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.
Energy Dissipation in Spacecraft Structures Incorporating Bolted Joints Operating in Macroslip
The energy dissipation capacity of bolted joints operating in macroslip in a simple satellite model was investigated. A novel way of modeling the macroslip behavior was introduced and it was shown to work well. The method was established by setting the behavior of an elastoplastic element pair to match the experimental response of a bolted joint in macroslip. This pair consisted of two elastoplastic solid elements forming a unit, appropriately constrained by multipoint constraints. Experiments carried out on real satellite joint coupons were used to define the response of the joints for use in the modeling work. The energy dissipated by the joints was obtained by calculating the area under the force-displacement hysteresis loop in the elements through nonlinear transient analyses. It was found that if some of the joints operated in the macroslip region the energy dissipating capability of the joints could be increased significantly. The response of the satellite with these joints was investigated and it was found that vibration levels were reduced significantly.
Non-linear vibrations of a simple–simple beam with a non-ideal support in between
Abstract A simply supported Euler–Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted.