Structural model and calculation of the aeroelastic response

It is implicitly assumed that the vibration amplitude remains within the bounds of linear behavior. The global aeroelasticity equations of a structural motion can be written as:

[M] {X} + [C] {X} + [K] {x} = {p (t) An} (7)

where M, C, K are the mass, stiffness and damping matrices, x is the dis­placement vector, p (t) is a vector of pressures, A being the application area and n is the normal unit vector of the blade’s surface. A standard structural finite element formulation is used to obtain the left-hand side while the fbw model above is used to obtain the unsteady forcing of the right-hand side. The free vibration problem can be solved to yield the natural frequencies иr and mode shapes [Фг] of the bladed disk assembly in vacuum, r being the mode index.

The aeroelasticity analysis can be conducted in two different ways, the so-called fully-coupled and the uncoupled approaches. In the fully-coupled method, the structural mode shapes are interpolated onto the aerodynamic mesh as the two discretization levels between structural and aerodynamic meshes are unlikely to be wholly coincident. To accommodate the structural motion, the fluid mesh is moved at each time step using a network of springs whose compression/extension is prescribed by the mode shape at the blade surface and becomes zero at the far field. The frequency of the motion is dictated by the natural frequency of the corresponding mode. An exchange of bound­ary conditions takes place at the fluid-structure interface at each time step. The main advantage of the fully-coupled method is the automatic inclusion of the aerodynamic damping while the main drawback is the computational overhead arising from the mesh movement. However, unless the change in aerodynamic damping is likely to be important, say in the case of a large blade vibrating at resonance, the forced response analysis can be simplified by con­sidering the unsteady fbw and the blade motion separately. Such an uncoupled analysis consists of computing the blade unsteady pressures without any blade movement. The aerodynamic damping may still be obtained from a transient aeroelasticity, or flitter, analysis, and such damping values can be used when calculating the blade’s response. A case study comparing the relative accuracy of both approaches is reported by Vahdati et al. (2003a).

A convenient indicator of the blade response is the modal force, which is a measure of the correlation between the structural mode shape and the unsteady pressure fhctuation, the other two key parameters being the amplitude and frequency of the flictuation. The modal force represents the strength of the unsteady forcing in a particular mode of vibration. In simple terms, the modal force can be thought as the product of the unsteady pressure and the structural mode shape. It can best be visualized by considering a rigid body motion where the blade is plunging only. In this case, the modal force is equivalent to the unsteady lift on the blade.