mechanics 1. Dynamics of a particle - revision dynamics The forces acting on rigid bodies.

Transcription

1 echnics. Dnics of pticle - eision sttics dnics The foces cting on igid bodies. The foces cting on oing bodies. The eltionship between foces nd otion The echnics of etenl foces (the echnics of igid bodies). The echnics of intenl foces (the echnics of fleible bodies). Jiří Podeš cult of Mechnicl Engineeing, VŠB Technicl Uniesit of Ost Ost, zech Republic

2 . Dnics of pticle - eision Isc Newton (64-77) Philosophie Ntulis Pincipi Mthetic (687). Newton s st lw lw of ineti bod sts t est o t constnt elocit if no foce cts upon it. Newton s nd lw lw of foce foce cting upon bod leds to chnge of elocit tht is diectl popotionl to the cting foce. The coefficient of popotionlit is the bod ss. ss cceletion foce Newton s 3 d lw lw of ction nd ection The ctions of two bodies upon ech othe e lws equl in gnitude nd opposite in diection.

3 . Dnics of pticle - eision Isc Newton (64-77) Philosophie Ntulis Pincipi Mthetic (687). Newton s lw of gittion n two objects eet gittionl foce of ttction upon ech othe. The gnitude of the foce is popotionl to the poduct of the gittionl sses of the objects, nd inesel popotionl to the sque of the distnce between the. G κ 6,67 - kg- 3 s- - κ On the Eth s sufce then : 5,98 4 kg k The gittionl foce is then : gittionl constnt, - the ss, - the ss, - the distnce between bodies. - the Eth s ss, - the Eth s dius. whee g is the gittionl cceletion : G g g κ, G 9 8 s G 3

4 The pticle - hs no diension, but hs cetin ss. Dnics of pticle - eision The igid bod - hs cetin diensions, is igid, undefoble The chin of bodies - echnis - the eltie position of one bod chnges with espect to nothe 4

5 . Dnics of pticle - eision dnics kinetics onl otion dnics otion, sses nd foces 5

6 . Dnics of pticle - eision The degee of feedo (DO) the possible, independent otion. z {,, z} φ + ± { } R R {independent coodinte} { φ} R sin φ 6 R cos φ

7 . Dnics of pticle - eision on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) 7

8 . Dnics of pticle - eision on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) the nube of DO deceses if the otion is esticted b joints 8

9 . Dnics of pticle - eision on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) the nube of DO deceses if the otion is esticted b joints 9

10 . Dnics of pticle - eision on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) the nube of DO deceses if the otion is esticted b joints diection tnsltion z diection tnsltion z is ottion

11 . Dnics of pticle - eision on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) NO eticl tnsltion independent hoizontl tnsltion independent ottion DO the nube of DO deceses if the otion is esticted b joints

12 . The dnics of pticle - epetition on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) NO eticl tnsltion the nube of DO deceses without sliding in the touch point if the otion is esticted b joints hoizontl tnsltion nd ottion elte one to the othe DO olling

13 . Dnics of pticle - eision on tck (D) in plne (D) in spce (3D) pticle DO DO ( tnsltions) 3 DO (3 tnsltions) bod 3 DO ( tnsltions nd ottion) 6 DO (3 tnsltions nd 3 ottions) the nube of DO deceses if the otion is esticted b joints 3

14 the otion of pticle. Dnics of pticle - eision tie t unit [s] units [in, h,...] tck, oute, coodinte s,,,... unit [] units [c, k,...] elocit unit [/s, s - ] units [k/h] cceletion unit [/s, s - ] 4

15 the otion of pticle. Dnics of pticle - eision ds dt d dt d dt s& & s d ds && s ( ) d ds these e the genell lid eltionships between tie, tck, elocit nd cceletion the elocit is the fist deitie of tck with espect to tie cceletion is the fist deitie of elocit with espect to tie cceletion is the second deitie of tck with espect to tie cceletion is the fist deitie of elocit with espect to tck, ultiplied b elocit cceletion is one hlf of the fist deitie of sque elocit with espect to tck 5

16 the otion of pticle. Dnics of pticle - eision ds dt d dt d dt s& & s d ds && s ( ) d ds these e the genell lid eltionships between tie, tck, elocit nd cceletion With espect to the behio of tck, elocit nd cceletion oe tie we cn distinguish ) Unifo otion elocit is constnt B) Unifol cceleted otion - cceletion is constnt ) Non-unifo otion eething chnges 6

17 the otion of pticle. Dnics of pticle - eision ) Unifo otion elocit is constnt d dt s t s s s s t the elocit is constnt, the cceletion is zeo t t t s - the instnt tck s - the initil tck the initil condition t - the instnt tie t - the initil tie, usull t s s ( t ) t s s t + s s these e the eltionships lid onl fo unifo otion t 7

18 the otion of pticle. Dnics of pticle - eision B) Unifol cceleted otion cceletion is constnt t + t s + t + s t t s s t ( s ) s + these e the eltionships lid onl fo the unifol cceleted otion the initil conditions : s - the initil tck - the initil elocit s 8

19 the otion of pticle. Dnics of pticle - eision B) Unifol cceleted otion cceletion is constnt The spots c cceletes fo zeo to k/h (7.8 /s) in tie t 5 s. (ssuing unifol cceleted otion) t cceletion is then 5.6 /s. s the tck is then s 7. t 9

20 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion honic otion the tck chnges honicll (se elocit nd ccel.) T φ φ ω T t sin ( ω t + φ ) ω ω f π T f π ω plitude [] cicul fequenc [s - ] fequenc [Hz] nube of ccles pe second peiod [s] tie of one ccle φ phse led [-]

21 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion honic otion the tck chnges honicll (se elocit nd ccel.) T φ φ ω T t sin ( ω t + φ ) & ω cos & ω sin ω ( ω t + φ ) ( ω t + φ ) ω plitude []. elocit [/s] ω. cceletion [/s ] the oscilltion of pticle ss on fleible link

22 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion,, otion within dping enionent d dt g β g β d g β dt ln β β ln g β t g β g t The solution with zeo initil conditions : β β d g β [ ln( g β ) ] t t [ ln( g β ) ln( g) ] t dt t g β ( βt e ) sted _ stte sted_ stte ( βt e ) g β

23 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion otion within dping enionent,, d dt g β g β d g β dt T β tie constnt [s] tngent T sted stte sted _ stte 95% sted stte βt ( e ) g β ( βt e ) sted _ stte ( βt e ) 63% sted stte tt t T t3 T t4 T t5 T t sted_ stte g β 3

24 the otion of pticle. Dnics of pticle - eision,, ) Non-unifo otion otion within dping enionent sted _ d stte d d dt sted _ stte t t t sted _ stte sted _ stte ( βt e ) ( βt e ) dt ( βt ) ( βt e dt ) sted _ stte e sted _ stte sted_ stte sted_ stte t t e β β t e β t e β βt t + β t ( ) βt 4 dt

25 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion otion within gittionl field G h G κ M κ M g R ( R + h) ( R + h) Eth R κ kg- 3 s- - M kg R k the gittionl constnt, - the Eth s ss, - the Eth s dius. on the Eth s sufce () : M G κ g R κ M R g 9.8 s κ M g R 5

26 6 G Zeě R, ( ) h R R g d d + ( ) h R R g G + h fee fll fo height of h ( ) + d h R R g d h R R g h R h R R g. Dnics of pticle - eision ) Non-unifo otion otion within gittionl field the otion of pticle Dnics

27 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion otion within gittionl field fee fll fo height of h, G h G g d d ( R + h ) g R R ( R + h ) Zeě R g R ( ) ( R + h ) ( R + h) the dop elocit : ( h ) g h R R + h h << R ( h ) g h 7

28 the otion of pticle. Dnics of pticle - eision ) Non-unifo otion otion within gittionl field eticl thow upwd, G Zeě R d G g g d d ( R + ) R R ( R + ) g R g R ( R + ) ( R + ) d [ ] ( R + ) g R g R ( R + ) g R R + 8 d

29 9, G Zeě R ( ) R R g R R R g + + R R g + R g R h s k R g / < s k R g / > ( ) R g stte sted li _ ( ) h the pticle stops t the height of h the pticle dws pt fo ee ( ) R R g G + eticl thow upwd. Dnics of pticle - eision ) Non-unifo otion otion within gittionl field the otion of pticle Dnics

30 the cued tjecto t tjecto (t) ( t ) the otion of pticle t ( t) (t+ t) ( + t) n n ( t+ t). Dnics of pticle - eision ( t+ t ) ( t ) + li t t + t n d dt & ( t) O ( t + t ) el t d dt ( t ) ( t + t) s n R R the dius of cutue 3

31 Newton s nd lw the lw of foce ss [kg] cceletion [/s ] foce [N] i the eqution of otion. Dnics of pticle - eision The eltionship between ss, foce nd otion. kg 3 N.5 /s 3

32 . Dnics of pticle - eision Newton s nd lw the lw of foce i the eqution of otion α G N T f i G + + N + T G sinα cos α T i G sin α cos α N f i N G cosα sinα N G cosα + sinα G sinα cosα f G ( G cosα + sinα) ( sinα f cosα) ( cosα + f sinα) 3

33 . Dnics of pticle - eision Jen Le Rond d lebet (77-783) d lebet pinciple.. D D + D i the equtions of equilibiu) D - D D 33

34 Newton s nd lw the lw of foce i the eqution of otion DO NOT MIX!. Dnics of pticle - eision Jen Le Rond d lebet (77-783) d lebet pinciple.. D D + D i the equtions of equilibiu) D - D D 34

35 . Dnics of pticle - eision G α T f N G ( sinα f cosα) ( cosα + f sinα) the kinetosttic tsk equied otion is gien, fo eple the cceletion, deteine the foce? needed to ech the equied otion D G ( sinα f cosα) cosα + f i sinα the equtions of equilibiu - lgebic G the dnic tsk the foce is gien, deteine the otion, the cceletion? ( sinα f cosα) ( cosα + f sinα) & s& the diffeentil equtions 35

36 36

37 d dt d( ) dt d dt ( ) p t d dt ( ) t I ( t) dt The phsicl quntities deied fo the eqution of otion. I. The lws of chnge. the oentu the foce ipulse if the foce is constnt : [kg s - ] [N s kg s - ]. The nlticl echnics the oentu chnge the chnge of ount, the chnge of diection I t p p p p the lw of the oentu chnge p p p I p - the oentu t the beginning, p - the oentu t the end of the eent. p p + p p 37

38 d ( ) d ds ( ) ds The phsicl quntities deied fo the eqution of otion. II. The lws of chnge.. The nlticl echnics d ( ) ds d ( ) ds s if the foce is constnt (both ount nd diection) : s E K ds s the kinetic eneg the wok [J kg s - ] [N kg s - ] the lw of kinetic eneg chnge E EK EK K E K the kinetic eneg t the beginning, E K the kinetic eneg t the end of the eent. 38

39 ds s The phsicl quntities deied fo the eqution of otion. II. the wok. The nlticl echnics the wok is scl poduct of foce nd tck, the ngle between the ust be tken into ccount : N δ δ < 9 δ 9 δ > 9 δ δ W s > s s cos s s cos δ woking coponent W cos9 cos cos δ P s cos δs ( δ > 9 ) < δ 8 s cos8 s cos δ > s cos δ < the scl poduct N sin δ positie wok wok done zeo wok not-done not-woking coponent negtie wok wok consued 39

40 ds s The phsicl quntities, deied fo the eqution of otion. II. the wok [N kg s - J]. The nlticl echnics P the powe d ds dt dt the elocit [N s - W] δ P cos δ N δ W W cos δ P W cos δ N sin δ 4

41 . The nlticl echnics EP ds s the potentil eneg h h d g d g d h g h 3 G g G G E P g h E P the zeo potentil eneg leel the choice the potentil eneg - positionl 4

42 EP ds E P s G G Eth R the potentil eneg the foce G ies with height M G κ κ κ 6,67 - kg- 3 s- - M 5,98 4 kg R k on the Eth s sufce : M G κ R h g d ( ) M. The nlticl echnics g R ( R + ) ( R + ) gittionl constnt, - the Eth s ss, - the Eth s dius, - the totl distnce fo the Eth s cente, - the height boe the Eth s sufce. g κ M g R 4

43 43 R ( ) ( ) + h h d R R g d ( ) h R R h g h R R h R g + + E P ds E s P + + h R R R g R R g h E g h R R h P + h«r h R R + h g E P E P the potentil eneg is equl to wok : fo sll height boe the Eth s sufce is ppo. : G G Eth the potentil eneg the foce G ies with height the potentil eneg - positionl. The nlticl echnics Dnics

44 . The nlticl echnics EP ds s the potentil eneg The foce cts on the fied be, the be defotion is. 3 l 3 E J k l the be length, E the Young odulus J the qudtic oent of ineti k the stiffness 3 E J k 3 l E P the potentil eneg is equl to wok : d k d k E P k the potentil eneg - defotion 44

45 . The nlticl echnics The lw of the totl echnicl eneg consetion E T EK + EP constnt the consetie sste E K E P g h E T E + E E + E K EK + EP constnt P K P h + g h + g h E K ½ E P E P The totl echnicl eneg consees. the zeo potentil eneg leel 45

46 46 (the foces which do not cete potentil eneg) α h s G T N konst + P K T E E E? The lw of the totl echnicl eneg chnge. The nlticl echnics the non-consetie sste E P g h E K ½ E P E K ½ E E T T + s T s h g α cos h g s T s α + + cos h g s T s α + cos E T E T α sin s h The chnge of the totl echnicl eneg is equl to the wok of non-consetie foces. α α + sin cos G N N f T Dnics

47 . The nlticl echnics The lw of the totl echnicl eneg consetion / chnge s h T α G N h the w of the dnic solution, bsed on the totl echnicl eneg nlsis, is clled eneg blnce solution 47

48 . The nlticl echnics the itul wok pinciple... in sttics - gien B deteine R? the fee-bod dig N B R _ i _ i M i N B the itul wok pinciple is the ltentie to the equtions of equilibiu 48

49 . The nlticl echnics the itul wok pinciple... in sttics δ B the itul otion δ B R δ R δ the itul wok δ E K E K B δ R δb t t P the lw of the kinetic eneg chnge δ the otion sttus does not chnge B t B the itul wok pinciple the powe P R B (the itul powe pinciple) 49

50 . The nlticl echnics the itul wok pinciple... in sttics ω π π B Bπ R B ω π R ω Bπ B ω B R π R Bπ π R Bπ R B N π M i π _ the kineticl ethod in sttics B R N B 5

51 . The nlticl echnics the itul wok pinciple... in sttics - gien deteine R? 5

52 . The nlticl echnics the itul wok pinciple... in sttics - gien the fee-bod dig 9 equtions of equilibiu 9 unknown foces 5

53 . The nlticl echnics the itul wok pinciple... in sttics δ the itul otion R δ R 53

54 54 R l l + + & & + R R φ tn R φ the itul wok pinciple... in sttics. The nlticl echnics Dnics

55 . The nlticl echnics δ δcos α the itul wok pinciple... in sttics α δ 55

56 . The nlticl echnics the itul wok pinciple... in sttics - gien B deteine R? 56

57 . The nlticl echnics the itul wok pinciple... in sttics B the itul otion R 57

58 . The nlticl echnics the itul wok pinciple... in sttics α B R B π ω α B R B + R B cos α + R cos8 B cos α R B ω Bπ B π π Bπ B cos α R Bπ R cos α π B π Bπ Bπ B π tn α cos α 58

59 . The nlticl echnics the itul wok pinciple... in dnics G M D D S T,, the wok of the d lebet foces will be tken into ccount φ, ω, ε D M D I S ε G D M D ω G D MD G D MD + IS G ω ε IS + G 59

60 . The nlticl echnics the Lgnge equtions of the nd kind d dt E K i E q K i + E q P i Q i sste with n degees of feedo i...n q i... the genelized coodinte i... the genelized elocit t... tie n independent coodinte deteining the position of the sste i q& i E K... the kinetic eneg E P... the potentil eneg Q i... the genelized foce Q i j q j j i j the el foces j... j the dius ecto of the point of ppliction of the foce j 6

61 . The nlticl echnics the Lgnge equtions of the nd kind T ω, ε φ ω l T π e φ ω φ the genelized elocit the genelized coodinte l sin φ Tπ e + e cos 9 Tπ e + l sin T ( φ) φ el sin ωtπ φ E K de dφ E P G E K dek [ ( e + l ( l e) φ) + I ] ω e + l ( l e) sin K l g e sin φ ( l e) sinφcos φω d Q l cos φ dφ de P dφ T g ecos φ + dω d dt d dt + de dω T + I T ω [ ( sin φ) + I ] ω K [ + I ] ( e + l ( l e) sin φ) [ l ( l e) sin φcos φφ& ] ω de dω K [ ( e + l ( l e) sin φ) + I ] [ l ( l e) sin φcos φ] ω T T T 6 ε + ε +

62 . The nlticl echnics the Lgnge equtions of the nd kind T ω, ε φ ω l T π e φ ω φ & l sin φ Tπ e + e cos 9 Tπ e + l the genelized coodinte the genelized elocit sin T ( φ) φ el sin ωtπ φ G [ ( e + l ( l e) sin φ) + I ] ε + [ l ( l e) sin φcos φ] ω + g ecos φ l cos φ d dt T de dω K [ ( e + l ( l e) sin φ) + I ] ε + [ l ( l e) sin φcos φ] ω ( l g e) cos φ T de dφ K de P dφ d Q dφ 6

63 . The nlticl echnics the Lgnge equtions of the nd kind T ω, ε φ ω l T π e φ ω φ & l sin φ Tπ e + e cos 9 Tπ e + l the genelized coodinte the genelized elocit sin T ( φ) φ el sin ωtπ φ G [ ( e + l ( l e) sin φ) + I ] ε + [ l ( l e) sin φcos φ] ω ( l g e) cos φ ε d φ dt T d φ dφ [ ( e + l ( l e) sin φ) + I ] + [ l ( l e) sin φcos φ] ( l g e) cos φ dω ε ω dφ dω [ ( e + l ( l e) sin φ) + I ] ω + ( ) T T dt dφ [ l l e sin φcos φ] ω ( l g e) 63 cos φ dt

64 . The nlticl echnics the Lgnge equtions of the nd kind l the pistic od ω, ε G T φ ω el/ ω dω dφ e l l e [ ( ) ] + I ε ( l g l) cos φ l T ε α cos φ 3 l ε α cos φ I g cos φ α T && φ α cos φ l ω ωdω φ α cos φdφ ω α sin φ α 3 l <> g g <> ω φ ( ) α sin φ 64

65 65 z, φ, ω ω G α S T b b-z cosα E P K I I E + ω + I d de K + I I d de dt d K + + & dz de K ( ) α sin z h g E P h α sin g dz de P α cos dz d Q i i P i K i K Q q E q E E dt d + α α + cos sin g I olling without sliding h b tgα + tg α + cosα the Lgnge equtions of the nd kind. The nlticl echnics Dnics

66 66

67 3. Vibtions with degee of feedo the stiffness φd φd k k the sping stiffness l hee : 3 8D n l 4 G d 4 G d l 8 D n 3 G the she odulus [P, MP], (the teil popet), d the wie diete [, ], D the sping diete [, ], n the nube of sping scews [-]. 4 G d l 8 D n k the stiffness 4 G d [N/, N/] 8D n 3 k 3 l k k l 67

68 the stiffness 3. Vibtions with degee of feedo k k the sping stiffness S l S k l The sping foce the ection, the esponse of the sping to the defotion. k l 68

69 potentil eneg (defotion eneg) 3. Vibtions with degee of feedo k k l ( ) d l k d k l l k l k l EP k l l S () k 69

70 tpes of ibtion 3. Vibtions with degee of feedo I. The fee ibtion no etenl foce s the eson fo ibtion. The foced ibtion - cused b n etenl foce. 7

71 tpes of ibtion 3. Vibtions with degee of feedo II. Undped ibtion no phsicl phenoenon, which will decese the ibtion, is pesent. The ibtion will lst foee. Dped ibtion due to soe phsicl phenoenon, the ibtion will decese until it nishes iscous liquid 7

72 tpes of ibtion 3. Vibtions with degee of feedo ibtion undped dped fee foced 7

73 k the fee undped ibtion 3. Vibtions with degee of feedo S i S k & + k k Ω iωt e sin Ω t + φ ( t ) ( ), && + Ω & && ( t ) ( t ) ( t ) e λt λ e λ λt e λt λ λ λ e λt + Ω + Ω Ω e λt Ω i Ω 73

74 k the fee undped ibtion 3. Vibtions with degee of feedo S i S k & + k k Ω, sin( Ω t + φ ) & Ω cos( Ω t + φ ) && Ω sin( Ω t + φ ) [ Ω sin( Ω t + φ )] + k sin( Ω t + φ ) [ ] k sin ( Ω t + φ ) + k [ sin( Ω t + φ )] 74

75 sin T ( Ω t + φ ) T t φ /Ω sin & k T ( Ω t + φ ) Ω cos( Ω t + φ ) the fee undped ibtion, t 3. Vibtions with degee of feedo Petes, esulting fo substitution : Ω k Ω f π π T f Ω the ntul cicul fequenc [s - ] the ntul fequenc [Hz] the nube of ccles pe second the peiod [s] the tie of one ccle The integtion constnts : the plitude [] φ the phse led [d] esult fo the initil conditions : t... the initil displceent, the initil elocit. Two goups of petes : + sin Ω Ω ( φ ) cos( φ ) φ Ω ctn 75

76 k ( Ω + φ ) the fee undped ibtion sin t the ltentie thetic desciption : sin Ω t + φ cos Ω t + B sin whee : sin φ B cosφ the integtion constnts e :, ( ) ( ) ( Ω t) sin sin ( Ω t + φ ) ( φ ) cos( Ω t) + cos( φ ) sin( Ω t) Ω sin( Ω t) + BΩ ( Ω t) & cos t... the initil displceent, the initil elocit. BΩ nd finll : + B B Ω φ ctn B 3. Vibtions with degee of feedo Petes, esulting fo substitution : Ω k Ω f π π T f Ω the ntul cicul fequenc [s - ] the ntul fequenc [Hz] the nube of ccles pe second the peiod [s] the tie of one ccle The integtion constnts : the plitude [] φ the phse led [d] esult fo the initil conditions : t... the initil displceent, the initil elocit. Two goups of petes : + sin Ω Ω ( φ ) cos( φ ) φ Ω ctn 76

77 k the fee undped ibtion 3. Vibtions with degee of feedo note bout the ctn function : The ctn function lws hs oots. E. : ctn(,5) 6,6º but lso : ctn(,5) 6,6º sin T ( Ω t + φ ) T T, t O : ctn(-) -45º but lso : ctn(-) 35º φ t φ /Ω B the ltentie thetic desciption : sin sin whee : sin φ B cosφ B Ω nd finll : + B φ ctn B ( Ω t + φ ) cos( Ω t) + B ( Ω t) φ > < B< B> 9º,8º,9º 77 8º,7º 7º,36º

78 the theticl pendulu : φ ω,ε S G sinφ n the fee undped ibtion t 3. Vibtions with degee of feedo The theticl pendulu (the ideliztion of el pendulu). The pticle of the ss on the weightless wie of the length. t ε ω ω φ & _ G sin φ t ε g sin φ ε + g sin φ φ && + g sin φ & φ + g φ t i n sinφ φ ) S G cos φ n n _ i ε & φ S G cos φ + ω φ [º] sin φ φ [d] eo º 5º 5º,745,8756,5889,7453,8766,6799,5 %,3 78 %,5 %

79 the theticl pendulu : φ ω,ε S n the fee undped ibtion t 3. Vibtions with degee of feedo The theticl pendulu (the ideliztion of el pendulu). The pticle of the ss on the weightless wie of the length. & φ + g φ & + k & φ + g φ G the initil conditions : t... φ φ the initil ngle ω ω the initil ngul elocit the nlogicl solution : sin ( Ω t + γ ) φ sin( Ω t + γ ) Ω γ k + Ω Ω ctn the cicul fequenc the plitude the phse led Ω γ g φ ω + Ω φ79 Ω ctn ω

80 k the fee dped ibtion S k b D b k stiffness b dping coefficient ( t ) e, ( iω δ) t δt e sin( Ω t + φ ) 3. Vibtions with degee of feedo Substitution : k S i D && + b & + k && + δ & + Ω & && ( t ) ( t ) ( t ) e λt λ e λ λt e λt b Ω δ λ e λt + δλ e λt + Ω e λ + δλ + Ω λ δ ± δ Ω δ ± i Ω λt Ω Ω 8 δ δ

81 e δt sin the fee dped ibtion k S k b D b ( Ω t + φ ), 3. Vibtions with degee of feedo Substitution : k S i D && + b & + k b Ω δ Ω k the ntul cicul fequenc of undped ibtion [s - ] (not pesent diectl in the solution) δ Ω citicl dping b δ the dec constnt [s - ] δ < Ω sub-citicl dping δ > Ω supe-citicl dping Ω Ω the ntul cicul fequenc of dped ibtion [s - ] δ f T Ω π f π Ω the ntul fequenc [Hz] nube of ccles pe second the peiod [s] tie of one ccle 8

82 e & δt e δt k sin the fee dped ibtion S k b D b, 3. Vibtions with degee of feedo Substitution : ( Ω t + φ ) [ Ωcos( Ω t + φ ) δ sin( Ω t + φ )] k S i D && + b & + k inll nd φ e integtion constnts, deteined fo initil conditions: t... the initil displceent, the initil elocit. sin ( φ ) [ Ωcos( φ ) δ sin( φ )] b Ω δ O ltentiel : δt e & e B [ cos( Ω t) + B sin( Ω t) ] δt [ ( BΩ δ) cos( Ω t) ( Bδ + Ω) sin( Ω t) ] + Ω δ + B BΩ δ φ ctn B 8

83 T the peiod e the fee dped ibtion k S k b D b, δt e δt sin ( Ω t + φ ) 3. Vibtions with degee of feedo Substitution : k S i D && + b & + k b Ω δ t φ /Ω t 83

84 the fee dped ibtion tngent e k S k b D b, δt e δt 3. Vibtions with degee of feedo Substitution : k S i D && + b & + k b Ω δ tτ τ 37% δ τ - the tie constnt t3 τ 5% t t5 τ,7% tτ t3 τ e e δτ δ3τ e δ δ e δ 3 δ e e, 37 3, 5 t5 τ e δ5τ e δ 5 δ e 5, 7 84

85 the fee dped ibtion tngent e k S k b D b, δt e δt 3. Vibtions with degee of feedo Substitution : k S i D && + b & + k b Ω δ tτ τ 37% δ τ - the tie constnt δ,; τ δ,8; τ,5 t 5 5 t3 τ 5% t t5 τ,7% sll dping slow dec, lge dping quick dec. e e e δτ δ3τ δ5τ e δ δ e e δ 3 δ δ 5 δ e e e, , 5, 7 85

86 sping ssebl 3. Vibtions with degee of feedo the pllel ssebl S k l S k l k k S + S k l + k l k T S S l ( k + k ) l k T l k k + k T k T l The totl stiffness is the su of ptil stiffnesses. the pllel ssebl the defotion l is coon fo both spings, the sping foces S nd S e sued 86

87 sping ssebl 3. Vibtions with degee of feedo the seil ssebl S k l S k l l k S l k S k l + l S S S l T l + l k T k S + k S k T k S k T k + k S l + l l T l + l k T k + k k k k + k k T l The ecipocl totl stiffness is the su of ecipocl ptil stiffnesses. the seil ssebl the defotions l nd l e sued, the sping foces S S e equl 87

88 sping ssebl 3. Vibtions with degee of feedo the pllel ssebl! S k l S k l k S + S k l + k l k T k S S l ( k + k ) l k T T l k k + k k T l The totl stiffness is the su of ptil stiffnesses. the pllel ssebl defotion l is coon fo both spings, the sping foces S nd S e sued 88

89 3. Vibtions with degee of feedo bending ibtion R 3 l 3 E J R k k bending bending 3 E J 3 l E the Young odulus [P, MP] (teil) J the sque oent of ineti [ 4, 4 ] (pofile) l the be length [, ] k bending the bending stiffness [N/, N/] k bending 89

90 3. Vibtions with degee of feedo bending ibtion R k bending R 3 l 48 E J k bending 48E J 3 l 9

91 the foced ibtion k S k b D b, 3. Vibtions with degee of feedo i D B + k b + && + b & + k constnt etenl foce honicll chnging etenl foce : konst, : sin(ω t) t 9

92 l k l stt l the fee length the sttic defotion the equilibiu position G l stt G k the ibtion unde the constnt foce G S stt S dn G l stt 3. Vibtions with degee of feedo G i S _ totl G k l totl ( l ) G k + stt G k l stt k S stt & + k G k l G _ stt D k l stt & + k S_ totl S _ stt + S _ dn S_ stt k l stt S_ totl S_ totl k l stt G + k stt + k ( t ) S dn ( t ) 9

93 Ω f the nube of ccles π pe second T - the peiod [s] T f π Ω honicll chnging eciting foce k b the petes of ntul (fee) ibtion Ω the ntul cicul fequenc [s - ] Ω k f the ntul fequenc [Hz] the tie of one ccle sin(ω t), these petes ust not be confused 3. Vibtions with degee of feedo T the plitude [N] ω the cicul fequenc of the eciting foce [s - ] f the fequenc [Hz] f the petes of the eciting foce ω π T the peiod [s] T f π ω T nube of chnges of the eciting foce fo positie to negtie nd bck pe second the tie of one chnge t 93

94 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) δt solution e sin( Ω t + φ ) ho, the plitude ω the cicul fequenc of the eciting foce && + b & + k sin the solution is the ssebl of hoogenous solution nd pticul solution the hoogenous && + b & + k k Ω b δ Ω Ω δ the hoogenous solution π T Ω the integtion constnts φ esults fo the initil conditions Ω the ntul cicul fequenc t 94

95 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) && + b & + k sin, the pticul solution sin ω t φ pt ω the cicul fequenc of the eciting foce the pticul solution the plitude ω the cicul fequenc of the eciting foce the solution is the ssebl of hoogenous solution nd pticul solution π T ω ( ) the plitude nd phse led φ of the pticul solution will be discussed lte t 95

96 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t), pt the plitude ω the cicul fequenc of the eciting foce && + b & + k sin the solution is the ssebl of hoogenous solution nd pticul solution the hoogenous && + b & + k the pticul solution δt e sin Ω t + φ sin ω t φ solution ( ) ho ( ) δt ( t ) ho + pt e sin( Ω t + φ ) + sin( ω t φ) the hoogenous solution the totl solution the pticul solution t the tnsient pocess the sted stte 96

97 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) ( t ) sin( ω t φ)?? ( t ) ωcos( ω t φ) ( t ) ω sin( ω t φ) && + b & + k sin & &&, the plitude ω the cicul fequenc of the eciting foce ( Ω ω ) + ( δω) δω φ ctn φ 8,, π Ω ω 97

98 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) && + b & + k sin δω φ ctn Ω ω, ( Ω ω ) + ( δω) t φ ω π T ω (t) (t) t, the esponse the eciting foce ( t ) sin( ω t φ) t [s] the plitude ω the cicul fequenc of the eciting foce the plitude iu displceent, phse del φ - epesents the tie del of the esponse (t) with espect to the eciting foce φ ω [d] 98

99 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) && + b & + k sin δω φ ctn Ω ω ( Ω ω ) + ( δω), ( t ) sin( ω t φ) k ξη φ ctn η the plitude ω the cicul fequenc of the eciting foce ( η ) + ( ξη) k stt the sttic defotion η ξ ω Ω δ Ω the tuning the dping tio ω ηω δ ξω Ω 99 k

100 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) && + b & + k sin δω φ ctn Ω ω ( Ω ω ) + ( δω), ( t ) sin( ω t φ) k ξη φ ctn η the plitude ω the cicul fequenc of the eciting foce ( η ) + ( ξη) k stt the sttic defotion The solution fo undped ibtion - δ, ξ (oe pecisel sll dped δ<<ω, ξ<<). Ω k η ω the displceent is in the se phse φ if ω < Ω η < with the eciting foce nebo the displceent is in the opposite phse φ π 8 if ω > Ω η > to the eciting foce

101 honicll chnging eciting foce k S k sin(ω t) 3. Vibtions with degee of feedo T T t b D b ( ω t) && + b & + k sin δω φ ctn Ω ω ( Ω ω ) + ( δω), ( t ) sin( ω t φ) k ξη φ ctn η the plitude ω the cicul fequenc of the eciting foce ( η ) + ( ξη) k stt the sttic defotion The solution fo undped ibtion - δ, ξ (oe pecisel sll dped δ<<ω, ξ<<). Ω k η ω the plitude is positie if ω<ω, η<, the se phse, φ the plitude is negtie ifω>ω, η>, the opposite phse

102 plitude nd phse chcteistics 3. Vibtions with degee of feedo the cuse the consequence ( t ) sin( ω t) ( t ) sin( ω t φ), ω, φ ( Ω ω ) + ( δω) φ δω ctn Ω ω

103 plitude chcteistic 3. Vibtions with degee of feedo ( ω) ( η) k ( Ω ω ) + ( δω) ( η ) + ( ξη) ible ible δ δ>. ω, η. ( η ) stt k 3.. ω Ω, η. The esonnce. st 3. ω>>ω, η>>, (η ) η η ω Ω.. the esonnce ω ηω 3

104 plitude chcteistic 3. Vibtions with degee of feedo ( ω) ( η) k ( Ω ω ) + ( δω) ( η ) + ( ξη)! ible ible. ω Ω, η. The esonnce ppes if the eciting fequenc is ne to the ntul fequenc. The plitude eches eteel high lue. δ δ> The totl iu of the chcteistic is t the tuning of : ηes ξ This ens slightl ω<ω, η<. st η η ω Ω ω ηω. the esonnce The plitude in esonnce is : st _ es ξ ξ 4

105 phse chcteistic 3. Vibtions with degee of feedo φ ω δω ctn Ω ω ( ) φ η ξη ctn η ( ) φ 8º δ 9º δ> η η ω Ω ω ηω 5

106 6

107 the fee ibtion 4. The ibtion with degee of feedo diection diection Ω k L Ω k R k L 9º k R Ω? 7

108 the fee ibtion 4. The ibtion with degee of feedo α k L k R l L '' sinα SL SR α l L ' cos α l l L R SL SL SL SR cos α + sin α k k L R l l L SR SR cos α + sin α sin α + cos α R sin α cos α l R '' cos α l R ' sinα 8

109 the fee ibtion 4. The ibtion with degee of feedo α k L k L k R SL SR α ( cos α + sin α) cos α + k R ( sin α + cos α) sin α ( cos α + sin α) sin α k ( sin α + cos α) cos α k L R k k + k cos α + k sin α + k k sin α cos α ( L R ) ( L R ) ( k k ) sin α cos α + ( k sin α + k cos α) + L R L R k k && + k + k && + + k + k && k k && k k M u& + K u M K u u& & l l L R SL SL SL SR cos α + sin α k k L sin α cos α k k k > k > the ss ti the stiffness ti the colun ti (ecto) of displceents9 the colun ti (ecto) of cceletions R l l L SR SR cos α + sin α sin α + cos α R

110 && + k && + k sin sin + k + k ( Ω t) ( Ω t) the fee ibtion u c sin 4. The ibtion with degee of feedo M u& + K u ( Ω t) c the colun ti (ecto) of plitudes && && Ω Ω sin sin ( Ω t) ( Ω t) u& & cω sin ( Ω t) Ω Ω sin sin ( Ω t) + k sin( Ω t) + k sin( Ω t) ( Ω t) + k sin( Ω t) + k sin( Ω t) ( k Ω ) + k k + ( k Ω ) ( Ω t) + K cω sin( Ω t) M cω sin ( K Ω M) c D c d d + d + d D d d d k Ω d k k Ω k

111 the fee ibtion 4. The ibtion with degee of feedo d d + d + d the tiil solution the non-tiil solution d d + d + d d + d d d d d d d d d d d the linel dependent equtions (the. eq. is the ultiple of the. eq.) the infinite nube of solutions fo nd eists, the onl we cn clculte is thei tio d d d d d d k Ω k the fequenc deteinnt k k Ω

112 the fee ibtion 4. The ibtion with degee of feedo k Ω k k k Ω the fequenc deteinnt ( Ω ) ( k Ω ) k k k ( k + k ) Ω + k k k k 4 Ω ( k + k ) λ + k k k k λ the bi-qudtic eqution whee λ Ω λ λ ( k + k ) ± ( k + k ) 4 ( k k k k ) ( + k ) ± ( k + k ) 4( k k k k ) k Ω, λ, Ω slle, Ω gete k k k k k k L L cos sin α + k ( k k ) L R α + k R R sin sin α cos α cos α α λ Ω k L k L λ Ω k R k R

113 the fee ibtion 4. The ibtion with degee of feedo diection α k L k P α diection Ω k L d d d d k Ω k Ω k R diection diection k k k k L cos α + k sin α ( k k ) sin α cos α L R R k Ω tn α k diection ode shpe k Ω k tn α diection ode shpe _ Ω _ Ω _ Ω _ Ω k Ω Ω Ω k Ω k k the odl ti 3

114 the fee ibtion 4. The ibtion with degee of feedo diection α k L k P α diection Ω k L d d d d k Ω k Ω k R diection diection the noliztion (scling) of ode shpes the noliztion to unit less _ thn _ the noliztion with espect to the ss ti { } M 4

115 the fee ibtion 4. The ibtion with degee of feedo diection α diection ( t ) κ sin( Ω t + φ ) + κ sin( Ω t + φ ) ( t ) κ sin( Ω t + φ ) + κ sin( Ω t + φ ) k L k P α κ κ κ κ diection diection the integtion constnts κ, κ, φ nd φ esults fo the initil conditions t... (t), (t), (t), (t) 5

116 the fee ibtion 4. The ibtion with degee of feedo diection α k L k P diection - ( t ) κ sin( Ω t + φ ) + κ sin( Ω t + φ ) ( t ) κ sin( Ω t + φ ) + κ sin( Ω t + φ ) 6 6 diection α diection t... (t) (t) 5 (t) -,5 /s (t), /s.5.5 ( t) ( t).5 6

117 the fee ibtion 4. The ibtion with degee of feedo diection α k L k P diection 4 ( t ) κ sin( Ω t + φ ) + κ sin( Ω t + φ ) ( t ) κ sin( Ω t + φ ) + κ sin( Ω t + φ ) 5 diection α diection t... (t) (t) 6,5 (t), /s (t),65 /s 7

118 the su : the fee ibtion 4. The ibtion with degee of feedo duing the ibtion with n degees of feedo n ntul fequencies ppes, these e nged in ode of thei gnitude f < f < f 3... (one ntul fequenc is siple nube) to ee one ntul fequenc one ode shpe coesponds, giing the tio of the single degees of feedo plitudes. (one ode shpe is the colun ti with n ows, equl to the nube of degees of feedo) the ntul fequencies nd ode shpes e the odl chcteistics, the esulting ibtion is the line cobintion of ll ode shpes, the coefficients of the line cobintion e deteined in dependence on the initil conditions. 8

119 9 k k c k b D Db Db Dc,, Db D + Dc Db D k k l ( ) b b b Db k k l c c c Dc k k l ( ) b k k + ( ) c b k k ( ) ( ) k k k k k k c b b b b k k k k k k c b b b b && && K M & + & 4. The ibtion with degee of feedo the fee ibtion Dnics

120 k k c k b D Db Db Dc,, k k k k k k c b b b b Ω + Ω + fekenční deteinnt ( ) ( ) k k k k k b c b b Ω + Ω + ( ) ( ) [ ] ( ) ( ) k k k k k k k k k b c b b c b b Ω Ω c b c b 4 + Ω Ω c 4 b b Ω, Ω + b b k k k c,, Ω + b b k k k c,, 4. The ibtion with degee of feedo the fee ibtion Dnics

121 the fee ibtion 4. The ibtion with degee of feedo the genelized eigenlue poble ( K Ω M) u the ntul cicul fequencies the eigenfequencies the ode shpes the eigenectos the nube of ntul fequencies Ω,,... nd the nube of ode shpes u,,... is equl to the nube of DO the specil eigenlue poble ( Ω ) u M K the odl ti U u u M u n,,, u u u,, M n, K K O K u u u, n, n M n, n n Ω Ω Ω n

122 the foced ibtion 4. The ibtion with degee of feedo k sin(ω t) k b sin(ω t) k c + && &&,, ( k + k b ) k b sin( ω t) k + ( k + k ) sin( ω t) && && b b k + k + k b b c k k b b + k c M & + K ω ω sin sin sin sin ( ω t) ( ω t) f ( ω t) ω ( ω t) ω sin sin ( ω t) ( ω t)

123 the foced ibtion 4. The ibtion with degee of feedo sin(ω t) sin(ω t) k k b k c,, ω ω sin sin ( ω t) ( ω t) k + k + k b b k k b b + k c sin sin ( ω t) ( ω t) sin sin ( ω t) ( ω t) k + k k b b k k b b + k c ω ( ) K ω M D f D the dnic stiffness ti f ( K Ω M) c 3

124 the foced ibtion 4. The ibtion with degee of feedo k sin(ω t) k b sin(ω t) k c,, the solution of the sted stte foced ibtion plitudes the sste of lgebic equtions the opetion shpe of ibtion ( k + k ω ) b k b k ( k b + k c ω ) ( K ω M) f ( K Ω M) c ω Ω the esonnce. the eciting foces he the diffeent phse ngle,. including dping. The solution in cople nubes b the ntul fequencies nd ode shpes 4

125 the foced ibtion 4. The ibtion with degee of feedo (t) sin(ω t) k k b ωnti ω Ω_ ω ω Ω_ nti-esonnce the zeo ibtion plitude in 5

126 6

127 the tpes of otion 5. Tnsltion nd ottion tnsltion ottion genel plne otion tnsltion spheicl otion spil otion genel spce otion the plne otion : ll points oe in the plnes pllel one to the othe the spce otion 7

128 the tpes of otion 5. Tnsltion nd ottion tnsltion D no lines chnge thei diection 8

129 the tpes of otion 5. Tnsltion nd ottion D one line does not chnge its position ottion 9

130 the tpes of otion 5. Tnsltion nd ottion D genel plne otion 3

131 the tpes of otion 5. Tnsltion nd ottion D genel plne otion 3

132 the tpes of otion 5. Tnsltion nd ottion no lines chnge thei diection tnsltion 3D 3

133 the tpes of otion 5. Tnsltion nd ottion one point does not chnge its position 3D spheicl otion 33

134 the tpes of otion 5. Tnsltion nd ottion the bod ottes bout specific is nd tnsltes in the diection of this is 3D otce spil otion posu 34

135 the tpes of otion 5. Tnsltion nd ottion 3D genel spce otion 35

136 the tnsltion - kinetics 5. Tnsltion nd ottion no lines chnge thei diection, o 3 degees of feedo η,,z the fied coodinte sste (not oing), its oigin is O O ζ Ω ξ ξ,η,ζ the bod coodinte sste fied to the bod nd oing with it, its oigin is Ω ξ//, η//, ζ//z z the coon point of the bod 36

137 z P Ω ζ η the tnsltion - kinetics no lines chnge thei diection, o 3 degees of feedo Ω Ω ξ 5. Tnsltion nd ottion Ω + Ω the dius ecto of the point with espect to z Ω the dius ecto of the point Ω with espect to z, the position of the bod in spce Ω the dius ecto of the point with espect to ξηζ, the position of 37 inside the bod

138 z P Ω ζ η the tnsltion - kinetics no lines chnge thei diection, o 3 degees of feedo Ω Ω ξ Ω + Ω the tie deitie & & & & Ω Ω Ω + Ω the tie deitie & & Ω Ω ll the points oe on the se tjecto, with the se elocit nd the se cceletion. 5. Tnsltion nd ottion Ω 38

139 the tnsltion - kinetics no lines chnge thei diection 5. Tnsltion nd ottion the diect line tnsltion ll the points oe on the se tjecto, with the se elocit nd the se cceletion. 39

140 the tnsltion - kinetics no lines chnge thei diection 5. Tnsltion nd ottion the cicul tnsltion R ll the points oe on the se tjecto, with the se elocit nd the se cceletion. 4

141 the tnsltion - kinetics no lines chnge thei diection 5. Tnsltion nd ottion the ccloid tnsltion ll the points oe on the se tjecto, with the se elocit nd the se cceletion. 4

142 the eqution of otion i the tnsltion - dnics 5. Tnsltion nd ottion the d lebet pinciple D + D i dg d d dg G d dg d dg D dd dd d d dd dd G d d G the point of ppliction of the dálebet foce is in the cente of git 4

143 the eqution of otion i the tnsltion - dnics b B 5. Tnsltion nd ottion b D T G B t G cos φ T φ G φ t ω ω ωdω φ φ ω g [ ω ] [ sin φ] φ φ ω ε g cos φ g ε cos φ dω g ω cos φ dφ g ωdω cos φdφ g cos φdφ g ( ) ω + ( sin φ sin φ ) ω φ 43 ( φ) ω( φ) ω + g ( sin φ sin φ )

144 the d lebet foce in the cente of git (tngentil nd nol coponent) D D t n ω t n + D ω g cos φ the tnsltion - dnics g G b T ( sinφ sinφ ) b B the d lebet pinciple D + D D S D 5. Tnsltion nd ottion D t D n i B S T G the thee equtions of equilibiu to sole : ) the eqution of otion, ) the ection foces. i i M i ε g cos φ S K S D K 44

145 the tnsltion - dnics 5. Tnsltion nd ottion i b B D + D i b D T G ssebl of equtions of otion centlize ll ss into the ss pticle. To sole foces (usull ection foces) use the d lebet pinciple to locte the d lebet foce within the cente of git, to sseble the equtions of equilibiu nd sole foces. o equtions of equilibiu the eqution of otion cn be deied. 45

146 ω, ε φ the ottion - kinetics 5. Tnsltion nd ottion one line does not chnge its position DO ee point uns on the cicul tjecto o of the dius R φ dφ ω φ& dt dω ε ω & dt d φ dt the ottion ngle [, d, eolute] the ngul elocit [d/sec, e/in] ( ω ) ω && d d φ ω dφ dφ the ngul cceletion [d/sec ] 46

147 the ottion - kinetics 5. Tnsltion nd ottion one line does not chnge its position DO ee point uns on the cicul tjecto o of the dius R ω, ε φ ω R φ ω the dius ecto the cicufeentil elocit t the tngentil cceletion n the nol cceletion ε dφ dt φ& dω ω& dt n t R φ, ω, ε S the ottion ngle [, d, eolute] the ngul elocit [d/sec, e/in] the ngul cceletion [d/sec ] ω R t εr s φ R n ω R ω t n ε ω 47

148 the ottion - dnics 5. Tnsltion nd ottion ω, ε the eqution of otion i I ε R M R _ i R the bod ss (not pesent in the eqution of otion) I R - the oent of ineti with espect to the cente of ottion R [kg ] ε - the ngul cceletion [d/s ] M the foce oent [N ] 48

149 the ottion - dnics 5. Tnsltion nd ottion ω, ε the eqution of otion i R D t T D n Gt Gn G M D, I R I ε R M R _ i the d lebet pinciple the d lebet foces (nd oent) M D D t I R ε Tt ε T D n Tn ω T 49

150 the ottion - dnics 5. Tnsltion nd ottion the pi, etenl foces (ctions) the eqution of otion ω, ε the ection foces R R R D n D t M D I ε R M R _ i the d lebet pinciple the d lebet foces (nd oent) M D D D t n I R ε Tt Tn ε ω T T i i M Ri the ection solution Σ including d lebet foces the eqution of otion E K IR ω the kinetic eneg 5

151 tnsltion the ottion - dnics nlog 5. Tnsltion nd ottion ottion the tck s,,... [, ] ~ the ngle φ [d, ] the elocit [/s] ~ the ngul elocit [d/s] s& ω φ& the cceletion [/s ] ~ the ngul cceletion [d/s ] & & s d ds dω ε ω & && φ ω dφ s t + t eple unifol cceleted otion + t + s ~ ~ ω ε t + ω φ ε t + ω t + φ 5

152 tnsltion the ottion - dnics nlog 5. Tnsltion nd ottion ottion the foce the ss the eqution of otion the kinetic eneg the wok the powe, G,... [N] ~ the foce oent M [N ] [kg] ~ the oent of ineti E K i [J] ~ ~ the eqution of otion d s [N ] ~ the wok P [W] ~ the powe the kinetic eneg chnge lw the kinetic eneg E EK EK Iε E K K I [kg ] Mi Iω M dφ P M ω [J] [N ] [W] [J ~ N ] 5

153 the ottion - dnics 5. Tnsltion nd ottion the phsicl pendulu : ω,ε φ The phsicl pendulu The bod of the ss nd oent of ineti I. The distnce between the cente of ottion nd the cente of git is. Iε G sin φ Iε + G sin φ Iφ && + G sin φ lineiztion sinφ φ G, I I & φ + G φ G sinφ 53

154 the ottion - dnics 5. Tnsltion nd ottion the phsicl pendulu : φ ω,ε G G the initil conditions : t... φ φ the initil ngle ω ω the initil ngul elocit, I The phsicl pendulu The bod of the ss nd oent of ineti I. The distnce between the cente of ottion nd the cente of git is. I & φ + G φ Ω γ & + k I & φ + G φ the nlogicl solution : sin ( Ω t + γ ) φ sin( Ω t + γ ) k + Ω Ω ctn the cicul fequenc the plitude the phse led Ω γ G I φ ω + Ω 54 φ Ω ctn ω

155 the ottion - dnics 5. Tnsltion nd ottion the phsicl pendulu : The phsicl pendulu The bod of the ss nd oent of ineti I. The distnce between the cente of ottion nd the cente of git is. ω,ε I & φ + G φ φ the clcultion of the oent of ineti I fo esued ibtion peiod T : G, I Ω π T G Ω G I I G Ω 55

156 the ottion - dnics 5. Tnsltion nd ottion the oent of ineti S d I d the thin ing const I d d 56

157 the ottion - dnics 5. Tnsltion nd ottion the oent of ineti S d I d the thin pistic od I d d d l d l d l d d I I l l d l l 3 3 l l l 3 l 3 d I 3 l 57

158 58 d I d S d I d d d d l l d d I / / / / l l l l l l d d l the thin pistic od I 3 3 l l l l l / / I l the ottion - dnics 5. Tnsltion nd ottion the oent of ineti Dnics

159 the ottion - dnics 5. Tnsltion nd ottion the oent of ineti I d h R the clinde 59

160 the ottion - dnics 5. Tnsltion nd ottion the oent of ineti d I d h d ρdv ρds h ρ ( π d) h the clinde ds π d 6

161 the ottion - dnics 5. Tnsltion nd ottion the oent of ineti h I the clinde R R d R R I d d ρdv ρds h ρ ( π d) h ρ V Sh πr h d π d h d πr h R R 3 d R 4 4 R R R 4 4 I R 6

162 the ottion - dnics 5. Tnsltion nd ottion the oent of ineti to the pllel shifted is I d I I G + e I e G I G I G - the oent of ineti to the is cossing the cente of git, I - the oent of ineti to the pllel shifted is. the Steine theoe 6

163 63 the thin cicul plte 4 T I b T b I _ z ( ) z T b I + _ T I _ ( ) 3 4 T I + the clinde 3 T I the cone the pid b ( ) T b I + 5 T I the ottion - dnics 5. Tnsltion nd ottion the thin ectngul plte the bll Dnics

164 the fi publiction the ottion - dnics 5. Tnsltion nd ottion 64

165 fi publiction the ottion - dnics 5. Tnsltion nd ottion 65

166 3D D odeling the ottion - dnics 5. Tnsltion nd ottion PRINT MSS PROPERTIES SSOITED WITH THE URRENTLY SELETED VOLUMES TOTL NUMBER O VOLUMES SELETED (OUT O DEINED) *********************************************** SUMMTION O LL SELETED VOLUMES TOTL VOLUME.537E+8 TOTL MSS.996E- ENTER O MSS: X-.4674E-3 Y. Z. *** MOMENTS O INERTI *** BOUT ORIGIN BOUT ENTER O MSS PRINIPL IXX IYY IZZ IXY.55354E E-3 IYZ.4695E E-4 IZX -.635E E-4 PRINIPL ORIENTTION VETORS (X,Y,Z): (THXY THYZ. THZX.) 66

167 kinetics 6. Genel plne otion tnsltion ottion genel plne otion the plne otion : ll points oe in plnes pllel to one nothe 67

168 kinetics 6. Genel plne otion, o 3 degees of feedo DO ottion DO tnsltion ottion tnsltion tnsltion 3 DO 68

169 kinetics 6. Genel plne otion φ, ω, ε ottion olling without slipping DO one independent otion olling without slipping,, tnsltion,, independent tnsltion nd ottion DO φ, ω, ε sliding in the touch-point two independent otions sliding in the touch-point 69

170 kinetics 6. Genel plne otion - the nlticl solution - the pole ethod - otion decoposition 7

171 the nlticl solution 6. Genel plne otion the bsic schee s& & & s d ds B + B l the geoet solution B, B B l, gien, deteine B, B, q( ) B B B B B l d dt d dt B B dp d dp d d d dp dt d dt B the elocit solution p( ) d dt l d + p dt + p + p q + p the cceletion solution 7

172 the nlticl solution 6. Genel plne otion s& & & s d ds B l cos φ B l sin φ B, B B l φ ω,ε, d dt d dφ l sin φφ& l ω sin φ dφ dt B B B db d dt dφ l cos φφ& l ωcos B φ dφ dt d l ω & sin φ l ωcos φφ& dt l ε sin φ l ω cos φ db l ω & cos φ l ω sin φφ& B dt 7 l ε cos φ l ω sin φ B

173 the nlticl solution 6. Genel plne otion B ( s) the geoet solution s f s, s B the genelized coodintes longitudinl o ngul B ds dt B ds ds B ds dt p ( s) ds ds B ds dt the elocit solution B p( s), B the genelized elocities longitudinl o ngul B B B d dt B dp ds q d ds dt ( p ) dt + p dp dt + p ( s) ( s) + p d dt dp q ds ( s), B the genelized cceletions longitudinl o ngul d ds s B the cceletion solution d dt 73

174 the nlticl solution 6. Genel plne otion φ,ω,ε ccloid cue φ 36º π 6,8 d φ φ olling without sliding, φ ω ε π 74

175 the pole ethod 6. Genel plne otion the pole the instntneous cente of zeo elocit B n B π B n B π ω B B l n B l n B B B ω π B ωbπ ω 75 B

176 the pole ethod 6. Genel plne otion the pole the instntneous cente of zeo elocit B n B π ω onl elocities! n n not cceletions! 76

177 the pole ethod 6. Genel plne otion the pole the instntneous cente of zeo elocit the B point oes on the line tjecto B B n B Bn l π n n point oes long the line tjecto onl elocities! the B point oes on the cicul tjecto not cceletions! B B Bn n B l ω π n n point oes long the cicul tjecto n R 77

178 the pole ethod 6. Genel plne otion the pole is t diffeent points t diffeent oents set of points epesenting pole in the pst, pesent nd futue is the pole cue B π (t- t) π (t) π (t+ t) 78

179 the pole ethod 6. Genel plne otion the pole is t diffeent points t diffeent oents set of points epesenting pole in the pst, pesent nd futue is the pole cue B π (t- t) π (t) B π (t) π (t+ t) the fied pole cue the oing pole cue set of points of pole in fied spce, in fied coodinte sste, the fied pole cue set of points of pole in oing spce, in bod coodinte sste, the oing pole cue 79

180 the pole ethod 6. Genel plne otion the pole is t diffeent points t diffeent oents set of points epesenting pole in the pst, pesent nd futue is the pole cue B π (t- t) π (t) the pole cues touch one nothe t the point whee the pole is t pesent π (t+ t) the fied pole cue the oing pole cue 8

181 the pole ethod 6. Genel plne otion the genel plne otion cn be intepeted s the olling of oing pole cue on fied pole cue (no tte if it coesponds to the el technicl eliztion) π (t- t) B D π (t) ω π (t+ t) E the oing pole cue the fied pole cue 8

182 the pole ethod 6. Genel plne otion the genel plne otion cn be intepeted s the olling of oing pole cue on fied pole cue (no tte if it coesponds to the el technicl eliztion) B π (t- t) D π (t) E π (t+ t) the oing pole cue the fied pole cue the B point uns on the line tjecto B olling the fied pole cue the oing pole cue the point uns on the line tjecto 8

183 the pole ethod 6. Genel plne otion olling without sliding the oing pole cue φ,ω ω the fied pole cue π onl elocities! π pole not cceletions! 83

184 otion decoposition bsic decoposition 6. Genel plne otion B B tnsltion tns B ot ottion B + the supeposition of tnsltion nd ottion gien :, the elocit nd cceletion of the point, deteine : B, B the elocit nd cceletion of the point B. 84

185 otion decoposition bsic decoposition 6. Genel plne otion B B tnsltion tns B ot B ottion B the efeence point the supeposition of tnsltion nd ottion B B B B_ tns + + B + B B_ ot 85

186 B B ω l otion decoposition bsic decoposition + B B B B tn φ sin φ B φ 6. Genel plne otion B B φ ω B l l sinφ 86

187 B B B φ l ω, ε + tn φ Bn + B B + Bn Bt B B Bn otion decoposition bsic decoposition sinφ + cos tn φ φ φ + Bn B B l ω l + Bn cos φ Bt sinφ sinφ + cos φ Bn Bt Bt 6. Genel plne otion Bn + sin φ tnφ φ Bt B B ε hoizontl eticl Bt l 87

188 the efeence point, φ ω b B otion decoposition ω ε + tnsltion + ottion B B B ψ B ( B) φ 6. Genel plne otion B B B ω b olling without sliding B B_ + ψ ctn B_ B_ B_ B_ + B _ B_ B _ ( + b φ) ω cos ω b + cos φ B_ B _ B_ B _ ωb sin φ b sin φ 88

189 the efeence point B γ, φ B_ + ctn B_ B_ b ω,ε olling without sliding B_ B otion decoposition ω ε B_ B_ γ B + B B B _ t _ + + tnsltion + ottion φ B B _ n B _ t B _ t B + B B _ n B _ t B _ n _ B_ B _ t _ + ( + b cos φ) ω b φ φ B _ n ε b ω b B _ n _ ε sin ε b sin φ + ω b cos φ B_ 6. Genel plne otion ε ω 89

190 the efeence point, φ b ω,ε olling without sliding B otion decoposition ω ε B B B ψ γ B _ t B _ n B B + + tnsltion + ottion ψ B B _ t 6. Genel plne otion + B _ n B _ t B _ n ε b ω b B γ B_ + ctn B cos _ t B B sin _ n B B_ B_ B_ ( γ ψ) ( γ ψ) 9

191 shoe otion decoposition genel decoposition 6. Genel plne otion the eltie cicling the fe otion the esulting otion the fe otion + the eltie otion 9

192 otion decoposition genel decoposition the esulting otion the fe otion + the eltie otion 6. Genel plne otion genel plne otion tnsltion + ottion eltiní posu bsic decoposition tnsltion ottion genel plne otion ottion + tnsltion unášiá otce eltiní otce genel plne otion ottion + ottion unášiá otce eltiní pohb - posuný příočý tnsltion tnsltion + tnsltion unášiý pohb - posuný kuhoý 9

193 otion decoposition 6. Genel plne otion genel decoposition the esulting otion the fe otion + the eltie otion fe el + ω fe fe + el fe ω fe fe el 93

194 otion decoposition 6. Genel plne otion genel decoposition the esulting otion the fe otion + the eltie otion fe_t + fe_n el el ω fe fe + el + + o fe fe_ n ω el + fe fe o _ t el the oiolis cceletion + el + o ε fe el fe _ t o fe_ n fe _ n fe_ t fe ε ω fe fe 94 the genel decoposition the oiolis decoposition

195 dφ dφ the oiolis cceletion ds dφds the eltie otion dφds dt dφds 6. Genel plne otion dt dφds dφ dt dt ωfe ds dt el o ω fe el o ω fe el ω fe the fe otion o el 95

196 the Resl ngul cceletion 6. Genel plne otion ω fe ω el ω ε esult esult ω ε fe fe + ω + ε el el + ε Resl ε Resl ω fe ω el 96

197 dnics bsic decoposition + the d lebet pinciple 6. Genel plne otion the d lebet foces the pi foces the ections M the equtions of equilibiu the d lebet pinciple the tnsltion nd ottion supeposition D t M π M D I ε Dt D n Gt Gn the eqution of otion B tnsltion εg ω D t G ottion G D n π M D the efeence point G the distnce between cente of git nd efeence point D t 97

198 bsic decoposition dnics 6. Genel plne otion the kinetic eneg B E k E k _ tnsltion + E k _ ottion cente of git G ω the efeence point the cente of git G E k G + I G ω I + ω 98

199 99 M i i i olling without sliding the genel plne otion with degee of feedo,, φ, ω, ε ε ω φ N L M D D D I M D G n G t D t ω ε ε the cente of git the efeence point π M i π_ the eqution of otion G D M p D α + sin G α α ε + sin G I S α + sin G I S α + sin G I S i i α cos G N D t G L α sin I I G L + α sin f N L f I I + tnα the not-sliding condition the ection solution D t G 5 I the bll α sin, G 4 6. Genel plne otion dnics Dnics

200 dnics 6. Genel plne otion olling without sliding the genel plne otion with degee of feedo the cente of git the efeence point φ, ω, ε,, Dt L D t π N G D t G G α φ ω ε M D D I M D t G Mi Dn ω G D n ε ε f, ( φ ω ) _ i _ i the eqution of otion ω 5 the diffeentil eqution of II. ode, non-line t non-unifo otion!

201 with sliding the genel plne otion with degees of feedo the cente of git the efeence point φ, ω, ε N f _ D t i N G G α the ection solution,, N G cos α φ ω ε - degees of feedo - independent tnsltion nd ottion - independent equtions of otion dnics D M D M D t G D ω t n D I ε ε G M _ i M D ε G cos α f I _ D t + G sinα i G 6. Genel plne otion ( sinα f cos α) _ i _ i M the equtions of otion i

202

203 kinetics 7. Spheicl otion one point does not chnge its position (sts otionless) the point is clled the cente of the spheicl otion konst střed sféického pohbu o k o 3 á 3 DO DO S o k 3

204 kinetics 7. Spheicl otion one point does not chnge its position (sts otionless) the point is clled the cente of the spheicl otion DO 4

205 kinetics the bsic kinetic quntities e ngul elocit ω nd ngul cceletion ε. 7. Spheicl otion both lue nd diection of these chnge duing spheicl otion the ecto of the instntneous ngul elocit ω deteines the instntneous is of zeo elocit (ee oent going though the cente of spheicl otion). the eleent otion cn be intepeted s the instntneous ottion bout the instntneous is of zeo elocit 5

206 kinetics the bsic kinetic quntities e ngul elocit ω nd ngul cceletion ε. 7. Spheicl otion the set of lines, being instntneous is of zeo elocit in pst, pesent nd futue, is the pole cone in fied spce fied pole cone in oing spce oing pole cone the instntneous is of zeo elocit the oing pole cone the fied pole cone the pole cones touch one the othe in the pesent instntneous is of zeo elocit the cente of the spheicl otion ω - the instntneous ngul elocit The spheicl otion cn be intepeted s the olling of the oing pole cone on the fied pole cone 6

207 kinetics 7. Spheicl otion the Eule s ngles fied coodinte sste z, bod coodinte sste ξηζ. ψ the ngle of pecession bout the z is. ϑ the ngle of nuttion bout the ξ is 3. φ the ngle of pi ottion bout the ζ is both S with coon oigin in the cente of spheicl otion. z ζ ζ ζ z ζ ζ ζ ζ z ζ ζ ϑ ϑ η φ η η ψ η η η ψ η η φ η ξ ξ ξ ξ ξ ξ uzloá přík ξ ξ ξ 7

208 kinetics 7. Spheicl otion the Eule s ngles fied coodinte sste z, bod coodinte sste ξηζ both S with coon oigin in the cente of spheicl otion.. ψ the ngle of pecession bout the z is the otion of pecession. ϑ the ngle of nuttion bout the ξ is 3. φ the ngle of pi ottion bout the ζ is the otion of nuttion the otion of pi ottion the pecession the pi ottion the nuttion the otion of pecession the fe ottion bout the fied is (z) the otion of nuttion the eltie ottion bout the is, doing pecession the otion of pi ottion 8 the eltie ottion bout is, doing pecession nd nuttion

209 kinetics 7. Spheicl otion the Eule s ngles fied coodinte sste z, bod coodinte sste ξηζ both S with coon oigin in the cente of spheicl otion.. ψ the ngle of pecession bout the z is the otion of pecession. ϑ the ngle of nuttion bout the ξ is 3. φ the ngle of pi ottion bout the ζ is the otion of nuttion the otion of pi ottion the pecession the pi ottion the otion of pecession the fe ottion bout the fied is (z) the otion of nuttion the eltie ottion bout the is, doing pecession the otion of pi ottion 9 the eltie ottion bout is, doing pecession nd nuttion

210 kinetics 7. Spheicl otion the Eule s kinetic equtions ω ω ω z φ& sin ϑ sinψ + ϑ & cos ψ φ& sinϑ cos ψ + ϑ & sinψ φ& cos ϑ + ψ& ω ω ω ξ η ζ the ngul elocit ω ψ& sinϑ sinφ + ϑ & cos φ ψ& sinϑ cos φ ϑ & sin φ ψ& cos ϑ + φ& ω ω + ω + ω z + ω + ω the diection ngles of the ecto : ω ξ η ζ K ω cos α cos β ω ω ω ψ& + ϑ& with espect to the is with espect to the is ω ω z z + φ& ω β γ + ψ& φ& cos ϑ ω ω α ωz cos γ ω with espect to the z is

211 kinetics 7. Spheicl otion the Eule s kinetic equtions ω ω ω z φ& sin ϑ sinψ + ϑ & cos ψ φ& sinϑ cos ψ + ϑ & sinψ φ& cos ϑ + ψ& ω ω ω ξ η ζ the ngul elocit ω ψ& sinϑ sinφ + ϑ & cos φ ψ& sinϑ cos φ ϑ & sin φ ψ& cos ϑ + φ& ω ω ω z ζ ξ η + ω + ω z ω the cclic chnge i ω ω ξ j ω + ω η k ω z z + ω ζ K ψ& + ϑ& + φ& + ψ& φ& cos ϑ ( ω z ω ) i + ( ω ω z) j + ( ω ω ) k ω z z ω z the cicufeentil elocit z z ω ω ωz ω z it jt k t ω ωξ ωη ωζ η ξ η ζ ω ζ ω η ω η ω ( ωη ζ ωζ η) it + ( ωζ ξ ωξ ζ) jt + ( ωξ η ω ξ) k t ξ η ζ η ωζ ξ ωξ ζ ζ ξ η ξ

212 kinetics 7. Spheicl otion the ngul cceletion ε ε ω & ε ε i + ε ε ω & ε ω& ξ it i t & ω ( ) ω i + ω j + ω k ω& i + ω& j + ω& k ε ε ε z j + ε z k z in fied coodinte sste z z ε ω& ε ε z ω& ω& z && φ sin ϑ sin ψ + φϑ & & cos ϑ sin ψ + φψ & & sin ϑcos ψ + ϑ && cos ψ ϑψ & & sin ψ φ && sin ϑcos ψ φϑ & & cos ϑcos ψ + φψ & & sin ϑ sin ψ + ϑ && sin ψ + ϑψ & & cos ψ & φ cos ϑ φϑ & & sin ϑ + ψ& in bod coodinte sste ξηζ ( ) ω it jt k & t it i & t jt j & ξ + ωη + ωζ ωξ + ωξ + ωη + ωη t + ωζ k t + ωζ k t + ω& η j t + ω& ζ k t + ω ξ & i t + ω the dius ecto of the point {,,} ω & & i + ω j & η & j t + ω ζ & & k t jt ( ) ω ωξ it + ωη jt + ωζ k t ω ω & k t & k {,,} {,,} it ω it jt ω jt t ω k t ξ t η t + ω ζ & k t ω ξ ( ω i ) + ω ( ω j ) + ω ( ω k ) t η t & ζ t &

213 kinetics 7. Spheicl otion the ngul cceletion ε ε ω & ε ε i + ε ε ω & ε ε ξ ( ) ω i + ω j + ω k ω& i + ω& j + ω& k ε ε ε z j + ε z k z z ε ω& ε ε z ω& ω& z && φ sin ϑ sin ψ + φϑ & & cos ϑ sin ψ + φψ & & sin ϑcos ψ + ϑ && cos ψ ϑψ & & sin ψ φ && sin ϑcos ψ φϑ & & cos ϑcos ψ + φψ & & sin ϑ sin ψ + ϑ && sin ψ + ϑψ & & cos ψ & φ cos ϑ φϑ & & sin ϑ + ψ& in bod coodinte sste ξηζ ( ) ω it jt k & t it i & t jt j & ξ + ωη + ωζ ωξ + ωξ + ωη + ωη t + ωζ k t + ωζ k t i t + ε η j t ε ξ ε η ε ζ + ε ζ k t & in fied coodinte sste z ε ξ ω& ξ ε ε ζ ω η ω& η ζ ψ&& sin ϑ sin φ + ψ& ϑ & cos ϑ sin φ + ψ& φ & sin ϑcos φ + ϑ && cos φ ϑφ & & sin φ ψ&& sin ϑcos φ + ψ& ϑ & cos ϑcos φ ψ& φ & sin ϑ sin φ ϑ && sin φ ϑφ & & cos φ ψ& cos ϑ ψ& ϑ & sin ϑ + & φ & & & 3

214 4 ( ) & & & + ω ε + ω ω ω ( ) ( ) ( ) k j z i z z k j i z z z ε ε + ε ε + ε ε ε ε ε ε z z ε ε z ε ε z z ε ε ( ) ( ) ( ) k j i k j i z z z z z z ω ω + ω ω + ω ω ω ω ω ω z z ω ω z z ω ω z ω ω k j i z + + z z z z ω + ω ε ε z ω + ω ε ε z z z z ω + ω ε ε 7. Spheicl otion kinetics the cicufeentil cceletion in fied coodinte sste z Dnics

215 kinetics 7. Spheicl otion the cicufeentil cceletion & ε i ε t ξ ξ it ω ω ξ ξ ( ω ) ω + ω ε + ω j ε t η η j ω t η η k ε ζ t ζ k ω t ζ ζ & & ( εη ζ εζ η) it + ( εζ ξ εξ ζ) jt + ( εξ η εη ξ) k t ξ εη ζ εζ η η εζ ξ εξ ζ in bod coodinte sste ξηζ εξ η εη ( ωη ζ ωζ η ) it + ( ωζ ξ ωξ ζ ) jt + ( ωξ η ωη ξ ) k t ξ ω ω η ζ ζ η η ζ ξ ζ ω ω ξ ζ ζ ξ ω ω ξ η η ξ ξ it + η jt + ζ k t ξ ε ζ ε η + ω ω η ζ η ζ ζ η η ε ξ ε ζ + ω ω ζ ξ ζ ξ ξ ζ ζ ε η ε ξ + ω ω ξ η ξ η η ξ 5

216 6 the Eule s equtions of otion ( ) ( ) ( ) ζ ξ η η ξ ζ ζ η ζ ξ ξ ζ η η ξ η ζ ζ η ξ ξ ω + ω ε ω + ω ε ω + ω ε i i i M I I I M I I I M I I I _ the kinetic eneg ζ η ηζ ζ ξ ξζ η ξ ξη ζ ζ η η ξ ξ ω ω ω ω ω ω ω + + ω + ω D D D I I I E K ξ ξ T D ( ) [ ] ( ) [ ] ( ) [ ] ξ η η ξ ζ ζ ζ ζ ξ ξ ζ η η η η ζ ζ η ξ ξ ξ ω + ω ε ω + ω ε ω + ω ε I I I M I I I M I I I M D D D _ η η T D ζ ζ T D the d lebet pinciple M M M M M M D i D i D i ζ ζ η η ξ ξ D D D i i i ζ ζ η η ξ ξ _??? ζ η ξ R R R 7. Spheicl otion dnics the d lebet foce... nd oent the Eule s equtions of otion Dnics

217 the specil cses dnics the he flwheel (the Lgnge flwheel) the bod is ill seticl, the set is is coincident with the is of pi ottion the pecession G 7. Spheicl otion the pi ottion suppoted in the cente of spheicl otion onl the gittionl foce G cts the zeo foce flwheel (the Eule flwheel) the bod is ill seticl, the set is is coincident with the is of pi ottion suppoted in the cente of spheicl otion which is coincident with the cente of git the pecession G the pi ottion 7

218 dnics 7. Spheicl otion the two unifo ottions bout two concuent is of constnt ngle the bod is ill seticl, the set is is coincident with the is of pi ottion the he flwheel ψ& const the pi ottion the pecession φ & const konst zeo nuttion ϑ const ϑ & 8

219 dnics 7. Spheicl otion ψ& G sinϑ the two unifo ottions bout two concuent is of constnt ngle const M G ω ψ the pecession ϑconst the bod is ill seticl, the set is is coincident with the is of pi ottion Gn I I G + G konst G D n I s D M n the pi ottion G I G M ω φ the kinetic eneg EK Iωψ sin ϑ + Is ωψ cosϑ + ω [ ( ) ] φ & const G φ G ωψ sinϑ Gn Is + M ω φ G I s ω ψ ω G ψ ( Is I) cos ϑ ωφ ωψ ω ω ω φ ω ψ φ ω ψ φ ) I s I b) ϑ 9º c) ω ψ << ω φ sin ϑ ω Is + s φ ψ ωφ ψ ( I I) cos ϑ ω ω sin ϑ M G I s ωφ ωψ sinϑ the he flwheel 9

220 dnics 7. Spheicl otion ψ& G sinϑ the two unifo ottions bout two concuent is of constnt ngle const M G ω ψ the pecession ϑconst the bod is ill seticl, the set is is coincident with the is of pi ottion Gn I I G + G konst G I s G φ & G I G ω φ D M n the pi ottion const G ωψ sinϑ I G G sinϑ M G G I G G s G s sin ϑ + Gn + ω ψ ( I I) cos ϑ ω ω sin ϑ s I s + ω φ ω ω ψ ( Is I) cos ϑ ωφ ωψ ω φ G ψ ( Is I) cos ϑ ωφ ωψ ω φ φ the he flwheel ψ

221 8. The echniss The echnis is the sste (chin) of igid bodies, linked one to the othe b joints. The echnis see the pupose to tnsit the foce nd to tnsfo the otion. tnsltion ottion

222 The bsic tes the ebe of echnis - the bod, do not chnge the shpe the fe the ebe fied to the gound (the Eth) 8. The echniss the kinetic pi the pi of ebes, linked togethe b joint the joint coodinte - the coodinte, deteining the eltie position of the ebes one to the othe the diing ebe, the poweed ebe - the ebe on the diing end of the chin - the ebe pefoing the function fo which the echnis ws designed the input ebe, the output ebe - the ebe on the beginning nd on the end of the chin the nube of degees of feedo (DO) - the nube of independent otions the echnis is ble to pefo the echnis coodinte - one o oe independent coodintes, deteining the position of the echnis the kinetic sche - the geoetic sketch of the echnis, siplified s oe s possible

223 8. The echniss The kinetic schee The geoetic sketch of the echnis, siplified s oe s possible... - to etin the diensions necess fo the kinetic function of the echnis, - to suppess the diensions not ipotnt fo the kinetic function of the echnis. the piston the fe the connecting od the connecting od length the cnk length the fe the cnk ojnice the pin The cnk echnis... nd its kinetic schee the cnk the eccenticit - - the function length of the cnk The cnkshft designed s the eccentic pin. The kinetic schee cn be e diffeent fo the el echnis design. 3

224 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss 3 4 the pln echnis the spce echnis 4

225 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss the clssifiction with espect to the nube of degees of feedo The instnt position of the echnis is unbiguousl deteined b the coodintes which nube is equl to the nube of the echnis degees of feedo (DO) The nube of echnis degees of feedo is equl to the nube of independent coodintes, deteining the echnis position the echnis coodinte one o oe independent coodintes, deteining the position of the echnis; the nube of the echnis coodintes is equl to the nube of the echnis DO 5

226 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss the clssifiction with espect to the nube of degees of feedo the echniss with DO posu otce 6

227 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss the clssifiction with espect to the nube of degees of feedo the c goes though the cue the echniss with DO the wheels otte with diffeent speed the echniss with DO the diing shft the diffeentil cge the diffeentil stellites 3 4 φ ω ω 5 ψ 5 the echnis with DO φ nd ψ two echnis coodintes the wheels the diffeentil ge 7

228 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss the clssifiction with espect to the nube of degees of feedo the echniss with DO the echniss with DO the echniss with oe DO the nube of the echnis DO is not liited the echnis with 7 DO 8

229 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss the clssifiction with espect to the nube of degees of feedo the echniss with DO the echniss with DO the echniss with oe DO... with espect to the die tio the echniss with constnt die tio the echniss with chnging die tio ω DR ω ω konst DR ω konst ω ω 9

230 The echnis clssifiction the pln echniss (D) ll ebes pefo the pln otion in ecipocll pllel plnes the spce echniss (3D) t lest one ebe pefo spce otion 8. The echniss the clssifiction with espect to the nube of degees of feedo the echniss with DO the echniss with DO the echniss with oe DO... with espect to the die tio 3 the echniss with constnt die tio the echniss with chnging die tio... with espect to the nube of ebes 3 the ebes echniss the 3 ebes echniss the 4 ebes echniss the oe ebes echniss 4 3

231 The nube of the echnis degees of feedo (DO) The nube of the echnis DO we cn deteine intuitiel o clculte. The nube of echnis DO is equl to the nube of the independent echnis coodintes. ψ φ 3 The onl one coodinte is independent, the othe two cn be clculted. One degee of feedo. B B sin φ B cos φ + B sin ψ B cos ψ The nube of DO foul i the nube of DO, n the nube of ebes (including the fe), c the nube of the st clss joints (fiing DO), c the nube of the nd clss joints (fiing DO). The spce echnis c j the nube of the j clss joints (fiing j DO). ( n ) c c ( 3 ) i 3 i 3 i 6 8. The echniss ( n ) 5 jc j j 3

232 The joints The joints e eithe pln (D) o spce (3D) 8. The echniss The pln joints (D) links two echnis ebes, pefoing pln otion in ecipocll pllel plnes The joints e eithe idelized o el. The idelized joints the fiction is negligible We e inteested in : The joint echnicl popeties with espect to sttics the tnsition of foces The spce joints (3D) links two echnis ebes, fo which t lest one pefo spce otion The el joints the fiction is tken into ccount The joint echnicl popeties with espect to kinetics the eltie otion of one ebe with espect to the othe the joint llows o fies The joint is of the j clss if it fies j eltie otions (fies j degees of feedo) 3

233 The joints The pln (D), idelized joints e : the 3 d clss joint the pefect fiing 8. The echniss fies 3 otions, does not llow neithe one fies both tnsltions nd ottion tnsits foces nd oent the pefect fiing the kinetic schee the fe the bod the bod the bod 33

234 The joints The pln (D), idelized joints e : the 3 d clss joint the pefect fiing the nd clss joints the pin joint 8. The echniss fies 3 otions, does not llow neithe one fies tnsltions, llows ottion fies tnsltions llows ottion 34

235 The joints The pln (D), idelized joints e : the 3 d clss joint the pefect fiing the nd clss joints the pin joint 8. The echniss fies 3 otions, does not llow neithe one fies tnsltions, llows ottion fies tnsltions tnsits foces llows ottion does not tnsit oent the pin joint the kinetic schee the bod the bod the bod the fe 35

236 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion fies tnsltion nd ottion llows tnsltion 36

237 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion fies tnsltion nd ottion tnsits foce nd oent llows tnsltion does not tnsit foce the sliding joint the kinetic schee 37

238 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, elted one to the othe no sliding in touch point φ fies tnsltion, llows ottion nd tnsltion, elted one to the othe φ ω ε 38

239 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, elted one to the othe φ no sliding in touch point φ tnsits foces, does not tnsit oent fies tnsltion, llows ottion nd tnsltion, elted one to the othe ω ε 39

240 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, the st clss joints the sliding pin joint fies tnsltion, llows ottion nd tnsltion, (independent) llows tnsltion nd ottion (independent) φ z fies tnsltion 4

241 The joints The pln (D), idelized joints e : the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, the st clss joints the sliding pin joint fies tnsltion, llows ottion nd tnsltion, (independent) llows tnsltion nd ottion (independent) does not tnsit foce nd oent 8. The echniss φ fies tnsltion tnsits foce z the sliding pin joint the kinetic schee 4

242 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, the st clss joints the sliding pin joint fies tnsltion, llows ottion nd tnsltion, the genel joint fies tnsltion, llows ottion nd tnsltion, (independent) llows tnsltion nd ottion (independent) φ sliding in touch point fies tnsltion 4

243 The joints The pln (D), idelized joints e : 8. The echniss the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, the st clss joints the sliding pin joint fies tnsltion, llows ottion nd tnsltion, the genel joint fies tnsltion, llows ottion nd tnsltion, (independent) llows tnsltion nd ottion (independent) φ sliding in touch point does not tnsit foce nd oent fies tnsltion tnsits foce 43

244 The joints The pln (D), idelized joints e : the 3 d clss joint the pefect fiing fies 3 otions, does not llow neithe one the nd clss joints the pin joint fies tnsltions, llows ottion the sliding joint fies tnsltion nd ottion, llows tnsltion the olling joint fies tnsltion, llows ottion nd tnsltion, the st clss joints the sliding pin joint fies tnsltion, llows ottion nd tnsltion, the genel joint fies tnsltion, llows ottion nd tnsltion, (independent) the joint tnsits such foce o oent, in wht diection it fies the tnsltion o ottion if the joint fies tnsltion in cetin diection, it tnsits ppopite foce if the joint fies ottion bout cetin is, it tnsits ppopite oent 8. The echniss if the joint llows tnsltion in cetin diection, it does not tnsit ppopite foce if the joint llows ottion bout cetin is, it does not tnsit ppopite oent not ectl the se w fo the el joints 44

245 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions tnsits 3 foces nd 3 oents 45

246 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions llows ottion tnsits 3 foces nd oents does not tnsit the oent to the is of ottion φ llows ottion 46

247 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions llows tnsltion tnsits foces nd 3 oents does not tnsit the foce z llows tnsltion 47

248 The joints The spce (3D), idelized joints e : the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions fies tnsltions nd 3 ottions the sliding joint the spil joint 8. The echniss fies tnsltions nd ottions llows tnsltion nd ottion eltes one to the othe z φ z s φ π s ω π s ε π s the spil led - the tnsltion coesponding to the 36º ottion ( π 6,8 d) 48

249 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions the 4 th clss joint the spil joint the sliding-otting joint fies tnsltions nd ottions llows fies tnsltions nd nd ottion ottions eltes llows independent one to the othe tnsltion nd ottion tnsits foces nd oents does not tnsit foce nd oent llows independent tnsltion nd ottion z φ 49

250 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions the spil joint fies tnsltions nd ottions the 4 th clss joint the sliding-otting joint llows fies tnsltions nd nd ottion ottions eltes llows independent one to the othe the double-otting joint fies 3 tnsltions nd tnsltion ottion nd ottion llows independent ottions tnsits 3 foces nd oent, does not tnsit oents φ ψ llows two ottions 5

251 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions the spil joint fies tnsltions nd ottions the 4 th clss joint the sliding-otting joint llows fies tnsltions nd nd ottion ottions eltes llows independent one to the othe the double-otting joint fies 3 tnsltions nd tnsltion ottion nd ottion the 3 d clss joint the spheicl joint llows independent ottions fies 3 tnsltions, tnsits 3 foces nd oent, llows 3 independent ottions does not tnsit oents tnsits 3 foces, does not tnsit 3 oents ψ, ϑ, φ llows thee ottions 5

252 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions the spil joint fies tnsltions nd ottions the 4 th clss joint the sliding-otting joint llows fies tnsltions nd nd ottion ottions eltes llows independent one to the othe the double-otting joint fies 3 tnsltions nd tnsltion ottion nd ottion the 3 d clss joint the spheicl joint llows independent ottions fies 3 tnsltions, the nd clss joint the sliding spheicl joint tnsits 3 foces nd oent, llows fies 3 tnsltions, independent llows ottions tnsltion does not tnsit oents tnsits nd 3 independent 3 foces, ottions does tnsits not tnsit foces, 3 does oents not tnsit the foce nd 3 oents z ψ, ϑ, φ llows tnsltion nd thee ottions 5

253 The joints The spce (3D), idelized joints e : 8. The echniss the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions the spil joint fies tnsltions nd ottions the 4 th clss joint the sliding-otting joint llows fies tnsltions nd nd ottion ottions eltes llows independent one to the othe the double-otting joint fies 3 tnsltions nd tnsltion ottion nd ottion the 3 d clss joint the spheicl joint llows independent ottions fies 3 tnsltions, the nd clss joint the sliding spheicl joint tnsits 3 foces nd oent, llows 3 independent ottions does fies not tnsltions, tnsit oents llows tnsltion the st clss joint the genel kinetic pi tnsits nd fies 3 independent tnsltions, 3 foces, ottions llows tnsltions does tnsits nd 3 not ottions tnsit foces, 3 does oents not tnsit the tnsits foce nd the foce, 3 oents does not tnsit the foces nd 3 oents llows ll otions ecept one tnsltion nol to the coon tngentil plne 53

254 The joints The spce (3D), idelized joints e : the 6 th clss joint the pefect fiing fies 3 tnsltions nd 3 ottions the 5 th clss joint the otting joint fies 3 tnsltions nd ottions the sliding joint fies tnsltions nd 3 ottions the spil joint fies tnsltions nd ottions the 4 th clss joint the sliding-otting joint llows fies tnsltions nd nd ottion ottions eltes llows independent one to the othe the double-otting joint fies 3 tnsltions nd tnsltion ottion nd ottion the 3 d clss joint the spheicl joint llows independent ottions fies 3 tnsltions, the nd clss joint the sliding spheicl joint tnsits 3 foces nd oent, llows 3 independent ottions does fies not tnsltions, tnsit oents llows tnsltion the st clss joint the genel kinetic pi tnsits nd fies 3 independent tnsltions, 3 foces, ottions llows tnsltions the joint tnsits such foce o oent, in wht diection it fies the tnsltion o ottion does tnsits nd 3 not ottions tnsit foces, 3 does oents not tnsit the tnsits foce nd the foce, 3 oents does not tnsit the foces nd 3 oents if the joint fies tnsltion in cetin diection, it tnsits ppopite foce if the joint fies ottion bout cetin is, it tnsits ppopite oent 8. The echniss if the joint llows tnsltion in cetin diection, it does not tnsit ppopite foce if the joint llows ottion bout cetin is, it does not tnsit ppopite oent not ectl the se w 54 fo the el joints

255 The tpes of echniss 8. The echniss the thee ebes echniss the c echnis 3 3 the theoeticl outline

256 The tpes of echniss 8. The echniss the thee ebes echniss the echnis with the genel kinetic pi 3 56

257 The tpes of echniss 8. The echniss the fou ebes echniss the cnk echnis 3 3 the centic cnk echnis 4 ecenticit the ecentic cnk echnis the ecentic cnk echnis 57

258 The tpes of echniss 8. The echniss the fou ebes echniss the fou-joints echnis B 3 tnsits ottion 4 in : tio B tnsltion 3 4 D D the plelog 58

259 The tpes of echniss 8. The echniss the fou ebes echniss the coulisse echnis the centic coulisse echnis the centic coulisse echnis the ight-ngled coulisse ecenticit the ecentic coulisse echnis

260 The tpes of echniss 8. The echniss the fou ebes echniss the Oldh clutch nesouosost tnsits ottion in : tio with cetin islignent of the is 6

261 The tpes of echniss 8. The echniss the oe-ebes pln echniss the sewing chine echnis 6

262 The tpes of echniss 8. The echniss the spce echniss 3 4 the 3D cnk echnis 6

263 The echnis die tio the nlticl solution of the othogonl coulisse echnis ω, ε φ, sinφ & φ& cos φ && && φ cos φ φ& φ & ω && φ ε ω cos φ & && ε cos φ ω the echnis with ible die tio ( φ) 9. The kinetics of echniss p f ω sinφ DR - the die tio sinφ the DR deitie 63

264 φ ω t,ε t tlíř φ & ω && φ ε t R s φ R t The echnis die tio the nlticl solution of the chin echnis ω t R ω k ψ kolečko ω k, ε k s ψ s ψ R ψ φ ψ & ω ψ && ε the echnis with constnt die tio k k 9. The kinetics of echniss R ψ φ R ψ & φ & R ωk ωt ωt p dψ R p const dφ R ω & k ω& t R εk εt εt p DR the die tio 64

265 The echnis die tio the nlticl solution of the chin echnis 9. The kinetics of echniss φ ω t,ε t tlíř R s φ R ω t R ω k The DR die tio is constnt, does not ψ kolečko ω k, ε k The cceletion of the diing nd dien echnis ebe e in the se tio (DR) s elocities dp q dφ R ψ φ R ψ & φ & R ωk ωt ωt p dψ R p const dφ R ω & k ω& t R εk εt εt p ε k ε t p + ω t q 65

266 The echnis die tio 9. The kinetics of echniss ω ω R ω, ε R ω, ε ω ε p ε R ω p p the diing wheel (sll) the dien wheel (lge) the deceletion die p < i R ω ω i > ω, ε R γ s δ ω ε p R ω ε p p s s sin sin γ δ sin γ sinδ ω, ε 66

267 the cobintion of die tios ω ω 3 R 3 3 ω R The echnis die tio ω ω 3 p ω ω R R R 9. The kinetics of echniss R 3 ω 3 p 3 ω R 3 R 3 p ω ε 3 3 p ω ε p p p 3 R R 3 the die tios e ultiplied 67

268 The echnis die tio 9. The kinetics of echniss ω ω R R the non sliding olling with fiction tnsision ω, ε ω, ε the teeth the echnicl estint ginst sliding ω ω the pitch cicle the pitch cicle ω ω ω ω R 68

269 the ge The echnis die tio the echnis with ing DR pω /ω 9. The kinetics of echniss pω /ω φ ω ω φ the pushing sufce the pushed sufce ω ω If the kinetic pi of pushing nd pushed sufce should epesent the constnt die tio echnis the both sufces ust he the specific shpe of epiccloids 69

270 the double ge The echnis die tio the stellite ge 9. The kinetics of echniss ω the oute wheel R 3 3 ω 3 ω 3 the stellite the cie 3 ω 3 R předloh ω 3 R ω ω 3 the centl wheel p ω ε 3 3 p 3 ω ε p p p p 3 R R R + + R + S π S p ω 3 ω ω ω 3 S 3 3 S S 7 3

271 The echnis die tio the stellite ge 9. The kinetics of echniss the oute wheel 3 ω 3 the stellite the cie 3 ω 3 R ω ω 3 the centl wheel S π S p ω 3 ω ω ω 3 S 3 3 S S 7 3

272 the pulle block The echnis die tio 9. The kinetics of echniss l l l 4 π, l l π 4, l l 4 4 /3 7

273 the itos The echnis die tio 9. The kinetics of echniss p ω ε ε ω ω ýstupní ýstupní ýstupní ýstupní stupní ω ε dp q dφ ε stupní konst stupní stupní p p + ω p stupní q 73

274 the nlticl solution s u the echnis the echnis s geoetic conete The echnis die tio u u 3 9. The kinetics of echniss echnis with DO - degee of feedo. the position poble u f ( s) the tel function. the elocit solution the genelized coodinte length o ngle s the diing ebe coodinte - the input coodinte - the coodinte of the echnis the nube of the coodintes of echnis is equl to the nube of the echnis DO u the dien ebe coodinte - the output coodinte the nube of the output coodintes is bit out du ds ds dt ( s) ( t ) out du p du ds ( s( t )) ( s) ( t ) dt p ( s) in ( s) in ds the genelized elocit dt DR the die tio 74

275 the nlticl solution s u the echnis the echnis s geoetic conete The echnis die tio u u 3 9. The kinetics of echniss echnis with DO - degee of feedo. the position poble u f ( s) the tel function. the elocit solution 3. the cceletion solution d d( p ) ( s) in dp( s) out out dt dp out ds ( s) ds dt dt in + p ( s) dt d dt in in + du out dt du din p( s) ( s) dt p ds dp ds ( s) ( t ) q( ) dsdt du ds ( s( t )) ( s) ( t ) ( s) s st ds dt DR the die tio the DR deitie out p + q ( s) in ( s) in the genelized cceletion ds dt ýst in p ( s) st d in dt the genelized elocit in 75

276 the nlticl solution s u the echnis the echnis s geoetic conete The echnis die tio 3. the cceletion solution out p + q ( s) in ( s) u u 3 in 9. The kinetics of echniss echnis with DO - degee of feedo. the position poble u f ( s) the tel function. the elocit solution out p ( s) in the conesion functions u f ( s) du ds dp ds ( s) ( s) p q ( s) ( s) the tel function DR - the die tio the DR deitie 76

277 the nlticl solution u the echnis s the echnis s geoetic conete The echnis die tio u u 3 9. The kinetics of echniss echnis with DO - degees of feedo. the position poble u f, ( s) the tel function. the elocit solution 3. the cceletion solution q q ýst + st st + p p + q + st st st p + st q (, s) (, s) q(, s) (, s) p s (, s) (, s) p q + p s (, s) du ýst u u (, s) (, s) du dt d + the totl diffeentite u d dt s ds u + s ds dt ýst st p, + p, p u (, s) (, s) p(, s) ( s) st ( s) u (, s) s 77

278 the nlticl solution the tigonoet ethod The tigonoet ethod ens the intuitie use of seel geoetic lws, ules nd solutions, pplied to the echnis geoet 9. The kinetics of echniss echnis with DO - degee of feedo. the position poble u f ( s) the tel function ω 3, ε 3 φ b 3 4, + b b cos φ ( φ) + b b cos φ 78

279 9. The kinetics of echniss the nlticl solution the ecto ethod The ecto ethod ens the eplceent of the kinetic schee b the ecto chin the closed loop. The equtions of the closed ecto chin epesents the position poble solution echnis with DO - degee of feedo. the position poble u f ( s) i φ i the tel function φ n n φ φ the kinetic schee - the chin of ebes n + + K+ i + K+ n i the closed ecto loop n i i sin φi i n i i cos φ i 79

280 9. The kinetics of echniss the nlticl solution echnis with DO - degee of feedo the ecto ethod. the position poble φ B φ u f ( s) the tel function H ω,ε ψ 4 3 ω 3, ε 3, z H ψ z tn ψ ψ ctn ω 4 sin φ H + cos φ sin φ H + cos φ pω ω 4, ε 4 ε 4 pε + q ω + z + H sinφ z sinψ cos φ z cos ψ + H p q ( φ) ( φ) dψ dp dφ ( φ) dφ d ( φ) ( φ) 8 ψ dφ

281 9. The kinetics of echniss the nlticl solution echnis with DO - degee of feedo the ecto ethod. the position poble φ B φ u f ( s) the tel function H ω,ε ψ 4 3 ω 3, ε 3, z H ψ z φ& cos φ z& sin ψ z ψ& cos ψ φ& sin φ z& cos ψ + z ψ& sin ψ φ & ω ω4 φ && cosφ φ& z & ψ& 34 sin φ && z sin ψ ω 4, ε 4 + z + H z& ψ& cos ψ z ψ&& cos ψ + z ψ& sin ψ φ && sin φ φ& cos φ && z cos ψ + + z& ψ& sin ψ + z ψ&& sin ψ + z ψ& cos ψ sinφ z sinψ cos φ z cos ψ + H & φ ε ψ& & ε && z

282 9. The kinetics of echniss the pole solution gien : ω π π find : ω 4 ω 3 n B n 3 B B 3 B B ω 4 ω 4? B ω ω 4 D B ω B Bπ π D D ω 3 ω B Bπ 3 π B π Bπ 8

283 9. The kinetics of echniss the pole solution B cos φ + B sin φ φ B B cos γ B sin γ γ 3 D S D V + D cos ψ D D sinψ D 4 ψ D sin s B 8 B φ+γ π 8º-(φ+ψ) ψ-γ [ ( φ + ψ) ] sin( φ + ψ) Bπ sin π ( ψ γ) sin( φ + γ) sin Bπ B sin ( ψ γ) ( φ + ψ) B sin π B sin ( φ + γ) ( φ + 83 ψ)

284 9. The kinetics of echniss the pole solution B ω B π n B 3 π n n D 4 n B D D n B B 84

285 9. The kinetics of echniss the otion decoposition gien : find : ω,ε B 3, 4 t the bsic decoposition + n B B Bt B B Bn + Bt + Bn the esulting otion the genel plne otion the cing otion tnsltion + the eltie otion ottion B B + B B B B B Bn Bn B B B B B B Bn B Bn Bt B Bt 85

286 9. The kinetics of echniss the otion decoposition gien : find : ω,ε B 3 B ψ φ, Bt Bt 4 sin φ + cos φ Bn Bn t + the bsic decoposition n B sinφ B cos φ B B Bt B B Bn sin φ cos φ Bt Bt + B B sinψ + cos ψ Bt + sinψ cos ψ Bn Bn Bn cos ψ sin ψ φ ψ B B φ Bn Bn B B B B Bn φ Bn ψ Bt ψ Bt φ 86

287 the otion decoposition B gien : 3 4 find : B,, Bt ω,ε D t B ω B 9. The kinetics of echniss the oiolis decoposition B D 4 : Bn B φ ω,ε B 4 ψ 34, 34, B Bt + 34 ω 4,ε 4 δ D B BD 4 δ φ B B ψ B B 3 : 4 + D ω 4 4 BD B B sinφ cos φ 4 4 sin δ + cos δ cos ψ sinψ 34 B D 87

288 the otion decoposition B gien : 3 4 find : B,, Bt ω,ε D t 9. The kinetics of echniss the oiolis decoposition B ε B B ω B Bt D Bn ω B Bn φ ω,ε Bt Bn Bn + B B Bt sin φ 4n sin φ + 4n ψ 34, 34, B Bt Bn cos δ Bt sin δ + 4 4n cos φ 4t cos φ 4t ω 4,ε 4 δ + + 4t sin δ + cos δ + o o D cos ψ + sin ψ + B o o B φ Bn φ Bt B sin ψ 4t BD cos ψ 4n ψ 34 δ δ o 4 : B B B B BD 3 : 4 o + ω 4n ω BD D D ω 4 ε 4 88 BD 4 BD 4t BD

289 the otion decoposition B gien : 3 4 find : B,, Bt ω,ε D t 9. The kinetics of echniss the oiolis decoposition B ε B B ω B Bt B D Bn ω 4 4 ω D cos ψ cos γ 34 4 ψ 34, 34 sin ψ ω 4,ε 4 D, 4 4t 4n γ o 34 D 4 sin γ γ ν ψ B 4 : 3 : 4 B B o + ω 4n ω BD D D ω 4 ε 4 89 BD 4 BD 4t BD

290 the otion decoposition B gien : 3 4 find : B,, Bt ω,ε D t 9. The kinetics of echniss the oiolis decoposition B ε B B ω B Bt B D Bn ω n 4t ε4 4n ω4 34 4t t D D o cos ψ cos γ 34 4n 4n ψ 34, 34 + ω 4,ε 4 o 4t D cos γ sin γ 4t 34 γ D, 4 4t 4n γ 34 γ 4n D sin γ + sin ψ + o ψ B o o 3 µ sin ψ cos ψ ψ o 4 : 3 : 4 B B B o + ω 4n ω BD D D ω 4 ε 4 9 BD 4 BD 4t BD

291 The cnk echnis 9. The kinetics of echniss the cnk - piston echnis 3 3 the centic cnk echnis 4 ecenticit the ecentic cnk echnis 4 9

292 The cnk echnis 9. The kinetics of echniss the cnk - piston echnis 9

293 The cnk echnis 9. The kinetics of echniss dφ ω dt dω ε dt the cnk - piston echnis φ d dt B d ω, ε b ψ ω, ε ( ) b sin φ cos φ dφ dφ dt d ( φ) p ( φ ) ω p ( ) d dp dω ω + p dt dt dt dp dφ dω ω + p dφ dt dt q ω + p φ ε q ( φ ) ( ) ( φ) φ dp dφ d ( φ) ( φ) dφ dφ, 5 4,5 4 3,5 3,5,5,5 d dt d dt sin φ b sinψ bcos ψ cos φ b sin φ cos φ b - cosφ φ 93

294 The cnk echnis 9. The kinetics of echniss the cnk - piston echnis dφ ω dt dω ε dt B φ ω, ε b ψ ω, ε dψ dt dω dt ω ε, d dt d dt sin φ b sinψ bcos ψ cos φ b sin φ cos φ sin φ b sin ψ b cos ψ cos φ cos φφ & b cos ψ ψ& & b sin ψ ψ & + sin φφ& cos φω ω b sin ψ ω ω ω b b cos ψ ω cos φ cos ψ + sin φω ( sin φ tn ψ cos φ) & cos φω sin φω b cos ψ ω b sin ψ ω ( cos φω& sin φφω & ) b ( cos ψ ω& sin ψ ψ& ω ) ( cos φφω & + sin φω& ) b( cos ψ ψ& ω + sin ψ ω& ) ( cos φε sin φω ) b ( cos ψ ε sin ψ ω ) ( cos φω + sin φε ) b ( cos ψ ω + sin ψ ε ) ε ( cos φε sin φω ) b cos ψ + b sin ψ ω 94

295 The cnk echnis 9. The kinetics of echniss the cnk - piston echnis dφ ω dt dω ε dt B φ ω, ε b ψ ω, ε dψ dt dω dt ω ε, d dt d dt sin φ b sinψ bcos ψ cos φ b sin φ cos φ sin φ b sin ψ b cos ψ cos φ cos φφ & b cos ψ ψ&,8,6 & b sin ψ ψ & + sin φφ&,4, cos φω b sin ψ ω b cos ψ ω -, ,4 ω -,6 -,8 - ω ω ω b cos φ cos ψ + sin φω ( sin φ tn ψ cos φ) & cos φω sin φω b cos ψ ω b sin ψ ω 4 3,5 3 ( cos φω& sin φφω & ) b ( cos ψ ω& sin ψ ψ& ω ) ( φφω & cos + sin φω& ) b( cos ψ ψ& ω + sin ψ ω& ) ( cos φε sin φω ) b ( cos ψ ε sin ψ ω ) ( cos φω + sin φε ) b ( cos ψ ω + sin-ψ ε ) - -3,5 -,5 - ε -,5 ε ( cos φε sin φω ) ε b cos ψ + b sin ψ ω

296 96

297 . The dnics of echniss the fee bod dig ethod the eduction ethod G G 97

298 the fee bod dig ethod. The dnics of echniss I α f G G? 98

299 the fee bod dig ethod. The dnics of echniss S α S I S S ε - the fee bod dig - the equtions of otion of the single bodies S G sin α T N cos α S G S S I G S G f N f G cos α f ( sin α + f cos α) f G N - the kinetic solution ε G the eqution of otion - the thetic solution S S G ( sin α + f α) + G cos I + + G G cos ( sin α + 99 f α)

300 the fee bod dig ethod. The dnics of echniss - the kinetic solution εecos φ ω e sin φ + e sin φ & ecos φφ & ωecos φ & eω & cos φ eω sin φφ& φ & ω ω& ε ω,ε e, φ +e sinφ M e sinφ 3

301 the fee bod dig ethod. The dnics of echniss - the kinetic solution εecos φ ω e sin φ - the fee bod dig - the equtions of otion of the single bodies e cosφ, R ω,ε e φ ε φ R M M Iε M R ecos φ R 3

302 the fee bod dig ethod. The dnics of echniss - the thetic solution εecos φ ω e sin φ R R + Iε M R ecos φ Iε M ( + ) ecos φ Iε + ecos φ M ecos φ ( I + e cos φ) ε e sinφcosφω M ecosφ, ω,ε e φ M 3

303 the fee bod dig ethod. The dnics of echniss the kinetosttic tsk ( I + e cos φ) ε e sinφcosφω M ecosφ gien : φ, ω, ε,. M ecosφ + I + e cos φ ε e sinφcosφω find : M. ( ) ω const, ε, const R [ N ], ω,ε e φ M [ N ] φ [ º ] - M 33

304 the fee bod dig ethod. The dnics of echniss the dnic tsk gien :, M. find : φφ (t), ωω (t), εε (t). ( I + e cos φ) ε e sinφcosφω M ecosφ ( I + e cos φ) φ && e sin φcos φφ& M ecos φ the nueicl solution t φ ω ε R, ω,ε e φ M 34

305 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to tnsltion eq eq 3 I, ω,,,, M G the thee links 35

306 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to tnsltion eq eq 3 I, ω,,,, k M the kinetic eneg E I ω + I ω elit eq + + I G 3 eq ltentie + I 3 the kinetic eltions ω ω

307 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to tnsltion eq eq 3 I, ω,,,, M the powe P M ω G eq eq elit M 3 G G ltentie the kinetic eltions ω ω

308 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to tnsltion eq eq eq I, eq M 3 + I M 3 G ω 3 + I G 3,, d,, the eqution of otion eq eq + eq const eq eq d eq d eq d + I I M G

309 the eduction ethod. The dnics of echniss the elit the ltentie the eduction to tnsltion the kinetic eneg chnge lw E K the wok E K P t t the powe de K dt P E K eq de dt K d dt eq + eq d dt eq + d d eq d dt eq + d d eq 3 eq + d d d dt eq 3 P eq eq + d d eq d dt eq d eq eq + d eq 39

310 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq I, 3 ω,, ω,ε M eq M G 3

311 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq I, 3 ω,, ω,ε M eq E k M I ω I eq + I ω elit 3 + G + I + I I eq ω ltentie ω ω ω 3 3

312 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq I, 3 ω,, ω,ε M eq M G P M ω G Meq ω elit ltentie Meq M G 3 ω ω ω 3 3

313 I, ω the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq I, I M M eq eq 3 ω 3 I + M G 3 G + I,, ω,ε the eqution of otion di dφ ed I ed ε + ω 3 I eq const : I ed ε M ed + I + I M ed di ed dφ M eq ε M G 3 33

314 the eduction ethod the elit. The dnics of echniss the ltentie φ the eduction to ottion M I eq ω, ε I ω,ε M eq, sin φ & cos φφ& ω cos φ E k Iω + I eq ω P M ω Meq ω ω cos φ ω cos φ I eq I + cos φ M eq M cos φ 34

315 M φ the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq I eq di dφ M eq I ω, ε, I + ed cos φ M cos φ cosφ sinφ ω,ε the eqution of otion M eq died I ed ε + ω Med dφ I + cos φ ε sin φcos φω M cos ( ) φ ( I + cos φ) φ && sin φcos φφ& M cos φ 35

316 M φ the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq ω, ε I ω,ε, The. tpe tsk - kinetosttic gien : φ (t), ω (t), ε (t), the eqution of otion M eq died I ed ε + ω Med dφ I + cos φ ε sin φcos φω M cos ( ) φ M ( I + cos φ) ε sin φcosφω + cos φ find : M M (φ) 36

317 M φ the eduction ethod. The dnics of echniss the elit the ltentie the eduction to ottion I eq ω, ε I, The. tpe tsk - dnic gien : find :, M the otion φ f (t), ω, ε ω,ε the eqution of otion M eq died I ed ε + ω Med dφ I + cos φ ε sin φcos φω M cos ( ) φ ( I + cos φ) φ && sin φcos φφ& M cos φ t φ ω ε ω [s - ] t [s]