33 Studies of a Geometrical Nonlinear Friction Damped System Using NNMs 345 The autonomous decay process is calculated as reference solutions y.t/ and Py.t/ usingNEWMARK-method with t D0.01s. Initiated on the isolated NNM an equivalent modal damping ıN is identified by the logarithmic decrement [11] for every oscillation period T D2 !R of the reference solution during the decay process ıN D 1 T lnˇ ˇ ˇ ˇ y.NT/ y.NT CT/ˇ ˇ ˇ ˇ (33.9) assuming that the modal damping is constant at least over one period. Due to the fact of an isolated resonance the calculation of the modal damping must be done for one DOF only. Because of the energy dissipation introduced by the friction damper the damping value is not constant in the whole decay process but only valid during the period it is identified. For decreasing energy level less and less Jenkin elements (see Fig. 33.2) are slipping, dependent on the sticking force distribution, and therefore the damping varies with energy and time. With the identified modal damping and the requested constant vibration frequency !R the general solution for a weak damped system can be written as qi.t/ De ıNŒt NT qi,0.N/ cos.!RŒt NT / C P qi,0.N/ CıNqi,0.N/ !R sin.!RŒt NT / (33.10) and Pqi.t/ De ıNŒt NT Pqi,0.N/ cos.!RŒt NT / ıNPqi,0.N/ CŒı 2 N C!2 R qi,0.N/ !R sin.!RŒt NT / (33.11) with NT t ŒNC1 T, N D0: : : tend T , qi,0.N/ D yi,0 )N D0 qi.NT/ )N >0 , Pqi,0.N/ D P yi,0 )N D0 Pqi.NT/ )N >0 (33.12) for every periodN. This is called the modal solution in this work. 33.3.2.3 Identification of Frequency-Independent System Configuration Thirdly, all modal decay processes qj.t/ and Pqj.t/, calculated with Eqs. (33.10) and (33.11), are compared to the j reference solutions of the decay processes yj.t/ and Pyj.t/. An optimal j and connected to that an optimal sticking force is identified by an error estimation between the reference and the modal solution. Therefore this comparison gives an indicator if the requested assumption of a constant oscillation frequency persists during the decay process. 33.4 Numerical Results In this section the described procedure is applied for different excitation amplitudes and linked to that for different energy levels. The system parameters of Table 33.1 are used. For every excitation amplitude (0.05, 0.1, 0.25 0,5 and 1 N) j D30 FRFs (number of harmonics in the MHBMnh D5) are calculated with varying normal force Fj H,tot. An normal force is found for F j H,tot D4.16N where the error between the reference solution and the model solution is minimal. The decay process initiated on the isolated resonance with an excitation amplitude of 1 N is shown in Fig. 33.4. For both degrees of freedom the displacement and velocity, calculated with NEWMARK integration (yi and Pyi) and the modal ansatz of Eqs. (33.10) and (33.11) (qi and Pqi) respectively, is shown. In two time zooms the congruence between the reference and the modal solution can be seen at different decay sectors (see Fig. 33.4b and c), including amplitude and time period. The modal damping identified during this process is shown in Fig. 33.5 where the ıN over the logarithm of the kinetic energy is diagrammed. With decreasing energy in the system the damping sinks as well due to the fact that less Jenkin elements
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