Part V: Velocity and Acceleration Analysis of Mechanisms

Transcription

1 Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton. Ths secton of notes wll be ve among the followng topcs: 1) Ovevew of velocty an acceleaton analyss of mechansms 2) Velocty analyss: analytcal technques 3) Velocty analyss: Classcal technques (nstant centes, centoes, etc.) 4) Statc foce analyss, mechancal avantage 5) Acceleaton analyss: analytcal technques 6) Acceleaton analyss, Classcal technques ME 3610 Couse Notes - Outlne Pat II -1

2 1) Ovevew of velocty an acceleaton analyss of mechansms: Impotant featues assocate wth velocty an acceleaton analyss: a. Knematcs: b. Types of equatons that esult c. Geneal appoach/stategy. Uses/Applcatons ME 3610 Couse Notes - Outlne Pat II -2

3 2) Velocty Analyss: Analytcal Technques The stana appoach to velocty analyss of a mechansm s to take evatve of the poston equatons w..t. tme. (Note, altenatve appoaches, such as those teme nfluence coeffcents, can be pefome by fst takng the patal evatve wth espect to an altenate paamete multple by the tme evatve of that paamete) The poston equatons that we conse ae peomnantly loop closue an constant equatons. The appoach wll take the evatves of these equatons, expan nto scala equatons an solve fo the unknowns (all poblems wll be lnea n the unknowns!). Example: 1) Devatve of a loop equaton: 2) Expan nto scala equatons: 3) Cast nto matx fom an solve: ME 3610 Couse Notes - Outlne Pat II -3

4 The fst step s to evew evatves of a vecto. = ˆ = ˆ v = t = t v = ˆ ω ˆ v = ˆ ω ( ) ( ˆ ) = ( ) ˆ ( ˆ ) Expan nto scala components: vx = cos( ) ω sn( ) v = sn ω cos y t ( ) ( ) t = e v = t = t e v = t e v = e e π v = e e ( ) ( ) () t ( e ) ( ) Expan nto scala components: v = cos sn v x y = sn ( ) ω ( ) ( ) ω cos( ) 2 ME 3610 Couse Notes - Outlne Pat II -4

5 . Expan nto scala components: v = cos sn v x y = sn ( ) ω ( ) ( ) ω cos( ) Expan nto scala components: v = cos sn v x y = sn ( ) ω ( ) ( ) ω cos( ) ME 3610 Couse Notes - Outlne Pat II -5

6 Explanaton of Velocty Tems o The Dynamcs of the Dukes: ME 3610 Couse Notes - Outlne Pat II -6

7 ME 3610 Couse Notes - Outlne Pat II -7 Fo a loop equaton: = e e e e e e e e =

8 Example 1: Velocty analyss ME 3610 Couse Notes - Outlne Pat II -8

9 1. Revew vecto moel an poston analyss: 2. Fom moblty: n=6, f1=7, f2=0,=> M=1 Ths means thee s one nput, so nput velocty an acceleaton ates ae known (fo the hyaulc cylne) 3. Ceate vecto moel (See fg.) 4. Count unk s: Fst, velocty: q2a, q2b, q3, q4, 5, q5 = 6 vaables, 5 = nput. 5. Take evatves of poston equatons, Solve ME 3610 Couse Notes - Outlne Pat II -9

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12 Example #2: ME 3610 Couse Notes - Outlne Pat II -12

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14 Example 3: ME 3610 Couse Notes - Outlne Pat II -14

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16 3) Velocty analyss: Classcal technques (nstant centes, centoes, etc.) Classcal technques fo velocty analyss consste peomnantly of gaphcal technques an etemnaton of nstant centes. The gaphcal technques nvolve awng velocty polygons that fom geometc equvalents of ou evatve loop closue equatons. Due to the fact that analytcal technques can be easly pogamme an fomalze, gaphcal technques have been lagely outate. Howeve, some of these technques pove a sgnfcant amount of nsght nto the poblem an wll be evewe befly hee. The technques evewe ae: Instant Centes Centoes Instant Centes of Velocty o nstant centes ae a pont common to two boes whch has the same velocty fo both boes (at that nstant n tme). The numbe of nstant centes fo a n- boy lnkage s: c=n*(n-1)/2. ME 3610 Couse Notes - Outlne Pat II -16

17 The nstant cente of two lnks connecte by a evolute s tval (t s that evolute). The nstant cente fo two lnks connecte by a sle s also smple (The cente of cuvatue of the sle axs). 1 2 I I 12 Fo boes not connecte mmeately by a lnk, the technque eles on Kenney s Theoem: Kenney s Theoem: Any thee boes n plane moton wll have thee nstant centes an they wll le on the same staght lne. ME 3610 Couse Notes - Outlne Pat II -17

18 Applyng Kenney s theoem to a fou ba: I 13 I 23 I 34 I 24 I 12 I 14 Applyng Kenney s theoem to a Sle-cank: ME 3610 Couse Notes - Outlne Pat II -18

19 I 24 I 13 I 23 I 34 I 14 I 12 A few examples of the use of nstant centes: ME 3610 Couse Notes - Outlne Pat II -19

20 Centoes: A Centoe s the cuve efne by the locatons of the nstant cente ove the ange of moton of a mechansm. Each possble nstant cente can ceate two centoes, foun by conseng the moton elatve to each of the two lnks efnng the nstant cente. One centoe wll be calle the fxe centoe an one a movng centoe. The centoes can then eceate the moton of the fouba by ollng (wthout slp) n contact wth each othe. As an example, foubas can be use to efne the pofle fo non-ccula geas (fo example ellptcal geas). ME 3610 Couse Notes - Outlne Pat II -20

21 4) Statc Foce Analyss (an Mechancal Avantage): Wth the ablty to analyze the velocty of a mechansm, a statc foce analyss can be ectly pefome. Ths pocess s base on the pncples of consevaton of enegy (o powe hee snce we assume the constants ae not tme-epenent) an supeposton. Fst, conse Mechancal Avantage (= ato of output foce to nput foce). Fo a consevatve system, (not a ba assumpton fo a well-esgne mechansm), the powe nto the mechansm equals the powe out: P n = P out Wth the powe nstantaneously efne as, P = F. V (Foce otte wth velocty) fo a tanslatng component o P = T. ω fo a otatng component. Then, the Mechancal Avantage s efne as: P n = P out T n. ω n = T out. ω out T out /T n =ω n / ω out Fo a otatng system (T an ω n the same ecton) an an F n. V n = F out. V out F out /F n =V n /V out Fo a tanslatng system (F an V n the same ecton) ME 3610 Couse Notes - Outlne Pat II -21

22 T n. ω n = F out. V out F out /T n = ω n /V out Fo a mxe system Example: Mechancal Avantage ME 3610 Couse Notes - Outlne Pat II -22

23 Statc Foce Analyss: To complete the statc foce analyss, fst apply the ea of mechancal avantage to evaluate the nput foce eque fo each apple loa. If loas ae gven on multple lnks, then evaluate the mechancal avantage fo these multple lnks. Secon, apply the pncple of supeposton (these poblems ae lnea n foce). Thus, the total nput foce s gven as the supeposton of the nput foces eque fo each of the apple loas. The last possble step to conse hee s constant foces (foces n the beangs). In lne wth the concept of statc foce analyss base on consevaton of enegy, one appoach woul be to epeat the pocess above fo evey beang, but nstantaneously elmnate the moton constant fom each beang, an solve fo the foce eque to enfoce ths constant. Fo example, to fn the x-component eacton of a beang, allow that beang to move (assgn t unt velocty) nstantaneously (.e., beang oes not change poston). Solve fo the mechancal avantage elatng the x-foce at that beang to all apple loas, an sum to get the total x-ecte eacton va supeposton. Ths metho s known as the metho of Lagange Multples. If only one o two eactons ae ese, t s elatvely easy to apply. If all eactons ae ese, t makes moe sense to apply the technques of knetostatc analyss (to be covee n upcomng topcs). ME 3610 Couse Notes - Outlne Pat II -23

24 5) Acceleaton Analyss: Analytcal Technques: As n velocty analyss, the acceleaton analyss of a mechansm s pefome by takng the evatve of the poston equatons w..t. tme. When acceleaton analyss s pefome, t s of patcula nteest to note the fame n whch evatves ae taken, snce Newton s secon equaton eques acceleaton wth espect to an netal fame (also calle Newtonan fame). In ou analytcal acceleaton analyss of mechansms, we wll ensue netal acceleatons by havng ou mechansm goune to an appopate netal fame. The secon evatve of the poston vecto equatons wll yel vecto equatons n acceleaton (secon evatves of constant equatons wll yel scala constants n acceleaton). The unknown acceleatons ae lnea an ae solve usng lnea algeba. The appoach pocees as follows: 1) Take secon evatve of a loop equaton: 2) Isolate unknowns on one se of equaton knowns on the othe 3) Expan nto scala equatons: 4) Cast nto matx fom an solve: ME 3610 Couse Notes - Outlne Pat II -24

25 ME 3610 Couse Notes - Outlne Pat II -25 The fst step s to evew the secon evatve of a vecto. ˆ ˆ = = ( ) ( ) () () ( ) ( ) a a a t t t t t t ˆ 2 ˆ ˆ ˆ ˆ 2 2 = = = = ω ω ω α ω ω ω = e ( ) ( ) () ( ) ( ) ( ) t t t t t t e e e e a e e e e a e e a = = = =

26 Expan nto scala components: Catesan: a = ˆ α a a x y = cos = sn ω ω 2ω ˆ ( ) ω sn( ) ( ) ω cos( ) Complex Pola: a = e e a a x y = cos = sn 2 e ( ) ω sn( ) ( ) ω cos( ) 2 e ME 3610 Couse Notes - Outlne Pat II -26

27 Explanaton of Acceleaton Tems o The Dynamcs of the Dukes: ME 3610 Couse Notes - Outlne Pat II -27

28 Fo a loop equaton: 2 3 = 1 4 ME 3610 Couse Notes - Outlne Pat II -28

29 Moe scusson of acceleaton: Examples of each tem n the acceleaton equaton. Lnea acceleaton tem Angula acceleaton tem Centpetal acceleaton tem Cools acceleaton tem ME 3610 Couse Notes - Outlne Pat II -29

30 Example 1: Acceleaton on 3-pt htch: ME 3610 Couse Notes - Outlne Pat II -30

31 Example 1 (cont.) ME 3610 Couse Notes - Outlne Pat II -31

32 Example 2: ME 3610 Couse Notes - Outlne Pat II -32

33 Example 2 (cont.) ME 3610 Couse Notes - Outlne Pat II -33

34 6) Acceleaton analyss: Classcal technques (acceleaton polygon metho) The classcal technques fo acceleaton analyss conssts peomnantly of gaphcal technques, ceatng a vecto acceleaton polygon to escbe acceleaton of multple ponts (an thee elatve acceleatons). Fom any gven polygon, two unknown scala components n acceleaton can be etemne (much lke ou loop equatons), an so the poblem pocees. Wth the ablty to solve systems of equatons on computes easly an accuately, ths metho s no longe use othe than to pove ntuton on the behavo of a system. ME 3610 Couse Notes - Outlne Pat II -34

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