1. A body will remain in a state of rest, or of uniform motion in a straight line unless it

Transcription

1 Pncples of Dnamcs: Newton's Laws of moton. : Foce Analss 1. A bod wll eman n a state of est, o of unfom moton n a staght lne unless t s acted b etenal foces to change ts state.. The ate of change of momentum of a bod acted upon b an etenal Foce (o foces) s popotonal p to the esultant etenal foce F and n the decton of that foce: F dmv ( ) dt Whee m s the mass and v s the veloct of the bod. Fo constant mass; m: F ma Whee a s the acceleaton of the bod. 3. To eve acton of a foce thee s an equal and opposte eacton. 1

2 Foce s a vectoal quantt F Paalelogam law of addton F Fo seveal concuent foces F F F F Fz F z

3 couple a) The moment vecto does not have a pont of applcaton. Hence, t s a fee vecto. b) The elatve poston vecto,, s n between an two ponts on the lnes of acton of the foces fomng the couple. c) The foce couple that ceates a moment M s not unque, e.g. thee ae othe foce couples that ma ceate the same moment. Moment of a foce M F 3

4 Resultant of nonconcuent foces F F M ( F) o 4

5 Foces n Machne Sstems Reacton Foces: ae commonl called the jont foces n machne sstems snce the acton and eacton between the bodes nvolved wll be though the contactng knematc elements of the lnks that fom a jont. The jont foces ae along the decton fo whch the degee-of-feedom ee eedo s estcted. F j - F jj 5

6 REACTION FORCES AT KINEMATIC PAIRS (JOINT FORCES) DE REE OF FRE EEDOM ROT TATIONAL FRE EEDOM TRA ANSLATIONA AL FRE EEDOM NAME JOINT SHAPE JOINT FORCE 5 3 Sphee between paallel planes z F Sphee n a clnde Clnde between paallel planes z z F M F F 3 0 Sphecal pa (Ball jont) z F z F F 3 1 Slotted sphee n a clnde z M z F F 1 Plane jont jt z M z M F 6

7 REACTION FORCES AT KINEMATIC PAIRS (JOINT FORCES) DEREE OF FREEDOM ROTATIONAL F REEDOM NAL T RANSLATIO FR REEDOM NAME JOINT SHAPE JOINT FORCE 0 Slotted sphee z F z F M F 0 Tous 1 1 Clndcal jont 1 1 Slotted clnde z z z F z F z F M F M F M F F M F M Revolute pa (tunng jont) Psmatc pa (sldng jont) z z M z F M F z F M M F F M 7

8 Foces n Machne Sstems a) Jont Foces b) Phscal Foces c) Fcton o Resstng Foce: d) Inetal Foces. 8

9 Fee-Bod Dagam 9

10 Statc Equlbum 0 F M 0 In space, these two vecto equatons eld s scala equatons: F 0 F 0 F z M 0 0 M 0 M z 0 In case of coplana foce sstems, thee ae thee scala equatons: F 0 F 0 M z 0 10

11 Two Foce Membe In a Two-Foce membe, the foces must be equal and opposte and must have the same lne of acton 11

12 Two foce and one moment membe: Two foces fom a couple whose moment s equal n magntude but n opposte sense to the appled moment 1

13 Thee Foce Membe 13

14 STATIC FORCE ANALYSIS OF MACHINERY Sstems wthout Resstng Foce 14

15 Lnk F 3 + l 0 (F 0) F 3 + l 0 (F 0) F 3 a cosθ l -F 3 a snθ l + T 1 0 (M Ao 0) Fo lnk 4: F 34 + l4 -F l4 cosη 0 (F 0) F 34 + l4 -F l4 snη 0 (F 0) F 34 a 4 snθ 14 + F 34 a 4 cosθ 14 + F 14 4 (cosηsnθ 14 -snηcosθ 14 )0 (M Bo 0) Fo lnk 3: Due to Thd Law: F 3 + F 43 0 (F 0) F + 0 F 3 - F 3 F 43 (F 0) 3 F 43 a 3 snθ l3 + F 43 a 3 cosθ l3 0 (M A 0) F 3 - F 3 F 43 - F 34 F 43 - F 34 15

16 F 3 F θ 13 F 43 -F 3 -F F 3 -F 14 cosη + F 34 cosθ cosφ 0 o -F 14 cosη + F 34 cosθ F 14 snη + F 34 snθ snφ 0 o - F 14 snη + F 34 snθ F 14 (cosη snθ 14 - snη cosθ 14 )+a 4 F 34 (snθ 13 cosθ 14 - cosθ 13 snθ 14 )0 And a F 3 (cosθ 13 snθ 1 -snθ 13 cosθ 1 ) + T 1 0 (F 3 - F 34 ) 16

17 In geneal: a) Fo two-foce membes ou don't have to wte the equlbum equatons. You can smpl state that the foces ae equal and opposte and the lne of acton concdes wth the lne jonng the ponts of applcatons. b) Fo two-foce plus a moment membes ou must wte the moment equlbum equaton onl. The two foces ae equal and opposte and the fom a couple equal and opposte to the moment appled. c) In case of thee o fou foce membes, the thee equlbum equatons ( ΣF 0, ΣF 0, Σ M 0) must be wtten. F Fv u M F ufv F(uv) u 1 α cos α + sn α j v 1 β cos β + sn β j u v (sn β cos α - cos β sn α) k O: u v sn (β - α) k M F F sn (β - α) k 17

18 a 4 u 4 F 34 u u 4 (-F14)v 1 (ΣM Bo 0) a u (-F 3 )u 3 + T 1 k 0 (ΣM Ao 0 ) Whch esult n: a 4 F 34 sn(θ 13 -θ 14 ) - 4 F 14 sn( η θ 14 ) 0 -a F 3 sn(θ 13 -θθ 1 )+T

19 aphcal Soluton 19

20 aphcal Soluton 0

21 1

22 Modes of Contact n Psmatc Jonts

23 3

24 Pncple of Supeposton 4

25 Sstems wth Resstng Foce 1. Statc Fctonal Foce : R 3 - μf 3 μ, s known as the coeffcent of statc fcton.. Sldng Fctonal Foce R 3 -μf 3, μ s the coeffcent of sldng fcton, whch s less than the coeffcent of statc fcton. Sldng fcton s also known as Coulomb fcton. Epements have shown that the statc o sldng fcton foce does not depend on the aea of contact. It depends on the tpes of mateals n contact, on the suface qualt n contact and the tpe of flm fomed between the contactng t sufaces. The coeffcent of statc t fcton s slghtl lage then the coeffcent of sldng fcton. 3. Vscous Dampng Foce R 3 -cv /3 Whee c s the coeffcent of vscous fcton and v 3/ s the elatve veloct. Vscous fcton assumes that thee s a flud flm between the two sufaces n contact. 5

26 fcton angle : fcton ccle. R 3 R F R + F f sn φ tan φ μ tan φ μ R M F F t T F 3 Fcton Toque 3 f 3 6

27 θ

28 8

29 -F 34 cos(φ)-μ 14 -F 14 0 ( F 0) -F 34 sn(φ) (F 0) μ 34 F 34 -M 14 -μ 14 a0 ( M B 0) -F 34 cos(φ) () -F 34 sn(φ) () 5F 34 -M (M n Nmm) Unknowns ae F 34, φ, 14 and M 14. () 9

30 F 43 -F 34 -F 3 (M A 0): - AB F 43 sn(φ θ 13 )-μ 3 F 3 -μ 34 F 43 0 o: -800F 43 sn(φ θ 13 ) - 5F 3-5 F 43 0 snce F 43 F 3 (n magntude): -800F 43 sn(φ θ 13 ) -30 F

31 F 3 -FF 3 -FF 34 FF F 3 (M Ao 0): A 0 A F 3 sn(φ θ 1 )+μ 3 F 3 + μ 1 1 +T 1 0 whch can be smplfed as (notng F 43 F 3 ) 00 F 43 sn(φ θ 1 )+30 F 43 +T 1 0 (v) 31

32 When thee ae seveal foces actng on dffeent lnks: 1. Pncple of supeposton cannot be used.. Some of the equatons ae not lnea fo the unknowns. Theefoe, numecal teatve solutons ae equed. In mechancal sstems, f the desgne has taken some good desgn measues, fcton can usuall be neglected n evolute jonts wth sze small compae to lnk length dmensons (ths usuall smplfes the soluton) 3

33 DYNAMIC FORCE ANALYSIS DYNAMIC FORCE ANALYSIS Cente of Mass s commonl known as the cente of gavt m Cente of Mass,, s commonl known as the cente of gavt, Σ m /m m Σ m /m Moment of neta + 0 m )m ( I m I k )m v (u I + m ) v (u I ( ) ( ) [ ] m v u m I ) ( m v m u m m v u I ) ( ) ( 0 ) m(k m I I Paallel As Theoem

34 Newton's Second Law of Moton fo a Rgd d Bod F d F j m a (m ) + j dt F j s commonl known as the ntenal foce. Fo all patcles: F + Fj m a j F F F j 0 j d ( dt m ) sum of all etenal foces actng. m m m Newton's Second Law of moton fo Lnea momentum d (m ) d(mv ) F ma dt dt 34

35 + j j m a F F a a 0 + a /0 a 0 + a /0 n + a /0 t Moment equlbum a /0 t α a /0 n -ω ( ) [ ] ω + α + 0 j j m a m a F F M F 0 M F j j 0 F j ( ) [ ] ω α + ω + α 0 0 m m a m m a 0 0 a m a m α α α 0 I m m 0 m ω α I a m M Newton's Second Law of Moton fo Angula Momentum ω θ ) d(i I d 35 α ω θ I dt ) d(i dt I d M

36 D'Alambet's Pncple FF ma 0 M Iα 0 F ma A nonestant (fcttuous) foce neta foce T Iα A nonestant (fcttuous) toque neta toque Consdeng Ineta foce and toque as f an etenal foce o toque F 0 M 0 D'Alambet's Pncple In a bod movng wth a known angula acceleaton and a lnea acceleaton of the cente of gavt, the vecto sum of all the etenal foces and neta foces and the vecto sum of all the etenal moments and neta toque ae both sepaatel equal to zeo. 36

37 Eample + θ a 3 e cos θ ω13 a a sn θ 3 3 a sn θ a ω cos θ 3 ω13 cosθ a sn θ α13 3 a ω sn θ + a α cos θ v a a a + e 3 θ ω 13 e θ 1 θ θ a e ( 3 α13 ω13 ) F 3 ma 3 T 3 I3α13 37

38 38

39 Fo Lnk 3 F 43 -F 3 0 R 3 -F F 43 sn( ) + 50(5.97)sn( )

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