36 Adaptive Harmonic Balance Analysis of Dry Friction Damped Systems 413 Table 36.2 Comparison of calculation times for different approach 1 AHBM/SAI calculations Tolerance Time (min) 8 10 2 3:2 1 10 2 5:0 4 10 3 6:8 −1.5 −1 −0.5 0 0.5 1 1.5 ·10−2 −2 −1 0 1 2 · 103 Relative displacement / mm Transmitted joint force / N HBM MHBM AHBM, tol =8·10-2 AHBM, tol =1·10-2 AHBM, tol =4·10-3 Fig. 36.10 Friction hystereses for approach 1 AHBM/SAI calculations with different tolerances 36.4 Conclusions In this contribution an Adaptive Harmonics Balance Method is presented in order to perform calculations on a jointed structure in the frequency domain. The AHBM combines the advantages of classical HBM and MHBM. Two different approaches for the application of the AHBM are shown with both delivering different results. Approach 1 tends to bring out ‘rattling’, nervous behavior of single higher harmonics, but also manages to detect modal interactions for a wide range of tolerances. Approach 2 has smooth harmonic curves and a very good reproduction of the resonance peak. For both approaches the calculation times can be decreased compared to a full MHBM. The combination of both, the general modeling approach via ZT elements and the efficient computation using the AHBM brings out a powerful methodology for the calculation of the dynamics of jointed structures in the framework of the FEM. References 1. Cameron, T.M., Griffin, J.H.: An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. J. Appl. Mech. 56(1), 149–154 (1989) 2. Cardona, A., Lerusse, A., Géradin, M.: Fast Fourier nonlinear vibration analysis. Comput. Mech. 22(2), 128–142 (1998) 3. Cochelin, B., Vergez, C.: A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J. Sound Vib. 324(1–2), 243–262 (2009) 4. Dunne, J.F., Hayward, P.: A split-frequency harmonic balance method for nonlinear oscillators with multi-harmonic forcing. J. Sound Vib. 295(3–5), 939–963 (2006) 5. Ferri, A.A.: On the equivalence of the incremental harmonic balance method and the harmonic balance-Newton Raphson method. J. Appl. Mech. 53(2), 455–457 (1986) 6. Geisler, J.: Numerische und experimentelle Untersuchungen zum dynamischen Verhalten von Strukturen mit Fügestellen. Ph.D. thesis, Friedrich-Alexander-Universität Erlangen (2010) 7. Grolet, A., Thouverez, F.: On a new harmonic selection technique for harmonic balance method. Mech. Syst. Signal Process. 30, 43–60 (2012) 8. Jaumouillé, V., Sinou, J.-J., Petitjean, B.: An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—application to bolted structures. J. Sound Vib. 329(19), 4048–4067 (2010) 9. Krack, M., Panning-von Scheidt, L., Wallaschek, J.: A high-order harmonic balance method for systems with distinct states. J. Sound Vib. 332(21), 5476–5488 (2013)
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