Problem 1

(a) For some trial functions, Rayleigh-Ritz and Galerkin methods calculated identical frequency values for same number of terms.

When the Galerkin method projects the problem into the same mode space occupied by the Rayleigh-Ritz method, it can result in the same frequency values for the same number of terms.

(b) One needs a large number of finite elements to calculate loads near structural discontinuities such as the case with flap flexure (virtual flap hinge).

Because of large stress gradients induced by the presence of a flap flexure (a local change in blade elasticity), a larger number of elements is required to capture the local variations without aliasing.

(c) Normal mode reduction is a key simplification in structural dynamics.

Normal mode reduction changes the motion and response of a structural problem from a physical problem to a summation of normal modes. Hence, the response of the problem can be analyzed solely via the excitation of the normal modes. Also, it can reduce a PDE to a series of ODEs, simplifying the solution.

(d) A fan plot helps to fine tune operating rotational speed.

The fan plot shows the normal mode harmonics, the frequencies being excited at the chosen design point, and the structural resonance frequencies simultaneously, allowing the designer to design operating points away from resonance frequencies.

(e) Can you apply orthogonality condition to damped free vibration modes?

Yes it can because damping only effects amplitude, not frequency. The latter of these is the basis of the orthogonality condition.

(g) For hingeless rotors, the flap frequency is placed between 1.08 and 1.1/rev.

A hingeless rotor utilizes blades with a flexure or some other sort of virtual hinge to allow flapping and lag motion with the proper damping, leading to a lesser blade response than typically found in articulated systems. As the flapping frequency is comprised of the ratio of the blades centrifugal and flapping inertia summed with the flapping stiffness, it can be inferred that the contribution from each term will be relatively low. For this smaller blade response, the contribution of centrifugal force to flap damping is decreased. Similarly, the flapping stiffness is still low (relative to a rigid blade). Hence, the flapping frequency is only slightly increased over an articulated rotor system.

(h) The harmonic balance method is used to calculate the periodic steady response of blade. It is not applicable to calculate blade response under gust loading.

A gust loading would not produce a steady state but a transient response. As the harmonic balance method computes steady state responses, it cannot be used to calculate a blade’s response under such a transitory load.