Lagrange Approach 1 Basilio Bona DAUIN Politecnico di Torino - - PowerPoint PPT Presentation

Lagrange Approach – 1

Basilio Bona

DAUIN – Politecnico di Torino

Semester 1, 2015-16

• B. Bona (DAUIN)

Lagrange-1 Semester 1, 2015-16 1 / 6

Initial steps

Find the k = 1, . . . , N rigid bodies composing the system and compute the n degrees-of-freedom. If necessary, define the various body frames Rk Define a set of complete and independent generalized coordinates qi(t), i = 1, . . . , n ≤ 6N Compute the position vectors of each center-of mass pk(q(t)), k = 1, . . . , N Compute the linear velocity vectors of each center-of mass vc,k(q(t), ˙ q(t)) and the angular velocities vectors ωk(q(t), ˙ q(t)) of each body

• B. Bona (DAUIN)

Lagrange-1 Semester 1, 2015-16 2 / 6

Energy computation

Compute the kinetic co-energy K∗

k of each k-th rigid body with mass

mk and inertia matrix (wrt the center-of-mass) Γk as K∗

k = 1

2mk vc,k2 + 1 2ωT

k Γkωk

Set the local gravity acceleration vector g and represent it in R0 Localize the Ne elastic energy storage elements and model them with “ideal springs” with elastic constants kℓ, ℓ = 1, . . . , Ne Compute the gravitational potential energy Pg,k and the elastic energy Pe,ℓ as Pg,k = −mkgTpk Pe,ℓ = 1 2kℓ e2 where e is the elongation (positive or negative) of the spring.

• B. Bona (DAUIN)

Lagrange-1 Semester 1, 2015-16 3 / 6

Lagrangian function

Compute the total energies K∗(q, ˙ q) =

N

• k=1

K∗

k

P(q) =

N

• k=1

Pg,k +

Ne

• ℓ=1

Pe,ℓ Compute the Lagrange function of the system L = K∗ − P Compute the generalized forces Fi Write the n Lagrange equations d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = Fi i = 1, . . . , n i.e., d dt ∂K∗ ∂ ˙ qi

• − ∂K∗

∂qi + ∂P ∂qi = Fi i = 1, . . . , n

• B. Bona (DAUIN)

Lagrange-1 Semester 1, 2015-16 4 / 6

Lagrange equations

If there are linear dissipative elements, model them with a linear dashpot, having friction coefficient βi. Compute the dissipative function Di = 1 2βi vf ,i2, where vf ,i is the velocity associated to the friction producing element. Upgrade the Lagrange equations as follows d dt ∂K∗ ∂ ˙ qi

• − ∂K∗

∂qi + ∂P ∂qi + ∂D ∂ ˙ qi = Fi i = 1, . . . , n What happens when nonlinear elastic or friction elements are present? One should directly introduce the resulting nonlinear elastic or friction forces in the Lagrange equations.

• B. Bona (DAUIN)

Lagrange-1 Semester 1, 2015-16 5 / 6

Advanced material

Part 2 will introduce advanced material

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Lagrange-1 Semester 1, 2015-16 6 / 6