The primary aim of the present chapter was to find an efficient model for the crack quantification in the beam. The feed-forward back propagation ANNs and the RFs were incorporated into the search for an accurate predictive model of the formulations. Another objective of the study was connected to the datasets, specifically to the feature vectors and the size of the training set. Two sets of data were calculated numerically. The first one contained the first eight DFPs; the other one - up to 64 HWTCs derived from the first mode shape. The hypotheses were evaluated on the testing set (not shown to the models in advance).
The results of one crack quantification showed that the ensemble of 50 ANNs produced as accurate predictions as the RFs. However, if two cracks had to be quantified, the RFs slightly outperformed the ensemble of the ANNs. Notably, the RFs had fewer hyperparameters to tune and the training process was remarkably shorter than in the case of the ANNs.
Analysing the results of the predictions, it was found out that the depth of cracks was more difficult to predict accurately than the location. The dataset of eight DFPs produced more accurate predictions of the crack depths, but not of the crack location. This was in line with the correlation analysis, Mahmoud and Kiefa [Mahm 99], and Ndambi et al. [Ndam 02] results: the natural frequencies could follow the damage severity but were not influenced by the crack damage locations. The hypothesis on the sensitivity of the Haar wavelet transform coeffi-cients towards the crack localisation was decisively confirmed; however, the Haar wavelet transform method could not follow the severity of the crack.
Chapter 6
Elastic supports
The present chapter focuses on the Euler-Bernoulli beam with elastic supports.
The supports are simulated by the spring model. The inverse problem of the stiff-ness parameter identification is solved using the modal domain, the feed-forward back propagation neural network (ANN), the random forests (RF) and the discrete Haar wavelet transform (HWT). In particular, the first mode shape is decomposed into the Haar wavelet transform coefficients (HWTCs) with a small level of res-olution. The obtained coefficients are used in the feature vector. In line, the machine learning methods are trained on the dimensionless natural frequency pa-rameters (DFPs). The results of the approaches are compared to each other. The foundation for this chapter can be found in [Fekl 13b, Fekl 14].
6.1 Related work
Structural supports play an important role in engineering. They do not only firmly carry a structure, but are used to improve the overall constructional performance.
For example, the fundamental frequency can be increased if a beam has additional internal point supports. If supports are rigid, the optimum locations of the sup-ports are the nodal points of the highest vibration mode [Cour 66]. If supsup-ports are elastic and the stiffness parameter of the supports exceeds a certain minimum value, the optimum locations of the supports are still the same as in the case of rigid supports - no decrease in the fundamental frequency is observed [Akes 88].
Compared to the rigid supports, the elastic supports have the advantage of hardening or softening the non-linear behaviour of the structure. Furthermore, both horizontal and vertical elastic supports reduce the dynamic coefficient of
dis-placement [Liu 13]. Therefore, beams with elastic supports are frequently used in spring-beam coupled systems, cable-arch structures, fluid-conveying pipes, frames and trusses, railway tracks, towers, piles, tall buildings, robot arms and other technical applications [Wu 06, Ronu 19].
It is theoretically and practically proved that a long-term performance of sup-ported constructions greatly depends on the stiffness parameters of their supports [Wang 06]. Due to environmental conditions or damage, the stiffness characteris-tics of supports can change. The decreased stiffness may lead to the collapse of the whole construction if no preventive maintenance measures are taken on time. To avoid loss of functionality or catastrophic failure, it is vital to monitor the stiffness characteristics of supports.
Different boundary conditions and supports have been analysed in a large number of papers. Most of the papers are devoted to the vibration of beams with classical boundary conditions. Fewer papers are focused on the beams with elastic supports. Some of them are [Sait 79, Glab 99, Nagu 02, Bane 04, Xant 07, Hsu 08, Silv 09, Wang 13, Lore 18]. The exact expressions for the natural frequencies and mode shapes of the beam with one end hinged and restrained by a rotational spring and the other end free were derived by Chun [Chun 72]. The effects of rotational and transversal supports at one end of the beam was studied in [Afol 86] and Lau [Lau 84]. Lau expanded Rutenbergs research on the free vibration of the uniform beams with a rotational constraint and presented a closed-form solution to the beam with rotational and translational supports at some point of the beam.
The frequency equation of the beam with an intermediate elastic support us-ing the continuity conditions at the supported point was presented by Chellapilla [Chel 89]. Albarracin et al. [Alba 04] described the effect of an intermediate support mathematically when the ends of the beam were elastically constrained.
Opposite, Rao and Mirza [Afol 86], and Li [Li 00] studied the vibrations of the beams restrained by two transverse springs and two rotational springs. The nu-merical results presented in the papers demonstrated that the natural frequencies and the mode shapes were sensitive to the position and stiffness of the intermedi-ate elastic supports. The authors also stintermedi-ated that the stiffness parameter of the elastic supports considerably caused the first few modes shapes and was negligible on the higher frequencies.
Sato et al. [Sato 08] demonstrated that a beam on equidistant elastic supports could be considered as a beam on an elastic foundation in static and free vibration problems. Nevertheless, the presented model was valid for a limited range of support stiffness, spacing and flexural rigidity of the beam. The proposed model
was suitable for simplified analysis of the global behaviour of some structures, such as floating tunnels.
The stiffness parameter identification from the governing equation of the free vibration of beams with elastic springs at the ends of the beam is an inverse prob-lem and cannot be solved analytically. Therefore, alternative methods have to be sought. ANNs and RFs are a promising tool in the search for the relationships between the dynamic response of the beam and the corresponding stiffness param-eters of elastic supports. The present chapter is focused on the Euler-Bernoulli beam with non-classical (elastic) supports. The supports are simulated by the elas-tic spring model. Two datasets are calculated numerically: the first eight dimen-sionless natural frequency parameter based dataset (DFP) and 16 Haar wavelet transform coefficient based dataset (HWTC). The HWTCs are obtained from the first mode shape. No alternative research has been found in the literature.