30 Identification of Nonlinear Characteristics of an Additive Manufactured Vibration Absorber 233 Fig. 30.3 Nonlinear stiffness (left) and damping (right) characteristics of the complaint mechanism. The nonlinear properties are obtained by collecting the data of the restoring force surface at .z =0 for the damping and at . ˙z =0 for the stiffness , , , , , , Fig. 30.4 Nonlinear equivalent SDOF oscillator (left) and experimental validation (right) of the identified model. The numerical solution is compared with the experimental data in the frequency range between 10.5 and 11.5 Hz for different excitation amplitudes 2.1. At each set-up, the input and output signals are recorded for about 10 seconds after reaching the steady-state condition. Then, the signal is filtered and integrated, and its peaks are averaged to estimate the amplitude of the system response. This process is repeated for each frequency and each input amplitude and it allows for measuring the nonlinear experimental TFs in different conditions. However, it is worth noticing that the output of the shaker, hence the base excitation of the system, is influenced by the dynamics of the complaint system. This results in the necessity of using the recorded experimental motion of the base as numerical input to numerically compute the dynamic response of the equivalent SDOF oscillator. Thus, the numerical integration is performed with the MATLAB function ode4 5 which is adopted to solve the following equation of motion, representing the nonlinear oscillator described in Fig.30.4: .m¨x +c1(˙x − ˙y) +c2,RFS(˙x − ˙y) 2 +c3,RFS(˙x − ˙y) 3 +k1,opt (x −y) +k2,RFS(x −y) 2 +k3,RFS(x −y) 3 =0 (30.4) where y represents the experimental base displacement; . ˙y denotes the experimental base velocity; .k1,opt and . c1 are linear coefficients of the secondary structure, described in Table 30.2 and Sect.30.3; .k2,RFS, .k3,RFS, .c2,RFS, and .c3,RFS represent the cubic and quadratic coefficients, identified in Table 30.3; and m is the dynamic mass of the oscillator. The numerical and experimental TFs are reported on the right side of Fig.30.4: the experimental data shows the typical reduction of resonance frequency at high excitation amplitudes, confirming the softening nature of the compliant mechanism stiffness. Moreover, the numerically computed nonlinear dynamic response is in good agreement with experimental data at both high and low response amplitudes; therefore, the identified equivalent oscillator and its coefficients can be considered validated.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==