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.
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.
TGV disc brake squeal
The discomfort generated by the noise emission of braking systems in trains has aroused recently many studies on the mechanical modelling of brake noise in France. A theoretical and numerical discussion on the phenomenon of brake squeal is given in this paper in relation with some experimental data. This study is based upon a flutter instability analysis giving unstable modes of the brake system under the contact and Coulomb friction.