**3. Results**

## *3.1. Experimental Results*

The results of the vibration tests for FFU composite beams under an ultimate load test and repeated load test for healthy and damaged conditions are presented in Tables 4 and 5 and Figures 7 and 8. The first five-mode shapes of vibration under ultimate and repeated load tests are shown in Tables 4 and 5, respectively. For all beams, the first natural bending mode in a vertical plane obviously controlled the first resonant mode of vibration both under an ultimate load test and repeated load test. In addition, the lowest frequency corresponds to the natural bending mode, the second frequency to the lowest torsional mode, the third frequency to the second bending mode, the fourth frequency to the second torsional mode, and the fifth mode to the third bending mode. Clearly, the internal dynamic properties of FFU composite beams can be changed when damages occur.

Table 4 exhibits the results of natural frequencies and damping values of the FFU composite beams under the ultimate load test. The differences between the natural frequencies of all mode shapes in healthy and failed conditions are 16.5%, 11.4%, 15.1%, 22.46%, and 25.27%, respectively. As shown in Figure 7, the frequencies of all five modes under failed conditions are lower than those under healthy conditions. For damping values, the value of the first mode damping values under failed conditions increased by 49%, compared with those under healthy conditions. There are several transverse damages on the beam surface for the first time under failure conditions, and there are cracks (30 mm in width). Nevertheless, the beam specimens remain the same and could completely recover without any load. After measurement and unloading, the residual bending deformation level of the material is only 2 mm.


**Table 4.** Frequencies, damping values, and mode shapes under ultimate load test for healthy and failed conditions.

**Figure 7.** Frequencies against damping values over mode shapes under ultimate load test for healthy and failed conditions.

The dynamic behavior of the FFU composite beams under the repeated load test is demonstrated in Tables 5 and 6 and Figure 8. In Table 5, it is clear that all modes have no obvious deviation before the load reached 100 kN. Beyond this load to the ultimate load, the frequencies of all five modes tend to reduce with percent variations of different mode shapes. The maximum difference of frequency is found in the first mode, approximately 27%, compared with the frequency under healthy conditions. Surprisingly, the frequencies of the fourth mode are unchanged under the different loading conditions. A comparison of natural frequencies and damping values between healthy and damaged conditions in all the five modes is presented in Table 6, which shows that there were maximum differences in natural frequencies and damping values between the healthy condition and the ultimate loading condition (170 kN). We note that the minimum difference in frequencies and damping values between the healthy condition and the damaged condition could be found under a load of 67 kN.

However, the different frequencies of the other modes reduce dramatically, as shown in Figure 8. In regards to damping values in Table 5, all the modes except the first and fifth mode are scant. Obviously, the difference of damping values between under 100 kN and 167 kN loading in the first mode is significantly high and increased two-fold from 3.94 to 8.24. In addition, the difference of damping values between healthy conditions and 67 kN loading in the fifth mode is considerable, which decreased by 36% from 2.39 to 1.53. It is clear that the dynamic modal parameters of FFU beams decrease when damages appear. These beams could reduce with the damage severity.

**Figure 8.** Frequencies against damping values over mode shapes under repeated load test for healthy and damaged conditions.


**Table 5.** Frequencies, damping values, and mode shapes under repeated load test for healthy and damaged conditions.

**Table 6.** Relative values of frequencies and damping to healthy condition.


#### *3.2. Numerical Results*

Based on the frequencies experimentally obtained by the impact hammer excitation technique, dynamic Young's modulus, E, can be computed using Equation (1). As shown in Table 7, the big difference of the Young's modulus values is significant in the first bending mode and relatively small when compared with the Young's modulus for FFU composite sleepers, E = 8.1 GPa, according to the reviews in [48,49].

In order to verify the model, the natural frequencies of a full-scale FFU beam in free-free conditions are calibrated against the existing experiments. The values of a dynamic elastic modulus in different bending modes obtained from Table 7 have been used in the finite element analysis. A comparison between numerical and experimental investigations for frequencies and mode shapes are given in Table 8, especially the experimental data based on the ultimate load test. The results are found to be in a very good agreement in all the first five modes. The maximum difference of frequencies between the numerical and experimental data is less than 4% in the second twisting mode, because of the effects of experimental disturbances in our laboratory. Additionally, there is a satisfied correlation between both results for the shifts in natural frequencies under free vibration. It is important to note that the numerical modal analysis of a FFU composite beam can only be achieved under free-free boundary conditions. This free-free boundary condition is commonly used for performance benchmarking of an individual component (i.e., like-for-like comparison), especially for railway sleepers and bearers, which are safety-critical components [50]. In the near future, the situ investigation into modal parameters of FFU composite beams can be further carried out in order to determine the effect of different boundary conditions (e.g., type of ballast aggregate, or resultant effects of particle size distribution, tamping) on vibration properties of the beams.

**Table 7.** Determination of dynamic Young's modulus from dynamic measurements in free-free end conditions.



**Table 8.** Natural frequencies of a conceptual FFU composite beam (Hz) in the free-free conditions.

#### **4. Conclusions**

Dynamic modal parameters of FFU composites are extremely significant for the development of a realistic dynamic model of a railway track capable of predicting its dynamic responses for predictive and preventative maintenance to ensure railway safety. The results of the experimental and numerical modal analysis for Fiber-reinforced Foamed Urethane composite beams in free-free boundary conditions are indicated in this study. For the purpose of like-for-like performance benchmarking for a particular component, the free-free boundary condition is considered to be more suitable since the test results will not be affected by uncertainties stemmed from supports (e.g., dimension and particle size distribution of ballast, tamping technique, ballast geological properties). Full-scaled experiments have been performed to artificially create damage and failure in accordance with European standards. Dynamic parameter tests have been conducted by using an impact hammer excitation technique throughout the frequency range of interest: from 0 to 2100 Hz. According to experimental results, it provides the correlations between modal parameters and structural damage. Furthermore, the dynamic parameters obtained are later used to extract the dynamic elastic moduli. The results of frequency parameters under free-free conditions are in a very good agreement between experimental and numerical data with less than 4% discrepancy. Further research could be conducted to investigate the vibration characteristics of Fiber-reinforced Foamed Urethane composite beams in situ conditions in order to consider the influence of various ballast conditions on the natural frequencies, modal damping values, and vibration mode shapes of FFU composite beams under the in-situ boundary conditions. Some interesting novel findings from this research can be concluded as follows:


**Author Contributions:** Conceptualization, P.S. and S.K.; data curation, P.S. and A.L.O.d.M.; formal analysis, P.S., C.N., A.L.O.d.M., and S.K.; funding acquisition, S.K.; investigation, P.S., C.N., A.L.O.d.M., and S.K.; methodology, P.S., C.N., and S.K.; project administration, S.K.; resources, S.K.; software, S.K.; supervision, S.K.; validation, P.S., C.N., and A.L.O.d.M.; visualization, P.S., C.N., and A.L.O.d.M.; writing—original draft, P.S., C.N., and A.L.O.d.M.; writing—review & editing, S.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors wish to gratefully acknowledge the European Commission for the financial sponsorship of the H2020-MSCA-RISE Project No. 691135 "RISEN: Rail Infrastructure Systems Engineering Network," which enables a global research network that tackles the grand challenge in railway infrastructure resilience and advanced sensing in extreme environments (www.risen2rail.eu). In addition, this project is partially supported by the European Commission's Shift2Rail, H2020-S2R Project No. 730849 "S-Code: Switch and Crossing Optimal Design and Evaluation". The APC is kindly sponsored by MDPI's Invited Paper Program.

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