.archivetemp4. Balancing of Masses 2.pptx

balancing of masses

1. Balancing of Masses By Prof. Charles M.M. Ondieki

2. B a l a n c in g The p r ocess of p r o v i d i ng the second m ass i n o r d er t o counte r act the e ff ect of the cent r if u g al f o r ce of t he fi r st m ass, i s ca ll ed ba l anc i ng of r ot a t i ng m asses. • T o e li m i n a te a l l shak i ng f o r ces and shak i ng m o m ents • A n y li nk or m e m ber th a t i s i n p u r e r ot a t i on can, theo r et i ca l l y , • be p e rf ect l y ba l anced to e li m i n a te a l l shak i ng f o r c es and shak i ng m o m ents. A r ot a t i ng m e m ber can be ba l anced e i ther st a t i ca l l y or d y na m i ca l l y . • Rot a t i ng p a r ts can, and g ene r a l l y shou l d , be d es ig ned to be i nhe r ent l y ba l anced b y the i r g eo m et r y . •

3. • H o w e v e r , t he v a g a r i e s ( un p r e d ic t ab le c han g e o r a c t i o n) o f p r o d uc t i o n to l er a nces g u a r a n t ee t h a t t here w il l s t il l b e s o me s m al l un b ala nce i n e a ch pa r t . Thus a b ala nc i ng p r o ce d ure w il l h a v e t o b e a ppli ed t o e a ch pa r t a f t er m a n u f a c t ure • The a m o unt a nd l o c a t i o n o f a n y i m b ala nce c a n b e me as ured q u i t e a ccur a t e l y a nd c o m p en s a t ed f o r b y a d di ng o r rem o v i ng m a t e r ia l i n t he c o r rect l o c a t i o n s. .

4. S T A T I C B ALA NC E • The r e q u ir e m ent f or st a t i c ba l ance i s s i mp l y th a t t he sum of f o r ces on the m o v i ng s y stem ( i nc l u d i ng d ‘ A l e m be r t i ne r t i al f o r ces) m ust be ze r o. a l l • D es p i te i ts na m e , st a t i c ba l ance d oes a pp l y to th i n g s i n m ot i on. • The unba l anced f o r ces of conce r n a r e d ue to the ac c e l e r a t i ons of m asses i n the s y ste m . • An other na m e f or st a t i c ba l ance i s s i n gl e- p l ane ba l anc e , w h i ch m eans th a t the m asses w h i ch a r e g ene r a t i ng t h e i ne r t i a f o r ces a r e i n, or nea r l y i n, the sa m e p l ane. • It i s essent i a l l y a t w o- d i m ens i onal p r o b l e m .

5. • S o me e x a m pl es o f c o mm o n d e v i ces w h i ch meet t h i s c r i t e r i o n, a nd t hus c a n s ucce ss fu l l y b e s t a t i c al l y b ala nce d , a re: • a si n g l e g e a r o r p u ll e y o n a s h a f t , • a • a b i c y c l e o r m oto r c y c l e t i re a nd w hee l , t h i n f l y w hee l , • a n ai r pla ne p r o p e ll e r , • a n i n d i v id u a l t ur b i ne b lad e - w heel ( b ut n o t t he en t i re t ur b i ne)

6. Th e c ommo n d e n omi n a t o r a mo n g t h ese d e v i ces i s t h a t t h e y a re a l l s h o r t i n t h e a xi a l d i rect io n c om pa red t o t h e r ad i a l d i rect io n , an d t hu s c a n b e c o n s i d ered to e x ist i n a s i n g l e p l an e. • A n au t omobil e t i re an d w h eel i s o n l y m a r g i na l l y s u i t e d st a t i c b a l an c i n g a s i t i s re a s o n a b l y t h i ck i n t h e ax i a l d i rect io n c om pa red to i ts d i a m ete r . to • Des p i te t h i s f a ct, au to t i res a re s om et im es st a t i c a l l y b a l an ce d . • M o re o f ten t h e y a re d y na mi c a l l y b a l an ced an d w il l b e d i sc u ssed und er t h a t t o p i c . •

7. N o t e t h a t t he o n l y f or ce s act i ng o n t h i s s y s te m a r e t he i n e r t i a f or ce s. F o r b a l a n c i n g , i t do e s n o t m a tte r w h a t e x te r n a l f or ce s m a y be act i ng o n t he s y s te m. Ex te r n a l f or ce s ca nn o t be b a l a n ce d b y m a k i ng a n y c h a ng e s t o t he s y s te m ' s i n te r n a l g e o m et r y . F o r b a l a n c i n g , i t a l so do e s n o t m a tte r h o w f a st t he s y s te m i s ro t a t i ng

8. D yna mic B a l anc e • A n y r o t a t i n g o b j ect o r a sse m b l y w h i ch i s re l a t i v e l y l o n g i n t h e a xi a l d i rect io n c om pa red to t h e r ad i a l d i re c t io n re qu i res d y na mi c b a l an c i n g f o r c om p l ete b a l an ce. • Dy na mi c b a l an ce i s s om et im es c a ll ed t w o p l an e b a l anc e • It re qu i re t h a t t w o c r i te r i a to b e m et • Th e s u m o f f o rces m u st b e z ero • A n d t h e s u m o f mom e n ts m u st a l so b e z ero

9. T o co r r ec t d y n a m i c i m b ala nc e r e q u i r e s e i th e r a dd i n g o r r em o v i n g the r i ght a moun t o f m ass a t the p r ope r a n g u lar l oc a t i on s i n t w o co r r ec t i o n p la ne s s ep a r a t e d b y s om e d i sta nc e al on g the sha f t. This will c r e a t e t he n ece ss a r y co un te r f o r ce s t o s t a t i c a l l y ba l a n c e t he s y s te m a nd a lso p r o v i d e a co un te r co u p le t o c a n ce l t he un ba l a n ce d m o m e n t .

10. S om e ex a mp l e s b ala nc i n g a r e : o f d e v i ce s w h i c h r e q u i r e d y n a m i c • roll e r s • c r a nk-sh aft s • ca msh aft s • a x l e s • c l us te r s o f m u l t ipl e prop e ll e r sh aft s. g ea r s, m o t o r ro t o r s, t u r b i n e s,

11. Ba l an c i n g o f S e v e r a l M asse s R o t a t i n g i n th e S a m e P l an e

12. E xa mple 21.1 . F ou r m ass e s m 1 , m 2 , m3 an d m4 a r e 20 k g , 30 k g , 24 k g an d 26 k g r e sp ec tiv e l y . T h e c o r r e spondin g r adi i o f r otatio n a r e . 2 m, . 1 5 m, . 2 5 m an d . 3 m r e sp ec tiv e l y an d th e a ngl e s b e t w e e n su c c e ssiv e an d 13 5 ° . m ass e s a r e 4 5 °, 75 ° F ind t he po si t i o n a nd mag ni t u d e o f th e balan c e m as s r e qui r e d , i f it s r adiu s o f r otatio n i s . 2 m

13. so l ut i on

14. S i n ce t h e m a g n i t ud e o f ce n t r i fu g a l f o rces a re p r o p o r t io na l to t h e p r o du ct o f e a ch m a ss an d i ts r ad i u s , Th ere f o re The p r o b l e m m a y , n o w , b e s o l v e d e i t h e r a n a l y t i c a l l y o r g r a p hi c a l l y . B ut w e sh a ll s o l v e t he p r o b l e m b y b ot h t he m et h od s o ne b y o n e .

15. ∴ R e s u l ta nt ,

16. W e kn o w t h a t S i n c e ′θ i s t he a ng l e o f t he r e su l ta nt R f ro m t he h o r i z o n ta l m a ss o f 20 k g , t h e r ef or e t he a ng l e o f t he b a l a n c i ng m a ss f ro m t he h o r i z o n ta l m a ss o f 20 k g ,

17. 1 . F i r st o f a ll , d r a w t h e s pa ce d i a g r a m s h o w i n g t h e p o s i t io n s o f a l l t h e g i v en m a sses a s s h o wn i n F i g ( a )

18. 2 . S i n ce t h e ce n t r i fu g a l f o rce o f e a ch m a ss i s p r o p o r t io na l to t h e p r o du ct o f t h e m a ss an d r ad i u s, t h ere f o re 3 . N o w d r a w t h e v ect o r d i a g r a m w i th t h e a b o v e v a l u es, to s om e s u i t a b l e sc a l e a s, s h o wn i n F i g . ( b ).

19. B a l a n c i n g o f S ev e ra l M a ss es R o t a t i n g i n Di f f e r e n t P l an es re f ere n ce p l an e: t h e p l an e pa ss i n g t h r o u gh a p oi n t o n t h e a xi s o f r o t a t io n an d p e r p e nd i c u l a r to i t. In o r d er to h a v e a c om p l ete b a l an ce o f t h e s e v er a l r e v ol v i n g m a sses i n d i ff er e n t p lan es, t h e f oll o w i n g t w o c o nd i t io n s m u st b e s a t i s f i ed : 1 . Th e f o rces i n t h e re f ere n ce p l an e m u st b a l an ce, i .e res u l t an t f o rce m u st b e z er o . t h e 2 . Th e c o up l es a bo u t t h e re f ere n ce p l an e m u st b a l an ce, i .e. t h e res u l t an t c o up l e m u st b e z ero .

20. Ex a m pl e 21. 2 A sh af t ca r r i e s f o ur m a ss e s A , B , C a nd D o f m a gn i t u d e 20 k g , 30 k g ,40 kg a nd 20 kg r e s p ect i v e l y a nd r ev olvi ng a t r a di i 8 mm, 7 mm, 6 mm a nd 8 mm i n pl a n e s m ea su r e d f ro m A a t 30 mm, 40 mm a nd 70 mm. The a ng l e s b et w ee n t he c r a nks m ea su r e d a n t i c lo c kw is e a r e A t o B 45 ° , B t o C 70 ° a nd C t o D 120 ° . The b a l a n c i ng m a ss e s a r e t o be pl ace d i n pl a n e s X and Y . The di s ta n c e b et w ee n t he pl a n e s A a nd X i s 10 mm, b et w ee n X a nd Y i s 4 mm a nd b et w ee n Y a nd D i s 20 mm. If t he b a l a n c i ng m a ss e s r ev ol v e a t a r a di us o f 10 mm, f i n d t h e i r m a g n i t ud es an d an g u l a r p o s i t io n s .

21. The di s ta n ce s o f t he pl a n e s t o t he r i ght o f pl a ne X a r e ta k e n a s + v e w h il e t he di s ta n ce s o f t he pl a n e s t o t he l ef t o f pl a ne X a r e ta k e n a s – v e .

22. 1 . F i r st o f a ll , d r a w t h e c o up l e p o l yg o n f r o m t h e d a ta g i v en i n T a b l e By m e a s u re m e n t, t h e an g u l a r p o s i t io n o f m Y

23. N o w d r a w t h e f o rce p o l yg o n f r o m t h e d a ta g i v en By m e a s u re m e n t, t h e an g u l a r p o s i t io n o f m X

24. E xa mple 21.4 . A, B , C an d D ar e fo u r m a ss e s ca r r i e d b y a r o t a ti n g s ha f t a t ra dii 100,125 , 20 an d 15 mm re s p ec t i v e l y . Th e pl ane s in w h i c h t h e m a ss e s r e v o l v e ar e s p ace d 60 mm a p a r t an d t h e m a s s o f B , C an d D ar e 1 k g , 5 k g , an d 4 k g re s p ec t i v e l y . Fi n d t h e re q u i re d m a s s A an d t h e re l a t i v e an g u l a r s e tti n gs o f t h e fo u r m a ss e s s o t h a t t h e s ha f t s ha ll b e in c o mpl e te ba l anc e

25. U N B A L AN C E LE C # 2 ) Th e i n e r t i a f o rce i s a n im a g i na r y f o rce, w h i ch w h en a cts up o n a r i g i d bo d y , b r i n gs i t i n a n e qu ilib r i u m p o s i t io n . ) T h e re s u l t an t o f a l l t h e f o rces a ct i n g o n t h e bo d y o f t h e e n g i n e d u e to i n e r t i a f o rces o n l y i s kn o wn a s un b a lan ced f o rce o r s hak i n g f o rce. ) i f t h e res ul t an t o f a l l t h e f o rces du e to i n e r t i a e f f ects i s z er o , t h e n t h ere w il l b e n o u n b a la n ced f o rce, b u t e v e n t h en a n un b a l an ced c o up l e o r s hak i n g c o up l e w il l b e p rese n t

26. ) t h e i n e r t i a t o r qu e i s a n im a g i na r y t o r qu e, w h i ch w h en a pp li ed up o n t h e r i g i d bo d y , b r i n gs i t i n e qu ilib r i u m p o s i t io n . ) Thu s, t h e pu r p o se o f b a l an c i n g t h e rec i p r o c a t i n g m a sses i s to e limi n a te t h e s hak i n g f o rce an d a s hak i n g c o up l e. ) In mo st o f t h e m ec ha n i s m s, w e c a n re d u ce t h e s hak i n g f o rce an d a s s h aki n g c o up l e b y a d d i n g a pp r o p r i a te b a l an c i n g m a ss, b u t i t i s u s ua l l y n o t p r a ct i c a l to e limi n a te t h em c om p l ete l y .

27. HORIZO N T A L RECI P R OC A T I N G E NG I N E MECH AN I S M

28. ) S i n ce F R an d F I a re e qua l i n m a g n i t ud e b u t o pp o s i te i n d i rect io n , t h ere f o re t h e y b a l an ce e a ch o t h e r . ) Th e h o r i z o n t a l c om p o n e n t o f F B ( i .e. F B H ) a ct i n g a lo n g t h e li n e o f rec i p r o c a t io n i s a l so e qua l an d o pp o s i te to F I . Th i s f o rce F BH = F U i s a n un b a l an ced f o rce o r s hak i n g f o rce. ) Th e f o rce o n t h e s i d es o f t h e c y li nd er w a ll s ( F N ) an d t h e v e r t i c a l c om p o n e n t o f F B ( i .e. F BV ) a re e qua l an d o pp o s i te an d t hu s f o r m a s hak i n g c o up l e o f m a g n i t ud e F N × x o r F BV × x .

29. ) E ff ect o f t h e rec i p r o c a t i n g pa r ts i s to p r o du ce a shak i n g f o rce an d a s hak i n g c o up l e. ) S i n ce t h e s hak i n g f o rce an d a s hak i n g c o up l e v a r y i n m a g n i t ud e an d d i rect io n du r i n g t h e e n g i n e c yc l e, t he re f o re t h e y c au se v e r y o b j ect io n a b l e v ib r a t io n s. ) b u t i t i s u s ua l l y n o t p r a ct i c a l to e limi n a te t h em c o m p l ete l y . In o t h er w o r d s, t h e rec i p r o c a t i n g m a sses a re o n l y p a r t i a l l y b a l an ce d .

30. L E C TU RE #0 3

31. P RIM A R Y AN D S ECO N D A R Y U N B A L AN CED FORCES OF RECI P R OC A T I N G M A SS ES Co ns i der a r ec i p r o c at i ng en g i ne mech a n i sm: a cce l er a t io n o f t h e rec i p r o c a t i n g pa r ts i s a pp r o xi m a te l y g i v en b y t h e e x p ress io n ,

32. I n e r t i a f o rce du e to rec i p r o c a t i n g pa r ts o r f o rce re qu i red to a cce l er a te t h e rec i p r o c a t i n g

33. )The p r i ma r y un ba l a n ce d f o r c e is max i m u m , w h e n θ = ° o r 180 °. )Thus, t he p r i ma r y f o r c e is max i m um t wi c e in o ne r e v o lu t i o n o f t he c ra nk. )The max i m um p r i ma r y un ba l a n ce d f o r c e is g i v e n b y

34. ) The s eco n d a r y un ba l a n ce d f o r c e is max i m u m , w h e n θ = °, 90 ° ,180 ° a nd 360 °. ) Thus, t he s eco n d a r y f o r c e is max i m um f o ur t i m e s in o ne r e v o lu t i o n o f t he c ra nk. ) The max i m um s eco n d a r y un ba l a n ce d f o r c e is g i v e n b y

35. ) F r om a b ove w e s e e t h at m ax im u m s ec o nd a r y unb a l a n ce d f o r c e i s 1 / n time s t h e m ax im u m p rim a r y unb a l a n ce d f o r c e . ) In c ase of m o d er a t e s p ee d s, t h e s ec o nd a r y unb a l a n ce d f o r c e i s so s m a l l t h at i t m ay b e n e g lecte d as c o m p a r e d t o p rim a r y unb a l a n ce d f o r ce . ) Th e unb a l a n ce d f o r c e du e t o r eci p r o c a ti n g m ass e s va rie s i n m ag n it ud e bu t c o n s t a n t i n d i r ecti on w h il e du e t o t h e r e vo l v i n g m ass e s, t h e unb a l a n ce d f o r c e i s c o n s t a n t i n m ag n it ud e bu t va rie s i n d i r ecti o n .

36. P a r tia l B alan c in g o f U nbalan ce d Pr i m a r y F o r c e i n a Rec ip r o c atin g E ngin e : T he pr i m a ry unb a l a nced fo r ce m a y be c o ns i de r ed a s t he c o mp o nent o f t he cen t r i f u g a l fo r ce

37. L e t T he m a ss B a t a r a d i us b b a l a nces t h i s unb a l a nce f o r ce w hen a c t ed i n o pp o s i t e d i r ec t i o n , p l a ced d i a me t r i c a ll y o pp o s i t e t o t he cr a nk p i n C . W e kn ow t h at ce n t ri f u gal fo r c e du e to ma s s B , h o ri z o n tal c om p o n e n t of t h i s fo r ce , T h e p ri ma r y fo r c e i s b a l a n ce d , i f

38. C onditio n fo r balan c in g th e p r i m a r y fo r c e

39. v ertic al c o m p o n e n t ce n trif u gal f o r c e : ) b u t t h e ce n t r i fu g a l f o rce ha s a l so a v e r t i c a l c om p o n e n t o f M a g n i t ud e: Th i s f o rce re m a i n s un b a l an ce d . .

40. ) T he m ax i mum va l ue o f t h i s fo r ce i s w hen : θ i s 90 ° a nd 2 7 ° ) w h i ch i s s a me a s t he m ax i mum va l ue o f t he pr i m a ry fo r ce

41. ) F r om t h e a b ove d i s c u ss i o n , ) In t h e fir st c as e , t h e p rim a r y unb a l a n ce d f o r c e a ct s a l o n g t h e li n e of s t r o k e . ) In t h e s ec o n d c as e , t h e unb a l a n ce d f o r c e a ct s a l o n g t h e p er p e nd ic u l ar t o t h e li n e of s t r o k e . ) Th e m ax im u m va l u e of t h e f o r c e r em a i n s sa m e i n b o t h t h e c as e s ) It i s t hu s o b v i o u s t h a t : ) a b ove met h od of b a l a n ci n g c h a n g e s t h e d i r ecti on of t h e m ax im u m unb a l a n ce d f o r c e f r om t h e li n e of s t r o k e t o t h e p er p e nd ic u l ar of li n e of s t r o k e .

42. A s a c o m p r o mi se le t a fr a cti on ‘c ’ of t h e r eci p r o c a ti n g m ass e s i s b a l a n ce d s u c h t h a t : Unb a l a n ce d f o r c e a l o n g t h e li n e of s t r o k e : a n d unb a l a n ce d f o r c e a l o n g t h e p er p e nd ic u l ar s t r o k e t o t h e li n e of R esu l ta nt unb a l a nced fo r ce a t a ny i ns ta n t :

43. I f th e balan c in g m as s i s r e qui r e d t o balan c e th e r e volvin g m ass e s a s w e l l a s r ec ip r o c atin g m ass e s , th e n :

44. E xa m p l e 22.1 A s i n g l e c y li nd e r r eci p r o c a ti n g e n g i n e h as ) s p ee d 240 r p m ) s t r o k e 300 m m ) m ass of r eci p r o c a ti n g p a rt s 50 k g ) m ass of r e vo l v i n g p a rt s at 150 m m r a d i u s 37 k g. If t w o -t h ir d of t h e r eci p r o c a ti n g p a rt s a n d a l l t h e r e vo l v i n g p a rt s a r e t o b e b a l a n ce d fi n d : 1. Th e b a l a n c e m ass r e qu i r e d at a r a d i u s of 400 mm . 2. Th e r e s i du al unb a l a n ce d f o r c e w h e n t h e cr a n k h as r o t a te d 60° f r om t o p d e ad ce n t r e

45. Re sid u a l : W h a t is l e ft o v e r