Eserciziario Fondamenti di meccanica teorica ed applicata

7/29/2019 Eserciziario Fondamenti di meccanica teorica ed applicata

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1

t = 0O

v = 2i + 4tj

x = x(t) y = y(t)

t = 2 s

O

x(t = 0) = 0, y(t = 0) = 0

vx = x = 2 e vy = y = 4t


14/308

P x(t)y(t)

dx = x dt

x x(t = 0) =t0

xdt =

t0

2 dt = 2t

dy = y dt

y y(t = 0) =t0

ydt =

t0

4t dt = 2t2

x(t) = 2ty(t) = 2t2

ty = f(x)

t = x2 y = x2

2

x = d

2x

dt2 = 0

y = d2ydt2

= 4


15/308

t = 2s

px = x = 4 m , py = y = 8 m;

vx = x = 2m

s, vyy = 8

m

s;

ax = x = 0m

s2, ay = y = 4

m

s2;

(P O) = p = 4i + 8j [m]

|(P O)| = |p| =

(px2 + py2) = 8.94 m

tan() =y

x=

8

4= 2 = atan(2) = 63o


16/308

x y

(P O) = p = 8.94ei63o [m]

v = 2i + 8j [m/s]

|v| =

(vx2 + vy2) = 8.25 [m/s]

tan() =vyvx

=8

2= 4 = arctan(4) = 76o

v = 8.25ei76o

[m/s]

a = 4j [m/s2]

|a| =

(ax2 + ay2) = 4 m/s2

a = 4ei2 [m/s2]

y

t n

t =v

|v| = 0.24i + 0.97j =

=vx

|v

|

i +vy

|v

|

j = cos()i + sin()j

xy v a


17/308

v a =

i j kvx vy 0ax ay 0

=

i j k2 8 00 4 0

= 8k

b

(v a) v =

i j k0 0 82 8 0

= 64i + 16j

n =(v a) v|v a| |v| = 0.97

i + 0.24j

= sin()i + cos()j

nt t n b

n = b tn

t = 2 s


18/308

at = a t = axtx + ayty = 4 0.97 = 3.88 ms2

an = a

n = axnx + ayny = 4

0.24 = 0.96

m

s2

a = att + ann


19/308

2

sT = 1870m

A B

C

D E

200

R1 = 400

R2 = 400

vMax = 60km/ha1 = 1m/s

2

a3 = 0.8m/s2


20/308

0.8m/s2

t0 = 0

t1 t2 t3v(t)

at(t)

v (t) = v0 +

tt0

at (t) dt

t = t1t1

v (t1) = 0 + t10

a1dt = a1t1 = vMax t1 = 16.67s

t3 = t3t2


21/308

vmax

0

0

t

t

v

at

t1 t3t2

a1

a3

v (t3) = vMax +

t3t2

a3dt = vMax + a3 (t3 t2) = 0

t3 = vMaxa3 = 20.84s

at(t)

v (t3) = v0 +

t30

at (t) dt = 0 a1t1 + a3 (t3 t2) = 0

v(t)

s (t) = s0 +

tt0

v (t) dt

s

sT = s1 + s2 + s3


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t2

s (t3) = s0 + t1

0 v (t) dt + t2

t1 v (t) dt +t3t2 v (t) dt = sT

vMaxt12

+ vMax (t2 t1) + vMax (t3 t2)2

= sT

t2 = t2 t1 = 93.4st3 = 130.9s

vMed =1

t3

t30

v (t) =sTt3

= 14.3m/s

an

v2/R

B,C,D tB = 20.3s tC = 58.0s tD = 115.2s

A

B

C

D

E

t

n

sE = 1870 mtE = 130.9 s

sD = 1770.8 mtD = 115.2 s

s2 = 1696.0 mt2 = 110.1 s

sC = 828.3 mtC = 58 s

sB = 200 mtB = 20.3 s

s1 = 138.9 m


23/308

t2 tD

0

500

1000

1500

2000

s[m]

0

5

10

15

20

v[m/s]

1

0.5

0

1

0,5

at

[m/s2]

0 10 t1 tB 30 40 50 tC 70 80 90 100 t2 tD 120 t30

0.25

0.5

0.75

1

an

[m/s

2]

t [s]

s = s(t)s

0 s1 sB 400 600 800sC 1000 1200 1400 1600 s2 sD1 80 0 s 30

5

10

15

20

v[m/s]

0 s1 sB 400 600 800sC 1000 1200 1400 1600 s2 sD1 80 0 s 31

0.5

0

0.5

1

at

[m/s2]

0 s1 sB 400 600 800sC 1000 1200 1400 1600 s2 sD1 80 0 s 30

0.2

0.4

0.6

0.8

an

[m/s2]

s [m]


24/308

an,Max = vMax2

R1= 0.69 m/s2


25/308

3

= 0.1 rad/s = 0.01 rad/s2

vr = 0.7 m/s ar = 0.1 m/s2

, .

vr, ar


26/308

3.9 m

(P O)

y Im

x Re

P

O

P

O

(P O) = ei

= (t)

= (t)

(P O)


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= 3.9 m

= 30

Re (P O) = cos()Im (P O) = sin()

(P O)

(P O) = cos() + i sin()

vP =d (P O)

dt= ei + i ei = ei + ei(+

2 )

i

(PO)

ei

0.5 m/s

vP

vP

Im

Re

ei( +

)2

.

P

O

vP


28/308

|vP| =

2 +

2

vP = + arctan

Re (vP) = |vP| cos(vP) = cos() sin()Im (vP) = |vP| sin(vP) = sin() + cos()

= vr = 0.7m/s

= = 0.1rad/s

vP |vP| = 0.80 m/s 59.1vP = 0.41 + i0.69

aP =d (vP)

dt= ei + i2 ei + i ei 2ei =

= ei + 2 ei(+2 ) + ei(+

2 ) + 2ei(+)

aP

2

ReaP = cos() 2 sin() + sin() 2 cos()Im aP = sin() + 2 cos() + cos() 2 sin()

aP

|aP| =

(Re(aP))2 + (Im (aP))

2

aP = arctanIm(aP)Re(aP)


29/308

Im

ReO

0.05 m/s2

aP

Pe

i( +)2

..

2ei( +

)2

..

ei..

ei( + )

.2

ar

aP = 0.037+i0.185|aP| = 0.189 m/s2 aP = 101.2

Oxy

Oxy O O x y

z

z

x

Oxy

vP = v(Tr)P + v

(Rel)P


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i

j

i

j

y

,

y

x

x

P

O O

O

vP = (P O) + vri = P O

j + vr

i

vP = v(Tr)P + v

(Rel)P

P O vr

P O P O

vP

y

x

P

O

vP(Rel)vP

(Tr)

0.5 m/s


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|vP| = 0.80 m/s 59.1v(Rel)P

v

(Tr)

P

vP(Rel)

vP(Tr)

vP

0.5 m/s

aP = a(Tr)P + a

(Rel)P + a

(Cor)P

a(Tr)P

(P

O) (P O)

(P O) a(Rel)P

a(Cor)P

v(Rel)P

aP = (P O)

2

(P O) + ari + 2 vr == P Oj 2P Oi + ari + 2vrj


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aP = a(Tr)P,t + a

(Tr)P,n + a

(Rel)P,t + a

(Rel)P,n + a

(Cor)P

P O 2P O ar 2vr

P O P O P O P O

y

xO

aP

P

aP,t(Rel)

aP,n(Tr)

aP,t(Tr)

aP(Cor)

|aP| = 0.189 m/s2 101.2

aP

aP,t(Rel)

aP,n(Tr) aP,t

(Tr)

aP(Cor)

0.05 m/s2


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4

O1A O1A = 0.2 m AB

0.8 m BBO2 0.6m

RSD = 0.15 m O2

O1

O2

A

D

GB

P

D RD = 0.15 m


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= 160 = 0.1 rad/s = 0 rad/s2

160

0.3 m 0.8 m

GO2 RSD/2 D

GO2 RSD/2

D

O1 ABBO2 D

O1O1

A ABO1

B ABO2 B

O2

O2


39/308

DP D D

O1

O2O2

O1ABO2lmin + lmax 0 BO2G

vG = BO2 (G O2)

= k GO2 cos+ 2i + sin+ 2j= 2.3 103ei(+2 ) m/s

(G O2) = GO2ei(+2 )

d

dt(G O2) = v G

D


45/308

O2

G

PRSD

RSD2

vP vG

|vD| = |vP| = |BO2 (P O2)| = P O2 = 4, 6 103 m/s

(P O2) ddt(P O2)

vPvG

x1OAy1A B B

O2 BBO2 O2

B

vB(Tr) = vA

O1

BB A B

AB AB

v(Ass)B = v

(Tr)B + v

(Rel)B


46/308

v(Ass)B v

(Tr)B v

(Rel)B

O2B (B O2) O1A (A O1) AB (B A)O2B O1A AB

O1

O2

A

GB

c

a

b

ca

b

B

5 103ms

aei(+2 )

v(Tr)B

+ bei(+2 )

v(Rel)B

= cei(+2 ) = cei(2 ) v(Ass)B

v(Ass)B B

v(Tr)

Bv(Rel)B


47/308

aei(+2 ) a2ei + bei(+2 ) b2ei+

+cei(+2 ) c2ei = 0

a sin a2 cos bsin b2 cos csin c2 cos = 0a cos a2 sin bcos b2 sin ccos c2 sin = 0

=

b sin c sin +b cos c cos

1

+a sin + a2 cos + b2 cos + c2 cos

a cos + a2 sin + b2 sin + c2 sin

=

1, 8 103 rad/s21, 4 103 rad/s2

G D

aG = BO2 (G O2) + BO2 (BO2 (G O2))aP = BO2 (P O2) + BO2 (BO2 (P O2))


48/308

O2

G

P

RSD2

RSD

2RSD2

2RSD

P G

|aGt| =k (G O2) = RSD2 ei

= 1.0 104 m/s2|aGn| =

k k (G O2) = 2 RSD2 ei(+2 ) = 6.9 105 m/s2

|aPt| =k (P O2) = RSDei = 2.1 104 m/s2

|aPn| =k

k (P O2)

=

2RSDe

i(+2 )

= 1.3 104 m/s2

P GD

aPt| aD| = |aPt| = (P O2)

aDD

B


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a(Ass)B = a

(Tr)B + a

(Rel)B + a

(Cor)B

a(Ass)B Ba(Ass)Bn a

(Ass)Bt

B O1a(Ass)Bn = BO2 (BO2 (B O2))

a(Ass)Bt = BO2 (B O2)

a(Tr)B BaAa

(Tr)Bn a

(Tr)Bt

a(Tr)Bn = AO1 (AO1 (A O1))

a(Tr)Bt = AO1 (A O1)

a(Rel)B Ba(Rel)Bn a

(Rel)Bt B

A

a(Rel)Bn = AB (AB (B A))a(Rel)Bt = AB (B A)

a(Cor)B = 2 v(Rel)B = 0

aei(+2 )

a(tr)Bt

a2ei a(tr)Bn

+ bei(+2 )

a(rel)Bt

b2ei a(rel)Bn

+ cei(+2 )

a(ass)Bt

c2ei a(ass)Bn

= 0


50/308

a(ass)Bn O2B (O2B (B O2)) BO2

a(ass)Bt

O2B (B O2) BO2a(tr)Bn O1A (O1A (A O1)) AO1a(tr)Bt

O1A AO1a(rel)Bn AB (AB (B A)) AB

a(rel)Bt

AB ABa(Cor)B

O1

O2

A

GB

c2

c

a2

b2b

B c2

c

a2

b

b2

5 104 ms2

B a(Ass)Ba(Tr)B

a(Rel)B b

2

10


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0 50 100 150 200 250 300 35020

0

20

40

60

80

100

120

140

Angolo di manovella []

Rotazionidelleaste[]

Asta AB

Asta BO2

0 50 100 150 200 250 300 350

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12


V

elocitangolari[rads

1]

Asta O1A

Asta AB

Asta BO2

0 50 100 150 200 250 300 3506

4

2

0

2

4

6x 10

3


VelocitcentrodiscoD[m

s

1]

= 160

D

= 160 = 0.1rad/s

|vD| > 0 52.2%2/

O1A


52/308

0 50 100 150 200 250 300 3504

3

2

1

0

1

2

3

4

5

6x 10

3


Acc

elerazioniangolari[rads

2]

Asta O1A

Asta AB

Asta BO2

0 50 100 150 200 250 300 3504

2

0

2

4

6

8x 10

4


Accele

razionecentrodiscoD

[ms

2]

D

= 160 = 0.1rad/sO1A

vmax


53/308

5

OA AB R

t

B P

= 3 = 1rad/s = 0 rad/s2

= 0 c = 123 + 1 m OA = 13 mAB = 1 m R = 0.2 m


54/308

O

AR

P

C

, ,

B

d 0, 3m

c

3

AO OAB


55/308

= 0 = 60

= 120 = 180

= 240 = 300


56/308

0 50 100 150 200 250 300 35020

0

20

40

60

80


Angolo[]

0 50 100 150 200 250 300 3500

0.5

1

1.5

2

Posizionec[m]


0 50 100 150 200 250 300 3500.8

0.6

0.4

0.2

0

0.2

0.4

0.6


Angolo

[rads

1

]

0 50 100 150 200 250 300 3500.8

0.6

0.4

0.2

0

0.2

0.4

0.6

Posizione

c[m

s

1]


0 50 100 150 200 250 300 3500.8

0.6

0.4

0.2

0

0.2

0.4

0.6


Angolo[rads

2

]

0 50 100 150 200 250 300 3501

0.8

0.6

0.4

0.20

0.2

0.4

Posizionec[ms

2

]



57/308

O

A B

P

C

, ,

H

R

O

BB

AB AOA

(B O)

(B O) = (A O) + (B A)

(B O)

(B O)


58/308

O

AB

O

A B

, ,

a

b

c

d

H

B

(B O) = (H O) + (B H)

a = (A O) a b = (B A) b =c = (H O) c d = (B H) d


59/308

O

A

B

, ,

a

b

c

d

H

a b d

c dB

P

a + b = c + d

a = aei

b = bei

c = cei

d = dei

aei + bei = cei+ dei

a cos + b cos = c cos + d cos a sin + b sin = c sin + d sin


60/308

B

a b

OA AB d

c

c

aei(+2) + bei(+

2) = cei

sin

+ 2

= cos cos

+

2

= sin

c a sin bsin = c cos a cos + bcos = c sin

cos b sin sin +b cos

c

=

a sin

a cos

c

Ax = b xi

x

xi =detAidetA

AiA b


61/308

O

A R

P

C

, ,

X

Y

x

y

B

= 0 = 0

c

= a sin

(a cos ) /b

= 0, 5 m/s

0, 29 rad/s

c O

AB

A X YB

B

Ao

B A

v(Ass)B =

v(Tr)B + v

(Rel)B


62/308

v(a)B v

(tr)B v

(Rel)B

c a b

X AO AB

vB(Ass)

vB(Tr)

vB(Rel)

v(tr)B

v(Rel)B

B

vB(Ass) = cei

vB(Tr) = aei(+/2)

vB(Rel) = bei(+/2)

cei

= aei

(+

2 ) + bei

(+

2 )


63/308

v

(Tr )B

= |a|

v

(Ass)B

= |c|

v

(Rel)B

=

b

AB

0, 1ms

iaei a2ei + ibei b2ei = cei

i2

aei(+2 ) + a2ei(+) + bei(+

2 ) + b2ei(+) = cei

a sin a2 cos bsin b2 cos = c cos a cos a2 sin + bcos b2 sin = c sin

cos b sin sin b cos c

= b2 cos + a sin + a2 cos

b2 sin + a2 sin a cos = 0 = 0

c


64/308

c

=

b2 a sin a2 cos (a2 sin a cos ) /b

=

0, 37 m/s20, 5 rad/s2

c BO

aB(Ass)

= aB(Tr)

+ aB(Rel)

+ aB(Co)

aB(Ass) B aB(Tr) B

aBn(Tr) aBt(Tr)

aB(Rel) BA a

(Rel)Bn

a(Rel)Bt

aB(Co) =

= 0

a(tr)Bt

= 0

a(rel)Bt


65/308

a(a)B a

(tr)Bn a

(tr)Bt a

(rel)Bn a

(rel)Bt a

(Cor)B

c a2 a b2 b 0

X AO AO AB AB

a

(Ass)B

= |c|

a

(Rel)Bt

=

b

a

(Rel)Bn

=

b2

a

(Tr )Bn

=

a2

0, 1 ms2

a(Ass)B = ce

i

a(Tr)Bn = a

2ei(+)

a(Tr)Bt = ae

i(+2 )

a(Rel)Bn = b

2ei(+)

a(Rel)Bt = be

i(+2 )

cei = aei(+2 ) + a2ei(+) + bei(+

2 ) + b2ei(+)

B


66/308

c c P

vB = (B C) = |c|i

= k

vB = k Rj = Ri = |c|iB

aB = k Rj = Ri = |c|i

= cR = 2, 5rad/s

= cR = 1, 86rad/s2

a = 1 m b = 2, 5 md = 0, 9 m

= 1 rad/s

O C


67/308

= 0 = 60

= 120 = 180

= 240 = 300


68/308

0 50 100 150 200 250 300 35020

10

0

10

20

30

Angolo di manovella [

]

Angolo[]

MOC

MOD

0 50 100 150 200 250 300 3501

1.5

2

2.5

3

3.5


Posizionec[m]

MOC

MOD

c

4 180

Ta =4 + 2 3

= s

Tr =3 4

= s

21


69/308

0 50 100 150 200 250 300 3501.5

1

0.5

0

0.5

1

1.5


Velocitac

1

2 3

4

MOC

MOD

Ta =1 + 2 2

= 2, 74 s

Tr =2 1

= 3, 51 s


70/308


71/308

6

O1B = 2.5 m OO1 = 1.41 m = 0 = 4

t = 3 sOB

OB(t) = b(t) = 3, 385 + 0, 07t + 0, 005t2

O1

O

B


72/308

O1B OB

O1B OB

3

9

O1

OBO

B O1BB

OO1


73/308

OBb(t)

b(t = 3) = 3, 385 + 0, 07t + 0, 005t2 = 3, 64 m

b(t = 3) = 0, 07 + 0, 01t = 0, 1 m/s

b(t = 3) = 0, 01 = 0, 01 m/s2

a b c

a

b

c

Re

Im

a = aei

b = bei

c = cei

ab c O1BOB OO1

c + a = b


74/308

B

cei+ aei = bei

ei = cos + i sin c cos + a cos = b cos c sin + a sin = b sin

sin = 0c sin = b sin = arcsin( cb sin ) = 0.27 rad

aecb

a c

OB

iae = bei + ibei

a a

b b

c c

i2 i = e

i2

aei(+2) = bei + bei(+

2)


75/308

sin

+ 2

= cos cos

+

2

= sin

a sin = b cos b sin a cos = b sin + bcos a sin b sin

a cos b cos

=

A

=

b cos

b sin

A1

=b cos b sin

b sin b cos

;

A2

=a sin b cos

a cos b sin

=det[A1]

det[A]= 0, 146 rad/s

=det[A2]

det[A]= 0, 096 rad/s

XOY O X Y

B X1O1Y1BO

BO1 B O1B

BX1 X1O1Y1


76/308

O1

O

B

X1Y1

X

Y

BX1 O

BO

c b O1B

c sin = b sin

= arcsin(

c

b

sin ) = 0, 27 rad

v(ass)B = v

(tr)B + v

(rel)B

v(Ass)B O1B (B O1)

v(Tr)B OB (B O)

v(Rel)B x1

x1OB O1B OB O1B


77/308

v(Ass)B v

(Tr)B v

(Rel)B

O1B(BO1) OB(BO) x1

O1B BO BO

wO1BO1B

05

wOBOB

X1

0,ms

v(a)B = ae

i(+2 )

v(tr)B = be

i(+2 )

v(rel)B = be

i

O1B =|vassB |O1B

= ...rad

s

OB =|vtrB |OB

= ...rad

s

O1B = ...k

OB = ...k

vtrB vassB


78/308

aei(+2 ) a2ei = bei + 2bei(+2 ) + bei(+2 ) b2ei

a sin + bsin = a2 cos + b cos 2bsin b2 cos a cos bcos = a2 sin + b sin + 2bcos b2 sin

a sin b sin a cos b cos

=

a2 cos + b cos 2bsin b2 cos a2 sin + b sin + 2bcos b2 sin

=

0, 04 rad/s2

0, 025 rad/s2

a(Ass)B = a

(Tr)B + a

(Rel)B + a

(Co)B

a(Ass)B Ba(Ass)Bn = O1B O1B (B O1) a(Ass)Bt =

O1B (B O1)


79/308

a(Tr)B Ba(Tr)Bn = OB OB (B O)

a(Tr)Bt = OB

a(Rel)

B B x1i1

a(Co)B = 2O1B vB(rel)O1B

vrelB = x1i1

a(Ass)Bn a

(Ass)Bt a

(Tr)Bn a

(Tr)Bt a

(Rel)B a

(Cor)B

2O1B(O1B) O1B(O1B) 2OB(OB) OB(OB) x1 2O1Bx1

O1B O1B OB OB OB OB

b

b2

a

a2 b

2b

0, 02ms2


80/308

a(Ass)Bn = a2ei

a(Ass)Bt = ae

i(+2 )

a(Tr)Bn = b2eia(Tr)Bt = be

i(+2 )

a(Rel)B = be

i

a(Co)B = 2be

i(+2 )

O1B =|aassB |O1B

= ...rad

s2

OB =|atrB |

OB

= ...rad

s2

O1B = ...k

OB = ...k

atrB aassB


81/308

7

AB = 3 mBC C

0, 6 mC

CxC(t) = 6 + 2, 5 sin(2t)

A

B

C

G

t = 0 s

AB

G


82/308

AB

G

A

B

C

G

xc

t = 0

xc

A

B

C

G

t = 0, 17

xc

A

B

C

G

t = 0, 34

xc

A

B

C

G

t = 0, 51

xc

A

B

C

G

t = 0, 68

xc

A

B

C

G

t = 0, 85

xC C


83/308

AB BCB

CBC


84/308

A

B

a

d

b

c

aei = bei + cei+ dei

d c 2

ae

i

= be

i(+2 )

+ ic + d


85/308

a AB

b BC

c c = R = 90d d = xC = 0

AB BC

AB BC = +

2

= =

b

a cos = b sin + da sin = b cos + c

b

b =d

a cos

sin

d cos + c sin a = 0

cos2 + sin2 = 1

cos =ad

4a2d2 a (c2 + d2) (a2 c2)(c2 + d2)cos

65, 87


86/308

b5, 23 m

iaei = bei + ibei + d

= i

i

aei bei = bei + daei bei

f

A

B

a

d

b

c

f

f ei(+2 ) = bei + d

b f

fsin = b cos + dfcos = b sin


87/308

b t = 0

b =

1, 23 rad/s

18, 0m

/s

d

b

f

2 ms

GBC

C G

vG = vC+ BC (G C) = d + GCei(+2 ) = vC+ vGC

vGCG C


88/308

vG

vC d

vGC GC GC

G

vGx = d GCsin = 14, 3 m/s

vGy = GCcos = 3, 1m

/s

G

|vG| =

v2Gx + v2Gy = 14, 65

m/s

A

B

C

G

vC

vGCvG

G


89/308

iaei 2aei = bei + ibei + ibei + ibei 2bei + d

= =

i aei bei 2 aei bei = bei + i2bei + daei bei

f = f ei

f ei(+2 ) 2f ei = bei + 2bei(+2 ) + d

b

fsin 2fcos = b cos 2b sin + dfcos 2fsin = b sin + 2b cos

b t = 0

b = 7, 59 rad/s214, 89 m/s

2


90/308

b

2b

f

2f

10 m/s

GBC

aG = aC+ GC (G C) atGC

+ GC (GC (G C)) anGC

aC Cxc(t = 0) = 0

atGCG

C

anGCG C


91/308

aG

aC xc

atGC GC GCanGC GC

2 GC

x y

aGx = d GCsin GC2 cos = 12, 4 m/s2

aGy = +GCcos GC2

sin = 17, 4m

/s2

G

|aB| =

a2Bx + a2By = 21, 43 m/s2

G t = 0

A

B

C

G

aGCn

aGCt

aG

G

B G


92/308

A

B

C

G

B G


93/308

8

R

C

x x

D

tx = +0, 4 m/s x = +0, 2 m/s2


94/308

= 6 =4

AB 0, 2 m R = 0, 025 mt

B

D

BD

39

A

2


95/308

1

t

B

A

C

O D

a

b

d

c

f

Im

Re

a OB = 90

b AB = 315

c OD = 0

d CD = 30

f AC = R = 120

b f


96/308

A

a + b = c + d + f

aei + bei = cei+ dei + f ei

B AB

a d c = 0, 4 m/s

a cos = c cos + d cos

a sin = c sin + d sin

a

d

=

0, 231 m/s0, 462 m/s

0 = c cos + d cos

d = c cos cos a = c sin c cos cos sin

D (A C) = DAC = vA = v(Rel)B D = |d|AC


97/308

B

aei = cei+ dei

a d

a cos = c cos + d cos a sin = c sin + d sin

a d c = 0.2ms2

a

d

=

0, 2 m/s2

0, 1 m/s2

d = c cos

cos a = c sin c cos

cos sin

D (A C) = DAC = aA = a(Rel)B D = |d|AC

XOY ??ternarel??D

B

v(Ass)B

v(Rel)B


98/308

v(Tr)C

v(Rel)C v

(Ass)B

0, 1ms

v

(Ass)

B yA OBv(Tr)B xD = 0, 4ms OD

v(Rel)B xC1 CD

c

v(Tr)BD

A B

v(Ass)B

D (A C) = v(Rel)B


99/308

v(Rel)B = d = d ACd = dAC

a(Ass)B B a

(Rel)B B

a(Tr)B B

a(Ass)B yA OB

a(Tr)B xD = 0, 2ms2 OD

a(Rel)B xC1 CD

a(Tr)C

a(Rel)C

a(Ass)B

0, 05 ms

B


100/308


101/308

9

AO = 0.4 mO AB = 1.4 m A AO

C

O

A

B

C

t

AC AC

= 45 = 25 rad/s = 0 rad/s2 = 170 AC = 0.6 m

t


102/308

AB AB

AB AB

vB B

aB B

O A

C

B

= 0

O

A

C

B

= 60

O

A CB

= 120

OA

CB

= 180

O

A

CB

= 240

O

A

C

B

= 300


103/308

AO O

AO AB A

C AB

AO OAB

x e y m O

O

A

B

i Re x

j

Im y

C


104/308

a

b

c

O

A

C

Re

Im

a a = AO = 0.4m = 45

b b = AC = 0.6m = 170

c c = CO

a = b + c

c


105/308

aei = bei + cei

AB AB

aei (+2 ) = bei + bei(+

2 )

a sin = b cos bsin a cos = b sin + bcos

b cos b sin sin b cos

b

=

a sin a cos

b

= 8.19 m/s9.55 rad/s

ABxOy

AB = k = 9.55k

bb t

AOA C

A x1O1y1


106/308

O

A

B

i x

j

y

i1

j1

y1

x1O1

O1 C ABA

v(Ass)A = v

(Rel)A + v

(Tr)A

v(Ass)A AOA O v

(Ass)A = AO

(A O) v(Rel)A A

O1 C A

AC

v(Tr)

A A

O1 C AB v(Tr)A = AB (A C)

A

v(Ass)A


107/308

v(Ass)A AOAO AO

v(Rel)A v

(Rel)A AC

v(Tr)A ABAC AC

AC

AC

v(Ass)A

1ms

v(Tr)A

v

(Rel)

A

v(Ass)A

1ms


108/308

v(Tr)A

| AB| = |v (tr)A

|AC = 9, 55

rads

AB

aei(+2 ) + a2ei(+) = bei + 2bei(+

2 ) + bei(+

2 ) + b2ei(+)

a sin a2 cos = b cos 2bsin bsin b2 cos a cos a2 sin = b sin + 2bcos + bcos b2 sin

cos b sin sin b cos

b

=

a sin a2 cos + 2bsin + b2 cos a cos a2 sin 2bcos + b2 sin

b

=

198 m/s2

602 rad/s2

AB

A Cb


109/308

A x1O1y1

a(Ass)A = a

(Rel)A + a

(Tr)A + a

(Cor)A

a(Ass)A A

Oa(Ass)At = AO (A O)a(Ass)An = AO

(AO (A O))

a(Rel)A AA

AB

a(Tr)

AA

CAB a

(Tr)At = AB (A C)

AB a(Tr)An = AB (AB (A C))

a(Cor)A

A a(Cor)A = 2AB

v(Rel)A


110/308

a(Ass)At AOAO AO

a(Ass)An

2AOAO AO

a(Rel)At a(Rel)At ACa(Cor)A 2ABv

(Rel)A AC

a(Tr)At ABAC AC

a(Tr)An

2ABAC AC

a(Ass)A

AC

a(Co)A

AC

a

(Tr)

An

50 ms2

AO

AO AB


111/308

50 ms2

a(Ass)A

a(Rel)A

a(Co)A

a(Tr)At

a(Tr)An

a(Tr)At

AB = 602k

| AB| = |a (tr)A

|AC

BB AB

A v(Ass)A AB

B A

vB = vA + AB (B A) = AO (A O) + AB (B A)x y (AO) (BA)


112/308

(A O) = AO cos i + AO sin j(B A) = BA cos( )i + BA sin( )j

(B A)AO = 25k AB = 9.55k

A

B

(B A)

AO (A O) =

i j k0 0 AO

AO cos AO sin 0

== AOAO sin i + AOAO cos j = 7.07i 7.07j

AB (B A) = i j k

0 0 ABBA cos( ) BA sin( ) 0 =

= ABBA sin( )i + ABBA cos( )j = 2.32i + 13.17jx y

vBx = 9.39 m/svBy = 6.1 m/s

B

|vB| =

v2Bx + v2By = 11.2 m/s


113/308

(vB) = arctan

vByvBx

= 33

vB

O

AC

B

vA vBA

vB

B

BB

(B O) = (A O) + (B A)

xB + iyB = a expi

+d expi

ad

f

O

A

B

Re

Im

B

xB + iyB = vBx + ivBy = aei(+2 ) + dei(+

2 )

= + /2 = e m


114/308

vBx = a sin dsin(+ ) = 9.39 m/svBy = a cos + d(+ ) = 6.1 m/s

BAB

aB = aA + AB (B A) + AB (AB (B A))A

AO = 25k AO = 0

aA = AO (A O) + AO (AO (A O)) = AO (AO (A O)) =

2AO (A O) = 2AOAO cos i 2AOAO sin j = 176.8i 176.8j

AB = 9.55k AB = 602k

AB

(B

A) =

i j k0 0 AB

BA cos( ) BA sin( ) 0 == ABBA sin( )i + ABBA cos( )j = 146.3i + 830j

AB (AB (B A)) = 2AB (B A) =

2ABBA cos( )i 2ABBA sin( )j = 125.7i + 22.2j

x y

aBx = 156.2 m/saBy = 675.4 m/s

B


115/308

|aB| =

a2Bx + a2By = 694

m/s2

x

(aB) = arctanaBy

aBx

= 103aB

O

AC

B

aAanBA

atBA

aB

B

xB

+iyB

=aBx

+iaBy

==a

2

e

i(+) +d

e

i(+2 ) +d

2

e

i(+)

e m = + /2 =

aBx = a2 cos + dsin + d2 cos = 156.2 m/s2aBy = a2 sin dcos + d2 sin = 675.4 m/s2

B

b 0 360


116/308

O

A

C

B

B

b AB

ABAC

b = 12

= 192 AOAB

AB


117/308

0 50 100 150 200 250 300 3500.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Rotazione dellasta OA []

LunghezzadelsegmenteoA

C

[m]

AC

0 50 100 150 200 250 300 35020

10

0

10

20

30

40


Angolo

[]

AB

0 50 100 150 200 250 300 35010

8

6

4

2

0

2

4

6

8

10


VelocitdiallungamentodelsegmentoA

B[m/s]

AC

0 50 100 150 200 250 300 35010

5

0

5

10

15

20

25


Velocitangolare[rad/s]

AB

0 50 100 150 200 250 300 350200

100

0

100

200

300

400

500


AccelelrazionedelsegmentoAB[m/s]

AC

0 50 100 150 200 250 300 350800

600

400

200

0

200

400

600

800


Accelerazioneangolare[rad/s]

AB


118/308


119/308

10

OA = a = 0, 4 m AB = b = 0, 4 m B 30

CB

O

A

30

B

C


120/308

= 30 = 10 rad/s = 100 rad/s2 = 330 CB = 0, 5 m

B

CB

B

a b c d

a = (A O) = OAei = aeib = (B A) = ABei = bei

c = (D O) = ODei

= ce

i

d = (B D) = DBei = dei


121/308

O

A

30

B

C

D

a

cd

b

a a = 0, 4m = 30

b b = 0, 4m = 330

c c = 300

d d = 30

a + b = c + d

c

O

aei + bei = cei+ dei


122/308

B

aei(+2 )

v(Tr)B+ bei(+

2 )

v(Rel)B= dei

v(Ass)B

A

v(Ass)B B

v(Tr)B

AO

v(Tr)B BA B

A

d

a sin bsin = d cos a cos + b cos = d sin d

cos b sin sin b cos

d

=

a sin

a cos

Bd

=

cos b sin sin b cos

1a sin

a cos

=

6, 92 m/s20 rad/s


123/308

v(Ass)B d

v(Rel)B b AB

v(Tr)B a = 4 m/s OA

v(Tr)B

vB(Tr )

vC

vB

1 m/s

(Rel)

(Ass)


124/308

dei

30

B

C

vB(Ass)

vpist = d sin = 3, 46 m/s

B

aei(+2 )

a(Tr )Bt

a2ei a(Tr)Bn

+bei(+2 )

a(Rel)Bt

b2ei a(Rel)Bn

= deia(Ass)B

A

a(Ass)B B


125/308

a(Tr)BnA O

a(Tr)

BtA O

AO

a(Rel)BnB A

a(Rel)BtB A

AB

a sin a2 cos bsin b2 cos = d cos a cos a2 sin + bcos b2 sin = d sin

d )

cos b sin sin b cos

d

=

a sin + a2 cos + b2 cos

a cos + a2 sin + b2 sin

d = 429, 28 m/s2892, 82 rad/s2

AB


126/308

a(Ass)B d

a(Tr)Bt a = 40 m/s2 OA

a(Tr)Bn a

2 = 40 m/s2 OAa(Rel)Bt b = 14, 64 m/s2 AB

a(Rel)Bn b

2 = 160 m/s2 AB

a(As s)

B

a(Re l)

Bn

a(Re l)

Bt

a(Tr )

Bn

a(Tr )

Bt50 m/s2


127/308

11

R1 = 0, 4 m R2 = 0, 5 mR2

30

ED ED = 0, 25 m

R1

mm EO s O

2 m

E (t) =

+ 2

+

6sin(2t)

t = 0

t = 0 s t = 0.1 s

(t)


128/308

E

s

O

Fm D

t = 0

E

s

O

Fm

D

R1R2

t = 0.17

t = 0 t = 0 s

t = 0 s OE = 2.75 m = 120t = 0.1 s OE = 2.72 m = 187.9

t = 0.1 s

dvE E

vm

daE E

am


129/308

t

O

D

E

t = 0

O

D

E

t = 0.17

O

D

E

t = 0.34

O

D

E

t = 0.51

O

D

E

t = 0.68

O

D

E

t = 0.85

4AO m 12

AO R2 2


130/308

m

m

1

E AO

O

m

AO


131/308

O

E

s

O

Fm D

Re

Im

E

c + d + e = f

cei+ dei + eei = f ei

c D O

d e f O E


132/308

c s + R2 = + /2

d d =

e e = ED f f

E

O

d

c

e

f

Re

Im

d(t) = R2 ((t) (0)) = R2

(t) 2

t = 0

O D

(0) = +

2= (0) = (0)

cei R2

+ 2

ei + eei = f ei


133/308

f(t)

t = 0.1 s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 150

0

50

100

150

200

250

300

Tempo [s]

Rotazioni[]

Asta OE []

Asta DE []

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.68

2.69

2.7

2.71

2.72

2.73

2.74

2.75

Tempo [s]

DistanzaEO

[m]

Asta OE [m]

R2ei + eei(+2 ) = f ei + fei(+2 )

R2 cos e sin = fcos f sin R2 sin + e cos = fsin + f cos

(0.1) = 2/3 cos(20.1) =2.66 rad/s R2 cos e sin cos

R2 sin + e cos sin

f

=

f sin f cos

f

=

11.38 rad/s0.46 m/s


134/308

d

xOy d = 11.38kf

E O

f

-R2

e

f

1 m/s

f

t = 0.1 sE

Ef

vE = f ei + fei(+

2 ) = R2ei + eei(+

2 )

vEx = R2 cos ED sin = fcos + f cos( + 2

) 4.53 m/svEy = R2 sin + ED cos = fsin + f sin( +

2) 5.66 m/s


135/308

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

15

10

5

0

5

10

15

Tempo [s]

Velocit[rad/s]

Asta OE [rad/s]

Asta DE [rad/s]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0

0.5

Tempo [s]

Velocit[m/s]

Asta OE [m/s]

E

|vE| =

v2Ex + v2Ey = 7.25 m/s

(vE) = arctan

vEyvEx

= 231.3

Ex1O1y1

O1 OOA E

v(Ass)E = v

(Rel)E + v

(Tr)E

v(Ass)E ExOy

ED

(vD = d (D H)) E


136/308

E

s

O

Fm

D

D (vED = d (E H)) E

v(Ass)E = d (D H) + d (E H) =

= d (E H)

EE H

v(Rel)E EO E

OA

v(Tr)E E

O v(Tr)E = AO (E O)

E


137/308

v(Ass)E dEH EH

v(Rel)E v

(Rel)E OE

v(Tr)E AOEO OE

v(Tr)E

1 m/s

vE(Tr)

EO

EH

v(Ass)E = v

(Rel)E + v

(Tr)E

v(Ass)E,x = v(Rel)E,x + v(Tr)E,xv(Ass)E,x = v

(Rel)E,x + v

(Tr)E,x

v(Ass)E


138/308

1 m/s

vE(Tr)vE

(Ass)

vE(Rel)

|D| = |v(EASS)|

EH= 11.38 rad/s

d = 11.38k

x1O1y1

R2e

i + eei(+2 )

v(Ass)E

= f eiv(Rel)E

+ fei(+2 )

v(Tr)E


139/308

mR1 P

P P

P

vP = d (P H)d = 11.38k

(P H) = P Hsin i + P Hcos j

P H = R1 + R2

vp = d (P H) =

i j k0 0 d

P Hsin P Hcos 0

== dP Hcos i dP Hsin j = 8.87i 5.12j

vp

vPx = 8.87m

/svPy = 5.12 m/s

P


140/308

|vP| =

v2Px + v2Py = 10.24

m/s

(vP) = arctanvPy

vPx

= 210vP

vP

vP

vE

E D

P

2ms

P


141/308

R2ei + eei(+2 ) e2ei

a(Ass)E

= feia(Rel)E

+ 2fei(+2 )

a(Cor)E

+ fei(+2 ) f2ei a(Tr)E

(R2 cos e sin ) e2

cos =

fcos 2

f sin + f sin f2 cos (R2 sin + e cos ) e2 sin =fsin + 2f cos +

+ f cos f2 sin

(0.1) = 23/3 sin(20.1) = 12.15 rad/s

R2 cos e sin cos R2 sin + e cos sin f ==

e2 cos 2f sin f sin f2 cos e2 sin + 2f cos + f cos f2 sin

f

=

16.7 rad/s20.74 m/s2

d = 16.7k

ff


142/308

f

t = 0.1s

5 m/s^2

f..

f. 2

e. 2

e..

R2..

f.. 2f

. .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1150

100

50

0

50

100

150

Tempo [s]

Accelerazioni[rad/s

2]

Asta OE [rad/s2]

Asta DE [rad/s2]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 18

6

4

2

0

2

4

6

Tempo [s]

Accelerazioni[m/s

2]

Asta OE [m/s2]

E


143/308

aE = R2ei + eei(+2 ) e2ei()

aEx = R2 cos Rp sin Rp2 cos = 38.71 m/s2aEy = R2 sin + Rp cos Rp2 sin = 12.77 m/s2

E

|aE| =

a2Ex + a2Ey = 40.76

m/s2

(vE) = arctanaEy

aEx

= 18.2

x1O1y1E

a(Ass)E = a

(Rel)E + a

(Tr)E + a

(Cor)E

a(Ass)E E

D

aD =d(D H)H

E

a(Ass)E = aD + aED =d (D H) +d (E D) +

+d d (E D)

=

=d

(E

H) +d

d

(E

D)

a(Rel)E E

AO


144/308

a(Tr)EE

OAO a

(Tr)Et = AO(E O)

AO a

(Tr)

En = AO (AO (E O)) a(Co)A

E a(Cor)E = 2AO

v(Rel)E

a(Ass)E1 dEH EHa(Ass)E2

2dED DEa(Rel)E a

(Rel)E OE

a(Cor)E 2dv

(Rel)E OE

a(Tr)Et AOEO OE

a(Tr)En

2AOEO OE

E1 2 d

d


145/308

EO

EH

5 m/s2aE,n

(Tr)

aE,t(Tr)

aE(Cor) aE2

(Ass)

5 m/s2 aE,n(Tr)

aE,t(Tr)

aE(Cor) aE2

(Ass)

aE1(Ass)

aEt(Rel)

a(Ass)E1

d =16.7k

|D| = |

a(EASS)

|EH =

E


146/308

R1

aP = aD + d (P D) + d (d (P D))

aPt = aD + PD (P D) = d (D H) + PD (P D) = d (P H)

apt = d (P H) =

i j k0 0 d

P Hsin P Hcos 0

=

= dP Hcos i dP Hsin j = 13.02i + 7.52j

aPtx = 13.02 m/s2

aPty = 7.52 m/s2

|aPt| =

a2Ptx + a2Pty = 15.04 m/s2


147/308

(aPt) = arctan

aPtyaPtx

= 30

aPt

E D

P

20 ms

aP

aE


148/308


149/308

12

B

DE

h

R

Rp

va

L/4

L/4

L/4

L/4

= 10


150/308

E B

h = 5 m R = 0.8 m L = 4 m Rp = 0.4 m

|a(t)| =

2 m/s2 0 t < 1,

0 m/s2 t 1.

t = 0.5 s

vEB t = 0.5 s

aEBt = 0.5 s


151/308

B

D

E

v =0.0ms

t =0.0s

=-0.0

t = 0s

B

D

E

v =0.8ms

t =0.4s

=-11.5

t = 0.4s

B

D

E

v =1.6ms

t =0.8s

=-45.8

t = 0.8s

B

D

E

v =2.0ms

t =1.0s

=-71.6

t = 1.0s

B

D

E

v =2.0ms

t =1.4s

=-128.9

t = 1.4s

B

D

E

v =2.0ms

t =1.8s

=-186.2

t = 1.8s


152/308

B E

|v(t = 0)| = 0

|v (t)| =t

0

|a (t)| dt

|v(t)| =

2t m/s 0 t < 1,

2 m/s t 1.

|s(t = 0)| = 0

|s (t)| =t

0

|v (t)| dt


153/308

|s(t)| =

t2 m 0 t < 1,

1 + 2(t 1) m t 1.

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

Tempo [s]

Accelerazione [ms2

]

Velocit [ms1

]

Spostamento [m]

x e y mH t = 0 s

a +b + c = d + e

aei + bei + cei = dei + eei


154/308

B

DE

a d

cb

e

a a = 0

b b = h = 90

c c

d d = 0

e e = Rp

= = 0 = 2

a + ib + cei = d + eei

d a L/4

s = sR


155/308

ib + cei =L

4+ eei

sR

c cos = L

4+ e cos

sR

b + c sin = e sin

sR

c

c =L4 + e cos

sR

cos

= arctan

e sin

sR

bL4

+ e cos s

R

s(t = 0.5s) = 0.25 m = 65.07

c = 5.64 m

vEBc

cei+ cei(+2 ) = eei(+

2 )

= sR

= vR

(0.5) = 1.25 rad/s


156/308

c cos csin = e sin c sin + ccos = e cos

c

cos c sin sin c cos

c

=

e sin e cos

c

=

0.366 m/s

0.060 rad/s

c

vecvBEvecc vecvBE = ce

i

Ex1O1y1 O1 B

BE


157/308

B

DE

h

R

Rp

v a

L/4

L/4

L/4

L/4

E

v(Ass)E = v

(Rel)E + v

(Tr)E

v(Ass)E E

E

E|D| = |v|R

v(Ass)E = vD + D (E D) = D (E P)

v(Rel)E EO1 B E

v(Tr)

E A

v(Tr)E = vB + BE

(E B)


158/308

v(Ass)E1

vD xv(Ass)E2

D (E D) EDv(Rel)E BE

v(Tr)E1

vB xv(Tr)E2

BE (E B) BE

vD vB

v

v(Ass)E2

= v(Rel)E + v

(Tr)E2

E

v(Ass)E2

v(Tr)E2

| BE| = |v (tr)E2

|BE = 0.060 rads

v(Rel)E

|vBE| = |v(Rel)E | = 0.366ms


159/308

BE

BE

v(Ass)E1

0.1ms

v(Tr)E2

v(Rel)E

v(Ass)E2

0.1ms

cei+ 2cei(+2 ) + cei(+

2 ) c2ei = eei(+2 ) e2ei

= aR(0.5) = 2.5 rad/s2


160/308

c cos 2csin csin c2 cos = e sin e2 cos c sin + 2ccos + ccos c2 sin = e cos e2 sin

cos sin sin cos

c

=

e sin e2 cos + 2csin + c2 cos e cos e2 sin 2ccos + c2 sin

c

=

0.32 m/s2

0.19 rad/s2

c

E x1O1y1

a(Ass)E = a

(Rel)E + a

(Tr)E + a

(Cor)E

a(Ass)

E E

ED a

(Ass)E1

=

aD = a = D (D P)


161/308

E D EDa(Ass)En2

= DD(E D)ED a

(Ass)Et2

= D (E D)

a(Rel)

E EE

BE

a(Tr)EE

a(Tr)E1

= a a(Tr)En2

=

BE BE (E B) a(Tr)Et2 =BE (E B)

a(Co)EE a

(Cor)E = 2BEv(Rel)E

a(Ass)E1

aD xa(Ass)En2

D D (E D) EDa(Ass)Et2

D (E D) EDa(Rel)E BE

a(Tr)E1

aB xa(Tr)En2

BE BE (E B) BE

a(Tr)E2 BE (E B) BE

a(Cor)E 2BE v(Rel)E BE


162/308

aD aB

a

a(Ass)E2

= a(Rel)E + a

(Tr)E2

a(Ass)Et2

a(Ass)En2

a(Cor)E

a(Tr)En2

BE

BE

0.2ms2


163/308

a(Ass)Et2a

(Ass)En2

a(Tr)En2

a(Tr)Et2

a(Rel)E

a(Cor)E0.2 m

s2

v

(Tr)

Et2

| BE| = |a (tr)

Et2|

BE = 0.19 rads2a(Rel)E

|aBE| = |a(Rel)E | = 0.32ms2


164/308


165/308


166/308


167/308

13

a = 0.3m l = 0.5m (t) =At2 + Bt + C A = 0.03 rads2 B = 0.04

rads C = 0.06rad

m = 2kg JG = 0.015

G

l

m,a

C

O

i

k

j

t = 2s

C

O


168/308

x

ij

k

Ga

Ld = (a2 )2 + l2G

+ =

|vG| = |d| = 0, 083ms

OG

|at,G

|=

|d

|= 0, 031

m

s2

|an,G| = |2d| = 0, 013 ms2


169/308

G

l

m,a

OC.I.R.

VG

.

at,Gan,G

G

l

m,a

O

C.I.R.

.

.

d d

X

Y

X

Y

EcEST IN T

EST + INT =dEc

dt

EST OC


170/308

G

l

m,a

C

OC.I.R.

.

d

H

V

mg

mdmd

. 2..

JG..

G

l

m,a

C

OC.I.R.

.

d

H

V

mg md

.

JG.

O

Ec =1

2mv2G +

1

2JG

2

Ec =1

2md22 +

1

2JG

2

dEcdt

= (md2 + JG)

EST

EST = (Ck ) (k ) + (mgj ) (d sin()j )

EST = C + mgd sin()

INT = 0 C

C = JG + md2 mgd sin() = 5, 32Nm


171/308

dQdt

=R

d 0dt

+ v0 Q = M0Q

R

0O M0

OG

O

G = (

r G

r O)

QG + JG

QG = m

v G = md = md cos()

i md sin()j

O

O = (d sin()

i + d cos()

j ) (md cos()i md sin()j ) JG

k

= md2 sin2()k md2 cos2()k JGk

= md2

(sin

2

() + cos

2

())k JGk

= md2k JGk

d Gdt

= md2k JGk

v 0 QG = 0v 0 = 0 O

O


172/308

MO = Ck + (d sin()i + d cos()j ) (mgj )

= Ck (d sin()mg)k

md2k JGk = Ck d sin()mgk


O

MEST0 +

CIN + (G O) FIN = 0

CIN =

JGi

FIN =

FIN


JG =ma2

12 = 0, 015kgm2


173/308

H V

x yx

Qx = m

v xG = md cos()i

Qxdt

= md cos()i md2 sin()i

x

Rx = Hi

H = md cos() md

2

sin() = 0, 039Ny

Q y = m

v yG = md sin()j

Qydt

= md sin()j md2 cos()jy

R y = mgj

V = mg md sin() md2 cos() = 19, 56N


174/308

O

Fsistx = 0Fsisty = 0

H = Fi,n sin() + Fi,t cos()V = Fi,n cos() Fi,t sin() + mg

Fi,n = md2 Fi,t = md

H = md cos() md2 sin() = 0, 039NV = mg md sin() md2 cos() = 19, 56N


175/308

14

C(t) = At(0) = 0

(0) = 0 m = 5kg R = 0.1mA = 0.008 t = 1s

(t)

H V

G

HX

Y

C

m,R

O

i

k

j


176/308

k

dEcdt

= EST + IN T

Ec EST

IN T

H

H

Ec =1

2mv2G +

1

2JG

2


177/308

vG = R

Ec =1

2

mR2 + JG

JH

2 =1

2JH

2 =1

2

3

2mR2

2

v G = RiH

ESTA =Ck

k

+mgj

v G = C

ESTR = HH

i

v H +

VH

j

v H = 0

i

k

j

G

H

X

Y

m,R

O

.

VG

aG

,. .

3

2mR2 = C = (t) = 2

3

C

mR2=

2

3

A

mR2t = 0.1

rad

s2

C1 C2C1 C2

(t) =1

2

2

3

A

mR2

t2 + C1 = 0.05

rad

s


178/308

(t) =1

6

2

3

A

mR2

t3 + C1t + C2

(0) = 0 = C1 = 0 (0) = 0 = C2 = 0

(t) =1

9

A

mR2t3 = 0.017rad

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

Andamento di nel tempo

tempo(s)

(rad)

dQdt

=R

d 0dt +

v 0 Q = M0Q

R

0O M0O


179/308

H

H = IHk = 3

2mR2

k

v 0 Q H

MH = Ck

32

mR2k = Ck

=2C

3mR2=

2

3

A

mR2t

OR in +

R +

R = 0

Min + (G O) R in + MO + MO = 0

R in = ma G = mRiMin = IG = 12mR2

k


180/308

i

k

j

G

m,R

JGC..

HH

VH

maG

mg

H

H

mRi + HH

i + (VH

mg)

j = 0

12mR

2k + Rj (mRi ) Ck = 0


181/308

Q = mvG

i = mR

i

R = mgj R = HHi + VHj

mRi = HH

i + (VH mg)j

x yHH = mR = 0.01N

VH = mg = 49N

H

12

ik

j

G

m,R

JGC

.

HH

VH

mVG

mg

H


182/308

x y

HH VH C


183/308

15

1

Mm

P1

G2 M2

M1,J1,R

O

A B

F

j

i

k

R1 = 0.2m M1 = 10kg J1 = 0.4kgm2(discononomogeneo)M2 = 3kg F = 30N = 45

AP1 = 0.3m AB = 1m Mm = 50Nm


184/308

(0) = 0rad(0) = 0rad/sO

i j x y

k

1 = kvB = vG = Ri

v0 = 0

1 = kaB = aG = Ri

a0 = 0


185/308

P1

G2 VG

O

,

A B

j

i

k

. ..

aG

VB

aB

EST + INT =dEc

dtEc

EST

IN T

IN T

OB

vB

EST = Mm 1 + F vB + M1g v0 + M2g vG


186/308

EST = Mm F R sin

vG g v0 = 0

O AB

Ec =1

2J1

21 +

1

2M2v

2G

Ecdt

=

J1 + M2R2

=Mm F

22

J1 + M2R2= 88

rad

s2

(t) = 88t rads(t) = 44t2rad

VO HO

dQdt

=R

d 0dt + v0 Q = M0Q R

0O MO


187/308

P1

G2 M2VG

O

J1

A B

j

i

k

.

F

M2g

M1g

H0

V0

VB

OBG1 G2

Q1 = 0 Q2 = M2v2 = M2v2i = M2Ri

d Q

dt= M2Ri

RATT x

RATT = F cos i = F

2

2i

x RRV

RRV = H0i

M2Ri = F

2

2i + H0i


188/308

H0 = M2R + F

2

2

B

B = J1k

Bdt

= J1k

vB Q2i

B

MATT0 = (M1g)j (B P1)i + (M2g)j (B G)i Mmk

MRV0 = HOi Rj + V0j (B P1)i

J1 = M1g (B P1) + M2g (B G) Mm + H0R V0 (B P1)HO VO

HO = 74N

VO = 83.4N


189/308

P1

G2M2aG

O

J1

A B

j

i

k

..

F

M2g

M1g

H0

V0

VB

Fsistx = 0 H0 M2R F22 = 0

MsistB = 0 M1g (B P1) + M2g (B G) Mm + H0R+V0 (B P1) J1 = 0

HO = 74N

VO = 83.4N


190/308


191/308

16

DR

m Fb(t)

O

R

r

D

C

B

b(t)

c34c

e

a

m

J

F

C

R = 6 m r = 1 m a =

3 m

c = 2 m e = 2 m C = 20 Nm

m = 100 kg J = 1 kgm2


192/308

m b(t) = 2+sin(2t)

t = 0, 7 s

bt = 0, 7 s

bt = 0, 7 s

F

D

O

D

C

B

a

b

c

e

d

aei + bei + cei = dei + eei


193/308

a a = 0

b b = 2c c = 2d R r e e

a ib ic = dei

+ eei

a = d cos + e cos b c = d sin + e sin

= arccosa d cos

e

b c = d sin + e

1 cos2

= d sin + e

e2 (a d cos )2

e2

0 arcsin b+c

d= 0, 6


194/308

a2 + b2 + c2 + d2 e2 + 2bc

K

= (2bd + 2dc)

A

sin + 2da

B

cos

K = A sin + B cos cos2 + sin2 = 1

sin

cos

A2 + B2

A2 cos2 2BK

A2cos +

K2 A2

A2= 0

cos 1,2 = 0, 742cos 3,4 = 0, 196

1 0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

cos

sin

cos1,2

=0.742

cos1,2

=0.196

1 = 42, 11 3 = 78, 68


195/308

O Ab(t)

C

B

D

1 = 42, 11

O Ab(t)

C

B

D

3 = 78, 68

=

78, 68

67, 95

ib = dei(+2 ) + eei(+2 )

0 = dsin e sin b = dcos + e cos

b

=b dcos

e cos

dsin eb dcos e cos

sin = 0


196/308

b

e

d

0, 5ms

O

C

B

D

b

e

d

= tan d sin d tan cos b = (b) b

= 0, 65 rad/s = 1, 73 rad/s

DC ib

dei(+2 )

C eei(+2 )

D

|vd| = (R r) = r

= Rrr

= tan d sin d cos tan

R rr

b = R rr

(b) b

b b(t = 0, 7 s) = 2 cos(2t) =1, 94 m/s


197/308

t = 0, 7s = 3, 27 rad/s

ib = dei(+2 ) d2ei + eei(+2 ) e2ei

0 = dsin d2 cos e sin e2 cos b = dcos d2 sin + e cos e2 sin

b

=b dcos + d2 sin + e2 sin

e cos

=b tan d2 (cos + tan sin ) e2 (tan sin + cos )

d sin

d tan cos

= 11, 12 rad/s2 = 30, 86 rad/s2

b [b]

= (b)

bb2 + (b) b

b b(t = 0, 7 s) = 42 sin(2t) =37, 54 m/s2


198/308

b

dd2

e

e210 m

s2

O Ab(t)

C

B

D

b

d

d2

ee2

DC ib

dei(+

2 )d2ei e2ei

eei(+2 ) C

= R rr

= R rr

(b)

bb2 + (b) b

t = 0, 7s 55, 63 rad/s2

F

Fb C = mbb + J

F =

CR rr + J

R rr

2 (b)

bb2 + (b) b

(b)

b

+mb


199/308

878N

DD F m

DC

O

D

C

B

a

b

c

e

d

DC

DC

V = F mb = ...

HDC Q


200/308


201/308

17

a = b = 1m d = 2m e = 0.7m f = 2m Acilindro = 0.5dm2

M1 = 20kg M2 = 30kg J1 = 10kgm2

c

F

a

f

b

e

d

A

B

C

DP

M2

G M1,J1

j

i

k


202/308

c =

3m

BD CP

CP

Q = 9m3

h

A

M2

D

BD BC + CD

a + b = c

aei + bei = cei

a cos + b cos = c cos

a sin + b sin = c sin


203/308

c

F

a

f

be

d

A

B

C

D

P M2

GM1,J1

j

i

k

a

b

c

C

D

B


204/308

a = (C B) a = 1m 90b = (D C) b = 1m

c = (B H) d 3m

= 2

b cos = c cos

a + b sin = c sin

cos2 + sin2 = 1

c2 = (b cos )2 + (a + b sin )2

c2 = b2 + a2 + 2ab sin

sin =c2 b2 a2

2ab = arcsin c

2 b2 a22ab

= arcsin1

2= 30

cos =b

ccos =

13

3

2=

1

2 = arccos 1

2= 60


205/308

bsin = c cos csin bcos = c sin + ccos

3 c c

Q =dVcilindro

dtVcilindro

Vcilindro = Acilindrohcilindro

Q = Acilindrodhcilindro

dt= Acilindro c

c =9

3600

0.5 102m3

s

m2= 0.5

m

s

2 b sin c sin b cos c cos

=

c cos c sin

3

3

3

2

3

2

=

3

4+

3

4 = 0.5rad

s


206/308

3 = 12

= 1rads

b sin c sin b cos c cos

=

b2 cos 2csin c2 cos b2 sin + 2ccos c2 sin

=

3

2

rad

s2

=

3

4

rad

s2

CP M2

(G C) = e ei

(P

C) = f ei

vG = ieei ; aG = iee

i e2ei


207/308

vP = ifei ; aP = ife

i f2ei

CP

C

aG,t = ieei ; aG,n = e2ei

aP,t = ifei ; aP,n = f2ei

MA

VA

HA

M1g

M2gF

aP,t

vP

aP,n

aG,t

aG,n

vG

..

.


208/308

R

FgasFgas

VR

VT

T

T R

Wgas = Fgas,T vT + Fgas,R vR = Fgas vT+ Fgas vR = Fgas (vT vR) = Fgas vrel

Fgas = pgasAcilindro

Fin,P,n = M2aP,n

Fin,P,t = M2aP,t

Fin,G,n = M1aG,n


209/308

MA

VA

HA

M1g

M2g

Fin G,t

Fin G,n

F

Fin P,t

Fin P,n

vP

vG

Cin

Fin,G,t = M1aG,t

Cin = J1

Fgas vrel,cilindro + ( F + M2g) vP + M1g vG++ Fin,G,n vG + Fin,G,t vG + Fin,P,n vP++ Fin,P,t vP + Cin = 0

Fgas vrel,cilindro + ( F + M2g) vP + M1g vG+

M1aG

vG

M2aP

vP

J1

= 0

Fgasvrel,cilindro (F + M2g)cos vP M1g cos vG == M1aG,tvG + M2aP,tvP + J1


210/308

Fgasc (F + M2g)cos f M1g cos e M1ff+ M2ee J1 = 0

Fgas =

c[(F + M2g)cos f + M1g cos e+

+ M1f2+ M2e

2+ J1]

Fgas = 4964 N pgas = FgasAcilindro

=

=4964

0.5 102N

m2= 992800

N

m2= 9928 bar

MA

VA

HA

M1g

M2g

Fin G,t

Fin G,n

F

Fin P,t

Fin P,nCin


211/308

M1g

M2g

Fin G,t

Fin G,n

F

Fin P,t

Fin P,n

Cin

VC

HC

Fgas

CC

Fgas sin( )b Cin Fin,G,te Fin,P,tf+ M1ge cos M2gfcos F fcos = 0

Fgas sin(

)b

J1

M1e2

M2f

2+

M1ge cos M2gfcos F fcos = 0

Fgas = 4964 N pgas = FgasAcilindro

=

=4964

0.5 102N

m2= 992800

N

m2= 9928 bar

A

Fx = 0 HA + Fin,G,t sin + Fin,G,n cos +

+ Fin,P,t sin + Fin,P,n cos = 0


212/308

HA + M1aG,t sin + M1aG,n cos +

M2aP,t sin + M1aP,n cos = 0

Fy = 0 VA Fin,G,t cos + Fin,G,n sin M1g+

Fin,P,t cos + Fin,P,n sin M2g F = 0

VA M1aG,t cos + M1aG,n sin M1g+ M2aP,t cos + M1aP,n sin M2g F = 0

MC = 0 HAd + MA J1 M1ge cos M2gf cos +

Fin,G,te Fin,P,tf F fcos = 0

HAd + MA J1 M1ge cos M2gfcos + M1e2 M2f2 F fcos = 0

HA = 55.53 N

VA = 1509 N

MA = 1544

3 Nm


213/308

18

A

B

D

C G

M,J

Cm

h

F

45

30

P

,.

j

i

k

AB = 0.5 m BC = 0.183 m CD = 0.353 mAD = 0.5 m h = CG = 0.353 m M = 10 kgJ = 0.1 Kg m2


214/308

= 37.32rad/s =110rad/s2 BC

AD CD

AD CD

G PF

Cm BCF = 50 N

A B

i

j

k

a +b + c = d

aei + bei + cei = dei


215/308

A

B

D

C

a

b

c

d

j

i

k

a a

b b

c c

d d

a cos + b cos + c cos = d cos a sin + b sin + c sin = d sin

iaei + ibei + icei = idei

aei(+2 ) + bei(+

2 ) + cei(+

2 ) = dei(+

2 )

a sin bsin csin = dsin a cos + bcos + ccos = dcos


216/308

= = 37.32 rad/s

0.183 37.32 ei(90) + 0.535 ei(405) = 0.5 ei(120)

0.353sin(315) = 0.5sin(30)0.183 37.32 + 0.535cos(315) = 0.5cos(30)

= 10 rads

= 10rad

s

+

+

b

c

d

+

.

.

.

__2

__2

__2


217/308

ibei b2ei + cei ic2ei = dei id2ei

bei(+2 ) + b2ei(+) + cei(+

2 ) + c2ei(+) = dei(+

2 ) + d2ei(+)

bsin b2 cos csin c2 cos = dsin d2 cos bcos b2 sin + ccos c2 sin = dcos d2 sin

= = 110rad/s2

0.353 sin(315) 0.5 sin(30)

0.353 cos(315) 0.5 cos(30) =0.183 37.322 + 0.353 (10)2 cos(315) 0.5 cos(30)0.183 110 + 0.353 (10)2 sin(315) 0.5 sin(30)

= 449449 rad/s2 = 10497 rad/s2


218/308

G PG

(G P) = (C B) + (G C)

B G

(C - B) (G - C)

(G - B)45

j

i

k

|G C| =

GC

2

2+

h

2

2= 0.25 m

(G P) = (C B) + (G C) = bei + |G C|ei(+45)

d

dt(G P) = vG = ibei + i|G C|ei(+45)

G

vG,x = 0.25sin(315 + 45) = 0 m/svG,y = 0.183+ 0.25cos(315

+ 45) = 4.33 m/s

vG =

v2G,x + v2G,y =

02 + 4.332 = 4.33 m/s

vel = 90


219/308

vel

b

GC .

.

vG

d

dtvG = aG = ibe

ib2ei+i|GC|ei(+45)|GC|2ei(+45)

aG,x = 0.1832 0.25sin(315 + 45)+0.252 cos(315 + 45) = 280 m/s2

aG,y = 0.183+ 0.25cos(315 + 45)+

0.252 sin(315 + 45) = 144.45 m/s2

aG = a2G,x + a2G,y =

2802 + 144.452 = 315 m/s2

acc = 152.7

P

(P A) = (D A) + (P D) = dei + |P D|ei(270)

|P

D

|=

h

2= 0.177 m

d

dt(P A) = vP = idei + i|P D|ei(270)


220/308

A

D

P

(P - A)

(D - A)

(P - D)

45

PD

d

.

.

vP

P

vP,x = 0.5sin(30) 0.177sin(315 270) = 1.25 m/s

vP,y = 0.5cos(30) + 0.177cos(315 270) = 3 m/s

vP =

v2P,x + v2P,y =

(1.25)2 + 32 = 3.25 m/s

vel = 112.6


221/308

G M PF

A

B

D

C

G

Cm

F

PMg

vG

aG vP

VB

HB

HA

VA

.

..

Cm + Mg vG + F vP + Fin vG + Cin = 0

Fin = MaGCin = J


222/308

A

B

D

C

G

Cm

F

PMg

vG

Fin

vP

VB

HB

HA

VA

Cin

Cm MgvG + F vP cos(135 112.6) ++MaGvG cos(152.7

90) + J = 0

Cm =1

(MgvG F vP cos(22.4) +

MaGvG cos(62.7) J) = 20.15 Nm


223/308

4 A B

A

B

D

C

G

Cm

F

P

Mg

Fin

VB

HB

HA

VA

Cin

A B

MA = 0

HBBA + Cm (Mg + Fin,y) AD cos(30) Fin,xAB Cin ++FAD cos(45 30

) +h

2 = 0HB = 2

Cm (Mg + MaG,y)

3

4 MaG,y

2 J+ F

0.96

2+ 0.177

=

= 4130 N


224/308

MB = 0

HABA + Cm

(Mg + Fin,y)BC + CG

2

2 Cin + FBCcos(45) +h

2HA = 2

Cm (Mg + MaG,y) (0.183 + 0.25) J+ F

0.183

2

2+

h

2

=

= 1364 N

Fy = 0 VA + VB Mg Fin,y + F sin(45) = 0

A

D

C

G

F

PMg

Fin

VC

HC

HA

VA

Cin


225/308

19

C R MF

m a b

Y

Xmg

Mg

F

y

x

M,RP

G

C

a

b

i

k

j

O


226/308

x(t)

F

P

bP

YX

x(t)

k


227/308

y

x

x

vG

aG

S

C PX Y

vC = k (C S) = xi

vP = k (P S) = yj


228/308

X

P

C

i

k

j

OvC

S

vP

x

x y

k (R cos j) = xik (Rseni) = yj

=x

R cos

y = R sin = R xR cos

sin = x tan

Gx

=x

R cos y = x tan


229/308

G C vASSCvTRC

vRELC C

vASSC = vTRC + v

RELC x = yei

2 + vRELei

vASSC

x = vREL cos 0 = y + vREL sin

vREL =x

cos =vREL

R =x

R cos y = vREL sin = x tan

2

y

x

x.

y

.

Vrel

G


230/308

x = y + a + b + c

Y

X

G

CO

a

y

b

x

c

x = yei2 + a + bei + cei(

32+)

x y bC G C G

a c


231/308

Y

X

,

G

CO

. ..

vCaC

vG

aG

dEcdt

= EST + INT

Ec EST

INT

Ec = Ec,disco + Tc,cuneo =1

2Mv2C+

1

2JG

2 +1

2mv2G

EST IN T


232/308

EST = Fi vC Mgj vC mgj vG= Fi xi Mgj xi mgj yj

IN T = 0

Y

Xmg

Mg

F

x

M,RP

G

C

a

b

i

k

j

O

y.

.

H1

H2

V

x.

dEc

dt= Mxx + JG + myy

Mxx + JG + myy = Fx mgyMxx +

MR2

2 + myy = Fx mgy

x

Mx +1

2

MR2x

R2

cos2

+ mx tan2 = F + mg tan

F

F =

M +

M

2cos2 + m tan2

x mg tan


233/308

G

Y

Xmg

Mg

F

x

M,RP

G

C

a

b

i

k

j

O

.

J.

H1

H2

V

my

mx.

yV

d Qydt

= R(yES T)

Qy = myj

Vj Mgj mgj

myj = Vj Mgj mgj


234/308

V = my + Mg + mg

x y

V = mx tan + Mg + mgF

P

V

ddiscoPdt

= Q(disco) vP + M(EST)PP

discoP = J(disco)P

k =3

2MR2k

Q(disco) = Mxi

PyP = R cos

vP = xi

P

P

M(EST)P = (C P) (V Mg)j + (C P) Fi =

= (R sin i R cos j) (V Mg)j ++(R sin i R cos j) Fi

32MR2k = F R cos k + (V Mg)R sin k

x F


235/308

F =1

R cos

3

2MR2

x

R cos (V Mg)R sin

=

=3M

2cos2 x

(V

Mg)tan =

=3M

2cos2 x (mx tan + mg)tan =

=3M

2cos2 x + mx tan2 mg tan

Y

Xmg

Mg

F

x

M,RP

G

C

a

b

i

k

j

O

.

H1

H2

V

my.

mx..

J..

Rin = maG MaC = myj Mxi = mx tan j MxiMin = JG = 12MR2k


236/308

V

(V

Mg)j

mgj

myj = 0

V = Mg + mg

mx tan

F

P

V

P

C

i

k

j

O

NP

TPF

MgV

Mx

J..

(C P) (F Mx)i + (C P) (V Mg)j JGk = 0F R cos MxR cos + V R sin MgR sin JG = 0

F = Mx (Mg + mg mx tan )tan + Mg tan +

= +12MR2 xR cos 1R cos F =

M +

M

2cos2 + m tan2

x mg tan


237/308

H1 H2

B

QC,x = Mxi

R(EST)x = H1i + H1i + Fi

Mxi = H1i + H1i + Fi

B

dBdt

= Q vB + M(EST)B

B = JCk + (C B) Mxi + (G B) myj ==

MR2

2k + x

i yBj Mxi +

a

3i b

3j my

j =

=

MR2

2 + yBMx +

a

3my

k =

=

MR2

2 + (yB x tan ) Mx + a

3my

k


238/308

dBdt

=MR2

2k + yBMxk x tan Mxk x tan Mxk + a

3myk

Q = myj + Mxi

B

vB = yj

B

M

(EST)

B = (C B) Fi + (C B) (V Mg)j ++(G B) (mgj) + (D B) H2i

MR2

2k + yBMxk x tan Mxk x tan Mxk + a

3myk =

=

myj + Mxi

yj +

xi yBj

Fi + Vj Mgj

+

+a

3

i

b

3

j mgj +

b

2

j H2i

MR2

2k + (yB x tan ) Mxk + a

3myk =

= x(V Mg)k + yBFk a3

mgk +b

2H2k

F Mx + H1 + H2 = 0V x Mgx + F yB mgxG + H2 b2 MR

2

2 MxyB myxG = 0


239/308

H1 H2

Fi Mxi + H1i + H2i = 0(C B) (V Mg)j + (C B) (F Mx)i++(G B) (mg + my)j + (D B) H2i JGk = 0

F Mx + H1 + H2 = 0(V Mg)x + (F Mx)yB (mg + my)xG + H2 b2 JG = 0

xG yGa

3

x

Y

X

x

M,RP

G

C

a

b/2

R

xP

xG

2/3 a

A O

b/2

BH1

H2

yB

yD

D

}R CosR Sin tg}

}

R Sin

R

C

yP = R cos

xP = x R sin yA = yP xP tan


240/308

yA

yB = yA + b = R(cos + sin tan )

x tan + b =

= yB x tan xG =

a

3

H2H1

H2 =

2b

(Mx F) yB + (mg + my) xG + (Mg V) x + JG

H1 = Mx F H2

x =a

2+ R sin

xP =a

2

yB = R cos + R sin tan a

2+ R sin

tan + b

b

a= tan

b

2= R cos

yB = R cos a2

tan + b = b

H2 =

2b

(Mx F) b + (mg + my) a3 + (Mg V)

a2 + R sin

+ MR

2

2

H1 = Mx F H2


241/308

H2 = 2b M

2cos2 + m tan2

xb + mgb tan + (mg mx tan ) a

3

+

+2

b

(mg + mx tan )

a2

+ R sin

+MR2

2

x

R cos

H2 = Mxcos2

2mx tan2 + 2mg tan + 23

mgtan

mx

+

+ (mg + mx tan )

1

tan + tan

+

Mx

2cos2

H2 =

M

2cos2 + m tan2 m

3

x + mg

tan 1

3

1

tan

H1 = Mx

M +M

2cos2 + m tan2

x + mg tan +

M

2cos2 + m tan2 m

3 x mg

tan 1

3

1

tan H1 = m

3x +

1

3

mg

tan

F


242/308

V


243/308


244/308


245/308


246/308


247/308


248/308

R 1, J 1 R 2, J 2

M

T

BB 1

fd

J m

, A

A

A = 100Nm R1 = R2 = 0.5m = 20

B = 0.1Nms2/rad2 d = r = 0.95 = 45

Jm = 6.25kgm2 m = 70kg

J1 = 2.5kgm2 fd = 0.2

J2 = 3.75kgm2 = 1/5

C


249/308

C

Wsc

WToutWTin

M

T

v av a

ma

mg sin mg sin

ma

2

J 22

J 11

1C

C

m

Cm

J 2m

Wsc

C2 2mg sin() v Wu

J111 J222 2mav WINu

= 0

C

C

2


250/308


251/308

Wu = 2mg sin() v

WIN = J111 J222 Jmmm 2mva = WINu + WINm1 2

m v

v

1 = v/R1

2 = v/R2

m = 2/ = v/ ( R2)

2mg sin() v + dCm v R2

dJm va2R2

2

+

J1R21

+J2R22

+ 2m

va = 0

Cm0 = A

a =Cm0d/ ( R2) 2mg sin()

J1/R21 + J2/R22 + dJm/(

2R22) + 2m= 0, 63 m/s2

WIN = 0


252/308

Cm

Cm = R2D 2mg sin() = 49, 44Nm

J m J u

Wu

mC m u

Wp

Wm

C u

W Tin WTout

vCm = A B2

m

m =

A Cm

B= 22, 48 rad/s

v

v = R2m = 2, 25 m/s


253/308

m

Cm

m

Cm

S

ma

MN

mg

v a

S

Ssin() ma mg sin() = 0


254/308

S

MN

mg

T

i

v a = 0

S =1

sin()[ma + mg sin()] = 395 N

Cm

Wu

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Eserciziario Fondamenti di meccanica teorica ed applicata