Computation of Frequencies and Nodal Diameters

Blade Row Frequencies Computation

For a case with $N$ blade rows, the complete set of blade passing frequencies for blade row $j$ is:

Figure 1. EQUATION_DISPLAY

ω^{j} = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{N} [m_{i} B_{i} (Ω_{j} {- Ω}_{i}) + f_{i}]

(5050)

where, for blade row or wake $i$ , $m_{i}$ is the blade row index and can take all values between $- M_{i}$ and $M_{i}$ . $M_{i}$ is the specified number of modes.

When using the Consider All Blade Rows option, $i$ represents any blade row, wake, time varying boundary, or flutter condition, except blade row $j$ . The relative rotation rate between any two blade rows is kept non-zero by adding a small fictitious rotation rate to the blade row if necessary; see Fictitious Rotation Rate Computation. When using the Consider Only Neighbors option, the only blade rows that are considered are the rows next to $j$ that share a common indirect interface with $j$ , along with any wake, time varying boundary, or flutter conditions that are directly attached to blade row $j$ .

The variables $B_{i}$ , $Ω_{i}$ , and $f_{i}$ have different meanings, depending on what kind of blade row they refer to. These meanings are described in the following table.


Type of Blade Row $i$	Meaning of $B_{i}$	Meaning of $Ω_{i}$	Meaning of $f_{i}$
Modeled blade row	The number of blades within the given pitch as specified in the Number of Blades property of the Blade Row n > Blades Per Pitch node. See Blades Per Pitch Properties.	Rotation rate of the modeled blade row, which is inferred from the motion specification of the blade row region. See Defining Region Motion.	Always zero. $f_{i}$ is not used for a modeled blade row.
Wake	The number of blades within the given pitch as specified in the Number of Blades property of the Wake > Blades Per Pitch node. See Blades Per Pitch Properties.	Rotation rate that is specified in the Rotation Rate property of the Wake node under Regions > Boundary > Physics Values. See Understanding Wake Options.	Always zero. $f_{i}$ is not used for a wake blade row.
Time-varying option	Set equal to $(B_{r} σ) / (2 π)$ , where $B_{r}$ is the number of blades in the modeled blade row to which this boundary condition applies, and $σ$ is specified in the Phase Angle property of the Time Varying Option Parameters node. See Time Varying Option.	Set equal to the rotation rate of the modeled blade row to which this boundary condition applies.	The fundamental frequency of variation in the boundary conditions, in radians/s. Specified in the Frequency property of the Time Varying Option Parameters node. See Time Varying Option.
Blade flutter	Set equal to $(B_{r} σ_{i b}) / (2 π)$ , where $B_{r}$ is the number of blades in the modeled blade row to which this flutter condition applies, and $σ_{i b}$ is the inter-blade phase angle that is specified in the Flutter Phase Angle property of the Flutter Parameters node. See Flutter Parameters Properties.	Set equal to the rotation rate of the modeled blade row to which this flutter condition applies.	The frequency of unsteadiness $ω$ used in Eqn. (5059). Specified in the Flutter Frequency property of the Flutter Parameters node. See Flutter Parameters Properties.

The maximum value for $ω^{j}$ is set with the Modes Table Filter Option. When this option is set to Cut-Off, the model computes the maximum value $ω_{\max}$ using the cut-off factor $γ$ :

Figure 2. EQUATION_DISPLAY

ω_{\max} = \underset{i}{M A X} {γ \underset{j}{M A X} [M_{j} [B_{j} (Ω_{i} - Ω_{j}) + f_{j}]]}

(5051)

When the option is set to Maximum Limits, the model sets $ω_{\max}$ to the value of the Maximum Frequency property.

Blade Row Nodal Diameter Computation

For a case with $N$ blade rows, the complete set of nodal diameters that correspond to each frequency computed above (in Eqn. (5050)) for blade row $j$ is:

Figure 3. EQUATION_DISPLAY

N^{j} = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{N} m_{i} B_{i}

(5052)

where, for blade row or wake $i$ :

$m_{i}$ is the modal index and can take all values between $- M_{i}$ and $M_{i}$ , where $M_{i}$ is the number of modes.
$B_{i}$ is the number of blades.

When using the Consider All Blade Rows option, $i$ represents all blade rows except blade row $j$ . When using the Consider Only Neighbors option, only the adjacent blade rows that share a common indirect interface with blade row $j$ are considered.

The maximum value for $N^{j}$ is set with the Modes Table Filter Option. When this option is set to Cut-Off, the model computes the maximum value $N_{\max}$ using the cut-off factor $γ$ :

Figure 4. EQUATION_DISPLAY

N_{\max} = \underset{i}{M A X} {γ \underset{j}{M A X} [M_{j} B_{j}]}

(5053)

When the option is set to Maximum Limits, the model sets $N_{\max}$ to the value of the Maximum Nodal Diameter property.

Fictitious Rotation Rate Computation

When you use Consider All Blade Rows as your Frequency Compute Option, every blade must be assigned a different frequency for the purposes of computation, even if some have identical frequencies in the User Specified Mode Table. The Fictitious Rotation Rate Factor $ε$ determines the offset from the originally specified rotation rate.

The offset is an integer multiple of $f$ where:

Figure 5. EQUATION_DISPLAY

f = ε Ω_{max}

(5054)

$Ω_{max}$ is the maximum user-specified rotation rate in the blade row. The default value of $ε$ is 0.001. You can set the value of $ε$ in the Fictitious Rotation Rate Factor property of the Blade Rows manager node, but it is best to make the value small, so that the departure from the original value stays small.