*3.1. Dynamic Response of the Contact System*

The aim of this section is to compare the numerical and experimental dynamic responses of the system (Figure 2), in time and frequency. For the sake of conciseness, the comparison of the time signals with the experiments is first reported only for the "Modified PL"; in the following section, the general results obtained by both laws will be compared with the experiments.

Experimentally, an impulsive force was applied to the upper side of the guide by an instrumented hammer (PCB Piezotronics-086C03). The case presented hereafter corresponds to an impulsive force of 32 N. The applied force and the acceleration are measured and shown in Figure 5a,b, respectively. For the numerical model, the measured experimental force has been interpolated (see Figure 5a) and

introduced as a boundary condition in the numerical simulation. The model presented in Section 2, with the "Modified PL", has been used to simulate the system response to the force.

**Figure 5.** (**a**) Force signals over time, for a single force; (**b**) Acceleration signals over time obtained with 'Modified PL'. Test performed with average contact force 32 N.

Figure 5b shows the respective experimental and numerical accelerations, due to the dynamic system response.

Experimental and numerical responses show good agreement in amplitude and time evolution. Figure 6 shows the Frequency Response Functions (FRF) [26], which provide the response of a system to an external excitation in the frequency domain. They are calculated from both numerical and experimental signals, to characterize the dynamics of the system. The numerical curves shown in Figure 6 correspond to the one obtained with a constant interface stiffness of 8.5 × 1011 Pa/m (dashed line) and the one obtained with the "Modified PL" presented in Figure 3. The equivalent constant stiffness of 8.5 × 1011 Pa/m was calculated to obtain the same frequency for the first harmonics of the "Modified PL", to highlight its nonlinear contribution to the system response.

73

**Figure 6.** FRFs of the system (receptance, displacement/force). Numerical with nonlinear stiffness, numerical with constant stiffness with Kc = 8.5 × 10<sup>11</sup> Pa/m and experimental. Test performed with average contact force 32 N.

Only the fundamental (f~800 Hz) and second harmonics (f~1600 Hz) of the mass-spring mode are investigated here. Their results are well decoupled from the rest of the system dynamics.

ଵ The numerical spectra, obtained with either constant or nonlinear stiffness, show a peak around frequency *f*<sup>1</sup> = 800 Hz, corresponding to the fundamental frequency of the system. This is the natural frequency of the mass-spring mode, where the mass is the guide, while the spring is the series of the two interfaces and the sample stiffness (Figure 2). The comparison shows good agreement between the numerical and experimental results in terms of frequency and width of the peaks, i.e., damping. The amplitude of the fundamental is well simulated as well, with a percentage error less than 10%.

<sup>ଶ</sup> <sup>ଵ</sup> Unlike the spectra obtained with the constant stiffness, those based on nonlinear stiffness show a peak around frequency *f*<sup>2</sup> = 2 *f*<sup>1</sup> = 1600 Hz, which is also recovered experimentally. This peak represents the second harmonic and correlates with the experimental second harmonic.

The presence of the second harmonic in the spectra is due to the nonlinear nature of the contact stiffness. This is confirmed by the absence of this harmonic in the numerical results obtained with the constant contact stiffness (Figure 6 dashed line), and thus the occurrence of the second harmonic in experiments can be directly correlated with the nonlinearity of the interface stiffness.

#### *3.2. Nonlinear Response of the Interface*

In order to discuss the proposed trends of numerical contact stiffness and evaluate it by comparison with experiments, a spectrum analysis of the acceleration signals is reported in this section, as a function of the amplitude of the impulsive force. It is assumed that an increase in the force, and then in the system response, increases the nonlinear contribution of the interface to the system response.

A comparison of experimental and numerical FRFs, derived from the "Modified PL", is carried out for different impulsive force amplitudes, ranging from 9 to 32 N (Figure 7).

**Figure 7.** Frequency response functions (FRFs) of the system (receptance, displacement/force). (**a**) Experimental frequency response; (**b**) Numerical frequency response plotted with "Modified PL". Test performed with average contact force ranging from 9 to 32 N.

When increasing the force amplitude, the overall average stiffness at the interface decreases, leading to a decrease in mode frequency (Figure 7). The nonlinearity of the interface stiffness is observable by the appearance of the second harmonic (frequency from 1400 to 1800 Hz, depending on the amplitude of the impulsive force) in the system response.

In the following, the numerical results obtained with the different contact laws, as presented in Figure 3, are compared with the experimental results in terms of the magnitude of the fundamental and second harmonics, as well as in terms of the frequency of the fundamental, as a function of the applied force.

Figure 8 shows the frequency evolution of the fundamental harmonic as a function of the amplitude of the applied force. As observed in the experimental results, a decrease in frequency was obtained in the numerical simulations with the implemented "Modified PL". This decrease is due to the decrease in the effective average stiffness when the oscillation at the contact increases. Experimentally, this trend has already been observed in [18]. It is worth mentioning that the decrease in frequency when the force amplitude increases (Figure 8) is also recovered by the "PL", but with a lower slope. A slight decrease in frequency (2%) can be observed in Figure 8 for the "PL", which is lower than for the experimental one (5%). Conversely, the "Modified PL" introduces a greater decrease in terms of stiffness, particularly for low contact pressures, as shown in Figure 9.

**Figure 8.** Frequency of the fundamental, as a function of the force amplitude [N]. Experimental measurements (red cross), PL (black circle) and Modified PL (blue triangle).

**Figure 9.** Numerical results for maximum force F = 32 N (red) and minimum force F = 9 N (black). (**a**) Numerical contact pressure as a function of time; (**b**) Numerical stiffness as a function of contact pressure. Test performed on aluminium with Modified PL.

It should also be noted that, while the trend of the frequency is correctly simulated by the proposed laws, an error in the absolute value of the frequency, around 13%, is observed. This is due to the non-infinite stiffness of the counterpart (tribometer disc), unlike the infinite stiffness in the simulation, which implies a lower experimental frequency. Thanks to the numerical results, the decrease in the frequency of the fundamental can be shown to be related to a decrease in the mean value of contact stiffness, with the increase in the applied force

$$\mathbf{K}\_{\text{avr}} = \frac{1}{\mathbf{T}} \sum\_{t=0}^{\mathbf{T}} \left| \mathbf{K}\_{\text{c}}(t) \right| \tag{8}$$

where *K<sup>c</sup>* is the stiffness and T is the time of simulation. ௧ୀ

In order to highlight the difference in average stiffness according to the force amplitude, Figure 9a shows the system response to two different force amplitudes, using the "Modified PL". An applied force of 9 N generates a maximum contact pressure of −0.078 MPa, resulting in an average contact stiffness of 0.4 × 1012 Pa/m, while a force of 32 N generates a maximum contact pressure of −0.025 MPa and an average contact stiffness of 0.12 × 1012 Pa/m. T − −

௩ =

1 T


()|

Figure 10 shows the evolution of average contact stiffness Kavr for the "Modified PL" and the "PL". It confirms the decrease in average stiffness when the applied force is increased, which explains the decrease in frequency (Figure 7).

**Figure 10.** Average contact stiffness as a function of the applied forces. The test performed with average contact force ranging from 9 to 32 N, with Modified PL (blue triangle) and PL (black circle).

Figure 11 shows the evolution of the amplitude of the fundamental, as a function of the applied force, for the different contact laws.

Considering the mean value of the fundamental harmonic over the considered range of pressure, the "Modified PL" produces amplitudes closer to the experimental ones for this set of experimental measurements.

Nevertheless, as shown in Figure 11, for both of the implemented laws, the slight experimental increase in the amplitude of the fundamental (A1), with respect to the force amplitude, is not retrieved numerically. As mentioned above, this could be explained by the different boundary conditions between the numerical and experimental systems. In fact, the experimental set-up is not completely rigid, due to the deformability of the bench components (disc, shaft, bearings, etc.). Despite using a massive disc to isolate the dynamics of the investigated system (air guide and samples in contact) from the rest of the set-up as much as possible, a slight error is introduced by the residual flexibility of the system.

Finally, Figure 12 shows the ratio (A2/A1) of the amplitudes of the second harmonic (A2) to the fundamental (A1), obtained both experimentally and numerically, for both contact laws. It can be noted that the amplitude of the second harmonic is normalized by the amplitude of the fundamental, which is dependent on the energy introduced by the external force at this frequency, in order to highlight the nonlinear contribution originated by the contact interface.

**Figure 11.** Magnitude of the FRF of the fundamental A1, as a function of the force amplitude. Comparison of the contact laws. Experimental measurements (red cross), Modified PL (blue triangle), and PL (black circle).

**Figure 12.** Ratio of magnitudes of the FRF of the second harmonic (A<sup>2</sup> ) to fundamental (A<sup>1</sup> ), as a function of the force amplitude. Experimental measurements (red cross), Modified PL (blue triangle) and PL (black circle).

The trends of the A2/A<sup>1</sup> ratio, calculated for all the tested contact laws, are similar to the experimental trend. In general, an increase in the amplitude of the applied force (x axis in Figure 12), and consequently a higher amplitude of the system vibrational response, generates a greater nonlinear contribution, both numerically and experimentally. In fact, a larger oscillation of the contact pressure (especially within the low-contact pressure range, absolute value [0; 0.14 MPa]) generates a more nonlinear response by the system, which leads to a higher distortion of the signals and then a higher second harmonic contribution. The higher amplitude observed for the second harmonic of the "Modified PL" is due to the higher nonlinearity of the stiffness around the equilibrium position, with respect to the "PL".

The overall comparison, based on the analysis reported above, highlights the fact that the contact interface response depends heavily on the stiffness trend at lower pressures (less than 1 MPa). This stiffness trend at lower pressures introduced by the "Modified PL" increases the nonlinearity of the response (second harmonic amplitude) and decreases the average stiffness, i.e., the frequency of the main harmonics.

These results demonstrate that the stiffness trend at lower pressures plays a vital role and should be clearly identified, as it has a huge effect on the nonlinear response of mechanical systems with contact interfaces.

#### **4. Conclusions**

The nonlinear normal stiffness of contact interfaces, due to surface roughness, is a topic of major interest in several areas of application. A consequence of such nonlinearity is the appearance of second harmonic terms, either in acoustic wave propagation through the interface or in the dynamic vibrational response of systems with contact interfaces.

While contact stiffness nonlinearity at higher pressures has been widely discussed in the literature, and generally approximated by a power law, the contact stiffness trend at lower pressures has not been clearly identified. In this paper, a classical power law, fitted from experimental data at high contact pressures, is compared with a modified power law implementing an inflection point at lower pressures, where experimental data are not available. The stiffness-pressure trend within the higher contact pressure range was approximated from experimental measurements performed on a dedicated test bench. Within the lower contact pressure range, data from the literature were used to assume the different possible trends.

The nonlinear response of the system, obtained experimentally when exciting a dedicated system with an impulsive force, was analysed and compared with the nonlinear response of the numerical model that was developed, with the contact interface modelled by the different contact laws.

From the numerical simulations it was possible to identify the effect of the contact nonlinearity on the dynamic response of the system. The decrease in the average contact stiffness with the increase in the impulsive force explains the appearance of the second harmonics and the decrease in the fundamental frequency. In addition, the amplitude of the second harmonic was simulated and explained by the stiffness trend at the contact interface during the system oscillations.

In the nonlinear system response, the key role of the contact stiffness trend within the lower pressure range is highlighted, demonstrating the need to identify such parameters with dedicated experimental tests. Nonlinear contact laws and their effect on the dynamics of a system ought to be further investigated by implementing the contact laws considered here in finite element codes, in order to consider more realistic structures and interfaces.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Author Contributions:** Methodology, D.N., A.M., L.B. and F.M.; visualization, D.N.; investigation, D.N. and D.T.; data curation, D.N. and D.T.; writing—original draft preparation, D.N.; writing—review and editing, A.M., L.B. and F.M.; validation, A.M., L.B. and F.M.; supervision, A.M. and F.M. All authors have read and agreed to the published version of the manuscript.
