Reduced-order aerodynamic models for aeroelastic control of turbomachines

A reduced order aerodynamic model is developed for aeroelastic analysis of turbomachines. The proper orthogonal decompostion technique is used to obtain the modal basis vectors of this model. Twodimensional frequency domain solutions are used to obtain the basis vectors efficiently, however the model itself is developed in the time domain and is cast in state-space form. The number of states of the model is less than ten per blade passage, making it appropriate for control applications. The aerodynamic model is coupled with a simple structural model that haa two degrees of freedom for each blade. Results are presented for unsteady inviscid flow through a single stage rotor that moves in both pitch and plunge. The technique is applicable to viscous and three-dimensional problems as well as multi-stage problems with inlet and exit disturbance flows. Introduction With the current trend towards increased operating speeds and more flexible blading, aeroelasticity has become a critical consideration in the design of compressors. Understanding and predicting aeroelastic phenomena are crucial to ensuring that a compressor will operate within stability boundaries, and thus has a large impact on the design process. Appropriate blade design, together with strategies for controlling the onset of instabilities, can significantly impact the stable operating range, potentially leading to better compressor performance. In addition, understanding high cycle fatigue is important to prolong engine lifetimes. ‘Research Assistant, Department of Aeronautics and Astronautics tPrincipal Research Engineer, Department of Aeronautics and Astronautics tAssociate Professor, Department of Aeronautics and Astronautics SAssociate Professor, Department of Mechanical Engineering and Materials Science. Associate Fellow AIAA Copyright @I999 by Massachusetts Institute of Technology. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Various possible flow control methods for extending the stable operating range of compressors are currently under investigation. For active control, a variety of actuation mechanisms could be employed, including air injection and piezo-electrics. Implementation of these actuators with the appropriate control law would be used to ensure that blade vibrations remain damped. It is also possible to extend the stability boundaries of the compressor by altering the aerodynamic or structural properties of the blades. Mistuning is an example of this approach : rotor symmetry is broken by making the blades different from one another, either structurally or aerodynamically, and thus introducing a distribution of blade frequencies in the cascade.’ Experimental and numerical results show that intentional mistuning of rotor blades can delay the onset of instabilities.2 Mistuning is a form of passive control for flutter or high cycle fatigue. Aeroelastic phenomena involve a complicated interaction between the aerodynamics and the structural dynamics of the blades. Typically, very simple aerodynamic models have been used for aeroelastic analyses. The flow is usually assumed to be two-dimensional, inviscid and incompressible.3 These methods are useful near design conditions but do not predict the flow well off-design where blade loading effects are important.4 They are also not applicable to transonic flows where shock dynamics play a significant role in determining the aerodynamic response. The non-linear problem can be solved using computational fluid dynamics (CFD) methods to perform simulations of the unsteady Euler or Navier-Stokes equations, however such techniques are computationally very expensive and have a large number of degrees of freedom, which means that they are not suitable for control design purposes. More efficient methods for time-varying flow can be obtained by considering the unsteady solution to be a small perturbation about a steady-state 3ow.s A set of linearised equations is then obtained which can be time-marched to obtain a flow solution at each instant. This approach is still computationally very expensive and is not really practicable for