5.1.3. Rigid Body#

Newton dynamical equations - strong form. Dynamical equations governing the motion of a rigid body, referred to its center of mass G read

{Q˙=ReΓ˙G=MGe ,

with momentum Q=mvG and angular momentum ΓG=IGω.

Weak form. Weak form of dynamical equations is derived with scalar multiplication of the strong form by an arbitrary test functions wt, wr

(5.4)#0=wt(mv˙GRe)+wr(Γ˙GMGe)wt,wr

Lagrange equations. Lagrange equations are derived from the weak form, with a proper choice of the weak test functions. The “translational part” is recasted after choosing

wt=rqk=vq˙k .

Following the same steps show to derive Lagrange equations for a point system, the translational part becomes

ddtKtrq˙kKtrqkUtrqk=Qktr ,

being Ktr=12m|vG|2 the contribution to kinetic energy of the velocity of the center of mass G deriving from the momentum equation, Utr the contribution to the potential energy U from the momentum equation, and Qktr the contribution to the generalized force from the momentum equation.

The “rotational part” is recasted after choosing

wr=θqk=ωq˙k

Angular velocity ω can be written w.r.t the inertial {e^i} or the material reference frame {E^i},

ω=ωie^i=σjE^j ,

and the inertia tensor as

IG=IijE^iE^j ,

being the components Iij constant.

0=ωq˙kddt(IGω)ωq˙kMGe=ddt(ωq˙kIGω)ddtωq˙kIGωθqkMGe

The first term becomes

ddt(ωq˙kIGω)=ddt(σaq˙kIabσb)=ddtq˙k(12ωIGω)=ddtKrotq˙k

The second term becomes

ddtθqkIGω=qkdθdtωIGω==ωqkIGω==qk(σaE^a)E^bIbcσc==σaqkE^aE^b=δabIbcσc+σaE^aqkE^b=0Ibcσc==qk(12σaIabσb)==qk(12ωIGω)=Krotqk .

The third term can be written as the sum of the derivative of a potential function and a generalized force,

θqkMGe=Urotqk+Qqkrot

The rotational part of the wak form becomes

ddtKrotq˙kKrotqkUrotqk=Qqkrot ,

being Krot=12ωIGω the contribution to kinetic energy of the rotation aroung the center of mass G deriving from the angular momentum equation, Urot the contribution to the potential energy U from the angular momentum equation, and Qkrot the contribution to the generalized force from the angular momentum equation.

Adding together the contributions of the momentum and the angular momentum equations, the Lagrange equation can be formally written with the same expression found for the system of points,

ddt(Lq˙k)Lqk=Qqk ,

being L=K+U the Lagrangian function of the system, and K=Ktr+Krot, U=Utr+Urot, Qqk=Qqktr+Qqkrot the kinetic energy the potential function and the generalized force of the system, defined as the sum of the contributions coming from the momentum and the angular momentum equations.