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.
Simulation of Payne effect of elastomeric isolators with a harmonic balance method
In the presented work, a non linear effect of rubber referred as Fletcher-Gent effect or Payne effect is investigated. It leads to a change in the rubber dynamic modulus with vibration amplitudes and, consequently, modifies resonance frequencies of mechanical systems including non linear elastomers. In this study a new methodology is developed to take into account Payne effect in a linear viscoelastic rubber material. Small vibration amplitudes around a no-preloaded state are predicted by considering frequency and amplitude dependencies of the material. This methodology has the advantage of using tabular experimental data from characterization tests which avoids the development of a complex model. In order to compute frequency responses, the non linear harmonic balance method is used and, for each iteration, new rubber properties are affected at each element according to its strain state. An equivalent strain measure is evaluated from the element strain energy density. This equivalent strain allows to associate dynamic properties of a material element subjected to multiaxial strain state with experimental dynamic properties of a material sample subjected to an uniaxial strain state. Practically, DMAP alter procedures are developed in order to evaluate energies in models defined with MSC.Nastran and the non linear solver is developed with Matlab. The method is applied on a satellite instrument isolator including four non linear rubber mounts. A non homogeneous spatial distribution of element equivalent strains is observed. Moreover, the maximum equivalent strain varies with frequency. These two observations validate the use of a specific methodology to deal with amplitude dependency of rubber.
A methodology for the determination of dynamic instabilities in a car disc brake
The dynamics of a car disc brake system is investigated by a combined analytical and numerical method. The disc is rotated past the stationary pads and calliper in sliding friction at constant speed. The modal data of the disc are obtained by means of modal testing whereby a dynamic model for the disc is derived based on the thin plate theory. Then the pads, calliper and mounting are analysed by means of the finite element method. Finally the equations of motion for the whole disc brake system are established through the interfaces between the pads and the disc. The stability of the vibrating system is studied by the method of state space.