GRANT OPUS 04 - 2012/07/B/ST8/03931

Title "Nonlinear vibrations of composite rotating beams with embedded active elements"
"Drgania nieliniowe kompozytowych belek wykonujących obrotowy ruch unoszenia z wbudowanymi elementami aktywnymi" (in Polish)

Coordinator-Principal Investigator: Prof. Jerzy Warminski
Co-Investigators: Assoc. Prof.  Andrzej Teter, Dr. Andrzej Mitura, Dr. Jaroslaw Latalski, Dr. Marcin Bochenski, MSc Eng. Zofia Szmit, MSc Eng. Jaroslaw Gawryluk
Laboratory Researchers: MSc Eng. Wojciech Smagowski, Eng. Bozena Madej-Pawlowska

Duration: 11.07.2013-10.04.2017
Budget: 1 124 800 PLN

Keywords:  nonlinear vibrations, control, vibration modes, dynamics of rotating structures, active piezo-composite beams

Project summary: The major outcome of the project is the development of an authorial analytical model of the rotating system consisting of a rigid hub and flexible thin-walled composite beams. The model is represented by a set of partial differential nonlinear equations and associated boundary conditions. The derived equations are general ones and may be used for describing dynamics of any cross-section blades, including both open and closed crosssection profiles and an arbitrary laminate stacking sequence. The model is also suitable for nonsymmetric and multimaterial composites and takes into account a case of non-constant angular velocity. The derived relations may be used for an optimum laminate design to obtain requested mutual couplings of beam deformations upon system rotation. Detailed analytical calculations have been performed concerning box beams exhibiting circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS) profile properties. In a case of CUS lamination scheme the impact of rotor speed and laminate reinforcing fibres orientation on system natural frequencies and mode shapes have been examined. The significance of a piezoelectric patch embedded into master blade and its possible influence on dynamical characteristics of the rotor structure has been evaluated. It has been concluded, that even relatively small electrically isolated (i.e. disconnected from the power supply) piezoceramic patches may crucially affect system modal properties especially with respect to the flap-wise and chord-wise bending modes. The computed Campbell diagrams confirmed that at certain rotating speeds the flap-wise and chord-wise bending natural frequencies cross-over effect occurs. In a case of the CAS configuration the orientation of composite fibres, corresponding to the maximum twisting-flapwise bending coupling has been determined. This configuration has been used in further studies regarding control algorithms in terms of developing the efficient control strategies capable of suppressing both twisting-bending vibration by means of just a single actuator. 
The elaborated model enables study of the rotor dynamics in case of combined rotation and in-plane hub translation. This leads to the nonlinear parametric resonances of the system that, under specific conditions, may result in high amplitudes of blades vibrations and system instabilities. The analytical model of the flexible beam has been extended to represent an advanced design of an electro-mechanical structure. An active transducer has been represented by an additional piezoelectric layer in the hosting beam laminate material. In the analytical formulation of the piezoceramics a mutual two-way coupling of electrical and mechanical domains has been taken into account as well as a material nonlinear constitutive relation with respect to the electrical field. This nonlinear effect has been observed at high structure deformations, specific for nearresonance conditions. The vibrations excited by periodic driving torque revealed a softening resonance curve.
The reduced models have been analysed taking into account hub dynamics, a number of blades (from one to three) and selected configurations of the composite beam structure. Analysis of the rotor with a nonideal energy source has been also included. An additional equation, modelling the dynamics of a DC motor and its output torque has been taken into account. The influence of the motor characteristics on the rotor dynamics and the significance of the motor sub-system performance has been shown. Bifurcation and stability analysis and chaotic zones of the hub-pendulum system has been determined. The presence of a non-ideal energy source has yielded smaller chaotic solution zones and a transition from chaotic to regular oscillations. Active vibration control strategies aimed at suppressing complex mutually coupled deformation modes, typical for rotating thin-walled composite beams has been proposed in the project. It has been proved the rotor hub inertia has a significant impact on the system dynamics, blade vibration modes and on the final choice of system control strategy. The control algorithms in which the boundary bending moment is a linear combination of the beam states at its root and free tip has been applied. In order to improve effectiveness of the control the nonlinear saturation controller has been proposed as well. This technique exploits the effect of an added virtual oscillator nonlinearly coupling output of the beam and input of the active element. The obtained results confirmed the efficiency of the nonlinear saturation control method for beam vibration suppression in near-by resonance zones. Another proposal of the control was the cubic velocity feedback method. To evaluate the effectiveness of the method two distinct measures have been proposed - namely the magnitude of vibrations amplitude reduction and, control system power required by MFC actuator.
The elaborated within the project scope analytical models have been verified on laboratory stands dedicated to rotating structures. To this aim multiple box beam specimens with laminate CUS and CAS configurations have been manufactured. To identify the material parameters and specimens structural properties a series of experimental modal analysis tests have been performed. The vibration modes and vibration amplitudes of two- and three-blades rotors at different loading conditions have been experimentally tested. Moreover, the efficiency of piezoelectric active elements embedded into hosting structure has been verified.