“…For flexural vibrations and at low frequencies, the Bernoulli-Euler formula holds and a resonant frequency f i of the i -th independent beam is inversely proportional to the square of its length d i . As the frequency increases, the dependence on d i becomes more complex, however, it is still well described on the average, by the inverse of the square of the length d i of the beam 30,34 .…”
Section: Resultsmentioning
confidence: 99%
“…It is remarkable that these two extreme regimes are indeed observed in the same structured beam. In order to understand how this is possible, we point out that above the crossover frequency f c , the complexity of the relation between the resonant frequencies and d i grows even more, since doublets appear: a series of resonant frequencies grouped in pairs where the members of each pair are very close in frequency, as shown in refs 30,35 .Figure 4Measured wave amplitudes as a function of the position along the beam z , corresponding to two normal modes of frequencies ( a ) f = 4.768 kHz and ( b ) f = 5.125 kHz, respectively.…”
Section: Resultsmentioning
confidence: 99%
“…Analytic expressions of κ for different cross sections are available in the literature 46–49 , which are obtained under the common assumption that the cross section of the beam remains flat under flexural oscillations. This assumption however, is not always true, as demonstrated by experimental measurements 30,35 , specially at relatively high frequencies. Therefore, a dynamic coefficient κ is required in order to describe more accurately the bending oscillations of a beam like the one shown in Fig.…”
Section: Methodsmentioning
confidence: 96%
“…The first type is described, as well as the torsional oscillations, by a wave equation, while the flexural type, is better described by fourth-order differential equations instead, according to the Timoshenko beam theory (TBT) 28 . They result from the coupling of two second order differential Navier equations and as a consequence of this coupling, flexural oscillations has several characteristics not observed in compressional or torsional vibrations, one of them is the existence of a crossover frequency, beyond that, vibrations exhibit two wavelengths 29,30 instead of one, as occurs for frequencies below the crossover one. Since the localization length has a strong dependence on the frequency in both classical and quantum cases, a natural question that arises is how the Anderson localization phenomenon is affected by the crossover frequency.…”
We study, both experimentally and numerically, the Anderson localization phenomenon in flexural waves of a disordered elastic beam, which consists of a beam with randomly spaced notches. We found that the effect of the disorder on the system is stronger above a crossover frequency
f
c
than below it. For a chosen value of disorder, we show that above
f
c
the normal-mode wave functions are localized as occurs in disordered solids, while below
f
c
the wave functions are partially and fully extended, but their dependence on the frequency is not governed by a monotonous relationship, as occurs in other classical and quantum systems. These findings were corroborated with the calculation of the participation ratio, the localization length and a level statistics. In particular, the nearest spacing distribution is obtained and analyzed with a suitable phenomenological expression, related to the level repulsion.
“…For flexural vibrations and at low frequencies, the Bernoulli-Euler formula holds and a resonant frequency f i of the i -th independent beam is inversely proportional to the square of its length d i . As the frequency increases, the dependence on d i becomes more complex, however, it is still well described on the average, by the inverse of the square of the length d i of the beam 30,34 .…”
Section: Resultsmentioning
confidence: 99%
“…It is remarkable that these two extreme regimes are indeed observed in the same structured beam. In order to understand how this is possible, we point out that above the crossover frequency f c , the complexity of the relation between the resonant frequencies and d i grows even more, since doublets appear: a series of resonant frequencies grouped in pairs where the members of each pair are very close in frequency, as shown in refs 30,35 .Figure 4Measured wave amplitudes as a function of the position along the beam z , corresponding to two normal modes of frequencies ( a ) f = 4.768 kHz and ( b ) f = 5.125 kHz, respectively.…”
Section: Resultsmentioning
confidence: 99%
“…Analytic expressions of κ for different cross sections are available in the literature 46–49 , which are obtained under the common assumption that the cross section of the beam remains flat under flexural oscillations. This assumption however, is not always true, as demonstrated by experimental measurements 30,35 , specially at relatively high frequencies. Therefore, a dynamic coefficient κ is required in order to describe more accurately the bending oscillations of a beam like the one shown in Fig.…”
Section: Methodsmentioning
confidence: 96%
“…The first type is described, as well as the torsional oscillations, by a wave equation, while the flexural type, is better described by fourth-order differential equations instead, according to the Timoshenko beam theory (TBT) 28 . They result from the coupling of two second order differential Navier equations and as a consequence of this coupling, flexural oscillations has several characteristics not observed in compressional or torsional vibrations, one of them is the existence of a crossover frequency, beyond that, vibrations exhibit two wavelengths 29,30 instead of one, as occurs for frequencies below the crossover one. Since the localization length has a strong dependence on the frequency in both classical and quantum cases, a natural question that arises is how the Anderson localization phenomenon is affected by the crossover frequency.…”
We study, both experimentally and numerically, the Anderson localization phenomenon in flexural waves of a disordered elastic beam, which consists of a beam with randomly spaced notches. We found that the effect of the disorder on the system is stronger above a crossover frequency
f
c
than below it. For a chosen value of disorder, we show that above
f
c
the normal-mode wave functions are localized as occurs in disordered solids, while below
f
c
the wave functions are partially and fully extended, but their dependence on the frequency is not governed by a monotonous relationship, as occurs in other classical and quantum systems. These findings were corroborated with the calculation of the participation ratio, the localization length and a level statistics. In particular, the nearest spacing distribution is obtained and analyzed with a suitable phenomenological expression, related to the level repulsion.
“…Successful implementation of theoretical model of Timoshenko beam for such structures brings prospective to use such model in the presented case [51,52]. Nevertheless, the boundary conditions in such model, as shown in Figure 3, are obtained from experimental results [5].…”
Section: Methodology Of Wire Rope Researchmentioning
Steel ropes are complex flexible structures used in many technical applications, such as elevators, cable cars, and funicular cabs. Due to the specific design and critical safety requirements, diagnostics of ropes remains an important issue. Broken wire number in the steel ropes is limited by safety standards when they are used in the human lifting and carrying installations. There are some practical issues on loose wires-firstly, it shows end of lifetime of the entire rope, independently of wear, lubrication or wrong winding on the drums or through pulleys; and, secondly, it can stick in the tight pulley-support gaps and cause deterioration of rope structure up to birdcage formations. Normal rope operation should not generate broken wires, so increasing of their number shows a need for rope installation maintenance. This paper presents a methodology of steel rope diagnostics and the results of analysis using multi-criteria analysis methods. The experimental part of the research was performed using an original test bench to detect broken wires on the rope surface by its vibrations. Diagnostics was performed in the range of frequencies from 60 to 560 Hz with a pitch of 50 Hz. The obtained amplitudes of the broken rope wire vibrations, different from the entire rope surface vibration parameters, was the significant outcome. Later analysis of the obtained experimental results revealed the most significant values of the diagnostic parameters. The evaluation of the power of the diagnostics was implemented by using multi-criteria decision-making (MCDM) methods. Various decision-making methods are necessary due to unknown efficiencies with respect to the physical phenomena of the evaluated processes. The significance of the methods was evaluated using objective methods from the structure of the presented data. Some of these methods were proposed by authors of this paper. Implementation of MCDM in diagnostic data analysis and definition of the diagnostic parameters significance offers meaningful results.
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