Non-linear system identification of the dynamics of aeroelastic instability suppression based on targeted energy transfers

Abstract

Hilbert transform x k (m) the m th component of the k th measurement in IMOs y,α,υ,(υ 1 ) heave, pitch, and NES (NES1) modes, respectively y km) (c m (k) ) the m th component of the k th measurement for an analytical (empirical) model θ k (m) instantaneous slow phase of an IMF c m (k) ζ k (m) damping factor of an IMO Λ k (m) nonlinear modal interaction θ k (m) (ω m (k) ) instantaneous phase (frequency) of an IMF c m (k) ϕ k (m) (A m (k) slow envelope for an analytical (empirical) model ψ k (m) (ψ m (k) ) complexification of the variables for an analytical (empirical) model ω m time-invariant fast frequency value for the m th component CX-A complexification-averaging technique DOF degree-of-freedom EMD empirical mode decomposition FEP frequency-energy plot IMF (IMO) intrinsic mode function (oscillator) ABSTRACTWe revisit our earlier study of targeted energy transfer (TET) mechanisms for aeroelastic instability suppression by employing timedomain nonlinear system identification based on the equivalence between analytical and empirical slow flows. Performing multiscale partitions of the dynamics directly on measured (or simulated) time series without any presumptions regarding the type and strength of the system nonlinearity, we derive nonlinear interaction models (NIMs) as sets of intrinsic modal oscillators (IMOs). The eigenfrequencies of IMOs are characterised by the 'fast' dynamics of the problem and their forcing terms represent slowly-varying nonlinear modal interactions across the different time scales of the dynamics. We demonstrate that NIMs not only provide information on modal energy exchanges under nonlinear resonant interactions, but also directly dictate robustness behaviour of TET mechanisms for suppressing aeroelastic instabilities. Finally, we discuss the usefulness of NIMs in constructing frequency-energy plots that reveal global features of the dynamics to distinguish between different TET mechanisms and to study robustness of aeroelastic instability suppression.

Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration

“…A new contribution in this context is that of Vakakis and co-workers [6][7][8]. There are, however, very few attempts to develop practical modal testing of nonlinear structures.…”

Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

“…Then, the original response can be reconstructed as the sum of all IMO solutions; that is, the following expression holds: Figure 5 compares the 10th and 5th IMFs with the corresponding IMO solutions, exhibiting good agreement. Finally, we consider the physical meaning of the complex-valued forcing function Λ m (t) for the mth IMO of the linear problem, since such a term is known to be associated with nonlinear modal interactions in nonlinear dynamical systems [28,29]. In our linear beam problem, the slowly-varying envelope m (t) and phasê θ m (t) in Eq.…”

“…The method requires no a priori system information but only measured (or simulated) time series; that is, it is purely an output-based approach. Applications of the proposed NSI methodology have been provided with studies of targeted energy transfers in a 2-DOF dynamical system [28], instability generation and suppression in a 2-DOF rigid aeroelastic wing model [29], and the dynamics of a rod coupled to an essentially nonlinear end attachment [30].…”

Adaptive backstepping flight control for a mini‐UAV

“…where˛andˇare the attack angle and the sideslip angle, respectively, ı r is the rudder deflection, b the wing span, and C L and C D the lift and the drag coefficients, respectively; they have the following forms [29]:…”