3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or inrtia forcs), Dtrmin th input forc of torqu rquird to achiv a spcifid motion, and Dtrmin th motion as a consqunc of a spcifid st of forcs and/or torqus. Th first two cass ar xampls of invrs kinmatics, whrin w start with th motion and dtrmin th forcs. Th last is dirct kinmatics, whr w start with th forcs and dtrmin th motion. Txt Rfrnc: Dynamic or forc analysis of linkas is covrd in Chaptr 5 of th txt. 3.4.1 Inrtia Forcs and Torqus Considr a riid body in th x-y plan. W dfin th cntr of ravity of th body as that point about which th followin statmnt is tru: m r = i i r i whr m i is th mass of an infinitsimal lmnt of th body, r i is th location of this lmnt with rspct to th cntr of ravity, and th summation occurs ovr th ntir body. Onc w hav stablishd th cntr of ravity for th body, Nwton s scond law for th riid body can b xprssd in th followin simplifid form: F = Ma and r T = I whr is th sum of all xtrnal forcs actin on th body, M is th total mass of th body, and a r is th (translational) acclration of th cntr of ravity of th body. Similarly, T r is th sum of all xtrnal torqus actin about th cntr of mass, I is th mass momnt of inrtia of th body about th cntr of mass, and α r is th anular acclration of th body. For any body, w must apply both quations. This situation is illustratd in Fiur 3.28, whr th two xtrnal forcs, 1 and 2, hav bn rsolvd into a sinl rsultant,, actin at a point. Th total torqu (in th absnc of any concntratd torqus) is thn ivn as: T = F whr is th prpndicular distanc from th rsultant forc to th cntr of mass. α r 45
1 α r a r 2 1 2 Fiur 3.28: Planaiid body with applid forcs and rsultin acclrations. W nrally procd with a solution to this typ of problm by applyin D Almbrt s principl. That is, w form a kintostatic problm by rpalcin th acclrations by quivalnt inrtia forcs, which, whn addd to th xtrnally applid forcs, rsult in no acclrations. To dtrmin th appropriat inrtia forcs, w rwrit Nwton s scond law as: F Ma = F + F = and T I α = T + T = whr th inrtia forcs ar: F = Ma and r T = α r This is illustratd in Fiur 2.29. Somtims, this is calld th principl of dynamic quilibrium. I α r a r T r Fiur 2.29: Dynamic (lft) and kintostatic (riht) rprsntations of th problm from Fiur 2.28. In th lattr cas, thr is no acclrations. As an altrnativ to havin a sparat inrtia forc and torqu, w could simply apply th inrtia forc at a distanc,, from th cntr of mass, as illustratd in Fiur 2.3. 46
α r a r Fiur 3.3: Dynamic quilibrium with an inrtia forc applid at a distanc from th cntr of ravity to liminat th nd for a sparat inrtia torqu. 3.4.2 Analysis of four-bar linka whn only on link has mass Considr th four-bar linka in Fiur 3.31. In this cas, only th couplr is assumd to hav mass, and a torqu is assumd to b applid to th input link, 2. W will assum that w aquird to hav a known anular vlocity and acclration of th input link, and our job is to dtrmin th rquird input torqu, T r 2 to achiv th dsird rsult. T r T r 2 Fiur 3.31: Four-bar linka with spcifid vlocitis and acclrations. W start th analysis by dcomposin th linka and drawin fr body diarams for ach link, Fiur 3.32. Th forc xrtd by link i on link j is dnotd ij. Of cours, ths forcs ar qual and opposit so that: F ij = F ji In this xampl, sinc w hav a non-zro applid torqu applid to link 2 (th input link), and link 3 (th couplr) has a non-zro mass, only th output link (link 4) is a twoforc mmbr. For any two-forc mmbr, th forcs must act alon its axis. Thrfor, w know th dirction of forc 14 and 34 (but not th manitud). This mans that w hav thr unknowns actin on th couplr: th dirction and manitud of and th 47
manitud of 43. W can solv for ths unknowns usin th thr quilibrium quations availabl to us for any planar problm. W may choos to solv this ithr raphically or usin vctor componnts. T r 34 32 T r 2 43 14 12 Fiur 3.32: Fr body diarams for ach link. Graphical Solution To prform a raphical solution, w rdraw th fr body diaram for th couplr and rplac th inrtia forc and torqu by th inrtia forc actin at a distanc from th cntr of mass, Fiur 3.33. 43 Fiur 3.33: Fr body diaram of couplr. W know th dirctions and th point of application for forcs and 43. W can dtrmin th dirction of by rconizin that th only possibl way to achiv momnt quilibrium for a st of thr forcs actin on a planar body is for th forcs to all pass throuh th sam point. Whn w apply momnt quilibrium to a body, w ar fr to choos any point about which to sum th momnts. Choos th intrsction of 48
and 43, Fiur 3.34. Thn, must b such that it passs throuh this intrsction point, othrwis thr will b a nt momnt. This construction rsults from applyin momnt quilibrium to th couplr. W can us forc quilibrium to solv for th two unknown forc manituds. This is illustratd in Fiur 3.34 (riht). 43 43 Fiur 3.34: Constructions to stablish th dirction of manituds of and 43 (riht, nlard for clarity). (lft) and dtrmin th Onc ths forcs ar known, w can procd with a similar analysis of th othr links to obtain all of th unknown forcs. Vctor Componnt Solution Altrnativly, w could xprss th unknown componnts of th forc vctors in trms of a common (usually lobal) coordinat systm, and writ th thr quilibrium quations. This ivs thr quations in thr unknowns, which w solv for th unknown forcs. Onc ths forcs ar known, w can procd with a similar analysis of th othr links to obtain all of th unknown forcs as abov, for th raphical mthod. 3.4.3 Analysis of four-bar linka whn mor than on link has mass Th abov analysis only workd bcaus w assumd that only on link had a non-zro mass. This allowd us to dtrmin th dirction of th forcs actin on th output link in this cas, and lft us with only thr unknowns whn considrin quilibrium of th couplr. If th output link had a non-zro mass, w would not know th dirction of th forcs in this link, and w could not solv this problm dirctly as abov. In such a cas, w hav two options: suprposition or th matrix mthod. Suprposition If th othr links hav mass, w start th suprposition tchniqu by assumin that only on link has mass. W thn solv for all th forcs in th linka as abov. W rpat th procdur aftr assumin a diffrnt link has a non-zro mass (and assumin th link 49
whos mass w considrd in th first stp has no mass). This is rpatd for all links, and th rsultin forc in ach link is th sum of all th forcs for ach of th sparat cass. In ach cas, w ar solvin for th ffct of a link s mass on th forcs within th linka. Matrix Mthod In this mthod, w considr all of th unknown forc componnts simultanously, and apply quilibrium to all links simultanously. 5