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Table of Contents

Blade Section Stiffness

To model flexible blades or analyse vibrational dynamics, it is necessary to define the stiffness distribution of the rotor blade. Begin by selecting the appropriate degrees of freedom to include in the model by enabling/disabling the following options in the Blades screen under the Mass and Stiffness tab:

  • Bending stiffness
  • Shear stiffness
  • Torsional degree of freedom
  • Axial degree of freedom

It is also possible to include bend-twist and bending-bending coupling terms, and these can be defined using Project info.

Constitutive Relationship

The blade sectional stiffness inputs all relate to the cross-sectional 6x6 stiffness matrix used in the Bladed beam model as shown in Equation \(\eqref{eq:beamconstitutiverelationship}\). This matrix is associated with the principal elastic axes coordinate system, which is defined by the Principal elastic axes orientation (xe, ye) and the Elastic centre as described in Blade Section Coordinate Systems. The constitutive relationship for the full beam model can be expressed as:

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]= \left[\begin{matrix} \bscalar{GA}_x & & & | & & & \\ \bscalar{GA}_{xy} & \bscalar{GA}_y & & | & & \bmatrix{sym} & \\ 0 & 0 & \bscalar{EA} & | & & & \\ - & - & - & - & - & - & - \\ 0 & 0 & 0 & | & \bscalar{EI_x} & & \\ 0 & 0 & 0 & | & \bscalar{C_{xy}} & \bscalar{EI_y} & \\ - \bscalar{GA}_x y_{cs} & \bscalar{GA}_y x_{cs} & 0 & | & \bscalar{C_{xz}} & \bscalar{C_{yz}} & \bscalar{GI}_z \\ \end{matrix}\right] \left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right] \label{eq:beamconstitutiverelationship} \end{equation} \]

where
\(\bscalar{GI}_z = \bscalar{GI}_z^\ast + \bscalar{GA}_x \cdot y_{cs}^2 + \bscalar{GA}_y \cdot x_{cs}^2\)
\(x_{cs}\) and \(y_{cs}\) are the offsets between the Elastic centre and Shear centre along the principal elastic axes.


Table 1: Overview of blade sectional stiffness inputs in Bladed, relating them to Equation \(\eqref{eq:beamconstitutiverelationship}\).
Property Variable Description Unit Input Location
Bending stiffness about xe \(EI_x\) Bending stiffness about the principal elastic x-axis \(\bunit{Nm^2}\) Blades screen
Bending stiffness about ye \(EI_y\) Bending stiffness about the principal elastic y-axis \(\bunit{Nm^2}\) Blades screen
Torsional stiffness about zs \(GI_z^\ast\) Torsional stiffness about the principal shear z-axis (shear centre) \(\bunit{Nm^2}\) Blades screen
Shear stiffness along xe \(GA_x\) Shear stiffness along the principal elastic x-axis \(\bunit{N}\) Blades screen
Shear stiffness along ye \(GA_y\) Shear stiffness along the principal elastic y-axis \(\bunit{N}\) Blades screen
Axial stiffness \(EA\) Axial stiffness of cross-section \(\bunit{N}\) Blades screen
FlapEdgeCStiff \(C_{xy}\) Bending-Bending coupling stiffness along principal elastic x- and y-axes \(\bunit{Nm^2}\) Project Info
TorsionFlapCStiff \(C_{xz}\) Torsion-Bending coupling stiffness along principal elastic x-axes \(\bunit{Nm^2}\) Project Info
TorsionEdgeCStiff \(C_{yz}\) Torsion-Bending coupling stiffness along principal elastic y-axis \(\bunit{Nm^2}\) Project Info
Note

The shear-shear coupling stiffness \(\bscalar{GA}_{xy}\) and shear-twist coupling stiffness terms \(\bscalar{GA}_x y_{cs}\) and \(\bscalar{GA}_y x_{cs}\) are calculated automatically by Bladed based on the input values.

Project Info Definition of Bending-Twist and Bending-Bending Coupling Terms

Bending-twist and bending-bending coupling terms can be specified through Project Info. More details on how the inputs are used in the simulation can be found in the theory article Bend-Twist Coupling Relationships in Beam Elements.

You need enough entries for the number of elements in the blade. Each node in the Bladed user interface defines the properties for both the end of one element and the beginning of the next, unless it is a split station so there must be \(2\bscalar{N_{el}}-2\) entries, where \(\bscalar{N_{el}}\) is the number of blade elements.

MSTART EXTRA
TorsionEdgeCStiff * list of values
TorsionFlapCStiff * list of values
FlapEdgeCStiff * list of values
MEND

For example, for the demo_a turbine (which has 10 blade stations):

MSTART EXTRA
TorsionEdgeCStiff 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0
TorsionFlapCStiff 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0
FlapEdgeCStiff 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0 10000.0
MEND

Last updated 05-12-2024