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# Rotating Reference Frame

Example files on this page

- HEX20 mesh — Salinas_rtest/training/exampleproblem/boundary/angularAcceleration/hex20Beam40x.g

- input — Salinas_rtest/training/exampleproblem/boundary/angularAcceleration/beam.inp

# 19. Rotating Reference Frame

For certain types of analysis it is useful to express structural deformations relative to a rotating reference frame. For example, while the displacements of a rotating vertical axis wind turbine are large in a fixed frame, they may be small when measured relative to a reference frame which rotates with the base of the turbine. A significant benefit of using a rotating reference frame is that a linear structural analysis, like what Sierra/SD offers, can be a well-suited and efficient alternative to a fully nonlinear analysis.

The effects of a rotating coordinate system on the equilibrium equations are twofold. First, loading terms involving the angular velocity \(\Omega\) and angular acceleration \(\dot{\Omega}\) are present. Specifically, angular velocity loads are proportional to \(\Omega^2\) while loads associated with angular acceleration are proportional to \(\dot{\Omega}\). Second, the rotating coordinate system affects the system matrices. In the case of angular velocity, there is a symmetric contribution to the stiffness matrix proportional to \(\Omega^2\) and a skew symmetric contribution (Coriolis) contribution to the damping matrix proportional to \(\Omega\). For angular acceleration, there is a skew symmetric contribution to the stiffness matrix which is proportional to \(\dot{\Omega}\). We note that the skew symmetric damping matrix does not lead to energy dissipation, but will result in complex eigenvectors for a modal analysis. We also note for a constant angular velocity the structure becomes preloaded for a steady state static analysis. This preload can affect the stiffness matrix (through stress stiffening effects), but we do not discuss this topic further here.

Including the effects of a rotating reference frame is done by including body loads as shown below. Here, the reference frame rotates about the origin of coordinate system `rotz`. The angular velocity and angular acceleration are 500 and 200, respectively, about axis 3. There are a couple of important points to mention here. First, separate body sections are needed to include both angular velocity and angular acceleration. Second, it’s important that the same coordinate system be used for both angular velocity and angular acceleration. Third, angular acceleration loads are appropriate only for models with sufficient essential boundary conditions to fully constrain away any rigid body motions.

    LOADS
      body
        angular_velocity = 0 0 500
        coordinate rotz
      body
        angular_acceleration = 0 0 200
        coordinate rotz
    END

Including the effects of angular acceleration is currently only supported for static analyses. Further, Sierra/SD can be used for a snapshot static analysis where both \(\Omega\) and \(\dot{\Omega}\) can be nonzero. Clearly, the angular velocity changes over time for nonzero \(\dot{\Omega}\). This would lead to changes in the stiffness and damping matrices over time, but those effects are ignored in the snapshot static analysis.

To illustrate a static analysis with nonzero \(\Omega\) and \(\dot{\Omega}\), consider the beam-like structure modeled with HEX20 elements shown in Figure 19.1. For this model all the nodes are constrained at the left end. This means that the displacements of these nodes are all zero with respect to the rotating frame. The values for \(\Omega\) and \(\dot{\Omega}\) are shown in the input block above. Axial and transverse displacements are shown in Figure 19.2. Not surprisingly, the beam stretches along its length and transverse displacements lag behind the direction of the angular acceleration. We note that the body load for a point with position vector \(\boldsymbol{r}\) relative to the origin of the rotating coordinate frame is given by

\[ \boldsymbol{b} = -\boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\boldsymbol{r}) - \dot{\boldsymbol{\Omega}}\times\boldsymbol{r} \]

A snapshot static analysis can be used to help understand the importance of \(\Omega\) and \(\dot{\Omega}\). Alternatively, one can get a rough idea of their importance by comparing to the smallest flexible circular frequency \(\omega_1\) (in radians per second). If \(\Omega\ll\omega_1\) and \(\dot{\Omega}\ll\omega_1^2\), then their importance is not likely to be significant.

Figure 19.1 HEX20 mesh used in statics example problem for rotating reference frame.

Figure 19.2 Axial and transverse displacements for statics example problem for rotating reference frame.

The input for this example is attached.