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Applied Mechanics Lab

Mechanics of Continua and Structures

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Kinetics: Euler and Inertia tensors

Relative position vectors

The relative position vectors are defined as follows: Π(X)=XˉX,πt(x)=xˉxt,

Euler tensors

The Euler tensors E0 and E are defined as follows:

E0=Ω0ΠΠρ0dΩ0Et=ΩtππρtdΩt

Inertia tensors

The inertia tensors J0 and Jt are defined as follows:

J0=tr(E0)IE0,Jt=tr(Et)IEt,

Computation of material Euler and inertia tensors

Ellipsoid example: Calculation in Mathematica

Computation of spatial Euler and inertia tensors

The spatial tensors Et and Jt can be computed from their material counterparts. Specifically, it can be shown (part of your HW 5 prblem set) that,

Et=RtE0RTtJt=RtJ0RTt

Let ˆEi be Cartesian basis vectors. Say that (J0)ij (resp. (E0)ij) are the components of J (resp. E0) w.r.t. ˆEi, i.e.,

(J0)ij=ˆEi(J0ˆEj)(E0)ij=ˆEi(E0ˆEj)

Following the dyadic representation of tensors, we can also write J0=(J0)ijˆEiˆEjE0=(E0)ijˆEiˆEj

Say ˆei are corresponding co-rotational basis vectors corresponding to ˆEi. It can be shown that Jt=(J0)ijˆeiˆejEt=(E0)ijˆeiˆej

Proof. Let Ft stand for either Et or Jt, and Let F0 stand for either E0 or J0.

Ft=RtF0RTt=Rt((F0)ijˆEiˆEj)RTt=(F0)ijRt(ˆEiˆEj)RTt=(F0)ijRt(ˆEi((RTt)TˆEj))=(F0)ijRt(ˆEi(RtˆEj))=(F0)ij((RtˆEi)(RtˆEj))=(F0)ij(RtˆEi)(RtˆEj)=(F0)ij(ˆei)(ˆej)=(F0)ijˆeiˆej

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