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2.1  Rotating bodies

• Problem definition in a rotating frame

2.1.1  Problem definition in a rotating frame

The developments of this section are derived from internal work on the SDT Rotor module which is currently only distributed to SNECMA. The results shown here can be seen as a summary of those found in Ref. [1] which treats the problem with a strong emphasis on the theoretical formalism. Other classical references that treat of the problem of rotating bodies are [2],[3], [4].

Particles located in point p of the body fixed frame are at location x at time t. One defines the displacement u by

(2.1)

At time t, a reference point of the rotating body is assumed to have a rigid rotation speed {ω} with respect to the reference frame (in the present study, this speed is related to a global rotation around a fixed axis characterized by angle θ). The velocity is thus given by

(2.2)

This expression can easily be derived by decomposing the position in body fixed coordinates {x}=x_i {e_bi} and noting that the derivatives the base vectors ∂ {e_bi}/∂ t= {Ω}(t)∧ {e_bi}. In implementations, one replaces the vector product ω(t)∧ by the product by the corresponding skew-symmetric matrix

(2.3)

The acceleration, derived from the velocity expression, is given by

(2.4)

where three contributions (rows of the equation) are typically considered : the acceleration in the rotating frame, the Coriolis acceleration and the centrifugal acceleration.

The virtual work of acceleration quantities is thus typically expressed as

(2.5)

with the following element level expressions. The displacement within an element is given by the position and the element shape functions [N] in the three directions xyz

The matrices and loads are integrated over the volume S₀ in the reference configuration and are given by

• [M] = ∫_S0 ρ₀ [N_xyz]^⊤[I] [N_xyz] dS₀ the mass in the rotating frame
• [D_g] = ∫_S0 2ρ₀ [N_xyz]^⊤[Ω] [N_xyz] dS₀ the gyroscopic coupling
• [K_c] = ∫_S0 ρ₀ [N_xyz]^⊤[Ω²] [N_xyz] dS₀ the centrifugal softening/stiffening
• [K_a] = ∫_S0 ρ₀ [N_xyz]^⊤∂[Ω]/∂ t [N_xyz] dS₀ the centrifugal acceleration
• f_c= ∫_S0 ρ₀ [N_xyz]^⊤[Ω²] {p}  dS₀ the centrifugal load
• f_g= ∫_S0 ρ₀ [N_xyz]^⊤∂[Ω]/∂ t {p}  dS₀ the Coriolis load

It is acknowledged that the notations used can be somewhat confusing. Indeed, in a discretized vectors DOFs are placed either sequentially x,y,z at all nodes of the element, or separated x at all nodes, ... while the operations [N_xyz]^⊤[I] [N_xyz] imply the use of vectors. This 2 dimensional product notation however directly reflects the numerical implementation as is thus deemed preferable.

In the applications considered in this study, one will use a fixed axis of rotation Ω=ω(t) {e_z}∧. The matrices and loads are thus proportional to the scalars ω, ω² and ω. One will thus simply use

which results in significant computational cost savings since the matrix only needs to be computed for a single velocity. One proceeds similarly for the other matrices and loads.