This paper introduces a model-order reduction technique for lightly damped nonlinear vibrating systems. By combining calculation details that are specific to the harmonic balance method, the asymptotic numerical method, and the normal form style parametrisation for invariant manifolds, a complete procedure that can cope with single-mode reduction is detailed. Introducing harmonic decomposition in the process allows for a different treatment of the temporal information of the solution, which comes with advantages as compared to normal form expansions based on polynomial expansions. The computation proceeds with two nested loops on both the harmonics and the polynomial degree expansion. A decisive advantage of the procedure is its ability to compute a new expansion from a known solution, which allows the derivation of amplitude-dependent piecewise reduced order models (ROMs), together with an integrated procedure that can switch from the invariant manifolds computation attached to either fixed points or limit cycles. Once the validity limit of a first expansion is met, the procedure can restart from a point where convergence is reached and produce a new ROM. This feature has the potential to overcome the well-known limitations of asymptotic expansions associated with the parametrisation method for invariant manifolds, and is derived here only for conservative systems. The whole analysis also clearly establishes the links existing between the normal form approach and computations based on the harmonic balance combined with the asymptotic numerical method. Examples of increasing complexity, starting from a Duffing equation, a two-degree-of-freedom system and a finite element beam model, are analysed, and comparisons with existing techniques are provided.

The parametrisation method for invariant manifolds is a powerful technique for deriving reduced-order models in the context of nonlinear vibrating systems with geometric nonlinearities, allowing accurate computations of nonlinear normal modes. Thanks to arbitrary order asymptotic expansions, converged results are within reach and directly applicable to finite element structures. However, since it relies on a local theory and asymptotic expansions, the results are only valid up to a given amplitude, which defines the convergence radius of the approximation. The aim of this contribution is to investigate the validity limits of the approach and review the existing error estimates, with the concrete objective of proposing a practical approach to estimate the validity range during the computation, thus producing safe bounds within which the reduced-order model can be used. Three different criteria are assessed. The first one uses the error in the invariance equation as the distance to the fixed point increases. The second one is adapted from an upper bound criterion derived for normal form transforms and based on the potential singularities of the homological operator. The third one uses validity limits of series expansion through Cauchy and d’Alembert rules, which can be tested either on the reduced dynamics coefficients or those of the nonlinear mappings. The criteria are tested on a number of different examples that are representative of the situations encountered when dealing with nonlinear vibrations. The Duffing equation serves as a first benchmark that allows considering conservative oscillations, forced systems at primary resonance, and superharmonic resonance. The investigations are then extended to a vibrating system with two degrees of freedom. Finally, the different criteria are assessed on a finite element beam structure, and guidelines are formulated to generalise their practical use and produce accurate and easy-to-use error bounds in the context of model order reduction for nonlinear vibrating structures.

This paper considers the computation of reduced-order models for systems of ordinary differential equations that include non-polynomial non-linearities. An targeted example is the case of a geometrically exact model of highly flexible slender structure, that includes, after space discretisation, trigonometric non-linear terms. With a suitable change of variables, this system can be rewritten in an equivalent one with polynomial non-linearities at most quadratic, at the price of introducing additional variables linked to algebraic equations, leading to a differential algebraic set of equations (DAE) to be solved. This DAE is reduced thanks to a normal form parametrisation of its invariant manifolds and selecting a set of master ones. Arbitrary order expansions are detailed for the coefficients of the change of variable and the reduced dynamics, using linear algebra in the space of multivariate polynomials of a given degree. In the case of a single non-linear mode reduction, a criterion to evaluate the quality of the normal form results is also proposed based on an estimation of the convergence radius of the polynomial asymptotic expansion representing truncated series. The method is then applied to compute a single mode reduction of three test cases -- a Duffing oscillator, a simple pendulum and a clamped clamped beam with von~K\'arm\'an model --, in order to investigate the effect of the algebraic part of the DAE on the quality of the model reduction and its validity range. Then, the more involved case of a cantilever beam modelled by geometrically exact finite elements is considered, underlining the ability of the method to produce accurate and converged results in a range of amplitude that can be bounded thanks to a convergence criterion.

This article deals with the experimental validation of a theoretical model describing the dynamics of a special class of centrifugal pendulum vibration absorbers, designed to reduce the torsional vibrations of rotating machines. The original architecture proposed in this work consists in cylindrical-shaped masses rolling onto each other and acting as double pendulum absorbers. The main interest of these double pendulums lies in the two antiresonances they generate on the main system, which allow to reduce the vibrations at two harmonic orders, unlike standard single pendulum absorbers. The measurements are conducted on an architecture with six double pendulums. They focus on the first rotor antiresonance in the linear and nonlinear regimes, and they are compared to theoretical results from the literature. To the authors' knowledge, this is the first experimental observation of one of the rotor's torsional antiresonances and the first experimental validation of a double pendulum absorber model.

Centrifugal double pendulum vibration absorbers (CDPVAs) can be used to reduce torsional vibrations of rotating machines. These passive devices are made of several double pendulums oscillating relatively to a rotor. This study extends former works on CDPVAs by accounting for the rotational inertia of the pendulums and providing detailed linear and nonlinear analyses of CDPVA dynamics. First, the eigenmodes are computed and an efficient design procedure based on the linear response is proposed. Then, the nonlinear behaviour is assessed using an analytical perturbation method. Of particular interest is the nonlinear antiresonance detuning of the double pendulums, which strongly influences vibration reduction. Moreover, CDPVAs are subjected to nonlinear energy localisation. This causes the double pendulums to oscillate differently, thus affecting the proper operation of the system. The analytical results led to new design guidelines that minimise the antiresonance detuning while avoiding instabilities leading to localised responses. These results are validated through a comparison with a numerical resolution of the system's dynamics. They are then visualised in the design space to help identify easily the optimal CDPVA designs.

This article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure’s amplitude.

Rotating machines are often subjected to fluctuating torques, which causes rotor vibrations, early wear and noise pollution. These vibrations can be reduced using centrifugal pendulum vibration absorbers (CPVAs), which are passive devices made of several bodies (pendulums) oscillating along a given path and rotating relatively to a rotor. Previous studies showed that the dynamics of these devices is subjected to instabilities leading to a localisation of the motion of the pendulums. In this paper, the localised behaviour of a CPVA made of two pendulums allowed to rotate about their centre of mass is investigated. To this aim, a dynamical model based on an analytic perturbation method is established. The aim of this model is to highlight some special features of the localised response, such as the appearance of quasi -periodic regimes. The case studies showed that in some special cases, localisation can improve the filtering efficiency as compared to a unison motion. The validity of the model was confirmed through a comparison with numerical resolutions of the system's dynamics.

In this paper, a novel method for computing the nonlinear dynamics of highly flexible slender structures in three dimensions (3D) is proposed. It is the extension to 3D of a previous work restricted to in-plane (2D) deformations. It is based on the geometrically exact beam model, which is discretized with a finite element method and solved entirely in the frequency domain with a harmonic balance method (HBM) coupled to an asymptotic numerical method (ANM) for continuation of periodic solutions. An important consideration is the parametrization of the rotations of the beam's cross sections, much more demanding than in the 2D case. Here, the rotations are parametrized with quaternions, with the advantage of leading naturally to polynomial nonlinearities in the model, well-suited for applying the ANM. Because of the HBM-ANM framework, this numerical strategy is capable of computing both the frequency response of the structure under periodic oscillations and its nonlinear modes (namely its backbone curves and deformed shapes in free conservative oscillations). To illustrate and validate this strategy, it is used to solve two 3D deformations test cases of the literature: a cantilever beam and a clamped-clamped beam subjected to one-to-one (1:1) internal resonance between two companion bending modes in the case of a nearly square cross section.

Centrifugal double pendulum vibration absorbers (CDPVAs) can be used to reduce torsional vibrations of rotating machines. These passive devices are made of several double pendulums oscillating relatively to a rotor. In this work, an original CDPVA architecture made of cylindrical-shape pendulums is proposed. Measurements on a CDPVA made of six double pendulums are performed around the first rotor antiresonance. To the authors' knowledge, this is the first observation of an antiresonance of the rotor and the first comparison of an analytical CDPVA model with experimental results. The first rotor antiresonance is observed where expected and its nonlinear detuning as the forcing amplitude increases is well predicted by the model. Discrepancies between the experimental and analytical results are also observed, mostly regarding the rotor's amplitude after the antiresonance. These are likely due to the slipping of the pendulums and a limitation of the test-bed used for the experiments.

In this paper, we generalize the Koopman-Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman-Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von K & aacute;rm & aacute;n beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

An original method for the simulation of the dynamics of highly flexible slender structures is presented. The flexible structures are modeled via a finite element (FE) discretization of a geometrically exact two-dimensional beam model, which entirely preserves the geometrical nonlinearities inherent in such systems where the rotation of the cross-section can be extreme. The FE equation is solved by a combination of harmonic balance (HBM) and asymptotic numerical (ANM) methods. The novel solving scheme is rooted entirely in the frequency domain and is capable of computing both the structure’s frequency response under periodic external forces as well as its nonlinear modes. An overview of the proposed numerical strategy is outlined and simulations are shown and discussed in detail for several test cases.

Centrifugal pendulum vibration absorbers (CPVAs) are often used by the automotive industry to reduce vibrations of the drivetrain. These passive devices consist of several masses oscillating along a given path relative to a rotor. Recent CPVA systems make use of rocking pendulums, meaning that the pendulums rotate about their centre of mass during their motion along their path. In this work, measurements on a new CPVA architecture made of ball-type rocking pendulums are performed in classical and subharmonic operations. To the authors' knowledge, they lead to the first comparisons between experimental and analytical results regarding the nonlinear detuning of the rotor's antiresonance and the saturation of the rotor's response. It is also the first time this saturation phenomenon is observed experimentally. The CPVA investigated is shown to have a high filtering efficiency but its pendulums are subjected to slipping, which limits the operating range of the system. Experimental and analytical investigations of the slipping are carried-out to estimate the limit of adherence of the system.

In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.

A novel method for the numerical computation of the nonlinear normal modes (NNMs) of a highly flexible cantilever beam is presented. The flexible cantilever is modeled using a 2D finite element discretization of the geometrically exact beam model, wherein geometric nonlinearities relating to the rotation are kept entirely intact. The model is then solved using the proposed solution method, which is fully frequency domain-based and involves a novel combination of a harmonic balance (HBM) Fourier expansion with asymptotic numerical (ANM) continuation for periodic solutions. The NNMs are also calculated experimentally using a flexible cantilever specimen mounted to a shaker table. The experimental NNMs can be compared to their numerical counterparts in order to validate the frequency domain numerical technique.

Centrifugal pendulum vibration absorbers (CPVAs) are passive devices used to reduce torsional vibrations in rotating machines. Previous works showed that a CPVA configuration with two pendulums oscillating in phase opposition and at half the excitation frequency is efficient in reducing the rotor’s vibrations. This paper deals with a new generation of CPVAs, in which the pendulums admit a rotational motion relative to the rotor in addition to the traditional translational motion. The aim of this study is to assess the dynamic stability of a particular subharmonic solution of CPVAs composed of several pairs of pendulum. To do so, a new method based on an analytical perturbation technique is proposed. It leads to more general conclusions than previous studies as the results are derived for CPVAs with any even number of rocking pendulums. The validity of the analytical model is confirmed through a comparison with numerical resolutions of the system’s dynamics, and new design guidelines are proposed.

The automotive industry uses centrifugal pendulum vibration absorbers (CPVAs) to reduce vibrations of the transmission system. These passive devices are made of several masses oscillating along a given path relative to a rotor. This work addresses a recent design of CPVA, in which the pendulums are allowed to rotate relatively to the rotor. The dynamic stability of this CPVA and the shifting of its operating point are investigated in this paper. These two aspects, crucial for an optimal vibration reduction, are assessed using an analytic dynamical model based on a perturbation method. The results obtained allow to propose new design guidelines. The validity of the model is confirmed through a comparison with a numerical resolution of the system's dynamics.

In this article, we focus on the study of a forced single degree of freedom Hill-type differential equation, which can be found in particular in the modeling of gear interactions. We consider that the variable part of the stiffness is composed of several harmonics, and that the forcing term contains a constant part and a single harmonic component. The response of the equation is estimated using methods such as the multiple scale method and the harmonic balance method, and the results are compared to the ones obtained using numerical time integration. The phenomenon of parametric amplification, classically observed when the second harmonic of the parametric driving interacts with the first harmonic of the direct driving, is here observed and quantitatively investigated in the case of more complex direct and parametric interactions. The simplest case is the one with a constant direct forcing, seldom described in the literature. Then, more complex situations with both harmonic and constant forcing leads to parametric amplification with more complex behavior.