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.
Identification of Properties of Nonlinear Joints Using Dynamic Test Data
This paper presents a general purpose technique for identifying (extracting) the dynamic properties of nonlinear joints using dynamic test data. No theoretical model is required, only experimental data are needed for the identification. Because no assumption on the properties of the nonlinear jointly is made, the method is applicable for identifying almost any type of nonlinear joints. In the identification method, the nonlinear force at a joint is treated as an external force and the principle of multi-harmonic balance is employed. The force-response relationship of a nonlinear joint is then obtained. Various measurement errors in experimental data can affect the accuracy of identification. In this paper, efforts are also devoted to understanding how the accuracy of identification is affected by measurement errors. Techniques for improving the accuracy of identification are also discussed. The method is verified by experimental studies. The nonlinear properties of an impact joint and a friction joint have been successfully identified.
Representation of systems disturbed by wide band noise
In this paper, a method of handling and working with wide band noise is developed. We represent wide band noise as a distributed delay of white noise and use it to reduce a nonlinear system disturbed by wide band noise to a nonlinear system disturbed by white noise. An application of this reduction to a nonlinear filtering problem under a wide band noise disturbance is discussed.