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• Continuation and bifurcation analysis of beams in confined axial flow. • Flutter instabilities can have both sub- and super-critical character. • Flow nonlinear effects are the root cause for sub-critical hopf bifurcations. • Proposed method for an "augmented" linear stability analysis. • Nonlinear dynamics include hysteresis, internal resonances and quasi-periodicity. The flutter of cantilevered beams in channel flow is a benchmark example of flow-induced vibrations and its fundamental behaviour is found in numerous practical applications. Experiments have shown that such systems present a wide variety of complex nonlinear behaviour. However, the plethora of previous studies is mostly concerned with linear stability analysis. In this work, we provide an initial impulse for a comprehensive nonlinear study of these systems through bifurcation analysis. We consider a one-dimensional problem, where a cantilevered beam is treated in a modal framework and the surrounding flow is modelled by bulk-flow equations. The system is discretized in space and time via Galerkin procedures (modal, Tau and harmonic balance) and the continuation of periodic solutions is achieved using the asymptotic numerical method. Additionally, a numerical method for an "augmented" linear stability analysis is proposed, allowing the continuation of Hopf bifurcation branches, including their sub- or super-critical nature. The nonlinear dynamics are explored with respect to various dimensionless parameters. Results illustrate a number of behavioural trends: sub-critical bifurcations and hysteresis loops, internal resonances, grazing boundaries (separation between limit cycles with and without intermittent beam-wall impacts) as well as torus bifurcations and quasi-periodic oscillations. [ABSTRACT FROM AUTHOR]

Periodic solutions of systems with friction are difficult to investigate because of the non-smooth nature of friction laws. This paper examines periodic solutions and most notably stick–slip, on a simple one-degree-of-freedom system (mass, spring, damper, and belt), with Coulomb's friction law, and with a regularized friction law (i.e. the friction coefficient becomes a function of relative speed, with a stiffness parameter). With Coulomb's law, the stick–slip solution is constructed step by step, which gives a usable existence condition. With the regularized law, the Asymptotic Numerical Method and the Harmonic Balance Method provide bifurcation diagrams with respect to the belt speed or normal force, and for several values of the regularization parameter. Formulations from the Coulomb case give the means of a comparison between regularized solutions and a standard reference. With an appropriate definition, regularized stick–slip motion exists, its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a methodology to map the various acoustic regimes of wind instruments. The maps can be generated in a multidimensional space consisting of design, control parameters, and initial conditions. The boundaries of the maps are obtained explicitly in terms of the parameters using a Support Vector Machine (SVM) classifier as well as a dedicated adaptive sampling scheme. The approach is demonstrated on a simplified clarinet model for which several maps are generated based on different criteria. Examples of computation of the probability of occurrence of a specific acoustic regime are also provided. In addition, the approach is demonstrated on a design optimization example for optimal intonation. [Copyright &y& Elsevier]

Abstract: Self-sustained musical instruments (bowed string, woodwind and brass instruments) can be modelled by nonlinear lumped dynamical systems. Among these instruments, flutes and flue organ pipes present the particularity to be modelled as a delay dynamical system. In this paper, such a system, a toy model of flute-like instruments, is studied using numerical continuation. Equilibrium and periodic solutions are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches, as well as “jumps” between periodic solution branches. The influence of a second model parameter (namely the inharmonicity) on the behaviour of the system is addressed. It is shown that harmonicity plays a key role in the presence of hysteresis or quasiperiodic regime. Throughout the paper, experimental results on a real instrument are presented to illustrate various phenomena, and allow some qualitative comparisons with numerical results. [Copyright &y& Elsevier]

Combining the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time-domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called asymptotic numerical method (ANM) continuation technique. Several classical examples are presented to illustrate the use of this numerical approach. [Copyright &y& Elsevier]

Some bowed string instruments such as cello or viola are prone to a parasite phenomenon called the wolf tone that gives rise to an undesired warbling sound. It is now accepted that this phenomenon is mainly due to an interaction between a resonance of the body and the motion of the string. A simple model of bowed string instrument consisting of a linear string with a mass–spring boundary condition (modeling the body of the instrument) and excited by Coulomb friction is presented. The eigenproblem analysis shows the presence of a frequency veering phenomenon close to 1 : 1 resonance between the string and the body, giving rise to modal hybridation. Due to the piecewise nature of Coulomb friction, the periodic solutions are computed and continued using a mapping procedure. The analysis of classical as well as non-smooth bifurcations allows us to relate warbling oscillations to the loss of stability of periodic solutions. Finally, a link is made between the bifurcations of periodic solutions and the minimum bow force generally used to explain the appearance of the wolf tone. [ABSTRACT FROM AUTHOR]

The continuation of quasi-periodic solutions for autonomous or forced nonlinear systems is presented in this paper. The association of the Asymptotic Numerical Method, a robust continuation method, and a two-frequency Harmonic Balance Method, is performed thanks to a quadratic formalism. There is no need for a priori knowledge of the solution: the two pulsations can be unknown and can vary along the solution branch, and the double Fourier series are computed without needing a harmonic selection. A norm criterion on Fourier coefficients can confirm a posteriori the accuracy of the solution branch. On a forced system, frequency-locking regions are approximated, without blocking the continuation process. The continuation of these periodic solutions can be done independently. On an autonomous system an example of solution is shown where the number of Fourier coefficients is increased to improve the accuracy of the solution. [ABSTRACT FROM AUTHOR]

Abstract: The high-order purely frequency-based harmonic balance method (HBM) presented by Cochelin and Vergez (2009) [1] and extended by Karkar et al. (2013) [2] now allows to follow the periodic solutions of regularized non-smooth systems (stiff systems). This paper compares its convergence property to a reference method in applied mathematics: orthogonal collocation with piecewise polynomials. A first test is conducted on a nonlinear smooth 2 degree-of-freedom spring mass system, showing better convergence of the HBM. The second test is conducted on a one degree-of-freedom vibro-impact system with a very stiff regularization of the impact law. The HBM continuation of the nonlinear mode was found to be very robust, even with a very large number of harmonics. Surprisingly, the HBM was found to have a better convergence than the collocation method for this vibro-impact system. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the method proposed by Cochelin and Vergez [A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324 (2009) 243–262] to the case of non-polynomial nonlinearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems. The principle remains to transform the original ODE system into an extended polynomial quadratic system for an easy application of the harmonic balance method (HBM). The transformation of non-polynomial terms is based on the differentiation of state variables with respect to the time variable, shifting the nonlinear non-polynomial nonlinearity to a time-independent initial condition equation, not concerned with the HBM. The continuation of the resulting algebraic system is here performed by the asymptotic numerical method (high order Taylor series representation of the solution branch) using a further differentiation of the non-polynomial algebraic equation with respect to the path parameter. A one dof vibro-impact system is used to illustrate how an exponential nonlinearity is handled, showing that the method works at very high order, 1000 in that case. Various kinds of nonlinear functions are also treated, and finally the nonlinear free pendulum is addressed, showing that very accurate periodic solutions can be computed with the proposed method. [Copyright &y& Elsevier]

The concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments. Using a modal expansion of the impedance of the instrument, and by projecting the equation for the acoustic pressure on the normal modes of the air column, a system of second-order ordinary differential equations is obtained. The equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed. The system is treated using an amplitude-phase formulation for nonlinear modes, where the frequency and damping functions, as well as the invariant manifolds in the phase space, are unknowns to be determined. The formulation gives, without explicit integration of the underlying ordinary differential equation, access to the transient, the limit cycle, its period and stability. The process is illustrated for a model reduced to three normal modes of the air column. [Copyright &y& Elsevier]

Nonlinear Energy Sinks (NESs) are a promising technique for passively reducing the amplitude of vibrations. Through nonlinear stiffness properties, a NES is able to passively and irreversibly absorb energy. Unlike the traditional Tuned Mass Damper (TMD), NESs do not require a specific tuning and absorb energy over a wider range of frequencies. Nevertheless, they are still only efficient over a limited range of excitations. In order to mitigate this limitation and maximize the efficiency range, this work investigates the optimization of multiple NESs configured in parallel. It is well known that the efficiency of a NES is extremely sensitive to small perturbations in loading conditions or design parameters. In fact, the efficiency of a NES has been shown to be nearly discontinuous in the neighborhood of its activation threshold. For this reason, uncertainties must be taken into account in the design optimization of NESs. In addition, the discontinuities require a specific treatment during the optimization process. In this work, the objective of the optimization is to maximize the expected value of the efficiency of NESs in parallel. The optimization algorithm is able to tackle design variables with uncertainty (e.g., nonlinear stiffness coefficients) as well as aleatory variables such as the initial velocity of the main system. The optimal design of several parallel NES configurations for maximum mean efficiency is investigated. Specifically, NES nonlinear stiffness properties, considered random design variables, are optimized for cases with 1, 2, 3, 4, 5, and 10 NESs in parallel. The distributions of efficiency for the optimal parallel configurations are compared to distributions of efficiencies of non-optimized NESs. It is observed that the optimization enables a sharp increase in the mean value of efficiency while reducing the corresponding variance, thus leading to more robust NES designs. [ABSTRACT FROM AUTHOR]