(RotatingFrames)=
# 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.

:::{output}
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 {numref}`%s <fig:hex20rotation>`. 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 {numref}`%s <fig:hex20rotationresponse>`. 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-md} fig:hex20rotation

![](omega_omegadot_viz.png)

[HEX20 mesh](/tests/Salinas_rtest/training/exampleproblem/boundary/angularAcceleration/hex20Beam40x.g) used in statics example problem for rotating reference frame.
:::

:::{figure-md} fig:hex20rotationresponse

![](omega_omegadot.png)

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

The [input](/tests/Salinas_rtest/training/exampleproblem/boundary/angularAcceleration/beam.inp) for this example is attached.
