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

Modal analysis of a rotating twisted and tapered Rayleigh beam

Free vibration analysis of a twisted, double-tapered blade mounted on a rotating disk undergoing overall motion is presented here. The Lagrangian approach is adapted to study the modal characteristics of the blade modeled as a rotating Rayleigh beam. The expressions for the kinetic energy and potential energy of the cantilever blade are derived using hybrid deformation variables. The continuous deformation variables in these equations are discretized using a series of basis functions that satisfy all boundary conditions of the cantilever beam. The equations governing the coupled stretch–bending–torsion motion of the rotating blade are derived using Lagrange’s approach. The equations are then transformed into a non-dimensional form which are then solved for the eigenvalue problem for the modal characteristics of the blade. The results of the present model are verified with the results available in the literature. The variation of the natural frequencies with the rotating speed, taper ratio and pre-twist angle is presented. The tuned angular speed of the blade at which the angular frequency matches with any of the natural frequency of the blade resulting in the resonance is investigated. The Campbell diagram is plotted for the specific problem to identify the resonance where the natural frequency matches with the harmonics of the rotating speed. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature

Quantification of projection angle in fragment generator warhead

Tactical Ballistic Missile (TBM) class target neutralization by the fragment spray of a Fragment Generator Warhead (FGW) calls for quantification of fragment projection angle scatter to finalize the end game engagement logic. For conventional axi-symmetric warhead, dispersion is assumed to be normal with a standard deviation of 30. However, such information is not available in case of FGW. Hence, a set of experiments are conducted to determine the dispersion of fragments. The experiments are conducted with a specific configuration of FGW in an identical arena to quantify the scatter and then verified its applicability to other configurations having a range of L/D and C/M ratios, and contoured fragmenting discs. From the experimental study, it is concluded that the scatter in projection angle follows normal distribution with a standard deviation of 0.75° at Chi-square significance level of 0.01(χ20.99)