This paper presents a new perception in evaluating fretting fatigue damage nucleation and propagation lifetime under periodically forced circulation. A new approach, which is proposed in this paper, is to measure the change of the central point of power spectral density (CPPSD) in different structural stiffness degradation stages. A notable aspect of this study lies in the combination between vibration amplitude and forced frequency of the fatiguecausing factors in beam structures. Additionally, it is found that randomization of the first phase from 0 to 2
The experiment on the effect of fatigue on material and working capacity of the structure is an important topic. Fatigue affects significantly the design and operation of the structure. In particular, the loadbearing parts of the structure have circulation because they are always under the periodical or changeable load. Practical structures are always designed to work within a certain lifespan. If lifespan is too short, a structure will not satisfy the requirements given, indicating the design does not meet the requirements or the operation goes beyond the requirements. If lifespan is too long, the structure will exceed in size or the weight will not meet the design. All of the mentioned scenarios show that the fatigue evaluation helps forecast the lifespan as well as ensures the working capacity of the structure. Although this evaluation is not a new topic, we still encounter various difficulties when it comes to practical application. Theoretically, fatigue is part of duration science, which focuses on the characteristics of the material as well as structure form under the changeable load over time. According to Burdekin and Stone [
The current research improves upon shortcomings of previous studies such as either separately evaluating inducing fatiguecausing factors as vibration amplitude, forced frequency, and first phase or using only numerical simulations to evaluate fatigue from multiple factors by the computer. A new paradigm is presented in investigating the decline of beams under harmonic constraining force through vibration analysis. The subject of investigation—beamtype models—is at the core of various reallife structures; yet, despite being a topic of major interest, it is rarely researched thoroughly due to the extensive and complicated fieldwork required. Therefore, this research highlights the characteristics of beam structures under the fatigue phenomenon. Through the three mentioned factors causing fatigue, the new CPPSD parameter is presented. Using vibration measurements collected from accelerometers, we composed the signal’s power spectrum across various phases (periods) of fatigue on the beam. Research demonstrated that changes in the CPPSD at each phase of fatigue can be used as a basis of evaluating decline in beams over time. Furthermore, shifts in CPPSD are found to be significantly more sensitive than previous parameters such as natural frequency, eigenvalues, or stress value. CPPSD reflects various degrees of decline in the beam under the effect of the periodical fatigue propagation during experimentation. This shall be the basis of applying fatigue to determine structures’ lifespans through experiments.
The experiment model investigates fatigue generated on the beam by harmonic constraining force. Thanks to this theory, we propose the application of the harmonic function as the input for fatigue process (vibration amplitude and forced frequency) of Hysteresis process. This is the basis to define the fatigue period as well as degradation level over time on the experimented beam.
We consider the beam model simulated as a single rest beam with 2 tips as shown in Figure
The simple rest beam model with moving load.
The expression of the beam bearing the bending is given as
The boundary and first conditions are given by
To solve equation (
With the boundary condition equation (
Substituting equation (
The specific vibration frequency of the structure according to the
To solve equation (
For the experimental model, one uses
The manuscript proposes the limitation and complexity level of harmonic force on the beam structure by the fatigue phenomenon, the first phase
Setting
When doing the reverse transformation, we have the original function
In this research, the manuscript uses the fatigue bending method which shows the changeable relation between stress and the relative stress periods. The changeable stress is the highest stress or stress amplitude. In case of tangential stress, if
Fatigue bending.
Depending on the data processing method in practice, so far, we have had lots of mathematical equations to express the fatigue bending according to equations (
Practically, the forcebearing period of the hysteresis process includes symmetric stress cycle (see Figure
Symmetric stress period (
Asymmetric stress period (
Different forms of stress period.
If
The hysteresis of 2 vibration amplitudes
In structural dynamics, the stimulating force and response of the system are considered as the input and output of the system. Figure
The spectral density function
For the system of one degree, the frequency response function is
The CPPSD can be calculated in the time domain, frequency domain, or both. Because the concept of the central point is quite flexible, responses in different measurement systems can be used. It brings an opportunity for application to status monitoring systems of discontinuous (nonperiodical fatigue forms) or continuous forcebearing structures (periodical fatigue forms).
The shift of this central position is quite suitable for both linear and nonlinear equations.
The CPPSD provides energy information of the frequency range during the operation of the beam.
The CPPSD does not depend on the vibrating phase effect (according to previous studies), thus preserves the important characteristics of the signal. Hence, the analysis does not have a significant effect on these changes.
Determination of the central position of the representative power spectral density.
In order to test in this study, we have built an experimental model generated at Faculty of Applied Science, Ho Chi Minh City University of Technology, as shown in Figure
Arrangement of the experiment model.
Fatigue test beam model.
Experimental beams and the cuts.
Inverter.
Transmission system.
The signal receiver cluster which consisted of 4 measuring heads was implemented by an acceleration sensor. The measuring system was mounted under the beam and divided the beam length into 5 equal parts as depicted in Figure
The signal received from the acceleration sensor.
The objective of the experiment was to investigate the reduction in bearing capacity of beams caused by fatigue effect. It was expressed by the number of forcedcirculation periods on the beam. To meet that requirement, in this experiment, we used a beam made with plywood (mediumdensity fibreboard (MDF)). This is a soft material with highfatigue bearing ability and wide application. The vibration of the beam under the impact of the stimulating model could easily respond to large amplitudes and high frequencies. Table
Number of the vibration period.
Symbol  Time (day)  Periods 

F1  0  0 
F2  1  10^{6} 
F3  2  2 × 10^{6} 
F4  3  3 × 10^{6} 
F5  4  4 × 10^{6} 
F6  5  5 × 10^{6} 
The specific frequency of the beam under the effect of the harmonic constraining force is determined by the power spectral density. The identifying method lets the beams vibrate freely under the effect of an instant force. The signal of the vibration amplitude obtained from the acceleration sensors is shown in Figure
Signal amplitude of the beam vibration.
Power spectral density of freely vibrating beams.
The experiment was conducted in 6 different fatigue states with an increasing period. The specific frequency values are obtained from Table
The frequencies of beams during 6 fatigue cumulative periods.
State  1  2  3  4  5  6 


Frequency (Hz)  13.8  13.8  13.7  13.6  13.5  13.3 
Power spectral density of beams corresponding to different fatigue states.
Table
Corresponding to 6 different fatigue states (from 0 to 6), the value of the frequency on the beam is almost unchanged (0.5 Hz) (according to Table
Corresponding to 6 different fatigue states (from 0 to 6), the value of the frequency on the beam is almost unchanged (0.5 Hz) (according to Table
Corresponding to 6 different fatigue states (from 0 to 6), the value of the frequency on the beam is almost unchanged (0.5 Hz) (according to Table
As the degradation in bearing capacity of beam increases, it results in a decrease in the area of the highfrequency domain of the power spectral density as shown in Figure
Steps to identify 7 centers.
The change of the central position of the spectrum through 4 measurement channels of 6 different fatigue states.
To be more visual in the evaluation process, this study uses a line graph to show shifts. By standard, 7 lines on the graph that show the shift of the central point are arranged in order from low to high (the central point lines from 1 to 7). This is the line that shows the central of the power spectral density through 6 different fatigue states of forcedcirculation vibration on the beam. We also agree that the points representing the reduced value are marked by
According to Figure
Line graphs showing the position of 7 central points of beams with 4 fatigue levels.
The distinctive feature of the central shift allows us to get clearer assessment in each state of the fatigue period when compared to the change of specific frequency value. Accordingly, there are states where the specific frequency remains constant (from fatigue state 1 to 2); however, the value of the central point still shows a shift to the left (decline in value). Thus, in states where the specific frequency has not changed yet, in fact, it still has a decrease in energy, especially in highfrequency areas. This energy degradation must reach a certain level for the new frequency to continue to decrease. However, the central value represents distribution of energy on the spectrum, and with only a small change, this value can be shifted. In brief, this value, which is a new parameter, has much higher sensitivity than others.
The comparison of the decrease level at the central position with a specific frequency expressed is shown in Figures
The declined rates in channel 1 with 7 central points in 4 fatigue levels.
The declined rates in channels 2 with 7 central points in 4 fatigue levels.
The declined rates in channels 3 with 7 central points in 4 fatigue levels.
The declined rates in channels 4 with 7 central points in 4 fatigue levels.
Figures
The distribution of central position’s characteristics of the power spectral density is different at each point. In particular, the distribution of channels 1 and 4 is similar and that of the channels 2 and 3 is similar. Based on this factor, we can see that the degradation on beams is uneven at different points. Therefore, this parameter can be applied to identify the most degrading points of the tested beams. In contrast, the specific frequency values at different points are equal. Therefore, the distribution of energy in terms of mechanics expressed by its own specific frequency value does not fully reflect the dispersion process. The forced circulation is the cause of fatigue on beams. However, the energy dispersion is not big enough for the specific frequency value to decrease quickly. Thus, the central position value of the power spectral density shows the process of dissipating energy on the beam in this experiment.
Central points 3rd, 4th, 5th, 6th, and 7th experience greater changes than 1st and 2nd. This indicates that there is a rapid change in the area with the higher frequency domain (the second frequency range varies from 27 Hz to 60 Hz). Conversely, this value moves less in the lower frequency domain. This also confirms the sensitivity of 1st area in comparison to 2nd area. On the contrary, frequency value of the first area usually contains the value of the specific frequency on the structure. All in all, this is the main cause of the nonsensitivity of specific frequency values in this study as well as in other authors’ studies.
The above results show the CPPSD distribution of the vibrating signal is more sensitive than the specific frequency. This is a new parameter to assess the reduction of bearing capacity of the beam under the effect of periodically forced circulation. The sensitivity of the parameter is expressed by not only the decrease rate in each measurement channel but also the level of degradation on the tested beam. The results show that even when the specific frequency value is constant, the central points still have a certain shift. It means the new parameters are sensitive enough with small changes in the mechanical characteristics of materials when causing fatigue on beams. In addition, the distribution of central position value of the power spectral density indicates the degradation of the beam at different positions. In the future, the authors will quantify the degradation level of the beam corresponding with different fatigue cycles. This is the premise to assist evaluation of the structure lifespan under the effect of fatigue.
All data generated or analysed during this study are included in this work.
The authors declare that they have no conflicts of interest.
This research was funded by Ho Chi Minh City University of Technology, VNUHCM, under grant number BKSDH20191880703.