232 C. Martinelli et al. Fig. 30.2 Comparison between the experimental and analytical TFs for the amplitude (left) and phase (right) of the complaint mechanism. The original analytical transfer function is computed with the coefficients of Table 30.2, while the optimised transfer function is computed with the same coefficients but the optimal stiffness. kopt,1 Table 30.3 Damping and stiffness coefficients identified from the RFS. The polynomial fitting is performed with a third-degree polynomial Parameter First degree coeff. Second degree coeff. Third degree coeff. Stiffness .k1,RFS =40.9N/m .k2,RFS =−248.3N/m. 2 .k3,RFS =−49271.0N/m. 3 Damping .c1,RFS =0.0548Ns/m .c2,RFS =0.0500 Ns/m. 2 .c3,RFS =0.0387Ns/m. 3 where x represents the absolute displacement of the compliant mechanism tip, z is the relative displacement of the compliant mechanism tip with respect to the base, .c(˙z) and .k(z) denote the nonlinear damping and stiffness coefficients, and m is the dynamic mass of the oscillator equal to .0.0083 Kg. This equation can be used to create the restoring force surface (RFS) from which it is possible to identify the nonlinear damping.c(˙z) and the nonlinear stiffness.k(z) by considering, respectively, the intersection of the RFS and the plane passing for .z =0 and for . ˙z =0. The RFS is obtained by considering 20 different experimental tests—each test is performed by providing a sinusoidal voltage input to the shaker with a frequency of 10.7 Hz and different voltage input amplitudes, namely, an input amplitude of 0.686 V with the following amplification factors: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0. For each input condition, an accelerometer measures the absolute acceleration of the tip mass, while the laser measures the absolute velocity of the compliant mechanism base with the National Instruments unit NI-9234. Then, the recorded signals are integrated, derived, filtered, and manipulated to obtain the absolute acceleration . ¨x, the relative velocity . ˙z, and the relative displacement z of the system. Such quantities can be used to create the RFS with the definition of Eq.30.3, and the resulting nonlinear properties are reported in Fig.30.3. The figure shows the presence of a symmetric cubic softening stiffness which reduces the stiffness of the system when large deformations are imposed. On the other hand, the damping reveals an asymmetric quadratic characteristic which produces high damping at positive relative velocities and low asymptotic damping at negative relative velocities. This behaviour could be triggered by the imperfect shape of the compliant mechanisms which is due to the fabrication process [6, 21, 22]: indeed, during the vibration, the system is not deforming in the same way in the two oscillation directions, inducing the generation of an asymmetric nonlinear characteristic in the system response. The identified nonlinear characteristics show a continuous shape, which is easily fitted with a third-degree polynomial using the MATLAB function polyfit. The results of the fitting process are numerically and graphically reported in Table 30.3 and in Fig.30.3. The values of the first-order parameters of Table 30.3, i.e. .k1,RFS and .c1,RFS, are very similar to the linear coefficients .k1,opt and . c1 previously identified for the compliant mechanism; this confirms the robustness of results of the RFSM and allows using the identified higher-order coefficients in the validation process. The nonlinear validation is performed by adopting sets of nonlinear experimental data which are completely different from the one adopted in the identification process. Such data represent the TFs1 between the shaker voltage input and the displacement output signal of the tip mass. A discrete frequency sweep from 10.5 Hz to 11.5 Hz with an increment of 0.1 Hz is considered for the analysis, along with different excitation amplitudes, precisely equal to 0.686 V with amplification factors of 0.9, 1.2, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, and 1 For transfer function of nonlinear system, we mean the frequency response curve of the system for a given input.
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