An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—Application to bolted structures
Aeronautical structures are commonly assembled with bolted joints in which friction phenomena, in combination with slapping in the joint, provide damping on the dynamic behavior. Some models, mostly nonlinear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute nonlinear response functions in the frequency domain. The basic idea is to develop the response as Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus this paper presents an original adaptive HBM which adjusts the number of retained harmonics for a given precision and for each frequency value. The new proposed algorithm is based on the observation of the relative variation of an approximate strain energy for two consecutive numbers of harmonics. The developed criterion takes the advantage of being calculated from Fourier coefficients avoiding time integration and is also expressed in a condensation case. However, the convergence of the strain energy has to be smooth on tested harmonics and this constitutes a limitation of the method. Condensation and continuation methods are used to accelerate calculation. An application case is selected to illustrate the efficiency of the method and is composed of an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. The practice of adaptive HBM shows that, for a given value of the criterion, the number of harmonics increases on resonances indicating that nonlinear effects are predominant. For each frequency value, convergence of approximate strain energy is observed. Emergence of third and fifth harmonics is noticed near resonances both on vibratory responses and on approximate strain energy. Parametric studies are carried out by varying the excitation force amplitude and the threshold value of the adaptive algorithm. Maximal amplitudes of vibration and frequency response functions are plotted for three different points of the structure. Nonlinear effects become more predominant for higher force amplitudes and consequently the number of retained harmonics is increased.
Nonlinear normal modes of a two degrees-of-freedom piecewise linear system
Abstract A study of the Nonlinear Normal Modes (NNMs) of a two degrees of freedom mechanical system with a bilateral elastic stop for one of them is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. The Harmonic Balance Method (HBM) with a large number of harmonics, combined with the Asymptotic Numerical Method (ANM), is used to solve the regularized problem. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non-regular problem. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period, internal resonance and instabilities. On the other hand, the second NNM presents a more simple, almost linear, dynamics. The two NNMs converge asymptotically (for an infinite energy) toward two other Linear Normal Modes, corresponding to the system with a gap equal to zero.
Brake squeal: Linear and nonlinear numerical approaches
“Brake squeal” groups a large set of high-frequency sound emissions from brake systems. They are generated during the braking phase and are characterized by a harmonic spectrum. The onset of squeal is due to an unstable behaviour occurring in linear conditions during the braking phase, and a general approach used by several authors to determine the system instabilities is the complex eigenvalues analysis. When the brake begins to squeal, the response of the system reaches a new limit cycle where the linear models cannot be used anymore. This paper presents the integration of two different numerical procedures to identify the mechanism bringing to squeal instability and to analyse its dynamics. The first approach is a finite element modal analysis of the brake system and is used to identify its eigenvalues and to relate them to the squeal occurrence. The second one is a specific finite element programme, Plast3, appropriate for nonlinear dynamic analyses in the time domain and is particularly addressed to study contact problems with friction between deformable bodies. This programme computes the contact stresses and permits to determine the dynamics of the system along the contact surface, both in the linear and nonlinear fields. The two models are compared and the onset of squeal is predicted both in the frequency domain by the linear model and in the time domain by the nonlinear one. The instability predictions, obtained by the two models, are discussed. To simplify the dynamics of its components, the study is carried out on a simple model, made of a disc, a small friction pad and a beam supporting the pad. The geometry of the model is related to an experimental set-up used to validate the models and to compare the numerical results with the experiments.