatin W slider mving up f cr (b) Rtatin W slider mving dwn f cr f cr (c) Rtatin W slider mving up (d) Rtatin W slider mving dwn 55

56 quick return mechanism f crank and sltted lever type shaping machine is shwn in Fig. the dimensins f varius links are as fllws. O 1 O = 800 mm, O 1 = 300 mm, O D = 1300 mm and DR = 400 mm The crank O 1 makes an angle f 45 with the vertical and rtates at 40 rpm in the W directin. Find: iii) cceleratin f the Ram R, velcity f cutting tl, and iv) ngular cceleratin f link D. Slutin: Step 1: Draw the cnfiguratin diagram. R Tl R 00 D n rank, O n D 45 D O Step : Determine velcity f pint. 56

57 V b = O x O O = NO1 60 x rad / sec V b = 4.18 x 0.3 = 1.54 m/sec Step 3: Draw velcity vectr diagram. hse a suitable scale 1 cm = 0.3 m/sec d b c r.a Step 4: prepare table shwing the acceleratin cmpnents Sl. N. 1. O DR Link Magnitude m/s Directin Sense f c b = r =5.4 Parallel t O O Parallel t r t f c ac = r f t ac = r f s bc =r f cc bc = v = f c bd = r = 0 f t bd = r Parallel t r t Parallel t DR r t D _ D _ 5. Slider R ft bd = r Parallel t slider mtin 57

58 r 1 f r 1 a 1 f t dr b 1 f c b f ad d 1 f c dr r 1 f t ab b 1 f b b 1 f s ab f cc bc b 1 cceleratin f Ram = fr = 1 r ngular cceleratin f link D bd = f bd D KLENIN S nstructin This methd helps us t draw the velcity and acceleratin diagrams n the cnstructin diagram itself. The crank f the cnfiguratin diagram represents the velcity and acceleratin line f the mving end (crank). The prcedure is given belw fr a slider crank mechanism. 58

59 00 ω 45º 800 O T draw the velcity vectr diagram: Link O represents the velcity vectr f with respect t O. V a = a = ω r = ω O. b b a 00 45º ω 800 a Draw a line perpendicular at O, extend the line t meet this perpendicular line at b. ab is the velcity vectr diagram rtated thrugh 90º ppsite t the rtatin f the crank. cceleratin diagram: The line representing rank O represents the acceleratin f with respect t O. T draw the acceleratin diagram fllw the steps given belw. Draw a circle with O as radius and as centre. Draw anther circle with as diameter. The tw circles intersect each ther at tw pints and D. Jin and D t meet O at b 1 and at E. O 1, a 1, b a1 and b 1 is the required acceleratin diagram rtated thrugh 180º. 59

60 b 1 f b O1 f t ab f ab f a ω a 1 b a1 f c ab a 00 45º ba1 800 O 1 b 1 60