Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber

The nonlinear tuned vibration absorber (NLTVA) is a recently-developed nonlinear absorber which generalizes Den Hartog's equal peak method to nonlinear systems. If the purposeful introduction of nonlinearity can enhance system performance, it can also give rise to adverse dynamical phenomena, including detached resonance curves and quasiperiodic regimes of motion. Through the combination of numerical continuation of periodic solutions, bifurcation detection and tracking, and global analysis, the present study identifies boundaries in the NLTVA parameter space delimiting safe, unsafe and unacceptable operations. The sensitivity of these boundaries to uncertainty in the NLTVA parameters is also investigated.Comment: Journal pape

Nonlinear Generalization of Den Hartog's Equal-Peak Method

This study addresses the mitigation of a nonlinear resonance of a mechanical system. In view of the narrow bandwidth of the classical linear tuned vibration absorber, a nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA), is introduced in this paper. An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is tailored according to the nonlinear restoring force of the primary system. The NLTVA parameters are then determined using a nonlinear generalization of Den Hartog's equal-peak method. The mitigation of the resonant vibrations of a Duffing oscillator is considered to illustrate the proposed developments

Nonlinear normal modes, modal interactions and isolated resonance curves

The objective of the present study is to explore the connection between the nonlinear normal modes of an undamped and unforced nonlinear system and the isolated resonance curves that may appear in the damped response of the forced system. To this end, an energy balancing technique is used to predict the amplitude of the harmonic forcing that is necessary to excite a specific nonlinear normal mode. A cantilever beam with a nonlinear spring at its tip serves to illustrate the developments. The practical implications of isolated resonance curves are also discussed by computing the beam response to sine sweep excitations of increasing amplitudes.Comment: Journal pape

The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems

The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to extend the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed. A new procedure for the tracking of Neimark-Sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed. As an application, the frequency response of a structure spacecraft is studied, together with two nonlinear phenomena, namely quasiperiodic oscillations and detached resonance curves. This example illustrates how bifurcation tracking using the HB method can be employed as a promising design tool for detecting and eliminating such undesired behaviors

Continuation of bifurcations of periodic solutions based on the harmonic balance method

This paper proposes to extend the harmonic balance method to perform the continuation of bifurcations of periodic solutions in a codimension-2 parameter space. Based on the framework of bordering techniques, the procedure utilizes Hill’s method to augment the system of equations in order to track bifurcations. As an application to validate the procedure, the design of a nonlinear vibration absorber attached to a Duffing oscillator is studied

Performance and Robustness of Nonlinear Systems Using Bifurcation Analysis

Nonlinear vibrations can be frequently encountered in engineering applications, and take their origin from different sources including contact, friction or large displacements. Other manifestations of nonlinearities are peculiar phenomena such as amplitude jumps, quasi-periodic oscillations and isolated response curves. These phenomena are closely related to the presence of bifurcations in the frequency response, which dictate the system's dynamics. While recent progress has been achieved to develop tools for nonlinear modal analysis of industrial applications, bifurcation analysis was still limited to reduced models and academic case studies. Along with the lack of an efficient algorithm to detect and study bifurcations, bifurcation analysis for design purposes also remained unexplored. The fundamental contribution of this doctoral thesis is the development of a new methodology for the detection, characterization and tracking of bifurcations of large-scale mechanical systems. To this end, an extension of the harmonic balance (HB) method is proposed. Taking advantage of the efficiency of the HB method for the continuation of nonlinear normal modes and frequency responses, this extension allows for robust computation of bifurcation curves in the system's parameter space. A validation of the methodology is performed on the strongly nonlinear model of an Airbus Defence & Space spacecraft, which possesses an impact-type nonlinear device consisting of multiple mechanical stops limiting the motion of an inertia wheel mounted on an elastomeric interface. The second main contribution is the development of a new vibration absorber, the nonlinear tuned vibration absorber (NLTVA), which generalizes Den Hartog's equal-peak method to nonlinear systems. The absorber is demonstrated to exhibit unprecedented performance for the mitigation of nonlinear resonances. In a second step, the HB-based bifurcation methodology is utilized to characterize the performance regions of the NLTVA, and to ensure its robustness with respect to parameter uncertainties

HARMONIC BALANCE COMPUTATION OF THE NONLINEAR FREQUENCY RESPONSE OF A THIN PLATE

The harmonic balance method (HBM) is used to investigate the dynamical behavior of the geometrical nonlinear plate. The middle plane displacements are included in the plate in which the equations of motion are developed by the principle virtual work. Moreover, the nonlinear frequency response curves, or NFRCs, are obtained by a continuation method