Area Under the Torque vs. RPM Curve: Average Power

Transcription

1 Are Uner the orque vs. RM Curve: Averge ower

2 Wht is torque? Some Bsics Consier wrench on nut, the torque bout the nut is Force, F F θ r rf sinθ orque, If F is t right ngle to moment rm r then rf

3 How oes the ire pply torque to the ro? Cr of mss M F r FM, is the ccelertion of the cr hen rfrm

4 V r F he liner velocity of center of the wheel n the cr s velocity Is given by VrZ Where Z is the ngulr or rottionl spee of the wheel in rins/sec or Z*S*RM/60

5 Now for bit of Clculus n bsic Formuls V t Using v V r t rm r FV r eqn. eqn. eqn.3 eqn. 4 eqn.5 Where : ccelertion orque V velocity r rius of wheel ngulr spee of wheel power, Force times velocity

6 Averge ower Over Given ime Intervl t t t t t t I will lter relte this to verge Accelertion n finl velocity It woul be useful to hve the verge power eqution in terms of the corresponing Shift points or n. hen we coul relte yno curve orque vs. or RM to the verge power & ccelertion. hen: Combining eqns t r M n 3 gives :

7 Averge ower in erms of Or RM t t Now recll tht : FV Fr An : F M hen : Mr, Using rm r M r, Mr t, Mr t t t, t Mr

8 he finl result is hen, t t

9 Now for n ssumption- Assume tht the torque is given by some verge torque n smll term which vries with or RM I will lter show tht this ssumption is not tht restrictive: Where :, n <<

10 Grphiclly his Is orque or RM

11 o exmine this term.

12 mens"pproximtely"or "close enough" symbol An the... when x is smll: use for Where the series expnsion is x x x x x

13 retin terms only lrger thn will I.0 think bout it. very smllnumber squre is tht is smll very smll is then is smll, Becuse

14 x for smll x or x Agin using the series expnsion for

15 sy 000RM n 6000 RM constnt once the shift points re etermine, is becuse Constnt given time intervl is the verge power over the esire result, his is or or the verge of is Where Now Reclling tht ϖ ϖ ϖ ϖ ϖ

16 ϖ o get n expression for the verge ccelertion, F M, rm rf rm, then note tht : his shows tht the verge ccelertion over time intervl is rm Constnt * So for given cr with given mss, wheel geometry n fixe shift points, the verge ower n the verge ccelertion is proportionl to the AREA UNDER HE ORQUE CURVE. Mximize the re uner the torque vs. RM curve n you Will mximize ccelertion n finl spee.

17 OSIIVE DISLACEMEN vs. CENRIFIGAL BLOWERS It is cler from the grphicl exmple tht positive isplcement blowers hve very smll term becuse they ten to prouce reltively flt torque curve over the RM rnge. However, it is not cler tht for Centrifugls hving torque curve builing linerly with RM tht the preceing erivtion is vli.so lets check it with n exmple- Hint: his is where I will show tht the ssumption tht torque is given by some verge torque n smll term which vries with w or RM is not restrictive n the Are Uner the torque rule curve pplies even for the Centrifugls

18 An Exmple Centrifugl orque Curve his exmple is tken from rel yno t poste on moulrfors.com, probbly mustng G from the cutoff RM of 6000 Fitte orque function: RM : RM RM RM RM Actul Dt RM : 000 RM : 6000 : 5 : 373 orque ft-lbs 00 Rtio of / Fitte Dt R RM

19 With n exct eqution, I cn clculte the ifference between the exct expression for verge power n the pproximte eqution bse on the re uner the toque curve: 60 : Reclling tht ϖ π ϖ ϖ RM Exct Approx ln RM RM RM RM Exct RM Approx

20 Approx Using R RM gives : Exct RM ln Approx RM R Exct RM R ln R hese o not look like the sme thing but re they close?...yes! Nee to show R/ ~R-/lnR for resonble vlues of R sy R-4

21 Nee to look for series expnsion for R-/lnR: R ln R : R R 4 R R 4 For resonble vlues of R this cn be pproximte by: R R R ln R So tht R/ ~R-/lnR for resonble vlues of R

22 Sme result! Exct n pproximte nswers re very close ~5% Even for torque rtios of 4! At torque rtios of 373/53 they re 0% ifferent but the pproximte gives the higher nswer more re uner the torque curve Bottom line- Are Uner the torque rule curve pplies even for the Centrifugls