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 bifurcation analysis of large-scale nonlinear mechanical systems
Abstract The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed. A new procedure for the tracking of Neimark–Sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed. The HB method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device.
Effects of the thyroid state on cholesterol metabolism in the rat.
An isotopic equilibrium method which permits the in vivo measurements of cholesterol turnover processes was applied to different groups of rats: (1) radiothyroidectomized, (2) low-iodine fed, and (3) L-thyroxin fed. Plasma cholesterol concentrations were enhanced after thyroidectomy and reduced by large dose of L-thyroxin. A low-iodine diet decreased plasma thyroxin level but did not affect plasma cholesterol. Thyroid levels in plasma modified the coefficient of intestinal absorption of cholesterol. After thyroidectomy or under conditions of reduced thyroxin formation this absorption coefficient was enhanced. The absorption coefficient of cholesterol was decreased in rats receiving 31 or 61.5 mug/day of L-thyroxin but was not changed in rats fed 110 mug/day L-thyroxin. The proportion of de novo biosynthetisized cholesterol eliminated into the feces (external secretion) was reduced while thyroxin levels were low. The fecal excretion of cholesterol and the in vitro exchange of cholesterol between erythrocytes and plasma were increased by L-thyroxin ingestion. The rate of cholesterol biosynthesis was decreased after thyroidectomy and enhanced by L-thyroxin feeding. In fact, changes in thyroid state modified indirectly the biosynthesis of cholesterol by its effects on metabolism and on the coefficient of intestinal absorption of cholesterol.