Guide de survie : Mécanique II

~xC=1
mΩZΩ
ρ(~x)~xdV
m٨ ~xC(t) = ~ Rext(t)
d
dt (~xC(t)~ω (t)) = ~
Mext
~xC(t)
d
dt (~xA~ω ) + mΩ(~xC−~xA)∧¨
~xA=~ Mext
~xA
~xB(t) = ~xA(t) + (t)(~pB−~pA)
T=
•~e ϕ
= (~e, ϕ) = cosϕ + (1 −cosϕ)~e ⊗~e +sinϕ ∧
kek= 1 ∧ ∧~c =
~e ∧~c∀~c
•(φ, θ, ψ)
(φ, θ, ψ) = (~e3, ψ) (~c1, θ) (
~
i3, φ)
~e3= (~c1, θ)
~
i3~c1= (
~
i3, φ)
~
i1
(~e, φ)
φ=arccos(tr −1
2)
∧=1
2sinφ (−T)
T=
Ω = ˙T
Ω
ω
Ω~c =ω∧~c
•~e0
ω(~e0,ϕ)= ˙ϕ~e0
ω(~e,ϕ)= ˙ϕ~e + (1 −cosϕ)~e ∧˙ ~e +sinϕ ˙ ~e
•
ω(φ,θ,ψ)=˙ φ
~
i3+˙ θ~c1+˙ ψ~e3
˙= Ω
~xB(t) = ~xA(t) + (t)(~pB−~pA)
˙
~xB=˙ ~xA+~ω ∧(~xB−~xA)
¨
~xB=¨ ~xA+˙ ~ω ∧(~xB−~xA) + ~ω ∧(~ω ∧(~xB−~xA))
¨
~xC
ω

~xA(t)
Ω
~xA(t) = ZΩ
(k~x −~xAk2−(~x −~xA)⊗(~x −~xA))ρ(~x)dV
~xA(t) = (t)~pA(t)T(t)
•~e0
~pC= (Ie−I⊥)~e0⊗~e0+I⊥
•
~pC=I
~pA=~pC+mΩ(k~pC−~pAk2−(~pC−~pA)⊗(~pC−~pA))
~xA(t)
~
Rext =ZΩ
~
fVdV +Z∂Ω
~
fsdS
~
Mext
~xA=ZΩ
(~x −~xA)∧~ fVdV +Z∂Ω (~x −~xA)∧~ fsdS
~
MB=~ MA+~ BA ∧~ R
~
R
•
•
~
MA= 0
Ω1Ω2
Ω0
Ω2
Ω1
Ω1
C1
1Ω0
Ω2λ(t)~e(t)
Ω1
Ω2
˙
~x(M, t) = ~ XΩ1\Ω0
|{z } Ω1\Ω0
+˙ λ(t)~e(t)
| {z }Ω2\Ω1
˙
~x(M, t) = ˙ ~xC1+~ω 1∧(~x −~xC1) + ˙ λ(t)~e(t)
Ω2ϕ(t)~e(t)
Ω1Ω2
Ω2
˙
~x(M, t) = ~ XΩ1\Ω0
| {z } Ω1\Ω0
+ ˙ϕ(t)~e ∧(~x −~xA)
| {z }
Ω2\Ω1
˙
~x(M, t) = ˙ ~xC1+~ω 1∧(~x −~xC1) + ˙ϕ(t)~e ∧(~x −~xA)
256
Ω1Ω2
Ω1

•
•
Ω0
λ, ϕ
•
•
h= + −6N
Ω0
Ω0
h= 0
h > 0
Ω0
•
h= 0
•
~xC(t) = PN
k=1 mk~xCk(t)
PN
k=1 mk
278

~p(s, χ1, χ2) = ~pG(s) + ~pΣ(χ1, χ2)
•
•
•
Σ(s)
~uG=uGe
|{z}
~e +~uGΣ
|{z}
•
Σ(s)
~
θ=θe
|{z}
~e +~ θΣ
|{z}
~u =~uG+~ θ∧~xΣ
= (u0
Ge +h~
θ0
Σ∧~xΣ, ~ei)~e ⊗~e + (~u0 GΣ−~
θΣ∧~e)⊗S~e +(θ0
e~e ∧
~xΣ)⊗S~e
θΣ=~e ∧~u0
GΣ
~u0
GΣ=θΣ∧~e
~u(s, χ1, χ2) = ~uG(s)+(θe(s)~e +~e ∧~u0
GΣ(s)) ∧~xΣ(χ1, χ2)
= (u0
Ge − h~u00 GΣ, ~xΣi)~e ⊗~e + (θ0 e~e ∧~xΣ)⊗S~e
σ=σee ~τ T
Σ
~τΣσΣ
σΣ= 0
σ~e =σee~e +~τΣ=σee~e +σχ1~eχ1+σχ2~eχ2
~
R(s, t) = ZΣ(s) σ~edS
~
R
~e