Stability Analysis of Nonlinear Rotating Systems Using Lyapunov Characteristic Exponents Estimated From Multibody Dynamics

Abstract

The use of Lyapunov Characteristic Exponents to assess the stability of nonlinear, time-dependent mechanical systems is discussed. Specific attention is dedicated to methods capable of estimating the largest exponent without requiring the Jacobian matrix of the problem, which can be applied to time histories resulting from simulations performed with existing multibody solvers. Helicopter ground resonance is analyzed as the reference application. Improvements over the available literature are: the problem is formulated in physical coordinates, without eliminating periodicity through multiblade coordinates; the rotation of the blades is not linearized; the problem is modeled considering absolute positions and orientations of parts. The dynamic instability that arises at some angular velocities when the isotropy of the rotor is broken (e.g., caused by the failure of one lead-lag damper, a design test condition) is observed to evolve into a large amplitude limit cycle, where the usual Floquet-Lyapunov analysis of the linearized time-periodic simply predicts instability.

The Dependent Coordinates in the Linearization of Constrained Multibody Systems: Handling and Elimination

“…The objective is to express the reduced linearized dynamic Eqs. (66) in terms of the independent variations x𝑎𝑖 and their time derivatives ̇x 𝑎𝑖 and ẍ𝑎𝑖 . For that purpose, the constraints at velocity and acceleration levels, given by Eqs.…”

“…( 82), ( 83) and (84), respectively, in the linearized dynamic Eqs. (66), and using Eq. ( 81), yields:…”

“…Starting from the linearized dynamic Eqs. (66), the objective of this approach is to express Eq. ( 66) in terms of the independent variations x𝑎𝑖 and the time derivatives ̇x 𝑎𝑖 , ẍ𝑎𝑖 .…”

“…Additional works include stability analyses of multibody systems as motorcycles [27,[46][47][48][49][50][51][52][53], the waveboard [54,55] and other vehicles [56][57][58][59]. The stability analysis of nonlinear multibody systems, as rotorcraft systems, can be found in several works of Masarati et al [60][61][62][63][64][65][66], which make use of Lyapunov characteristic exponents.…”