Experimental investigation of hysteretic stiffness related effects in contacttype nonlinearity
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Abstract
The paper presents experimental works related to contact nonlinearities. The research is focused on effects derived from hysteresis stiffness characteristics and vibroimpacts generated during the relative movement of two surfaces. The modeling of the contact nonlinearities was divided in two parts. First, the parameters of the system were identified based on modal analysis test. Next, the model was created and verified with experimental data. The experimental works were performed on steel samples with prepared contact surfaces. Electromagnetic shaker was used to produce relative motion between surfaces in contact. The response of the system was acquired by noncontact laser vibrometer. Both displacement and velocity of vibration were measured. Additionally, the impedance head measures the force and acceleration. The experimental data were used to validate the created models.
Keywords
Contact nonlinearities Contact model Structural dynamics Hysteresis stiffness1 Introduction
The inspiration for elaborating the model with hysteresis stiffness and impact were phenomenon occurring in vibroacoustic modulation tests described widely in many works [3, 16]. It should be noted that works in this field are mainly experimental. There is no model that would allow a clear explanation of nonlinear effects resulting from the interaction lowfrequency excitation, ultrasonic wave and contact interface. The paper focus is on the elaborating model with hysteresis stiffness, vibroimpacts and experimental validation of the proposed model. In this paper only lowfrequency excitation is investigated.
The remaining part of the paper is organized as follows. Section 2 contains theoretical background on stiffness characteristics related to contact. Section 3 the assumed numerical model is presented. In Sect. 4 numerical model is validated with experimental data. The discussion of results is also provided in that section. The paper is concluded in the last section.
2 Theoretical background

Closing–opening action, where the crack faces move apart from each other.

Sliding action, where the crack faces slide relative to each other (the movement is perpendicular to the crack leading edge).

Tearing action, where crack faces move relative to each other in direction parallel to crack leading edge.
The presented above stress–strain characteristic are of course ideal. It means, for example, that after every loading and unloading cycle, structure returns to the same state at the same point. For real structures, the characteristics can be modified due to additional phenomena like cyclic hardening and creeping. The real hysteretic characteristics can also deviate from the theoretical and can change for example depending on the strain rate sign.

the different dimensions of the contact (small for thin fatigue cracked plate and large for delaminated composite materials)

the impulse excitation of the structure by roughness inducted vibration

the timevarying size of the contact area due to low and high frequency excitation

the friction and adhesion forces acting at the contact interface and provide microslip and stick and slip processes

the strain and temperature dependence of material properties.
3 Numerical model
Numerical model description
4 Experimental setup
Description of the test samples
Two test samples made from C45 steel were prepared for testing. The overall dimensions of the test sample are \(40\times 40\times 45\) mm with the contact surface of the area \(10\times 10\) mm. The contact surfaces are grounded with a sand paper with the grit size P40 to obtain rough surface. Figure 8 presents sample used in test.
Analysis of transfer function of the system
To preidentify the parameters of the analyzed system, the modal analysis test was carried out. The investigated object has been freely suspended and piezoelectric actuator NOLIAC NAC 2013 is attached to object to excite the structure in highfrequency regions. Excitation signal was a chirp signal with frequency from 5 to 20 kHz. Fourier transform of the signal is shown in Fig. 9 with blue solid line. The three modes have been identified based on Frequency Response Function (FRF). For this modes damping ratio (using halfpower method) and natural frequency has been estimated and the results are summarized in Table 1. These modes have been introduced to the model. The comparison of FRF for both numerical model and tested sample for the same excitation parameters is presented in Fig. 9.
It is worth noting that the estimated parameters were used in the simulation as initial parameters. Due to boundary conditions during the test, the values of frequency and damping can significantly change. In subsequent stages, they were tuned on the basis of experimental data.
Test rig description
In order to investigate phenomena related to the nonlinear stiffness contact, an experimental setup is created. It consists of modal shaker K2007E01 as an excitation source. One sample was mounted at the end of threaded rod connected to the shaker and the other was firmly attached to the steel base plate. In this configuration, the relative motion in a single axis could occur between the samples. The top sample is loaded by an extender mass 200 g. The experimental setup is shown in Fig. 10a. The data measurement was conducted using impedance head PCB 288D01 to obtain acceleration and force data and the Doppler laser vibrometer to record displacement and velocity data. The excitation signal is supplied by the means of Agilent 33500B Series generator controlled by Matlab script and the acquisition was performed using Polytec PSV software. Sampling frequency is 51.2 kHz with total acquisition time 9 s. A series of measurements are conducted with sine excitation with frequency ranging from 10 to 500 Hz, with step of 10 Hz, so in total 50 measurements are obtained. Amplitude is linearly increasing during each experiment, to observe the nonlinear behavior.
Results and discussion
To analyze the particular stick and slip phase, the time domain signal was zoomed to one period of motion and is presented in Fig. 14. Then, the one period window of the signal was investigated.
The four different motion phases are visible when Fig. 14 is analyzed—two for stick and two for phase motion. Additionally, the highfrequency structural components can be observed (Fig. 15a). The nature of this component is different than in the numerical model. The reason for this may be that the measurement data is collected at a certain distance from the contact interface. This means that there is a transfer function between the contact area and the measuring point that can modify the response signal. The model does not take into account the geometry of the structure.
Frequency and damping ratio values for identified highfrequency modes
Mode number  Frequency (Hz)  Damping (%) 

1  8120  0.07 
2  9393  0.05 
3  10,225  0.03 
It can be also noticed that there is a good agreement between model and experimental data results. For both the two phase of motion can be clearly recognized. The frequency components are very similar when Figs. 16 and 17 are compared. Moreover, the experimental data includes additional components (about 20 kHz), probably related to harmonics of natural frequency. This problem is not investigated in this paper.
Frequency dependence of motion types
It is shown in previous section that the type of nonlinearities present actually depend on the type of motion occurring, and this is related to relative displacement of two surfaces in contact. As it is shown in Fig. 14a, there are different phases of motion that depend on the value of relative displacement, i.e., additional highfrequency components appear in the signal, when certain displacement threshold value has been crossed, as seen in Fig. 15b.
Theoretical model assumes constant value of \(\varepsilon _{1}\) (according to Fig. 4), on the basis of which type of motion changes, while the analysis of Fig. 18a suggests frequency dependent behavior. To investigate this problem, the additional modal analysis test was performed to find lowfrequency structural components. Figure 18b presents FRF (red solid line) for frequency range up to 500 Hz. When the displacement threshold and FRF are compared, then the consistency of both can be noticed. It follows that the structural resonance of the sample (about 150 Hz) affects the moment of motion changes. Generated resonant vibrations cause additional displacement and in this way changes the moment of appearance of highfrequency structural components. Hence, the conclusion that in order to properly adjust the excitation level in nonlinear acoustics tests, both high and lowfrequency structural components should be taken into account.
5 Conclusions
Experimental and numerical investigation of hysteretic stiffness for contacttype nonlinearity was presented. The numerical model with nonsymmetrical, amplitude dependent hysteresis stiffness was developed. The two steel samples with contact were used for experimental work. The monoharmonic signals with a linearly increasing amplitude were used for both cases as excitation. Additionally, the structural dynamics was included in numerical model. The study involved the analysis of two motion types: microslip and stick and slip. The experimental results demonstrate that the displacement level caused motion phase change is closely related to the resonance vibrations of the structure. For these reasons, it is important to consider the appropriate frequency of excitation when nonlinear effects related to different types of motion/contact are analyzed. The time and frequency domain analyses confirm that inclusion of additional highfrequency structural components improves the quality of the model. In particular, this applies to the frequency domain. Further work assumes to consider highfrequency acoustic wave and observation of vibroacoustic modulation effects.
Notes
Acknowledgements
The work presented in this paper was performed within the scope of the research project 2015/17/B/ST8/03399 financed by the Polish National Science Centre.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest concerning the publication of this manuscript.
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