Thibaut Detroux

Nonlinear generalization of Den Hartog׳s equal-peak method

Abstract 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

Abstract 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 balance 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.

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

Abstract 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.

The harmonic balance method for advanced analysis and design of nonlinear mechanical systems

As a tool for analyzing nonlinear large-scale structures, the harmonic balance (HB) method has recently received increasing attention in the structural dynamics community. However, its use was so far limited to the approximation and study of periodic solutions, and other methods as the shooting and orthogonal collocation techniques were usually preferred to further analyze these solutions and to study their bifurcations. This is why the present paper intends to demonstrate how one can take advantage of the HB method as an efficient alternative to the cited techniques. Two different applications are studied, namely the normal modes of a spacecraft and the optimization of the design of a vibration absorber. The interesting filtering feature of the HB method and the implementation of an efficient bifurcation tracking extension are illustrated.

Experimental demonstration of a 3D-printed nonlinear tuned vibration absorber

Engineering structures are designed to be lighter and more flexible, hence reducing the extent of application of linear dynamic models. Concurrently, vibration mitigation is required for enhancing the performance, comfort or safety in real-life applications. Passive linear vibration absorbers are purpose-built, often designed using Den Hartog’s equal-peak strategy. However, nonlinear systems are known to exhibit frequency-energy-dependent oscillations which linear absorbers cannot effectively damp out. In this context, the paper introduces a new nonlinear tuned vibration absorber (NLTVA) whose nonlinear functional form is tailored according to the frequency-energy dependence of the nonlinear primary structure. The NLTVA design aims at ensuring equal peaks in the nonlinear receptance function for an as large as possible range of forcing amplitudes, hence generalizing Den Hartog’s method to nonlinear systems. Our focus in this study is on experimental demonstration of the NLTVA performance using a primary structure consisting of a cantilever beam with a geometrically nonlinear component at its free end. The absorber is implemented using a doubly-clamped beam fabricated thanks to 3D printing. The NLTVA performance is also compared with that of the classical linear tuned vibration absorber.

Experimental study of isolated response curves in a two-degree-of-freedom nonlinear system

In the present paper, the observation and characterization of isolated response curves (IRCs) are experimentally reported in the case of a nonlinear system consisting of two masses sliding on an horizontal guide. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. The existence of an IRC is related to a 3:1 internal resonance between the two modes of the system. The observed IRC is studied in detached and merged conditions using swept-sine excitations and system perturbations.

Isolated Response Curves in a Base-Excited, Two-Degree-of-Freedom, Nonlinear System

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulations are carried out to study the conditions of existence of isolated solutions, their bifurcations, their merging with the main response branch and their basins of attraction. This is achieved using tools including nonlinear normal modes, energy balance, harmonic balance-based continuation and bifurcation tracking, and global analysis.Copyright

The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems

Because nowadays structural engineers are willing to use or at least understand nonlinearities instead of simply avoiding them, there is a need for numerical tools performing analysis of nonlinear large-scale structures. Among these techniques, the harmonic balance (HB) method is certainly one of the most commonly used to study finite element models with reasonably complex nonlinearities. However, in its classical formulation the HB method is limited to the approximation of periodic solutions. For this reason, the present paper proposes to extend the method to the detection and tracking of codimension-1 bifurcations in the system parameters space. As an application, the frequency response of a 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.

Prediction of Isolated Resonance Curves Using Nonlinear Normal Modes

Isolated resonance curves are separate from the main nonlinear forced-response branch, so they can easily be missed by a continuation algorithm and the resonant response might be underpredicted. The present work explores the connection between these isolated resonances and the nonlinear normal modes of the system and adapts an energy balance criterion to connect the two. This approach provides new insights into the occurrence of isolated resonances as well as a method to find an initial guess to compute the isolated resonance curve using numerical continuation.The concepts are illustrated on a finite element model of a cantilever beam with a nonlinear spring at its tip. This system presents jumps in both frequency and amplitude in its response to a swept sinusoidal excitation. The jumps are found to be the result of a modal interaction that creates an isolated resonance curve that eventually merges with the main resonance branch as the excitation force increases. Excellent insight into the observed dynamics is provided with the NNM theory, which supports that NNMs can also be a useful tool for predicting isolated resonance curves and other behaviors in the damped, forced response.Copyright

Tracing a Prescribed Force-Displacement Curve Using Topology Optimization

The objective of this study is to develop an optimization methodology to trace a prescribed force-displacement curve. For this purpose, we employ the topology optimization to find the layout of a structure that minimizes the difference from the predefined force-displacement curve. In addition, we propose a constraint that is necessary for the initial design of the structure to escape from its inherent nonlinearities. The effectiveness of the proposed method is verified by performing a simulation with a thin plate.