# Rotor Dynamics

Rotor Dynamics is the analysis of structures containing rotating components.

The dynamic behavior of such structures is influenced by the type and angular velocity of rotating components and their locations within the model. Rotor dynamics is available in OptiStruct for modal frequency response, complex eigenvalue, static, linear direct transient and small displacement nonlinear direct transient analyses.

## Motivation

In Figure 1, the rotating components of the structure are the shafts on which gears are mounted. The design of the rotors and their angular frequencies can affect the dynamic response of the structure. Any design will most likely lead to asymmetrical mass distribution about the rotor axes. This unbalanced mass, even if it is not significant, can result in deflection of the rotor depending on various factors. The magnitude of these deflections will be augmented when the rotating speed of the shafts equals the natural frequency of the structure (Resonance), and can lead to catastrophic failure of the system.

## Implementation

The Rotor Dynamics functionality is activated in OptiStruct with the
use of the RGYRO Subcase
Information Entry
(RGYRO=`ID`).
This RGYRO entry references the
identification number of a
RGYRO Bulk Data Entry. Related
Bulk Data Entries, RSPINR,
UNBALNC,
ROTORG and
RSPEED are defined in the model
for Rotor Dynamics. Parameters
PARAM,`GYROAVG`,
PARAM,`WR3`,
and
PARAM,`WR4`
are also used.

## Whirl

A rotor is a structure that rotates about its own axis at a specific angular velocity. If a lateral force is applied to the rotor, it will deform in the lateral direction. This deformation is dependent on various factors, such as, magnitude of the applied force, rotor material properties, stator stiffness, and damping within the system. Due to rotor rotation, the deformed rotor will also whirl about an axis.

## Synchronous and Asynchronous Analysis

## Forward Whirl and Backward Whirl

The type of whirl depends on the spin direction of a rotor. If the rotor spin direction is the same as that of its whirl direction, then it is termed as forward whirl. If the rotor spin direction is opposite to the whirl direction, it is termed as backward whirl. In complex eigenvalue analysis, you can determine and differentiate between the modes of a structure undergoing backward whirl and forward whirl.

## Supported Solution Sequences

OptiStruct supports the Rotor Dynamics functionality in the following solution sequences.

### Frequency Response Analysis

The response of a structure with rotating components to a specified external excitation can be determined using the rotor dynamics functionality in frequency response analysis.

Asynchronous analysis (RGYRO=ASYNC)

If ASYNC is specified in the RGYRO Bulk Data Entry, the rotors within the structure have user-defined spin rates. The excitation frequency (FREQi entries) is independent of the reference rotor speed defined in the RGYRO Bulk Data Entry.

Synchronous analysis (RGYRO=SYNC)

If SYNC is specified in the RGYRO Bulk Data Entry, the reference rotor spin rate is equal to (or synchronous with) the excitation frequency. The reference rotor speed is not input via the RGYRO Bulk Data Entry and the FREQi entry values are used in this analysis.

### Complex Eigenvalue Analysis

The eigenvalues and critical speeds of a structure with rotating components can be determined using the rotor dynamics functionality in complex eigenvalue analysis.

Asynchronous analysis (RGYRO=ASYNC)

If ASYNC is specified in the RGYRO Bulk Data Entry, the rotors within the structure have user-defined spin rates via the RSPEED entry and the Campbell Diagram can be plotted to find the critical speeds. Additionally, since the calculated eigenvalues are complex, you can determine unstable modes by studying the real parts of the calculated eigenvalues. If the real part of a complex eigenvalue is positive, then the corresponding system mode is unstable.

Synchronous analysis (RGYRO=SYNC)

### Frequency Response Analysis (ASYNC)

The response of a system with rotating components to an external load in the frequency domain is calculated based on Equation 1.

### Frequency Response Analysis (SYNC)

The response of a system with rotating components to a rotor imbalance which is considered as a force acting in the frequency domain is calculated based on Equation 2.

### Frequency Response Analysis with WR3, WR4 and WRH (ASYNC)

`WR3`, PARAM,

`WR4`, and PARAM,

`WRH`can be used to avoid frequency dependent calculation of the rotor damping and circulation terms in systems with multiple rotors. The frequency values in the circulation damping terms are replaced with the values of the parameters as shown in Equation 3. PARAM,

`GYROAVG`should be set to -1 to be able to bypass frequency dependent look-up and use the

`WR3`,

`WR4`, and

`WRH`values.

### Frequency Response Analysis with WR3, WR4 and WRH (SYNC)

`WR3`, PARAM,

`WR4`and PARAM,

`WRH`can be used to avoid frequency dependent calculation of the rotor damping and circulation terms in systems with multiple rotors. The rotor speeds can be calculated as a linear function of the reference rotor spin rate (see description of terms below). The reference rotor spin rate values in the circulation damping terms are replaced with the values of the parameters as shown in the equation below. PARAM,

`GYROAVG`should be set to -1 to be able to bypass frequency dependent look-up and use the

`WR3`,

`WR4`, and

`WRH`values.

### Complex Eigenvalue Analysis with WR3, WR4 and WRH (ASYNC)

`WR3`, PARAM,

`WR4`, and PARAM,

`WRH`can be used to replace the values of

`WR3`,

`WR4`, and

`WRH`in Equation 5.

### Complex Eigenvalue Analysis with WR3, WR4 and WRH (SYNC)

`WR3`, PARAM,

`WR4`, and PARAM,

`WRH`can be used to replace the values of

`WR3`,

`WR4`, and

`WRH`in Equation 6.

### Static

### Linear and Small Displacement Nonlinear Direct Transient Analyses

Where,

$${\Omega}_{ref}$$ is the reference rotor spin rate

$${\Omega}_{Rj}({\Omega}_{ref})$$ is the spin rate of rotor "$$j$$" as a function of the reference rotor spin rate.

Where,

$${a}_{j}$$ and $${\beta}_{j}$$ are scaling factors calculated from the relative spin rates defined in the RSPINR Bulk Data Entry.

$$\left[M\right]$$ is the structural mass

$${\alpha}_{j}$$ is the viscous damping of the support

${C}_{R}$ is the rotor viscous damping

${C}_{RH}$ is the rotor hybrid viscous damping

${M}_{R}$ is the rotor mass

${K}_{R}$ is the rotor stiffness

${C}_{RM}$ is the rotor material damping

${C}_{RH}$ is the rotor hybrid material damping

${C}_{R}^{C}$ is the circulation, due to rotor viscous damping

${C}_{RH}^{C}$ is the circulation due to rotor hybrid viscous damping

${M}_{R}^{C}$ is the circulation, due to rotor mass

${K}_{R}^{C}$ is the circulation, due to rotor structural stiffness

${C}_{RM}^{C}$ is the circulation, due to rotor material damping

${C}_{RH}^{C}$ is the circulation, due to rotor hybrid material damping

${K}_{S}$ is the stiffness of the support

${C}_{SM}$ is the material damping of the support

$$N$$ is the number of rotors in the model

$$u(\omega )$$ is the displacement as a function of frequency

$$u({\Omega}_{ref})$$ is the displacement as a function of reference rotor spin rate

$$f(\omega )$$ is the external excitation as a function of frequency

$$f({\Omega}_{ref})$$ is the unbalanced load as a function of reference rotor spin rate (via DAREA or UNBALNC entries)

$$G$$ is the structural damping value of the support
defined using PARAM,`G`

$$GR$$ is the structural damping value of the rotor defined
using PARAM,`G`

$$\overrightarrow{\omega}$$ is the angular velocity vector obtained from a pertinent RFORCE entry

`WR3`, `WR4`, and `WRH` are
defined via the parameters PARAM, WR3, PARAM,
WR4, and PARAM, WRH. They may also be rotor
dependent and specified on RSPINR and RSPINT
Bulk Data Entries. These parameters allow you to bypass frequency-dependent looping
by specifying the equivalent “average” excitation frequencies when PARAM,
GYROAVG, -1 is specified.

- $D$
- Regular damping matrix
- $T$
- Skew-symmetric rotation matrix defined as follows in the rotor coordinate system

- $${C}_{RG}$$
- Gyroscopic matrix defined in a rotor coordinate system
as:
(12) $${C}_{RG}=\left[\begin{array}{l}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\\ 0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}\text{\hspace{0.17em}}{I}_{33}\\ 0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{1em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{I}_{33}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}\right]$$

## Model Guidelines

### 1D Rotor Model

The OptiStruct rotor dynamics feature currently supports only 1D
rotors. Rotor shafts modeled with 1D elements like CBEAM,
CBAR, or CBUSH only can be used.
CONM1 or CONM2 entries should be used to
define the mass and inertia of the rotors. Grid points are necessary for the
definition of mass and inertia via CONM1 or
CONM2. All grid points that belong to rotors should be listed
in the ROTORG entries and only grids listed in the
ROTORG entries are included in the calculation of gyroscopic
terms. The `Ixx` fields on the CONM2 entry
should contain meaningful values as only the inertia about the local X, Y, or Z axes
plays a role in the gyroscopic forces (Supported Solution Sequences section). If
CONM1 entries are used, the `Mxy` mass
values should be specified such that the moments of inertia about the local X, Y, or
Z axes are meaningful.

### Detached Rotor Model

The rotor should be detached from the rest of the structure. Only rigid elements (RBEi) can be used to attach rotors to the ground or to flexible bearings. If any connection exists between the rotor and other parts of the structure using elements other than RBEi, then the program will error out.

### Symmetric Rotor in a Fixed Reference Frame

Rotor dynamics analysis in OptiStruct is performed based on assumption that the rotor is symmetric. Therefore, the rotor model is required to be symmetric about the rotation axis. The implementation is based on equations of motion formulated in a fixed reference frame. Asymmetric rotors in a rotating reference frame is planned to be implemented in future versions of OptiStruct.

### Rotor-axis Guidelines

- These grids are permitted to have a user-defined input coordinate system. As the rotor axis is dependent on the input coordinate system, this definition would change the direction of the rotor axis.
- These grids can also have a user-defined output coordinate system. The output coordinate system does not affect the direction of the rotor axis.

## Multiple Rotors

During synchronous analysis, the calculations are performed with respect to the reference rotor. In synchronous frequency response analysis, the reference rotor is rotating at the frequency of the unbalanced load and in synchronous complex eigenvalue analysis, the reference rotor is rotating at the whirl frequency of the system. The interpretation of results in a multiple rotor system should always be done with respect to the reference rotor. Any deduction of results from the behavior of rotors other than the reference rotor will be inaccurate and can lead to incorrect results. If the behavior of a rotor other than the reference rotor is to be studied, a different analysis should be run with the rotor of interest as the reference rotor.

- Co-axial
- Rotors share the same axes of rotation
- Multi-axial
- Rotors have different axes of rotation

## Campbell Diagram

The critical speeds of a rotating structure should be calculated and the design parameters can then be altered if necessary to restrict the operating speeds of the structure from attaining those resonant speeds.

- Synchronous Complex Eigenvalue AnalysisThe RGYRO=SYNC option in Complex Eigenvalue Analysis can be used to determine the exact critical speeds of the rotating structure. During a synchronous analysis, the rotor speed is equal to the whirl frequency of the structure, which by definition, are the critical speeds of the structure that should be avoided during its operation.
- Asynchronous Complex Eigenvalue AnalysisThe RGYRO=ASYNC option and the RSPEED Bulk Data Entry in Complex Eigenvalue Analysis can be used to determine the whirl frequencies (backward whirl and forward whirl) of the structure. These Whirl frequencies can be calculated for a sequence of rotor spin rates. Forward Whirl and Backward Whirl frequencies can then be plotted against the range of rotor spin rates (Figure 4). The critical speeds can be calculated by superimposing the "Rotor Spin Rate = Whirl Frequencies" line on the plot. The points of intersection are the critical speeds.Note: The rotor speeds specified on the RSPEED entry should be input with sufficiently fine resolution to be able to capture the critical speeds. If the specified rotor speeds are too far apart, the critical speeds may be missed.

### Campbell Diagram in HyperGraph 2D

The procedure to create the Campbell Diagram in HyperGraph 2D is:

## Rotor Superelements

Rotors in frequency response and complex eigenvalue solutions can be replaced using superelements.

`METHOD`field in the CMSMETH Bulk Data Entry to generate the superelements. The superelement replaces the beam elements used to model the rotor. The ASET grid points should correspond to the

`GRIDi`points specified on the ROTORG Bulk Data Entry. The interface grid points of the superelement used to model the rotor should be exactly the same grid points defined on the ROTORG entry. The General Modal Formulation (GM method) cannot be used to generate superelements for Rotor dynamics.

## Output

```
Subcase: 1
Campbell Diagram Summary
Mode #: 1
-------------------------------------------------------------------------------
Step Rotor speed Eigenvalue Eigenvalue Frequency Damping Whirl
(RPM) (Re) (Im) (Hz)
-------------------------------------------------------------------------------
1 0.000E+00 -1.92148E-01 -3.81017E+02 6.064E+01 1.009E-03 LINEAR
2 2.000E+02 -1.92108E-01 -3.81011E+02 6.064E+01 1.008E-03 BACKWARD
3 4.000E+02 -1.91987E-01 3.80993E+02 6.064E+01 1.008E-03 BACKWARD
4 6.000E+02 -1.91788E-01 -3.80964E+02 6.063E+01 1.007E-03 BACKWARD
5 8.000E+02 -1.91513E-01 3.80924E+02 6.063E+01 1.006E-03 BACKWARD
6 1.000E+03 -1.91163E-01 3.80873E+02 6.062E+01 1.004E-03 BACKWARD
7 1.200E+03 -1.90742E-01 3.80810E+02 6.061E+01 1.002E-03 BACKWARD
Mode #: 2
-------------------------------------------------------------------------------
Step Rotor speed Eigenvalue Eigenvalue Frequency Damping Whirl
(RPM) (Re) (Im) (Hz)
-------------------------------------------------------------------------------
1 0.000E+00 -1.92148E-01 3.81017E+02 6.064E+01 1.009E-03 LINEAR
2 2.000E+02 -1.92108E-01 3.81011E+02 6.064E+01 1.008E-03 BACKWARD
3 4.000E+02 -1.91987E-01 -3.80993E+02 6.064E+01 1.008E-03 BACKWARD
4 6.000E+02 -1.91788E-01 3.80964E+02 6.063E+01 1.007E-03 BACKWARD
5 8.000E+02 -1.91513E-01 -3.80924E+02 6.063E+01 1.006E-03 BACKWARD
6 1.000E+03 -1.91163E-01 -3.80873E+02 6.062E+01 1.004E-03 BACKWARD
7 1.200E+03 -1.90742E-01 -3.80810E+02 6.061E+01 1.002E-03 BACKWARD
```

- In rare cases, when the job is run in different machines, a given model might show flip in the sign of imaginary eigenvalues between a pair of modes. This is due to numerical differences while ordering the modes.
- As the roots are complex conjugates, if the sign changes for a particular step (that is, rotor speed) in a mode, then the same step will have opposite sign in a consecutive mode.