Single-Plane Auto-Balancing of Rigid Rotors

Transcription

1 TECHNISCHE MECHANIK Bad 4 Hef (4) -4 Mauspegag: 4. Novebe 3 Sgle-Plae Auo-Balacg of gd oos L. Spelg B. h H. Ducse Ths pape peses a aalcal sud of sgle-plae auoac balacg of sacall ad dacall ubalaced gd oos cosdeg also he effec of paal ubalace copesao ad vbao educo. We cosde a oo equpped wh a self- balacg devce cossg of a ccula ac wh ovg balls o copesae fo oo ubalace. The vesgaos clude a aalss of he equaos of oo ad deeao of codos fo esece ad sabl of schoous oos. Dffee soluos fo he esece codos coespod o dffee pes of schoous oos cludg copesao oos wh he elees posos povdg coplee o paal copesao of ubalaced foces as well as educo of vbaos. A sabl aalss seves o deee he acual agula poso of elees a a oaoal speed ad o fd he speed age wh sable ubalace copesao. Nuecal sulaos cof he aalcal esuls ecep fo hose he edae vc of ccal speeds. Ioduco I 93 Theale oduced a balace sse equpped wh a pa of feel-ovg balacg balls. Aaged a he plae of ubalace of a sacall ubalaced oo he balls auoacall gavae owads he poso ude specfc codos ha copesaes fo he ubalace. Sce he sgle-plae auoac balacg has becoe a well-ow ehod. Howeve ecel hs ehod has aaced ceased aeo boh fo he heoecal po of vew ad fo he applcao po of vew. The ehod s advaageous paculal fo oos wh vaable ubalace such as washg aches cefuges had-held powe ools ad CD-OM dves. A ube of eseach goups vaous coues ae a pese vesgag hs ehod deal. ecel seveal publcaos have evealed soe poa aspecs of auo-balacg. The publcao Chug ad o (999) fo sace aalsed he dac sabl ad e espose fo a auoac sgle-plae balace as a fuco of he sse paaees. Kag e al. () evaluaed he pefoace of a ball-pe balace sse salled hgh-speed opcal ds dves. The esablshed aheacal odel was aalsed b he ehod of ulple scales. Geeal desg gudeles wee suggesed o he bass of possble sead-sae soluos ad he esuls of sabl aalses. Huag ad Chao () also placed ephass o he desg of a ball-pe balace sse fo a hgh-speed ds dve ad vesgaed he depedece of posoal eos o he uwa eccec ollg essace ad he dag foce due o dac eaco bewee he ball ad flud-flled uwa; he esuls of epees wee also dscussed. Howeve s a esablshed fac ha a geeal gd oo has sac ad dac ubalaces. Hece 977 Hedaa ad Shap (977) geealsed he ball balace devce b poposg a devce coag wo plaes wh wo balls each. Followg he equaos of oo a sabl aalss was peseed ad seveal eleva eds esablshed fo paaec sudes. Sla esuls wee also obaed b Ioe e al. (979). I Böv ad Högfos (986) he auhos vesgaed a eaple of a o-plaa oo sse faclag he oo of wo elees he balacg devce also he aal deco. The wo-plae auo-balacg devce fo gd oos was fuhe vesgaed b Spelg e al. The appled he ehod of dec sepaao of oo (see fo eaple Bleha ) o develop he codos fo esece ad sabl of he balls oo schoous wh ha of he oo. The coespodg sable phases wee also deeed. The esuls wee cofed ad suppleeed b copue sulao. Spelg e al. () whch balls wee eaed as pacles povded he fs sple aalcal esul deosag he fac ha copesao of boh sac ad dac ubalaces he sogl pos-ccal age (whee all spg foces a be egleced) s ol possble fo log oos.e. hose wh a pola oe of ea salle ha ha abou he asvese as hough he ass cee. Ths esul was eeded Spelg e al. () o he case of balls wh a fe oe of ea ollg aoud he ac whou slppg. Ths publcao cluded devao of he full sse of equaos of oo wh o-oag (bu vbag) asses also ae o accou a aalcal appoao ad esuls of uecal sulaos. Fuheoe aalsed he fluece of sse paaees such as dapg o he opeao of he devce. Fall a odfed veso of he Soefeld effec was deosaed wheeb he balls aa oo a a speed coespodg o he oo s

2 egefequec. Wh he spg foces also beg ae o accou he aalcal pa of he pape b Spelg e al. () shows ha full copesao of oo ubalaces s possble ol fo oos wh a pola oe of ea salle ha he asvese oe (.e. fo oos wh a secod ccal speed) he fequec age beod he secod ccal speed. Sulaos llusae he oo u-up o oal speed wheeb full ubalace copesao s acheved. The codos fo full ubalace copesao b eas of he above-eoed wo-plae balacg devces cause ajo escos fo applg he ehod pacce. Theefoe he pese pape s dedcaed o he poeal of paal ubalace copesao ael a fs ol usg a sgle-plae devce. The dffeeal equaos of oo ae eploed fo a odel oo wh balls dsbued o a ube of plaes o deve he geeal codos of esece ad sabl fo he balls oos schoous wh he oo s agula speed. Vaous cases of paal copesao usg sgle-plae balaces ae vesgaed ad dscussed. I pacula he vesgao evealed ha ude cea codos auo-balacg devces ca educe he appae ubalaces ad vbaos wh a fequec age beod he fs ccal speed eve f he oo s pola oe of ea eceeds he asvese oe. The esuls of he aalcal vesgao wee vefed b uecal sulaos fo oo speeds a suffce dsace fo he ccal speeds. Fo he aeas ea ccal speeds he sulaos show pcal oschoous ase oos whch as a ule ece ceased vbaos. Model Fg. shows a gd oo wh a sgle-plae auo-balacg devce. The asec oo has ass ad oes of ea a wh espec o he cee of ass S he o-oag veco fae e e e wheeas q V [ ] T ae he coodaes of he vbaoal oo (see Fg. a). The oo has ubalaces he plaes ( ) heeafe efeed o as hee ubalaces dealsed as pacles wh asses ad ecceces κ. The agula veloc of he oo s ; he agula posos of he hee ubalaces ae α κ. As log as ou a s o cosde eveuall vaous pes of devces we assue ha he geeal devce odel coas decal balls dsbued wh a ube of plaes (au ube ). The balls ae chaacesed b he asses he ad he ecceces ad he plae posos. The oo of he balls he auo-balacg plaes s descbed b he agula coodaes ad ϑ (see Fgs. ad b). Assug ha he balls oll aoud he acs whou slppg (see Fg. b) we a we ϑ ( ).... () e S e e Fgue. gd oo wh a sgle-plae auo-balacg devce

3 a) e e e b) ϑ e e e Fgue. Ma sse vaables The dapg oe caused b he vscous edu acg o he ball s wh d ( ϑ ) d ( l d ) () esulg fo he vscos of he edu. The coespodg geealsed foces fo he oo ad ball ae he oes Q ( ) β β ( Q ) whee β d. (3) Tag eeal dapg also o accou he oveall dapg oe acg upo he oo s M d β β ( ) β β whee β β β. (4) 3 Equaos of Moo The oo s assued o be oued o wo soopc elasc daped suppos wh sffesses ad dapg facos c c c wh espec o he vbaoal co-odaes q V. Usg he abbevaos M b... (5) 5 a a b b (6) ~ b ~ (7) 7 b 5 ad cludg he oo dvg oque L ( ) L... (8) we oba he followg Lagage s equaos fo he sse ude vesgao leased he vbaoal co-odaes q V (see Spelg e al. ) 3

4 (9) ( c c M cos s ) ( ) [ ] ( ) ( ) cos s ~ ~ s cos ~ ~ cos s a c c () () ( c c M s cos ) ( ) [ ] ( ) ( ) s cos ~ ~ cos s ~ ~ s cos a c c () (3) [ ] ( ) L cos s β β [ ] β β cos s... (4) wh.... (5) 4 Geeal Codos fo Esece ad Sabl of he Balls Moos Schoous wh he oo s Agula Speed Assug a cosa oo agula veloc cos. he equaos of oo of he balls (4) becoe β B β [ ] B... cos s. (6) Followg he ehod of dec sepaao of oo we suppose ( ) ξ α () () (7)... wh he slowl vag e () α () α << ad he π - peodc fas e wh a vashg aveage ove a peod of he fas e. ( ξ ) Thus Eq. (6) elds V α β α (8)... wh he so-called vbaoal oes ( ) π B d π V (9) 4

5 whee B [ s ( α ) cos( α )].... () To oba a appoao fo sead-sae vbaos we eglec he ubalace asses ad ea oes as well as he dapg es Eqs. (9) - () ad subsue Eq. (7) eglecg fas es o ge (wh sead of ~ o splf oao) M f cos( α ) () a ( α f cos ) () ( α M f s ) (3) a ( α f s ) (4) wh he cefugal foces f.... (5) Eqs. () (4) eld he saoa obal oo of he cees of he ccula ball pahs f A cos ( α ) f A s α ) (6) ( o cos s cos s (7) ξ η η ξ wh he copoes a fae ξ η oag wh he oo ξ f A η ad he haoc fluece coeffces whee A M cosα f A sα... (8) ( ) [( M ) ] { a } (9) 4 ( ) [ M ( ) ] a Thus we oba fo Eqs. (9) () (6) (7) V B f f A s a ( α α ) f ( sα cosα ) ξ η. (3).... (3) The balls schoous oos of ees have cosa phases α α codos fo such oos followg fo Eqs. (8) ad (3) ae.... Hece he esece V ( α α... α α ) f f A s( α α )... (3) 5

6 whee α α.... Obvousl because of Eq. (3) oe soluo pe of he esece codos (3) s deeed b he codos ξ η... (33) fo whch fo Eq. (7) edael follows..... (34) Ths eas ha fo hs pe of soluos he cees of he devce plaes ea a es. Theefoe we call hs pe of soluos copesao pe soluo. Noce ha he case of a sgle-plae devce hs pe of soluos ol esues he dsappeaace of vbaos he plae of he devce whle he case of a wo-plae devce he copesao of he ubalaces s coplee ad vbaos ae heoecall equal o eo fo he whole oo. To evaluae he sabl of he vaous soluos o Eq. (3) he equaos vaaos α β α f f A cos( α α )( α α ) α f A cos( α α )... followg fo Eqs. (8) ad (3) eed o be aalsed. Subsuo of Eq. (33) ude cosdeao of Eq. (8) elds he followg sple fo of he equaos vaaos fo soluos of copesag pe α β α f f A cos( α α ) α. (36) Assug ha β > he posve defeess of he sffess a of Eq. (35) (o of Eq. (36) he specal case of copesag soluos) s a ecessa ad suffce codo fo he aspoc sabl of a soluo of Eq. (3). We ca sugges also aohe vew o copesao codos wg sead of Eq. (8) f A cos ˆ ξ α ξ (35) η f A sα ˆ η... (37) whee ξ ˆ cos ˆ α f A η ˆ cosα ˆ ˆ s ˆ α f A sα. (38) To defel deee αˆ we pedefe ˆ >. Specfcall ˆ s he cosa oag dsplacee he plae of he -h ball caused b he hee (pa) sac ad dac oo ubalaces as a whole. Hece he esece codos fo schoous oos ae f A s ( α α ) ˆ s( α ˆ α ).... (39) Because of f A cos ( α α ) ( ) ˆ cos α ˆ α (4) we oba fo Eq. (35) he aleave fo of he equaos vaaos α β α f f A cos ( α α )( α α ) α ˆ cos( α ˆ α ).... (4) We would le o po ou ha he soluos of he esece ad sabl codos ae vald ol a a suffce dsace fo he ccal oo speed. 6

7 5 Sgle-plae Balace 5. A Sgle Ball A sgle ball aloe s o able o guaaee a soluo of he copesao codos. The esece codo followg fo Eq. (39) ( α αˆ ) ˆ s (4) has ol he wo soluos ˆ ˆ α α α α π. (43) The equao vaao ( α ˆ α ) α β α f ˆ cos α (44) whee f s he cefugal foce of he ball elds he sabl codo ( αˆ ) cos α >. (45) ˆ ˆ Thus he soluo α α s alwas sable (see Fg. 3) ad he soluo α α π s alwas usable (see Fg. 4). Ths eas ha he ball alwas follows he vbao he plae of he devce.e. he agula poso of he ball cocdes wh he phase of vbao because accodg o Eqs. (39) ad (4) he vbaoal oe egedeed b he ball s eo. a) b) < κ < < ˆ < κ < < ˆ α α c) <κ d) κ < > ˆ > ˆ α α α Fgue 4. Usable ball poso fo a sgle-ball devce Fgue 3. Dffee cases of sable ball poso fo a sgle-ball devce. Fo he sable soluo we oba ( fa ˆ ) cosα ( fa ˆ ) s ˆ α (46) ξ ˆ η η fa ξ ˆ. (47) The codo fo deceasg vbaos he plae of he devce s ˆ ˆ fa < 7

8 o ˆ f < A <. (48) Ioducg he fuco whee ( ) ( ) ( ) ( ) b fa κ ( ) (49) ˆ ˆ ( ) fa ( ) b s he sged defleco caused b he foce f we oba he codo fo sable deceasg of he vbao due o he sgle-plae self-balacg devce wh a sgle ball he fo (see Fg. 3 a) ad b)) ( ) < κ <. (5) Wh ohe ages of oaoal speed he vbaos wll cease (see Fg. 3 c) ad d)). (5) 5. Sgle-plae Devce wh Two Idecal Balls I he case of a sgle-plae devce ( A A... ˆ ˆ ˆ α ˆ α ) wh wo decal balls ( f f f ) he esece codos fo copesao soluos (33) a be ewe as o ξ fa( cosα cosα ) ˆ cos ˆ α fa ( sα sα ) ˆ s ˆ α η cos s ( α ˆ α) cos( α ˆ α) ( α ˆ α ) s( α ˆ α ) wh he ol soluo ˆ fa (5) (53) ˆ α ˆ α γ α α γ Ths soluo ess ol f ˆ γ accos. (54) fa ˆ fa. (55) The equaos vaaos (36) becoe α β α f Acos( α α ) α. (56) Thus he sffess a fo hese equaos s S f A wh he deea cos 4 cos( α ) ( ) α α α ( α α ) de S f A s. (58) (57) 8

9 Ude he codo s( α ) α Eq. (58) elds de S > ad we oba he ecessa ad suffce sabl codo A [ ( a ) ( M ) ] <. (59) Ioducg he fuco κ ( ) ( ) ( ) ( ) b fa (6) ˆ ˆ hs codo ogehe wh codo (55) elds (see Fg. 5 b)) κ ( ). (6) I he specal case of ol sac coceaed hee ubalace he devce plae he equal A < (see Eq. (59)) eas ha he deco of ˆ s oppose o ha of he cefugal foce due o hee ubalace. All ohe soluos of he geeal esece codos fa fa s s ( α α ) ˆ s( α ˆ α) ( α α ) ˆ α ˆ α ) ae of he pe s( α ) α. The geeal equaos vaaos (4) ae α β α f α β α [ fa cos( α α )( α α ) ˆ cos( α ˆ α) α] f [ fa cos( α α )( α α ) ˆ cos( α ˆ α ) α ]. (6) (63) a) Soluo α ˆ α α ˆ α The copoes (37) he oag fae becoe Hece ( fa ˆ ) cosα ( fa ˆ ) s ˆ α. (64) ξ ˆ η fa ˆ η ξ. (65) The vbaos decease whe o (see Fg. 5 a)) < κ < < ad κ A ( ) > ( ). (66) Copaed o codo (6) hs codo eas ha he ball asses ae suffce o copleel copesae he hee ubalace. The sffess a fo he equaos he vaaos becoes S f fa fa ˆ The cosdeed soluo s sable f fa. (67) fa ˆ fa ad ( fa ˆ ) (68) ˆ > ˆ > o f fa ˆ (69). e. > κ ( ) >. (7) 9

10 Hece he case (66) wh deceasg vbaos s alwas sable. Howeve he poso of he balls deeed b he codo < κ ad causg ceased vbaos s also sable (see Fg. 5 c)). ˆ b) Soluo α ˆ α π α α π The sffess a fo he equaos he vaaos becoes S f fa fa ˆ The sabl codos ˆ > fa. (7) fa ˆ fa ( fa ) ˆ (7) ˆ > coadc each ohe. Thus hs pe of ball oo s alwas usable (see Fg. 6 a). a) < κ < < b) κ < ˆ a) α α γ γ α c) <κ ˆ > b) α α Fgue 5. Dffee cases of sable posos of he balls fo a wo-ball devce. Fgue 6. Usable posos of he balls fo a wo-ball devce. ˆ ˆ c) Soluo α ˆ α π α α (o α ˆ α α α π ) The sffess a fo he equaos he vaaos becoes fa S f fa ˆ Ths a s posve defe whe fa. (73) fa ˆ fa >. (74) ˆ > ˆ Because he secod codo s possble hs pe of oo s also alwas usable (see Fg. 6 b).

11 6 Ivesgao of he Codo A < Fo he gd oo ha geeal s boh sacall ad dacall ubalaced ad has a sgle-plae selfbalacg devce (a ) a suve wll be ade of he vaous cases of sasfg he sabl codo (59) N A > (75) fo he phasg of wo balls fo whch he vbaos he balacg plae wll be copesaed - ehe copleel o fo balls of suffce ass paall (see Eq. (66)). Ths s a he sae e he ecessa codo fo deceasg vbaos b eas of ol a sgle ball (see Eq. (48)). The followg foulae ad gaphcs should asss egees volved auoac balacg of oos classfg he specfc oos ad desgg appopae balacg devces. The deoao ad ueao (75) ae M N 4 ( ) [ M ( ) ] a a (76) ( ) ( M ). (77) a Ioducg he squae of he oaoal ccal speed a (78) he squae of he aslaoal ccal speed (79) M ad he squae of he eacoal speed we oba M ( ) a (8) 4 ( a )[( )( ) ] M (8) ( )( ) M ( ) M ( ) N. (8) a a We assue. I s advaageous o use he followg paaees: sffess ase whee σ (83) < σ < σ eccec of he balacg devce ; (84) ; (85) ad oo-pe paaee a µ. (86) M

12 Thus we oba 4 [( )( µ ) σ ] M (87) N [( µ ) ( σ) ]. (88) Fo he codo we oba he squaes of he ccal speeds ± ( ) µ µ 4µσ µ (89) whee he lowe algebac sg of he oo efes o ad he uppe sg o. Codo (75) s fulflled he followg wo cases: Case : > ad N > o (9) Case : < ad N <. (9) The codos ude Case ae decal wh he sabl codos of copesao phasg of wo balls each wo plaes (see Spelg e al. () Spelg e al. ()). The ae sasfed f ad ol f > >. (9) a I he followg addoal codos have o be developed whch esul fo he codos ude Case. The log oo The log oo s defed as oe wh > µ >. (93) a Ude hs codo he oo has wo ccal speeds: < < (94) ad he uppe ego of sable copesao does es. The fs codo of (9) < s fulflled fo < <. The coespodg lowe ego of sable b b copesao < < s bouded above b he bouda speed fo whch we oba fo N σ b. (95) µ Because of σ < hs epesso s alwas posve: Fuheoe because of ( ) ( σ) b >. (96) µ b σ ad followg fo (97) ( ) ( µ ) (98) sg σ we oba ( µ )( ) N sg σ µ

13 b sg σ µ (99) µ ( µ )( ) N sg σ µ b sg σ µ. () µ The ecceces ad fo whch b ad b especvel esul µ µ sg σ µ µσ σ σ ( µ ) 4 µ µ sg σ µ 4 σ σ ( µ ) µ σ The ase σ σ fo whch b ( σ σ) o b ( σ σ ( µ ) ( µ ) 4 σ µσ (). () ) esuls fo as σ ( µ ). (3) µ If he epesso σ ( ) ( µ )( µ ) µ µ s posve σ s σ ohewseσ s σ. Fg. 7 shows he µ plae all egos fo whch σ o σ especvel es ad sulaeousl codo (84) s sasfed. We ca oba he σ egos fo µ < he sae wa because fo all paaee values of pos hese egos he deoao µ Eq. (99) s posve (see Fg. 7). The sgfcace of Fg. 7 s cofed also b he paaee values udelg he followg fgues. - - σ σ σ σ σ σ µ Fgue 7. egos of esece of σ ad of σ Illusao he coodae plae. Secall suppoed oo Wh σ (4) 3

14 we have b (5) µ see Fgs Fo he secall-suppoed oo s equpped wh he balacg devce he plae of he ass cee. The sabl age > s decal wh he oe fo he plae ubalaced oo. The vbaos he ceal plae ae suppessed. If oeove he oo s ol sacall ubalaced wh copesao he ceal plae he he oo s copleel balaced. A sla suao wll occu he followg fo µ ad fo µ <. Fo µ > he case of a eccecall-oued devce wo sabl ages alwas es. 4 4 b b Fgue 8. a > µ σ Fgue 9. a > µ.5 σ. Asecall suppoed oo Because of σ (6) he bouda speed b s deeed b Eq. (95) see Fgs.. Fo he eccec values ad a each case he wo sabl ages ae educed o coeced o oe sabl age. Illusao he o σ coodae plae. Cecall oued devce Wh we have (7) b (8) µ see Fgs. 3. Fo o-vashg sffess ase wo sabl ages alwas es. Fo he paccal po of vew we po o he fac ha he bodele case σ has o paccal ees. I wll be ealsed a os appoael. 4

15 6 b 4 b Fgue. a > µ σ.5 Fgue. a > µ.5 σ -.5 σ 5 b σ 5 b Fgue. a > µ Fgue 3. a > µ.5. Eccecall-oued devce Wh (9) σ 5 b he squae of he bouda speed ca be we as b b b µ < σ < µ σ () -5 - b σ see Fgs. 4-6 ad Fg. 7. Fgue 4. a > µ 3 5

16 σ 5 b σ 5 b σ b b Fgue 5. a > µ.5 Fgue 6. a > µ.5.7 The sphecal oo The sphecal oo s defed b µ. () a Eqs. (78) (8) (89) eld. () Fo o fo 4 [ ( ) σ ] M µ ( µ ) µσ l µ 4 µ we oba he squae of he ccal speed ( σ ). (3) The bouda speed s deeed b σ b σ. (4) Illusao he coodae plae. Secall-suppoed oo Wh codo (4) we have he sple foulae b see Fg. 7. (5) 4 - b -4 Fgue 7. a µ σ 6

17 . Asecall suppoed oo 4 Fo he oo wh o-vashg sffess ase see foula (6) Fg. 8 shows a eaple fo he sabl age whee fo Eq. (4) b /σ. (6) - -4 Fgue 8. a µ σ.5 Illusao he σ coodae plae. Cecall-oued devce Wh codo (4) we have σ 5 b (7) see Fg Fgue 9. a µ. Eccecall-oued devce Wh codo (9) he squae of he bouda speed ca be we as b b b < σ < σ (8) see Fg.. Fo he sffess ase σ fo whch b we oba fo Eq. (4) σ. (9) σ 5-5 σ b b The dsc-shaf oo - Fgue. a µ The dsc-shaf oo s defed b he codo a < µ <. () Eqs. (78) ad (89) eld < <. () The ol ccal speed s see Eq. (89): 7

18 µ µσ > µ ( µ ) 4 Fo he bouda speed we oba fo Eq. (95). () b > fo µ > fo µ <. (3) b < Illusao he coodae plae. Secall-suppoed oo Wh codo (4) we have b µ see Fg.. Fo Eq. (4) we oba fo aspoes µ ad µ. b (4) he 6 3 µ µ -3 b -6 Fgue. a < µ -.5 σ. Asecall suppoed oo Fo he oo wh o-vashg sffess ase see foula (6) Fg. shows a eaple fo he sabl age. Fo Eq. (95) we have he sae aspoes as fo he secall suppoed oo. 6 3 µ µ b -3-6 Fgue. a < µ -.5 σ -.5 Illusao he σ coodae plae. Cecall-oued devce Wh codo (7) as fo he sphecal oo aga Eq. (7) s vald; see Fg. 3.. Eccecall-oued devce Wh codo (9) he squae of he bouda speed ca be we as b b µ σ b < σ < (5) µ see Fg. 4 ad Fg. 7. 8

19 κ () σ σ 5 5 σ b b Fgue 3. a < µ -.5 Fgue 4. a < µ Sulao esuls Soe sulao esuls llusag he pocesses of ubalace copesao b eas of sgle-plae auobalacg devces ae peseed below. Sulaos wee pefoed eplog he Advaced Couous Sulao Laguage (ACSL). We vesgaed ase pocesses of oo u-up o he oal speed hghe ha ccal speeds. Two oo sses wee cosdeed: log ( a > ) ad ds-shaf ( a < ). The fs oo sse has a ass of 3.5 g ad oes of ea a.74 g.89 g. Is ccal speeds ae 7 ad/s ad 35 ad/s; paaees σ -.55 µ.68 ae close o hose peseed Fg.. We used wo values of paaee : advaageous.44 ad auspcous Fgs. 5-7 deosae he pefoace of a sgle-plae auo-balacg devce wh a sgle ball fo he case of.44. Fg. 5 shows he depedece κ ( ). I accodace wh codo (5) he codo fo deceasg vbaos due o a sgle-plae self-balacg devce wh a sgle ball s < κ < ( ). (6) As ca be see fo Fg. 5 (he sae esul follows fo Fg. ) heoecall he aea of sable copesao oo begs beod he fs ccal speed has a sall eupo ea he secod ccal speed ad he coues whou lao. aalcal sulao ccal speeds α Fgue 5. Log oo sse. Devce wh a sgle ball. Depedece κ fo he case of.44. ( ) Fgue 6. Log oo sse. Devce wh a sgle ball. Ball agula poso fo he case of.44. Fg. 6 peses he agula poso of he ball dug u-up copaso wh he aalcal pedco. Fo he sa he ball falls behd he oo so he agula poso of he ball befoe he fs ccal speed does o ea cosa. Nea he fs ccal speed we obseve a ase pocess wh fas-phase oscllaos. Ths 9

20 κ () o-schoous oo of he balls ea he ccal speed s of he sae aue as he well ow Soefeldeffec ubalaced oo sses wh a led dvg oe (h e al. ad h e al. ). Beod he fs ccal speed he ball schoes wh he oo povdg paal copesao fo ubalace. Theoecall he copesao phasg a be dsubed ea he secod ccal speed. Howeve due o he fluece of dapg hs dsubace sulaos was uch salle ha pedced aalcall. Fg. 7 shows he aplude of oo vbaos he plae of he devce. Oe ca see ha fo a advaageous choce of paaees he auo-balacg devce dshes vbaos he ego beod he fs ccal speed. ccal speeds wh a/b devce whou a/b devce Fgue 7. Log oo sse. Devce wh a sgle ball. Aplude of oo vbaos he plae of he devce fo he case of.44. Fgue 8. Log oo sse. Devce wh a sgle ball. Depedece κ fo he case of ( ) The esuls of sulaos fo he case of -.44 ae peseed Fgs Theoecall (Fg. 8) hee should be a sall aea of copesao beod he fs ccal speed ad a uled copesao aea afe he secod ccal speed. I sulaos we obseved ol oe copesao ego whch les beod he secod ccal speed. aalcal sulao α wh a/b devce whou a/b devce Fgue 9. Log oo sse. Devce wh a sgle ball. Ball agula poso fo he case Fgue 3. Log oo sse. Devce wh a sgle ball. Aplude of oo vbaos he plae of devce fo he case Fgs deosae he esuls of sulaos fo he sae oo sse he case of a sgle-plae auobalacg devce wh wo balls. Such a devce povdes dshg vbaos he plae of he devce fo he codo κ ( ) < ; whe κ ( ) < vbaos he plae of he devce should be equal o eo a leas heoecall alhough pacce hee ae alwas soe esdual vbaos. Fo advaageous posog of he auo-balacg devce plae (case. 44 ) aalcal vesgao pedcs sable ubalace copesao he aea beod he fs ccal speed wh a sho eupo ea he secod ccal speed as Fg. 3.

21 κ () ccal speeds aalcal sulao α α Fgue 3. Log oo sse. Devce wh wo balls. Depedece fo he case.44. κ ( ) Fgue 3. Log oo sse. Devce wh wo balls. Ball agula posos fo he case.44. Low dapg. I sulaos we obseve a slghl dffee pcue. Afe passg he ccal speeds balls schoe wh he oo sag heoecall usable ude codo κ ( ) < posos α ˆ α α ˆ α (Fgs. 3 33). Ol lae he aea of copaavel hgh speeds ad low acceleaos balls sepaae ad loo fo he copesao posos as ude (54). The sepaao po depeds osl o he dapg paaees β : he lowe β he eale balls sepaae. O he ohe had low dapg a polog he aea of o-schoous ball oos wh ceased vbaos ea ccal speed (see Fgs ). aalcal α α sulao whou a/b devce wh a/b devce Fgue 33. Log oo sse. Devce wh wo balls. Ball agula posos fo he case.44. ahe hgh dapg. Fgue 34. Log oo sse. Devce wh wo balls. Aplude of oo vbaos he plae of devce fo he case.44. Low dapg. Fo he case of -.44 he copesao aea begs afe he secod ccal speed. As above we obseve o-schoous oo of he balls hs e osl ea he secod ccal speed ad he ego whee he balls eep heoecall usable posos α ˆ α α α (Fgs ). ˆ

22 κ () κ () ccal speeds wh a/b devce whou a/b devce Fgue 35. Log oo sse. Devce wh wo balls. Aplude of oo vbaos he plae of devce fo he case.44. ahe hgh dapg. Fgue 36. Log oo sse. Devce wh wo κ fo he case balls. Depedece ( ) aalcal α α sulao whou a/b devce wh a/b devce Fgue 37. Log oo sse. Devce wh wo balls. Ball agula posos fo he case Fgue 38. Log oo sse. Devce wh wo balls. Aplude of oo vbaos he plae of devce fo he case sulao ccal speed aalcal Fgue 39. Ds-shaf oo sse. Devce wh a sgle ball. Depedece. κ ( ) Fgue 4. Ds-shaf oo sse. Devce wh a sgle ball. Ball agula poso.

23 κ () The secod ds-shaf oo sse has a ass of.5 g ad oes of ea a.936 g.77 g. Is ol ccal speed s 6 ad/s; paaees σ µ ae σ -.98 µ The sable copesao pcue s sla o ha peseed Fg.. We cosde ol oe value of paaee.5 whch povdes he uled aea of sable paal copesao he ego beod he ccal speed. Fgs deosae he esuls of copuaos fo a sgle-plae devce wh a sgle ball. The depedece κ ( ) s peseed Fg. 39. As pedced Fg. he aea of sable copesao begs beod he ccal speed ad coues whou lao. The poso of he ball ad oo vbaos dug u-up ae peseed Fgs Oe ca see ha afe he ball schoes wh he oo he pos-ccal aea he devce dshes he oo vbaos. ccal speed whou a/b devce wh a/b devce Fgue 4. Ds-shaf oo sse. Devce wh a sgle ball. Aplude of oo vbaos he plae of devce. Fgue 4. Ds-shaf oo sse. Devce wh wo balls. Depedece. κ ( ) Fgs deosae he pefoace of a sgle-plae devce wh wo balls. I hs case vbaos he plae of devce he pos-ccal aea becoe equal o eo alhough ohe oo plaes hee ae soe esdual vbaos (paal copesao of ubalace). The effecs descbed above of o-schoous oos ea he ccal speed ad of he heoecall usable phasg α ˆ α α ˆ α wh a cea age of oo speeds beod he ccal speed a be cleal obseved. aalcal α α sulao whou a/b devce wh a/b devce Fgue 43. Ds-shaf oo sse. Devce wh wo balls. Ball agula posos. Fgue 44. Ds-shaf oo sse. Devce wh wo balls. Aplude of oo vbaos he plae of devce. 3

24 8 Coclusos Aalcal vesgaos have evealed ha ude cea codos sgle-plae auo-balacg devces ae suable fo povdg paal copesao of sac ad dac ubalaces ad fo educg vbaos o a ajo ee. Codos fo a sable paal ubalace copesao have bee deved fo dffee pes of oos cludg oos wh a pola oe of ea geae ha he asvese oe. I pacula he possbl of paal copesao he fequec age beod he fs ccal speed has bee evealed. The aalcal coclusos wee vefed b uecal sulaos. Sulaos cof he esuls peseed bu also deosae ha he aeas ea ccal speeds he auo-balacg devce a egede ceased vbaos due o o-schoous ball oos. To avod hs udesable effec s ecessa o caefull selec he devce paaees. The possbl of paal copesao cosdeabl eeds he poeal age of applcaos of auoac balacg. I fuue eseach wo he auhos ed o vesgae a paal ubalace copesao b woplae auo-balacg devces. Acowledges The auhos would le o epess gaude o he Deusche Foschugsgeeschaf fo he facal suppo (No. SP 46/7-3). The auhos ae gaeful o Pofesso Mhal F. Debeg fo helpful dscusso dug hs sa ou Uves as a Fulbgh eseache/lecue. The ecelle oppou fo uual eseach as povded b he Fulbgh Cosso s os hghl appecaed. efeeces Bleha I.I.: Vbaoal Mechacs. Wold Scefc Sgapoe New ese Lodo Hog Kog Böv P.; Högfos C.: Auobalacg of oos.. of Soud ad Vbao 3 (986) Chug.; o D.S.: Dacal aalss of a auoac dac balace fo oag echass.. of Soud ad Vbao 8 5 (999) Hedaa M.T.; Shap.S.: A aalss of a ew pe of auoac balace.. Mechacal Egeeg Scece 9 5 (977) - 6 Huag W.-Y.; Chao C.-P.; Kag.-.; Sug C.-K.: The applcao of ball-pe balaces fo adal vbao educo of hgh speed opc dves. oual of Soud ad Vbao 5 3 () Ioue.; ouch Y.; Kubo S.: Auoac balaces (I apaese). Tasacos of he SME Se. C 49 (979) 4-48 Kag.-.; Chao C.-P.; Huag C.-L.; Sug C.-K.: The dacs of a ball-pe balace sse equpped wh a pa of fee-ovg balacg asses. Tasacos of he ASME 3 () h B. Ae T. Ducse H. ad Spelg L.: Zu Soefeldeffe be selbsäge Auswuche ee Ebee Techsche Mecha Vol. No. 4 () 97-3 h B. Spelg L. ad Ducse H.: Dspla of he Soefeld-Effec a gd oo Oe-Plae Auobalacg Devce Poc. of XXX Sue School Advaced Pobles Mechacs S. Peesbug () Spelg L.; Mee F.; Ducse H.: Self-schoao ad auoac balacg oo dacs. I.. oag Mache 6 4 () Spelg L.; h B.; Ducse H.: Two-plae auoac balacg. Mache Dacs Pobles 5 3/4 () 39-5 Spelg L.; h B.; L Ch.; Ducse H.: Sulao of wo-plae auoac balacg of a gd oo. Maheacs ad Copues Sulao () Theale E.L.: A ew pe of dac-balacg ache. Tasacos of he ASME 54 (93) 3-4 Addess: Pof. D.-Ig. habl. Lu Spelg D.-Ig. Bos h ad D.-Ig. Hee Ducse Isu fü Mecha Oo-vo-Guece Uvesä Magdebug Uvesäspla 396 Magdebug. E-al: Lu.Spelg@b.u-agdebug.de Bos.h@b.u-agdebug.de Hee.Ducse@b.u-agdebug.de 4