ANÁLISE DINÂMICA DE TRANSMISSÕES POR CORRENTE UTILIZANDO UMA ABORDAGEM MULTICORPO

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ANÁLISE DINÂMICA DE TRANSMISSÕES POR CORRENTE

UTILIZANDO UMA ABORDAGEM MULTICORPO

CÂNDIDA PEREIRA MALÇA

ISEC/INSTITUTO POLITÉCNICO DE COIMBRA

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TRANSMISSÕES MECÂNICAS

TRANSMISSÕES POR CORRENTE

FORMULAÇÃO MULTICORPO

ANÁLISE DINÂMICA

ANÁLISE DINÂMICA DE TRANSMISSÕES POR CORRENTE

UTILIZANDO UMA ABORDAGEM MULTICORPO

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TRANSMISSÕES MECÂNICAS

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Zoom1

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Pitch Inner Link Outer Link Pin Zoom1: Bushing Roller Clearance Pin/Bushing Clearance Bushing/Roller

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Pitch η_sr η_i θ·_i R_s ξ_sr ξ_i θ_i θ_s α X Y r_i r rr_cr sr_cr Pitch η_sr η_i θ·_i θ·_i R_s ξ_sr ξ_i θ_i θ_s α X Y r_i r r_i r rr_cr rr_cr sr_cr sr_cr

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DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS Body 1 Body 1 Revolute joint Body 2 Body i Body 3 Multi-revolute joint with clearance Actuator Spherical joint

Spring/Damper Applied forces

Flexible body Translational joint

Contact bodies Body n Gravitational acceleration field Spring Body j Lubricated joint Ground body Applied Torque Revolute joint with clearance Body K

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MECANISMO BIELA - MANIVELA

DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS

Clearance

1

η

1

ξ

4

η

3

η

2

η

2

ξ

3

ξ

4

ξ

X

Y

1

2

3

4 Clearance

1

η

1

ξ

4

η

3

η

2

η

2

ξ

3

ξ

4

ξ

X

Y

1

2

3

4

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Pankoke et al, 1998 Silva et al, 1997

MODELAÇÃO DO MOVIMENTO HUMANO

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III XII 5 V VI X VII II IV IX XII VIII XI XIV 1 8 2 4 7 11 9 3 6 12 10 13 I

Body no. 50% Human Male Li [m] 50% Human Male Mass [Kg] * I 0.260 14.2 II 0.250 24.9 III 0.230 4.24 IV 0.320 1.99 V 0.260 1.84 VI 0.320 1.99 VII 0.260 1.84 VIII 0.410 9.84 IX 0.385 4.81 X 0.410 9.84 XI 0.385 4.81 XII 0.160 1.06 XIII 0.053 1.62** XIV 0.053 1.62 ** L3-5 (a) 0.375 - L6-7 (a) 0.188 - L1-2 (b) 0.199 - L1-3 (b) 0.155 - MODELO BIOMECÂNICO

MASSAS E DIMENSÕES DOS CORPOS RÍGIDOS

Source: Silva et al. (1997)

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











−

=

























Φ

γ

g

λ

q

0 Φ

Φ

M

q

2 T

2 α &

β

&

CLEARANCE REVOLUTE JOINTS

PERFECT KINEMATIC JOINTS

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Read input data Is t >tend ? t t t ==== ++++ ∆ Yes No STOP START Evaluate g Generalized forces, Φ q Jacobian matrix, M

System mass matrix,

γ Φ Constraint functions, , 0 q q ==== t 0 = == = t t = = = = & & q0 q t

Solve linear equations of motion for and_q&_& _λ

T  _{    } =               q q M Φ q g Φ 0 λ γ && Form the auxiliary vector T T T ] [q y& ==== & t q&& Integrate the auxiliary vector ∆ y ==== + + + + t t T T T ] [q q&

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DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS X Y η ηη η_i ξ ξξ ξ_i (i) O_i (j) η ηη η_j ξ ξξ ξ_j O_j P_i P_j Q_j Q_i

r

_P i

s

r

_P j

s

r

e

r

_P i

r

_P j

r

i

r

j

r

δ

r

n

r

t

1. ECCENTRICITY VECTOR P P j i

=

−

e

r

2. ECCENTRICITY T

e

= e e

3. PENETRATION (C - CLEARANCE)

δ

= −

e c

4. NORMAL AND TANGENT VECTORS

e

=

n

e

T y x

n





=

_

−

_

t

5 . CONTACT FORCE

(

)

(

)

* 2 3 4 ( )

0.49∆R + 0.1 E

1

1 ∆R

−









=





_

+

−

_









f

&

n

&

144424443

n n r

l

c

K

δ

( )

1 t

c c f

f d n

v

T T −

= −

f

v

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DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS 1 2 6 7 4 5 3 a* b* c* _c b a d* cc _d b_c* c_c* b_c o_c RIGHT SIDE LEFT SIDE

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IF AND

THERE IS CONTACT WITH THE SEATING CURVE

1 1 2

θ

2

θ

−

_o

≤

_e

≤

_o

(

)

0

e

R

t

R

r

δ

=

e

−

>

IF THERE IS NO CONTACT AT ALL

(

)

0

e

R

t

R

r

δ

=

e

−

≤

2R_r θ_e R_t c_c* c_c η η η η_st θ_o o_c ξ ξξ ξ_st

δ

r

_e cr

s

r

e

r

oc

s

r

_Q t Qr

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DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS Strand A Strand B (K) Strand C (i) (j) P_i Strand A Strand B (K) Strand C (i) (j) P_i

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(

)

1a 1b 1a 1b 1 × ϕ ≥ ϕ  =  + × ϕ < ϕ  strand n Pitch if L n Pitch if 1a 1b _ _ 1a 1b 1 2 + ϕ ≥ ϕ  =  + ϕ < ϕ  Pins in strand n if N n if i j P - P = ×n Pitch dp   =     n integer Pitch X (n+1)×Pitch Y n×Pitch r_j r_iP_{- r} j φ_1a φ_1b P_j* (position1b) r P_i R_j (i) (j) r r rr_iP P_j* (position1a) X (n+1)×Pitch Y n×Pitch r_j r_iP_{- r} j φ_1a φ_1b P_j* (position1b) r P_i P_i R_j (i) (j) r r rr_iP P_j* (position1a)

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DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS δ_1a δ_1b Pk (position1b) X α_j n×α_j (n+1)×α_j (j) ψ P_j Strand B Strand A Y δ_1a δ_1b P_{k (position1a)} r_j r r_jP r u_jP r r_jk r sr_k sr_P δ_1a δ_1b Pk (position1b) X α_j n×α_j (n+1)×α_j (j) ψ P_j Strand B Strand A Y δ_1a δ_1b P_{k (position1a)} r_j r r_jP r u_jP r r_jk r sr_k sr_P j n integer  _ψ  = _ _ α   T p k 2 j cos R ψ = s s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) / 4 3 / 4 2 5 / 4 2 2

arccos cos if cos α_{nd sin} arcsin sin if

arcsin sin if cos α_{nd sin} arccos cos if

arccos cos if cos α_{nd sin} arcsin sin ψ = ψ ψ ≥ 0 ψ ≥ 0 ψ = ψ ψ < π ψ = ψ ψ < 0 ψ ≥ 0 ψ = π − ψ ψ > π ψ = π − ψ ψ ≤ 0 ψ < 0 ψ = π − ψ ψ > π ψ = π − ψ ψ > 0 ψ < 0 ψ = π + ( ψ) if (ψ > π7 / 4) x y y x p k p k 2 j sin R ψ = s s - s s k k

=

j j

s

r - r

P P

=

j j

s

r - r

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DINÂMICA DE SISTEMAS DE CORPOS MÚLTIPLOS R_c β β β β _β (K) Strand C P_i X Y c ψ r P_{m (position 1a)} P_{m (position 1b)} (i) r_c r u_k P_{n (position 1a)} P_{n (position 1b)} r_m r r_n r R_c β β β β _β (K) Strand C P_i P_i X Y c ψ r P_{m (position 1a)} P_{m (position 1b)} (i) r_c r u_k P_{n (position 1a)} P_{n (position 1b)} r_m r r_n r