The Effect of Blade Deformation on Rotorcraft Acoustics

The aeromechanical environment of rotorcraft is extremely complex, involving complex aerodynamics, rigid blade motion, and complex elastic deformation. Until recently rotorcraft noise prediction has included the rigid blade motions only, neglecting blade flexibility. Some recent research has attempted to include the effects of the elastic deformation, but this paper presents the first rigorous study of the impact of including flexibility on the acoustics. A derivation of Farassat’s Formulation 1A is presented which includes two additional terms related to the surface deformation. The acoustics code PSU-WOPWOP has been modified to include these terms: several cases are analyzed and the impact of flexibility, including the effect of the changing surface area, is examined. The main case considered is a utility helicopter in high-speed forward flight. The aerodynamic inputs to the acoustics code were calculated using an elastic blade definition. The effect of making a rigidblade approximation in the acoustics is investigated, and it is determined that the effect on thickness noise can be reduced by proper positioning of the blade tip. It is then shown that loading noise presents a larger challenge, and the rigid blade approximation can lead to significant errors at some observer locations. Finally, the thickness noise of a speculative advanced “smart materials” blade is examined to set an upper limit on the impact of the new formulation on rotorcraft acoustics. It is found that for present rotorcraft, the additional terms may be neglected safely, although the other effects of blade flexibility should be included. Notation 2 d’Alembertian or wave operator, [ (1/c2)(∂2/∂t2 ] −∇2 a Mean sphere radius, m c Speed of sound (quiescent medium), ms−1 dS Differential element of surface area, m2 ∗Graduate Research Assistant; chennes@psu.edu †Associate Professor; ksbrentner@psu.edu Presented at the 31st European Rotorcraft Forum, Florence, Italy, September 13–15, 2005. Copyright c © 2005 by Christopher C. Hennes and Kenneth S. Brentner. Published by the European Rotorcraft Forum with permission. f = 0 Function that describes the integration surface (i.e. the rotor blade) g Retarded time function, g = τ− t + r/c H Heaviside function, H( f ) = 0 for f < 0 and H( f ) = 1 for f > 0 J Jacobian of transformation `i Components of loading force acting on the fluid, Nm−2 `r `i r̂ i , loading force in the radiation direction (summation implied), Nm−2 `M `iM̂i (summation implied), Nm−2 ̇̀r r̂ i∂`i/∂t (summation implied), Nm−2 s−1 M Mach number of source, v/c Mi Components of the Mach number of the acoustic source, Mi = vi/c Mr Mi r̂ i , Mach number of source in radiation direction (summation implied) n̂ Outward-facing unit normal vector to surface f = 0 p′ Acoustic pressure, p−p0, Pa pT Thickness noise contribution to p ′, Pa pL Loading noise contribution to p ′, Pa Q Right-hand side term (source term) in an inhomogeneous wave equation r x−y, Distance between observer and source, m t Observer time, s Ti j Lighthill stress tensor u1,u2 Time-independent coordinates on source surface Ω vn vi n̂i (summation implied), Normal velocity of blade surface, ms−1 v̇n n̂i∂v̂i/∂t (summation implied), ms−2 vṅ vi∂n̂i/∂t (summation implied), ms−2 xi Observer location, m yi Source location, m Greek symbols δ( f ) Dirac delta function μ Advance ratio, ratio of forward flight speed to blade tip speed ω Pulsation rate, rad/s Ω Time-independent integration surface