Proiect Mecanisme I

7/21/2019 Proiect Mecanisme I

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UNIVERSITATEA MARITIMĂ CONSTANŢAFACULTATEA DE ELECTROMECANICĂ

PROIECT DE SEMESTRU

LA

M E C A N I S M E

STUDENT: ŞTEFĂNIŢĂ ALEXANDRU MIHAI

GRUPA: FM 2.3 IFR

2013 – 2014



T!" #$. I

Să se efectueze calculul cinetostatic al mecanismului cu bare articulate a cărei schemă

cinematică este reprezentată mai jos.

Se cunosc:

Lungimile elementelor cinematice:

- l OA=190 mm

- l O1 B=290 mm

- l BC =800mm

- l OO1

=0 mm

- !lementele cinematice notate cu 1"#"$" sunt bare omogene confec%ionate &in o%el cu

ρ=.800 Kg /m# 'i au sec%iunea trans(ersală constantă A=1 cm

2 .

- )ulisele 2 'i * au &imensiuni neglijabile" ele put+n& fi asimilate cu puncte materiale plasate ,n 'i

)" cu masele m2=m

*=0"2 Kg

- ∢ϕ=#0

- ura%ia elementului cinematic 1 /con&ucător n1=100 rot /min

- or%a tehnologică /&e rezisten%ă acti(ă F *=−200 N

- omentul M =−80 N

2

3

1

#

2

4 5

z

41

*

)

6

*

$

4



Se cer:

1. amilia mecanismului

2. 7ra&ul &e mobilitate

#. Schema structurală

$. naliza pozi%ională

*. eterminarea (itezelor prin meto&a ciclurilor in&epen&ente

. eterminarea accelera%iilor prin meto&a ciclurilor in&epen&ente

. )alculul torsorului &e iner%ie

8. )alculul greută%ilor elementare cinematice

9. )alculul for%elor &e legătură /reac%iunilor &in cuplele cinematice

1 – F"!%&%" !'"#%(!)&)%

naliza structurală a sistemului ,n raport cu sistemul &e a5e Oxyz .

r. element

cinematic

mobil

7ra&e &e libertate

v x v y v zω x ω y ω z

1 - - - - - 52 5 5 - - - 5

# - - - - - 5

$ 5 5 - - - 5

* 5 - - - - -

Se constată că # gra&e &e libertate" respecti( v z ,ω x ,ω y " sunt suprimate pentru toate cele

* elemente cinematice.

;ezultă că mecanismul este &e familie f =# /mecanism plan.

2 – G$"*)& * !+,%&%-"-

7ra&ul &e mobilitate se &etermină cu rela%ia M =#⋅n−2⋅C *−C $

,n care:

n < numărul elementelor cinematice

C $=0 numărul cuplelor &e clasă $ /cuple &e clasă superioară

C *= numărul cuplelor &e clasă * /cuple inferioare

#



M =#⋅*−2⋅−0=1

ecanismul are un singur gra& &e mobilitate 'i un singur element con&ucător /elementul 1.

M >0⇒mecanism DEZMODROM /mecanismul func%ionează

3 – S'!" (-$)'-)$"&/ " !'"#%(!)&)%

eterminarea numărului &e cicluri in&epen&ente N

N = l−n+1=−+1=2

,n care:

- l= - numărul &e laturi ale ciclului < numărul cuplelor cinematice

(O ,O 1 , A R , AT , B , C R ,C T )

- n= - numărul &e no&uri ale grafului < numărul elementelor cinematice /0"1"2"#"$"*.

7raful asociat mecanismului /schema ciclomatică

A R=c!la A , rota"ie = AT =c!la A , transla"ie

C R=c!la C ,rota"ie = C T =c!la C , transla"ie

in schema ciclomatică se obser(ă că no&urile 0 'i #" care corespun& elementelor

cinematice 0 'i #" sunt conectate fiecare la c+te # elemente cinematice" &eci sunt elemente &e rangul

#=# .

Schema structurală

$

*

$

0

#

1

2

0

0

1 0

0

2

$

#

# *

)iclul >

)iclul >>

>> >

0 ; 01

01 6 ); )1

0

* 1

$

#

2



ia&ă ;; ani(elă ia&ă ;;

Structura mecanismului &in punct &e (e&ere al grupelor cinematice

4 – A#"&%" +%%+#"&/

eto&a coor&onatelor carteziene

• eterminarea coor&onatelor punctului /centrul &e rota%ie &in :

?roiec%ie pe 45: x A=OA⋅cos#0 =190⋅√ #2 =1$"*mm

?roiec%ie pe 43: y A=OA⋅sin #0 =190⋅1

2

=9* mm

• )oor&onatele punctului O1

xO 1=0 mm

yO1

=−0 mm

• Stabilirea ecua%iei ghi&ajului O1 B " y=m⋅ x+n

m= y A− yO$

x A− xO1

=9*−(−0)1$"*−0

=1 - panta &reptei ce trece prin O1 'i A

@aloarea lui n se &etermină punăn& con&i%ia ca punctul O1 să se afle pe &reaptă:

yO1

=m⋅0+n⇒n= yO1

=−0 mm

!cua%ia ghi&ajului O1 B este y=1⋅ x−0

• eterminarea coor&onatelor punctului B

y B=1⋅ xB−0

( x B− xO 1

)2+( yB− yO1

)2=O1

B2⇒

y B= xB−0

( x B−0)2+[ yB−(−0)2]=2902⇒

y B= xB−0

xB

2+( y B+0)2=2902

x B

2

+( xB−0+0)2

= xB

2

+ xB

2

=2⋅ x B

2

=2902

⇒ x B1"2=±

√290

2

2 =±20*

legem solu%ia care (erifică rela%ia x B> x A> xO1 <A x B=20* mm

*

12

#

41

4

41

66

)

$

*

0



y B= x B−0=20*−0=1#* mm

• eterminarea coor&onatelor punctului C

( xC − xB)2+( yC − y B)

2=BC 2

( xC −20*)2

+(−0−1#*)2

=8002

<A xC 2

−2⋅20*⋅ xC +(−20*)2

−8002

=0

xC 2−$10⋅ xC −*9.9*=0⇒ xC

1"2

=−(−$10)±√ (−$10)2−$⋅1⋅(−*9.9*)

2⋅1

xC 1=1.00* = xC 2

=−*9*

legem solu%ia care (erifică rela%ia xC > xB <A xC =1.00* mm

yC =−0 mm

)oor&onatele tuturor cupelor cinematice:

Coor%onate ∖C!l& O A B C O1

x [mm ] 0 1$"* 20* 1.00* 0

y [mm] 0 9* 1#* -0 -0

– A#"&%" %-&+$

eto&a ciclurilor in&epen&ente

C%'&)& I

Se intro&uc (itezele unghiulare relati(e

ω21 ,

ω0# 'i (iteza relati(ăv 2#

' pentru mi'carea

&e transla%ie a culisei .

41

3

1

#

2

4 5

z



OA×ω21+OO

1×ω

0#+v

#2

' =0

ω10+ω21+ω0#=0

∣i # ( x A y A 0

0 0 −ω21∣+∣

i # ( xO

1

yO1

0

0 0 −ω0#∣−v#2

' ⋅sinα⋅i −v#2

' ⋅cosα⋅ #=0

−ω10⋅( −ω

21⋅( −ω

0#⋅( =0

O1 A2=O 1O

2+OA2−2⋅O1 O⋅OA⋅cos (O1OA)

O1

OA=90 +ϕ=90 +#0 =120

O1 A=√ 0

2+1902−2⋅0⋅190⋅cos120 =2## mm

O1 A

'

=∣ y A∣+∣ yO 1∣=9*+0=1* mm Bn ΔO1 A

' A cu : O

1 A=2## mm

O1 A' A=90

sinα=1*

2##=0"081*$⇒arcsin 0"081*$=$*"08⇒

α≈

4

y A⋅(−ω21)⋅i− x A⋅(−ω21)⋅ #+ yO1⋅(−ω0#)⋅i − xO

1⋅(−ω0#)⋅ #−v#2

' ⋅sinα⋅i−v#2' ⋅cosα⋅ #=0

−ω10⋅( −ω21⋅( −ω0#⋅( =0

(− y A⋅ω21− yO1⋅ω0#−v#2

' ⋅sin α)⋅i +( x A⋅ω21+ xO1⋅ω0#−v#2

' ⋅cosα)⋅ #=0

−ω10⋅( −ω21⋅( −ω0#⋅( =0∣⋅(−1)

− y A⋅ω21− yO1

⋅ω0#−v#2

' ⋅sinα=0

x A⋅ω21+ xO 1

⋅ω0#−v

#2

' ⋅cosα=0

ω10+ω 21+ω 0#=0

4

41

)

6

C

3

5



ω10=ω1=2⋅π⋅n

0=

2⋅π⋅100

0=10"$[ra% / s ]

−0"09*⋅ω21+0"0⋅ω0#−v#2' ⋅sin $*=0

0"1$*⋅ω21+0⋅ω0#−v#2' ⋅cos$* =0

ω21+ω 0#=−10"$

)oor&onatele punctelor au fost intro&use ,n metri.

Δ=∣−0"09* 0"0

−√ 22

0"1$* 0 −√ 2

21 1 0

∣=0"1$*⋅1⋅(−0"0)+0"0⋅1⋅(−0"0)−1⋅(−0"09*)⋅(−0"0)=−0"2##

Δω21=

∣ 0 0"0

−√ 22

0 0 −√ 22

−10"$ 1 0 ∣=0"0⋅(

−√ 22 )⋅(−10"$)=0"*18

Δω0#=∣−0"09* 0

−√ 22

0"1$* 0 −√ 2

21 −10"$ 0

∣=0"1$*⋅(−10"$)⋅(−√ 2

2 )−(−0"09*)⋅(−10"$)⋅(

−√ 22 )=1"921

Δ v#2' =∣

−0"09* 0"0 00"1$* 0 01 1 −10"$∣=−0"0⋅0"1$*⋅(−10"$)=0"12

ω21=Δω

21

Δ =

0"*18

−0.2##=−2"22[ra% / s ]

ω0#=Δω0#

Δ =

1"921−0"2##

=−8"2$ [ra% /s ]

v#2

' =Δ v12

'

Δ = 0"12

−0"2##=−0"*1 [m /s ]

@alorile negati(e pentru ω21 ,ω0# , v2#

' arată că sensul acestora este in(ers fa%ă &e cel

reprezentat con(en%ional pe &esen.

@itezele unghiulare absolute ale elementelor:

ω2=ω

1+ω

21=10"$⋅( −2"22⋅( =8"2*⋅(

ω#=ω2+ω#2=8"2*⋅( =ω0#

8



C%'&)& II

Se intro&uc (itezele relati(e ω#0

,ω$#

,ω*$ 'i v

0* pentru cupla &e transla%ie.

OO1×ω#0+OB×ω$#+OC ×ω$#+OC ×ω*$+v0*=0ω#0+ω$#+ω*$=0

∣

i # (

xO1

yO1

0

0 0 −ω#0

∣+

∣

i # (

xB yB 0

0 0 −ω$#

∣+

∣

i # (

xC yC 0

0 0 −ω*$

∣+v0*⋅i=0

−ω#0⋅( −ω$#⋅( −ω*$⋅( =0

yO1

⋅(−ω#0)⋅i− xO1

⋅(−ω#0)⋅ #+ y B⋅(−ω$#

)⋅i− xB⋅(−ω$#)⋅ #+ yC ⋅(−ω*$

)⋅i− xC ⋅(−ω*$)⋅ #+v

0*⋅i=0

ω#0⋅( +ω$#⋅( +ω*$⋅( =0

(− yO 1⋅ω#0− yB⋅ω$#− yC ⋅ω*$+v0*)⋅i +( xO1

⋅ω#0+ xB⋅ω$#+ xC ⋅ω*$)⋅ #=0

ω#0+ω$#+ω*$=0

− yO 1⋅ω#0− y B⋅ω$#− yC ⋅ω*$+v0*=0

xO1⋅ω#0+ xB⋅ω$#+ xC ⋅ω*$=0

ω#0+ω $#+ω *$=0

−(−0"0)⋅8"2*−0"1#*⋅ω$#−(−0"0)⋅ω*$+v 0*=0

0⋅8"2*+0"20*⋅ω$#+1"00*⋅ω*$=0

8"2*+ω$#+ω

*$=0

0"1#*⋅ω$#−0"0⋅ω*$−v0*=0"*0"20*⋅ω$#+1"00*⋅ω*$=0ω$#+ω *$=−8"2$

Δ=

∣0"1#* −0"0 −1

0"20* 1"00* 0

1 1 0 ∣=0"20*⋅1⋅(−1)−1⋅1"00*⋅(−1)=0"8

9

5

3

4

41

)

6#

$

*



Δω$#=∣0"* −0"0 −1

0 1"00* 0

−8"2* 1 0 ∣=−(−1)⋅1"00*⋅(−8"2*)=−8"291

Δω*$=

∣0"1#* 0"* −1

0"20* 0 01 −8"2* 0 ∣

=0"20*⋅(−8"2*)⋅(−1)=1"91

Δ v0*=∣0"1#* −0"0 0"*0"20* 1"00* 0

1 1 −8"2*∣

Δ v0*=0"1#*⋅1"00*⋅(−8"2*)+1⋅0"20*⋅0"*−1⋅1"00*⋅0"*−0"20*⋅(−0"0)⋅(−8"2*)=−1"99

ω 43=Δω

$#

Δ =−8"291

0"8= 10536[ra% / s]

ω4=Δω*$

Δ =1"91

0"8 =2511[ra% / s]

v 0=Δ v

0*

Δ =−1"99

0"8= 2512[ra% / s ]

@iteza unghiulară relati(ă ω$# 'i (iteza relati(ă a cuplei &e transla%ie v0* au sensuri

in(erse fa%ă &e cele alese pe figură.

@itezele unghiulare absolute:

4

=ω#

+ω$#

=8"2*⋅( −10"#⋅( = 2511⋅k

=ω$+ω

*$=−2"11⋅( +2"11⋅( =0

eterminarea (itezelor absolute ale punctelor caracteristice pentru ciclul > 'i >>:

)onsi&erăm vO=0 'i vO 1=0 - O )i O

1 !ncte fixe

v A=vO+ω1×OA=

∣

i # (

0 0 −ω1

x A y A 0

∣= x A⋅(−ω1)⋅ #− y A⋅(−ω1

)⋅i= ya⋅ω1⋅i− xa⋅ω1⋅ #

v A=0"09*⋅10"$⋅i−0"1$*⋅10"$⋅ #=0"99$⋅i−1"22⋅ #

∣v A∣=√ 0.99$2+(−1"22)2=1"98⇒ v A=1578 [m⋅s−1]

v B=vO1+ω#×O1 B=∣

i # (

0 0 −ω#

xB− xO1 yB− yO1

0 ∣=( xB− xO1)⋅(−ω#)⋅ #−( yB− yO1

)⋅(−ω#)⋅i

v B=(0"20*−0)⋅(−8"2*)⋅ #−[0"1#*−(−0"0)]⋅(−8"2*)⋅i=1"91⋅i−1"91⋅ #

∣v B∣=√ 1"91 2+(−1"91)2=2"#9⇒v B=2537[m⋅s−1]

10



vC =v B+ω$×BC =1"91⋅i −1"91⋅ #+∣ i # (

0 0 ω$

xC − xB yC − yB 0 ∣

vC

=1"91⋅i−1"91⋅ #+( xC

− xB

)⋅ω$⋅ #−( y

C

− yB

)⋅ω$⋅i

vC =[1"91−( yC − y B)⋅ω$]⋅i+[(−1"91)+( xc− xB)⋅ω$]⋅ #

vC =[1"91−(−0"0−0"1#*)⋅2"11]⋅i +[−1"91+(1"00*−0"20*)⋅2"11]⋅ #

vC =2"12#⋅i−0⋅ #

∣vC ∣=√ 2"12#2=2"12⇒ v C =2512[m⋅s−1]

?entru elementele cinematice 1" #" $" notăm cu C 1

, C #

, C $ , centrele &e masă aflate la

mijlocul fiecăruia &intre elemente.

)onsi&erăm vO=0 )i vO 1

=0 - O )i O1 !ncte fixe

)alculul (itezelor centrelor &e masă:

vC 1

=vO+ω1×OC 1=

∣

i # (

0 0 −ω1

x A

2

y A

20

∣=

x A

2⋅(−ω1)⋅ #−

y A

2 ⋅(−ω 1)⋅i=

y A⋅ω1

2 ⋅i−

x A⋅ω1

2 ⋅ #

vC 1=

0"09*⋅10"$2

⋅i −0"1$*⋅10"$

2 ⋅ #=0"$9⋅i−0"81⋅ #

∣vC 1∣=√ 0"$92+(−0"81)2=0"99⇒v C 1=0577 [m⋅s

−1]

vC #=vO1

+ω#×O1 C #=∣ i # (

0 0 −ω#

x B− xO 1

2

y B− yO 1

20 ∣= xB− xO1

2 ⋅(−ω#)⋅ #−

y B− yO1

2 ⋅(−ω#)⋅i

vC #=

0"20*−0

2⋅(−8"2*)⋅ #−

0"1#*−(−0"0)2

⋅(−8"2*)⋅i=0"8$*⋅i−0"8$*⋅ #

∣vC #∣=√ 0"8$*2+(−0"8$*)2=1"19⇒v C 3=1517[m⋅s

−1]

vC $

=vB+ω$×BC $=∣ i # (

0 0 ω$

xC − xB

2

yC − yB

20 ∣= xC − x B

2 ⋅ω$⋅ #−

yC − yB

2 ⋅ω$⋅i

vC $=1"91⋅i−1"91⋅ #+

xC − x B

2⋅ω

$⋅ #−

yC − yB

2⋅ω

$⋅i

11



vC $=(1"91−

y C − y B

2 ⋅ω$)⋅i +(−1"91+

xC − xB

2 ⋅ω$)⋅ #

vC $=(1"91−−0"0−0"1#*

2⋅2"11)⋅i +(−1"91+

1"00*−0"20*

2⋅2"11)⋅ #=1"90⋅i−0"8$⋅ #

∣vC $∣=√ 1"902+(−0"8$)2=2"08⇒ v C 4=2508[m⋅s−1]

6 – A#"&%" "''&$"%%&+$

eto&a ciclurilor in&epen&ente

C%'&)& I

)onsi&erăm ε10=0 &eoarece ω10=const*

OA×ε21+OO1×ε0#−ω 12⋅OA−ω#

2⋅ AO1+a#2

' +a#2' =0

ε10+ε21+ε0#=0

∣i # (

x A y A 0

0 0 −ε21

∣+∣ i # (

xO1

yO1

0

0 0 −ε0#

∣−ω1

2⋅( x A⋅i+ y A⋅ #)−ω#

2[( xO 1

− x A)⋅i+( yO1

− y A)⋅ # ]–

– a#2

'

⋅sinα⋅i−a#2

'

⋅cosα⋅ #+2⋅

∣ i # (

0 0−ω#

v#2

' ⋅sinα v#2

' ⋅cosα 0∣=0

−ε21⋅( −ε0#⋅( =0

12

41

3

1

#

2

4 5

z



y A⋅(−ε21)⋅i− x A⋅(−ε21)⋅ #+ yO1⋅(−ε0#)⋅i− xO

1⋅(−ε0#)⋅ #−ω1

2⋅ x A⋅i−ω1

2⋅ y A⋅ #−ω#

2⋅( xO1− x A)⋅i –

−ω#

2⋅( yO1

− y A)⋅ #−a#2

' ⋅sin $* ⋅i−a#2

' ⋅cos $* ⋅ #+2⋅v#2

' ⋅sin$* ⋅(−ω#)⋅ #−2⋅v

#2

' ⋅cos $* ⋅(−ω#)⋅i=0

ε21⋅( +ε

0#⋅( =0

[− y A

⋅ε21− y

O 1

⋅ε0#−ω

1

2⋅ x A

−ω#

2⋅( xO 1

− x A

)−a#2

' ⋅sin $*+2⋅ω#⋅v

#2

' ⋅cos$* ]⋅i

[ x A⋅ε21+ xO 1⋅ε0#−ω1

2⋅ y A−ω#2⋅( yO1

− y A)−a#2' ⋅cos$* −2⋅ω#⋅v#2

' ⋅sin $* ]⋅ #=0

−ε21⋅( −ε0#⋅( =0

− y A⋅ε21− yO1⋅ε0#−ω1

2⋅ x A−ω#2⋅( xO

1− x A)−a#2

' ⋅sin $*+ 2⋅ω#⋅v#2' ⋅cos$* =0

x A⋅ε21+ xO1⋅ε0#−ω1

2⋅ y A−ω#2⋅( yO1

− y A)−a#2' ⋅cos$* −2⋅ω#⋅v#2

' ⋅sin$* =0

ε21+ε0#=0

−0"09*⋅ε21−(−0"0)⋅ε0#−10"$2⋅0"1$*−8"2*2⋅(0−0"1$*)−a#2' ⋅√ 2

2 +2⋅8"2*⋅0"*1⋅√ 2

2 =0

0"1$*⋅ε21+0⋅ε0#−10"$2

⋅0"09*−8"2*2

⋅(−0"00−0"09*)−a#2'

⋅√ 2

2 −2⋅8"2*⋅0"*1⋅√ 2

2 =0ε21+ε0#=0

−0"09*⋅ε21+0"0⋅ε0#−0"0⋅a#2' =0"88

0"1$*⋅ε21−0"0⋅a#2' =*"1##

ε21+ε0#=0

Δ=∣−0"09* 0"0 −0"0

0"1$* 0 −0"0

1 1 0 ∣=0"1$*⋅1⋅(−0"0)+1⋅0"0⋅(−0"0)−1⋅(−0"09*)⋅(−0"0)=−0"2##

Δε21=∣

0"88 0"0 −0"0

*"1## 0 −0"0

0 1 0 ∣=*"1##⋅1⋅(−0"0)−0"88⋅1⋅(−0"0)=−#

Δε0#=∣−0"09* 0"88 −0"00"1$* *"1## −0"0

1 0 0 ∣=0"88⋅1⋅(−0"0)−*"1##⋅(−0"0)=#

Δa#2

' =

∣

−0"09* 0"0 0"88

0"1$* 0 *"1##

1 1 0

∣=0"1$*⋅1⋅0"88+1⋅0"0⋅*"1##−1⋅*"1##⋅(−0"09*)=0"992

ε21=Δε

21

Δ = −#

−0"2##=12589 [ra% /s

2]

ε03=Δε0#

Δ =#

−0"2##= 12589 [ra% /s

2]

a32

' =Δa#2

'

Δ = 0"992

−0"2##= 452 [ra% /s

2]

ccelera%iile ε0# 'i a#2' au sensuri in(erse fa%ă &e cele alese ini%ial.

1#



ccelera%iile unghiulare absolute:

εi=εi−1+εi , i−1

2=ε1+ε21=12589⋅k (ω1=const*⇒ε1=0)

3=ε2+ε#2=12589⋅k (

ε#2=0)

C%'&)& II

OO1×ε#0+OB×

ε$#+OC ×

ε*$−ω#

2

⋅O1 B−ω$

2

⋅BC +

a0*

'

=0ε#0+ε$#+ε*$=0

∣ i # (

xO1

yO1

0

0 0 −ε#0

∣+∣ i # (

x B yB 0

0 0 −ε$#

∣+∣ i # (

xC yC 0

0 0 −ε*$

∣−ω#

2⋅[( x B− xO 1

)⋅i +( yB− yO1

)⋅ # ]–

–ω$

2⋅[( xC − xB)⋅i +( yC − yB)⋅ # ]−a0*

' ⋅i=0

−ε#0⋅( −ε

$#⋅( −ε

*$⋅( =0

yO1⋅(−ε#0)⋅i− xO

1⋅(−ε#0)⋅ #+ y B⋅(−ε$#)⋅i − x B⋅(−ε$#)⋅ #+ yC ⋅(−ε*$)⋅i− xC ⋅(−ε*$)⋅ #−ω#

2⋅( x B− xO1)⋅i –

–ω#2⋅( y B− yO 1)⋅ #−ω$

2⋅( xC − x B)⋅i −ω$2⋅( yC − yB)⋅ #−a0*

' ⋅i=0

ε#0⋅( +ε

$#⋅( +ε

*$⋅( =0

[− yO

1⋅ε#0− y B⋅ε$#− yC ⋅ε*$−ω#

2⋅( x B− xO1)−ω$

2⋅( xC − x B)−a0*' ]⋅i +

+[ xO1⋅ε#0+ x B⋅ε$#+ xC ⋅ε*$−ω#

2⋅( yB− yO1)−ω$

2⋅( yC − yB)]⋅ #=0

ε#0+ε$#+ε*$=0

− yO1⋅ε#0− yB⋅ε$#− yC ⋅ε*$−ω#

2⋅( xB− xO1)−ω$

2⋅( xC − x B)−a0*' =0

xO1⋅ε#0+ x B⋅ε$#+ xC ⋅ε*$−ω#

2⋅( yB− yO1)−ω$

2⋅( yC − yB)=0

ε#0+ε $#+ε*$=0

1$

54

41

)

6#

$

*



−(−0"0)⋅12"8−0"1#*⋅ε$#−(−0"0)⋅ε*$−8"2*2⋅(0"20*−0)−2"112⋅(1"00*−0"20*)−a0*' =0

0⋅12"8+0"20*⋅ε$#+1"00*⋅ε*$−8"2*2⋅[0"1#*−(−0"0)]−2"112⋅(−0"0−0"1#*)=012"8#+ε$#+ε*$=0

−0"1#*⋅ε$#+0"0⋅ε*$−a0*

' =1"1#

0"20*⋅ε$#+1"00*⋅ε*$=1#"0$

ε$#+ε

*$=−12"8#

Δ=∣−0"1#* 0"0 −1

0"20* 1"00* 0

1 1 0 ∣=0"20*⋅1⋅(−1)−1⋅1"00*⋅(−1)=0"8

Δε$#=∣1"1# 0"0 −1

1#"0$ 1"00* 0

−12"8# 1 0 ∣=1#"0$⋅1⋅(−1)−(−12"8#)⋅1"00*⋅(−1)=−2*"9#$

Δε*$=∣−0"1#* 1"1# −10"20* 1#"0$ 0

1 −12"8# 0 ∣=0"20*⋅(−12"8#)⋅(−1)−1⋅1#"0$⋅(−1)=1*"

Δa0*' =∣−0"1#* 0"0 1"1#

0"20* 1"00* 1#"0$1 1 −12"8#∣=(−0"1#*)⋅1"00*⋅(−12"8#)+0"20*⋅1⋅1"1#+0"0⋅1#"0$⋅1 –

–1⋅1"00*⋅1"1#−(−0"1#*)⋅1⋅1#"0$−0"20*⋅0"0⋅(−12"8#)=−8"92

ε43=Δε

$#

Δ =−2*"9#$

0"8 = 32541[ra% / s

2]

ε4=Δε*$

Δ =1*"0"8

=1758 [ra% /s2]

a0

' =Δa0*

'

Δ =−8"92

0"8= 10586[ra% / s2]

ccelera%iile ε$# 'i a0*' au sensuri in(erse fa%ă &e cele alese ini%ial 'i reprezentate pe

figură.

ccelera%iile unghiulare absolute:

εi=

εi−1+

εi , i−1

4=ε#+ε$#=12"8⋅( −#2"$1⋅( = 1754⋅k

=ε$+ε*$=−19"*$⋅( +19"*8⋅( =0503⋅k

ccelera%iile punctelor "6"):

aO=0" ε1=0 / O este punct fi5

a A=a0+ε1×OA−ω12⋅OA=−ω1

2⋅ x A⋅i −ω12⋅ y A⋅ #=−10"$2⋅0"1$*⋅i −10"$2⋅0"09*⋅ #

a A=−18"0#⋅i −10"$1⋅ #

1*



∣a A∣=√ (−18"0#)2+(−10"$1)2=20"82⇒

a A=20582[m⋅s−2]

aO1=0 / O1 este punct fi5

aB= aO 1+ε#×O1 B−ω#

2⋅O1 B=∣ i # (

0 0 −ε#

x B− xO 1 y B− yO1

0 ∣−ω#2⋅[( x B− xO1

)⋅i +( y B− yO1)⋅ # ]

aB=∣ i # (

0 0 −12"80"20* 0"20* 0 ∣−8"2*2⋅(0"20*⋅i +0"20*⋅ #)=0"20*⋅(−12"8)⋅ #−0"20*⋅(−12"8)⋅i –

– 1#"9*2⋅i−1#"9*2⋅ #=−11"#1#⋅i −1"*9⋅ #

∣aB∣=√ (−11"#1#)2+(−1"*9)2=20"08⇒

a B=20508[m⋅s−2]

aC = aB+ε$×BC −ω$2⋅BC

aC =−11"#1#⋅i−1"*9⋅ #+∣ i # (

0 0 ε$

xC − xB yC − y B 0∣−ω$2⋅[( xC − xB)⋅i +( yC − yB)⋅ # ]

aC =−11"#1#⋅i−1"*9⋅ #+∣ i # (

0 0 19"*$0"8 −0"20* 0 ∣−2"112⋅(0"8⋅i−0"20*⋅ #)

aC =−11"#1#⋅i−1"*9⋅ #+0"8⋅19"*$⋅ #−19"*$⋅(−0"20*)⋅i−#"*1⋅i +0"912⋅ #

aC =−10"88⋅i

∣aC ∣=√ (−10"88)2=10"8⇒

a B=10586[m⋅s−2]

ccelera%ia centrelor &e masă ale elementelor 1"# 'i $:

aO=aO1

=0 )onsi&erăm O )iO1 ca fiin& puncte fi5e.

aC 1=aO−ω1

2⋅OC 1=−ω12⋅

x A

2 ⋅i−ω1

2⋅ y A

2 ⋅ #=−10"$2⋅

0"1$*2 ⋅i−10"$2⋅

0"09*2 ⋅ #

aC 1=−9"01⋅i −*"20⋅ #

∣aC 1∣=√ (−9"01)2+(−*"20)2=10"$1⇒ aC 1

10541 [m⋅s−2]

aC #=aO1

+ε#×O

1C

#−ω

#

2⋅O1

C #=∣ i # (

0 0 ε# xC #

yC #0∣−ω#

2⋅ xC #⋅i−ω

#

2⋅ yC #⋅ #

1



aC #=∣

i # (

0 0 ε#

x B− xO1

2

y B− yO 1

20∣−ω#

2⋅ xB− xO1

2 ⋅i−ω#

2⋅ y B− yO 1

2 ⋅ #

aC #=∣ i # (

0 0 12"8

0"20*−0

2

0"1#*−(−0"0)2

0 ∣−8"2*2⋅

0"20*−0

2⋅i−8"2*

2⋅0"1#*−(−0"0)

2⋅ #

aC #=∣

i # (

0 0 12"80"102* 0"102* 0 ∣−8"2*2⋅0"102*⋅i−8"2*2⋅0"102*⋅ #

aC #

=0"102*⋅12"8⋅ #−0"102*⋅12"8⋅i−"9⋅i −"9⋅ #=−8"29*⋅i−*"*⋅ #

∣aC #∣=√ (−8"29*)2+(−*"*)2=10"0#⇒ aC 3

10503 [m⋅s−2]

aC $=aB+ε$×BC

$−ω

$

2⋅BC $=∣ i # (

0 0 ε$ xC $

yC $0∣−ω$

2⋅ xC $⋅i−ω

$

2⋅ yC $⋅ #

aC $=−11"#1#⋅i−1"*9⋅ #+

∣

i # (

0 0 ε$

xC − x B

2

yB− yC

2 0

∣−ω$

2⋅ xC − x B

2 ⋅i −ω$

2⋅ y B− yC

2 ⋅ #

aC $=−11"#1#⋅i−1"*9⋅ #+∣

i # (

0 0 19"*$0"$ 0"102* 0 ∣−2"112⋅0"$⋅i −2"112⋅0"102*⋅ #

aC $=−11"#1#⋅i−1"*9⋅ #+0"$⋅19"*$⋅ #−0"102*⋅19"*$⋅i−1"8⋅i−0"$*⋅ #

aC $=−1*"09⋅i −9"2#⋅ #

∣aC $∣=√ (−1*"09)2+(−9"2#)2=1"9⇒ aC 4 19567[m⋅s

−2]

9 – C"&')&)& -+$(+$)&)% * %#$%

ρ=.800 Kg⋅m−#

A=1 cm2=10−$

m2

l1=l OA=0"19 m

l #=l O1 B=0"29 ml

$=l BC =0"8 m

1



!lementul cinematic nr. 1:

m1=ρ⋅l 1⋅ A=.800⋅10−$⋅0"19=0"1$8 Kg

+ ΔC 1

=m1⋅l 1

2

12=

0"1$8⋅0"192

12=0"$$⋅10

−#[ Kg⋅m−2]

aC 1=−9"01⋅i −*"20⋅ # ε1=0

F i1

: F i1

=−m1⋅aC 1

M i1=−+ ΔC 1

⋅ε1

F i1

: F i 1

=−0"1$8⋅(−9"01⋅i −*"20⋅ #)

M i1

=−0"$$⋅10−#⋅0

F i1

: F i 1

=1"##$⋅i +0"⋅ #

M i1

=

0

!lementul cinematic nr. #:

m#=ρ⋅l #⋅ A=.800⋅10−$⋅0"29=0"22 Kg

+ ΔC #

=m#⋅l #

2

12=

0"22⋅0"292

12=1"*8⋅10

−#[ Kg⋅m−2]

aC #=−8"29*⋅i−*"*⋅ #

ε#=12"8⋅(

F i#

: F i#

=−m#⋅aC #

M i#=−+ ΔC #

⋅ε#

F i#

: F i #

=−0"22⋅(−8"2*9⋅i −*"*⋅ #)

M i#=−1"*8⋅10−#⋅12"8⋅(

F i#

: F i #

=1"8⋅i +1"28⋅ #

M i#=−0"02⋅(

!lementul cinematic nr. $:

m$=ρ⋅l $⋅ A=.800⋅10−$⋅0"8=0"2$ Kg

+ ΔC $

=m$⋅l $

2

12=

0"2$⋅0"82

12=##"28⋅10

−#[ Kg⋅m−2]

aC $=−1*"09⋅i −9"2#⋅ # ε$

=−19"*$⋅(

F i$

: F i $=−m$⋅aC $

M i$=−+ ΔC $

⋅ε$

18



F i$

: F i$

=−0"2$⋅(−1*"09⋅i −9"2#⋅ #)

M i$=−##"28⋅10−#⋅(−19"*$)⋅(

F i$

: F i$

=9"$19⋅i+*"*⋅ #

M i$

=0"*⋅(

!lementul cinematic nr. 2:

m2=0"2 Kg

&imensiuni neglijabile" put+n& fi asimilat cu un punct material plasat ,n <A + ΔC 2

neglijabil

aC 2=a A=−18"0#⋅i−10"$1⋅ #

F i2

: F i2=−0"2⋅(−18"0#⋅i −10"$1⋅ # )

M i2=0

F i2

: F i2

=#"0⋅i +2"082⋅ #

M i2

=0

!lementul cinematic nr. *:

m2=0"2 Kg &imensiuni neglijabile" put+n& fi asimilat cu un punct material plasat ,n <A + Δ

C *

neglijabil

aC *=aC =−10"88⋅i

F i*

: F i *

=−0"2⋅(−10"88)⋅i

M i*=0

F i*

: F i *

=2"1#⋅i

M i*

=0

8 – G$)-/%& &!#-&+$ '%#!"-%'

i=−mi⋅g⋅ # ⇒∣ i∣=mi⋅g [ N ]

1=−0"1$8⋅9"81⋅ #=−1"$*1⋅ # ⇒∣1∣=1"$*1[ N ]

2=−0"2⋅9"81⋅ #=−1"92⋅ # ⇒∣

2∣=1"92[ N ]

#=−0"22⋅9"81⋅ #=−2"21⋅ # ⇒∣#∣=2"21[ N ]

19



$=−0"2$⋅9"81⋅ #=−"121⋅ # ⇒∣$∣="121[ N ]

*=−0"2⋅9"81⋅ #=−1"92⋅ # ⇒∣ *∣=1"92[ N ]

7 – C"&')&)& +$&+$ * &;/-)$/ <$"'%)#%= *%# ')&

- esfacem cupla &in 6:

"

F Δ

($+*)=0 eca"ia %e !roiec"ie !e %irec"ia g-i%a#li (Δ)

" M C ($)=0 eca"ia %e momente fa"& %eC

x B+ x$+ x*+ F *=0

CB×( x B+ yB)+CC $×( x$+ y$+ $)+ M i$=0

x*= F i*=2"1#⋅i

F *=−200⋅i

x B+9"$19⋅i +2"1#⋅i−200⋅i =0 ⇒

x B=188"$⋅i

∣ i # (

−0"8 0"20* 0188"$ y B 0∣+∣

i # (

−0"$ 0"102* 09"$19 −0"#1 0

∣+0"*⋅( =0

−0"8⋅ yB⋅( −188"$⋅0"20*⋅( +(−0"$)⋅(−0"#1)⋅( −9"$19⋅0"102*⋅( +0"*⋅( =0

(−0"8⋅ yB−#8"89)⋅( =0 ⇒−0"8⋅ yB−#9"08=0 ⇒ y B=−#8"89

0"8=−$8"$8

∣ R B∣=√ 188"$2+(−$8"$8)2=19$"*# N

- esfacem cupla &e rota%ie &in ) (C R) :

20

6

)

)$

$

* 0

/D



" F Δ(*)=0 eca"ia %e !roiec"ie !e %irec"ia g-i%a#li (Δ)

" M B($)=0 eca"ia %e momente fa"& %e B

xC + x*+ F *=0

BC ×( xC + yC )+BC $×( x$+ y $+$)+ M i$=0

xC +2"1#⋅i−200⋅i =0 ⇒ xC =19"82⋅i

∣ i # (

0"8 −0"20* 019"82 yC 0∣+∣

i # (

0"$ −0"102* 09"$19 −0"#1 0

∣+0"*⋅( =0

0"8⋅ yC ⋅( −19"82⋅(−0"20*)⋅( +0"$⋅(−0"#1)⋅( −9"$19⋅(−0"102*)⋅( +0"*⋅( =0

(0"8⋅ yC +$2"021)⋅( =0 ⇒0"8⋅ yC +$2"021=0 ⇒ yC =−$2"021

0"8=−*2"*2

∣ RC ∣=√ 19"822+(−*2"*2)2=20$"8 N

- esfacem cupla &e transla%ie &in ) (C T ) :

" M B

($+*)=0 eca"ia%e momente fa"& %e B

BC ×( R+*+ x*+ F *)+BC $×( x$+ y$+$)+ M i$=0

21

6

)

)$

$

* 0

/D



∣ i # (

0"8 −0"20* 0−19"82 R−1"92 0∣+∣

i # ( 0"$ −0"102* 0

9"$19 −0"#1 0∣+0"*⋅( =0

0"8⋅( R−1"92)⋅( −(−19"82)⋅(−0"20*)⋅( +0"$⋅(−0"#1)⋅( −9"$19⋅(−0"102*)⋅( +0"*⋅( =0

(0"8⋅ R−$0"*)⋅( =0 ⇒0"8⋅ R−$0"*=0 ⇒ R=$0"*

0"8=*0"82

R=*0"82⋅ # ⇒∣ R∣= R=*0"82 N

- esfacem cupla &e rota%ie &in ( A R)

" F BO1

(2 ) =0 eca"ia%e !roiec"ii !entr elementl 2 !e %irec"ia BO1

" M O(1)=0 eca"ia %e momente fa"& %e O

22

6

)

)$

$

*0

/D

1

#

2

6

4

41

$*E

)1



x2+ x A+ y2+ y A+2=0

OA×( x A+ y A)+OC 1×( x1+ y1+ 1)+ M =0

( x2− x A)⋅cos $* ⋅i+( y2+ y A+2)⋅sin $* ⋅ #=0

∣ i

#

( 0"1$* 0"09* 0

x A y A 0∣+

∣ i

#

( 0"082 0"0$* 0

1"##$−0"81 0∣−80⋅( =0

(#"0− x A)⋅cos$* +(2"082+ y A−1"92)⋅sin $* =0

0"1$*⋅ y A⋅( − x A⋅0"09*⋅( +0"082⋅(−0"81)⋅( −1"##$⋅0"0$*⋅( −80⋅( =0

(#"0− x A)⋅√ 22+(2"082+ y A−1"92)⋅√ 2

2=0 ∣#√ 2

2

0"1$*⋅ y A−0"09*⋅ x A−80"119=0

− x A+ y A=−#"20"1$*⋅ y A−0"09*⋅ x A=80"119

y A= x A−#"2

0"1$*⋅( x A−#"2)−0"09*⋅ x A=80"119

0"1$*⋅ x A−0"1#−0"09*⋅ x A=80"119

0"09*⋅ x A=80"#2 ⇒ x A=80"#2

0"09*=1.11"1

y A=1.11"1−#"2=1.1*"88$ ∣ R A R

∣=√ 1.11"1 2+1.1*"88$2=1.$0"1# N

- esfacem cupla &e transla%ie &in ( AT )

2#

1

#

2

6

4

41

$*E

)1



" M O(1+2)=0 eca"ia %e momente fa"& %eO

OA×( R+ x2+ y2+2)+OC 1×( x1+ y1+1)+ M =0

∣ i # (

0"1$* 0"09* 0 R⋅sin $*+#"0 0"12− R cos$* 0∣+∣

i # (

0"082 0"0$* 01"##$ −0"81 0∣−80⋅( =0

0"1$*⋅(0"12− R⋅cos$* )⋅( −( R⋅sin$* +#"0)⋅0"09*⋅( +0"082⋅(−0"81)⋅( −1"##$⋅0"0$*⋅( – –80⋅( =0

(0"019−0"11⋅ R−0"0⋅ R−0"#$2−0"0**−0"#−80)⋅( =0

−0"18#⋅ R−81"008=0

R=−81"008

0"18#=−$$2" N

∣ R AT ∣=$$2" N

− R⋅sin $* =−#1#"012

R⋅cos$* =#1#"012

R AT =−#1#"012⋅i+#1#"012⋅ #

- esfacem cupla &in O1

"

F BO1

(#)

=0 eca"ia !roiec"iilor for"elor !e %irec"ia BO1 "

M O(#+2)=0 eca"ia %e momente fa"& %e A

2$

01

)#

6

2



( xO1

+ x#+ x2+ xB)⋅cos$* ⋅i +( yO1

+ y#+#+ yB)⋅cos$* ⋅ #=0

AO1×( xO 1

+ yO 1

)+ AC #×( x

#+ y

#+

#)+ AB× RB+ M i#

( xO1+1"8+#"0+188"$)⋅√ 2

2 +( yO 1

+1"28−2"21−$8"$8)⋅√ 22 =0 ∣#√ 2

2

∣ i # (

−0"1$* −0"1* 0 xO

1 yO

10∣+∣

i # (

−0"082 −0"082 01"8 −0"9# 0∣+∣

i # (

0"0$ 0"0$ 0188"$−$8"$8 0∣−0"02⋅( =0

xO1

+ yO1

+1$$"$*#=0

−0"1$*⋅ yO1

⋅( −(−0"1*)⋅ xO1

⋅( +(−0"082)⋅(−0"9#)⋅( −1"8⋅(−0"082)⋅( +0"0$⋅(−$8"$8)⋅( –

– 188"$⋅0"0$⋅( −0"02⋅( =0

xO1=− yO1

−1$$"$*#

−0"1$*⋅ yO1+0"1*⋅ xO

1−9"2*=0

xO1=− yO1

−1$$"$*#

−0"1$*⋅ yO1+0"1*⋅(− yO 1

−1$$"$*#)−9"2*=0

−0"#29⋅ yO1−##"099=0

yO1

=−##"099

0"#29=−100"

xO1

=−(−100")−1$$"$*#=−$#"8*#

RO 1

=−$#"8*#⋅i −100"⋅ #

RO 1

=∣ RO 1

∣=√ (−$#"8*#)2+(−100")2=107594 N

- esfacem cupla &e rota%ie &in ( A R)

2*

1

#

2

6

4

41

$*E

)1



" F BO1

(1+2)=0 eca"ia %e !roiec"ii !e %irec"ia BO1

" M A(1)=0 eca"ia %e momente fa"& %e A

( xO+ x1+ x2)⋅cos$* ⋅i +( yO+ y1+1+ y2+ 2)⋅cos$* ⋅ #=0

AO×( xO+ yO)+ AC 1×( x1+ y1+

1)+

M =0

( xO+1"##$+#"0)⋅√ 22 +( yO+0"−1"$*1+2"082−1"92)⋅√ 2

2 =0 ∣#√ 2

2

∣ i # (

−0"1$* −0"09* 0 xO yO 0∣+∣

i # (

−0"082 −0"0$* 01"##$ −0"81 0

∣−80⋅( =0

xO+ yO+$"#9=0

(−0"1$*)⋅ yO⋅( −(−0"09*)⋅ xO⋅( +(−0"082)⋅(−0"81)⋅( −1"##$⋅(−0"0$*)⋅( −80⋅( =0

xO=− yO−$"#9−0"1$*⋅ yO+0"09*⋅ xO−9"8=0

−0"1$*⋅ yO+0"09*⋅(− yO−$"#9)−9"8=0

−0"09*⋅ yO−80"19=0

yO=−80"19

0"09*=−1.1*#"899

xO=−(−1.1*#"899)−$"#9=1.1$9"*2

RO=1.1$9"*2⋅i−1.1*#"899⋅ #

RO=∣ RO∣=√ 1.1$9"*22+(−1.1*#"899)2=1.628596 N

Proiect Mecanisme I

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