Modeling of geometrical stiffening in a rotating blade-A review

The present work reviews different approaches adopted for modeling the geometrical or centrifugal stiffening of a beam due to rotation about an axis perpendicular to its longitudinal axis. The longitudinal displacement of the beam consists of three components: the axial displacement of the neutral axis (elastic extension), displacement associated with rotation of the plane section and the displacement due to the foreshortening effect. A widely used approach for modeling the geometrical stiffening is based on the foreshortening effect, which essentially is the longitudinal shrinkage due to the transverse motion of the beam. This approach uses nonlinear strain-displacement relations. As a result, the equations of motion and associated boundary conditions are nonlinear. The geometric stiffening terms in the nonlinear models are fundamentally a linear/quadratic function of the high-frequency axial elastic deformation. Various nonlinear models are discussed and summarized based on the different approximations of the strain-displacement relation. The solution procedure of these nonlinear models is complicated and computationally expensive due to coupling between high-frequency axial and low-frequency bending modes. Simplifying the model by direct linearization of the equations of motion eliminates the geometrical stiffening term resulting in an incorrect model. Different approaches to include geometrical stiffening terms in the linear model are discussed. One of the approaches is linearizing the nonlinear terms arising from the coupled axial-transverse motion around the steady-state axial solutions. The steady-state axial equilibrium equation can be linear or nonlinear depending on the type of strain measure employed. A comparison of the solution of these different linear/nonlinear steady-state axial equilibrium equations is presented. The applicability of these models based on the steady-state axial equations is tested, and the rotation speed limit within which these models are valid is also discussed. In another approach, the equations of motion are derived using a time-independent centrifugal force. The resulting equations are equivalent to those governing the transverse vibrations of beams subject to an external axial force. Nevertheless, another approach proposed by Kane et al. (1987) uses stretch as a variable in the formulation instead of the axial displacement. The linear geometrical stiffening models are discussed in detail. Further, the effects of geometric properties of the blade, such as taper, twist angle, pre-setting angle and asymmetry in cross-section on the modal characteristics are brought out. A comparison of the different beam theories used in studying the dynamics of rotating blades is also presented.

Automated summary

This paper reviews different modeling approaches for the geometrical or centrifugal stiffening of beams due to rotation. The longitudinal displacement of the beam consists of three components: axial displacement, rotation-induced displacement, and displacement due to foreshortening effect. Nonlinear strain-displacement relations are used to model geometrical stiffening, resulting in nonlinear equations of motion. Various nonlinear models and their solution procedures are discussed based on different approximations of the strain-displacement relation. The applicability of these models is tested, and the impact of geometric properties on the modal characteristics is analyzed.