Non-stationary bending vibration of a helicopter rotor blade under distributed aerodynamic load. Part 2. Forced vibrations of the blade

Abstract

Each technical problem statement requires an adequate physical model. This model must take into account the basic physical properties and bring the physical problem statement closer to the technical problem. Part 1 of this work presents experimental instantaneous photographs of the bending vibrations of a helicopter rotor blade, which are similar to the small bending vibrations of a cantilever-clamped thin plate. It is known that helicopter blades are made rigid. Therefore, to model small bending vibrations, we apply Kirchhoff-Love's hypothesis and Lagrange-Sophie Germain's equation in this work.

Currently, existing approaches to solving the problem of natural bending vibrations of a thin rectangular plate under cantilever clamping conditions lead to uncertainty and non-uniqueness of the solution. Therefore, in Part 2 of this work, in order to obtain a unique solution to the problem, a non-zero distribution of normal forces is set at the two edges of the plate. These forces are taken into account only in the boundary conditions, and the rest of the blade surface is free. Therefore, no forces are added to the right-hand side of the Langrange-Sophie Germain equation. This refinement of the boundary conditions brings the physical formulation of the problem closer to the real technical problem: during rotation, forces act on the blade, causing a lifting force, which determines the load distributed normal to the blade surface.

The change in boundary conditions at the two edges of the blade resulted in the system of linear algebraic equations changing from homogeneous to non-homogeneous. This made it possible to find a unique solution to the problem based on Cramer's rule and to unambiguously determine the desired integration constants. Since the blade motion is periodic, the problem is solved under the condition of a time-harmonic load on the plate.

To obtain an analytical solution of the problem for forced vibrations, the normal coordinate method is used. It allows using the solution of the problem for natural vibrations, its stationary part, and on its basis, obtaining an equation for the normal coordinate that describes the non-stationarity of the general solution of the problem. Using L'Hôpital's rule, the obtained general solution is divided by frequency into non-resonant and resonant regions.

As an example of numerical calculation, the amplitudes of the bending of the natural vibrations of a plate, a helicopter blade, are calculated for different frequencies of its rotation. It was found that the amplitude of the bending has a wave character, slightly increasing towards the outer end of the blade with an increase in the rotation frequency of the helicopter rotor.