*3.3. Acquisition Equipment*

The equipment for acquisition of the rock bolt's dynamical response consisted of several elements, Figure 8, including a custom made nut with accelerometer, a hammer for generation of impulse and a laptop with developed state machine and a vibration input module.

**Figure 8.** Elements of acquisition equipment: (**A**) a custom made nut with accelerometer; (**B**) a hammer for generation of impulse and (**C**) a laptop with developed state machine and a vibration input module.

At one end of the model, a nut is fitted on which an accelerometer for detection of longitudinal waves is mounted. The sensitivity of the accelerometer is 10.2 mV/(m/s2), with a frequency range of 5 to 15,000 Hz, which has proved to be sufficient for this research. To generate an impulse at the top of the rock bolt, a soft-steel hammer was used. The impulse generated by a hammer is not an ideal delta function but has a certain time duration, which is, along with the shape of the frequency function, determined by the mass and rigidity of the hammer and the steel bar. Therefore, a soft-steel head proved to be the optimal form among an array of materials (hard steel, lead, rubber, wood, plastic). The registered signal goes through a vibration input module, which has built-in anti-aliasing filters that automatically adjust to define the sample rate. After arriving at the developed state machine, the signal is transformed from the time to the frequency domain by using the Fourier transformation.

Considering that even the accelerometer positioning nut itself influences the frequency response, it is necessary to determine such undesirable frequencies in order to eliminate them from the frequency spectrum. Therefore, a calibration system for the direct positioning of the accelerometer on the rock bolt head using a magne<sup>t</sup> was developed. Even though, in this case, the impact of the positioning equipment is minimized, such positioning cannot be considered as acceptable in real case conditions, because it would be impossible to generate an impact on the rock bolt head. By testing laboratory models, a magne<sup>t</sup> can be positioned at one end of the bar and the impulse imposed on the other end of the bar. Thus, the frequency response can be compared with the frequency response of the system using the nut as an integral part of the equipment. By overlapping these two spectra, conclusions as to the nut's 'contribution' from the aspect of the additional, undesirable, frequencies in the spectrum can be deduced.

It was found that the nut, after generating an impact on the rock bolt head, vibrates at higher frequencies than those of interest in this research. By overlapping the frequency spectrums, it can be concluded with a high level of reliability that the first three frequencies from the spectrum correlate relatively well, whereas at higher frequencies, especially at those greater than 4000 Hz, there is a significant difference in the frequency response. If only first three natural frequencies are considered, the proposed mode of accelerometer positioning using a nut can be considered acceptable. This was confirmed for all 94 tested combinations.

### *3.4. The State Machine*

A state machine, developed within this research, provides the ability to collect data in the time domain and their real-time transformation into the frequency domain, thus gaining insight into the frequency spectrum already during the investigation. As a programming tool, the LabVIEW platform was used, where the concept of so-called 'state machines' is implemented. This concept relies on three elements necessary for proper functioning [23]: States, events, and actions. Several states were developed including 'acquire' state, 'analyses' state which includes implementation of the fast Fourier transform (FFT), 'save' state, 'wait' state, and 'stop' state. All these states provide a faster and more efficient data collection and analysis. After the user has defined all the states, the user interface is visible, where the user has, at any time, access to representations of the signal in the time domain and in the frequency domain. Additionally, a stacking procedure involving averaging three frequency logs for each tested configuration was conducted, Figure 9.

**Figure 9.** Frequency spectrum averaging procedure.

### *3.5. The Relevant Frequency Range*

As a basic point for investigation of dynamical response, a frequency range needs to be determined. Taking into consideration Equations (7) and (8) and the relation between angular and ordinary frequency ω = 2π *f*, a following arises:

$$f\_n = n \frac{v}{2L'} \text{ where } \mathbf{n} = 1, 2, 3 \dots \tag{9}$$

The wave velocity in a steel bar can be calculated from the ratio of the elasticity and the density of the steel. It is 5048 m/s. In this case, the first three vibration frequencies of the 2100 mm steel rod are 1202, 2404, and 3060 Hz, as shown in Table 1. Comparing these first three calculated frequencies with the measured values in the laboratory shows only slight di fferences (up to 2.70%). Furthermore, the average wave velocity of a 2000 m long grou<sup>t</sup> (length of bar minus the threads) is 3824 m/s (determined by conducted ultrasound laboratory tests) so the first three frequencies, using Equation (9), have values of 956, 1912, and 2868 Hz.

**Table 1.** Comparison of calculated and measured first three natural frequencies of steel bar and fully grouted model.


In the case of the fully grouted model (fg\_model), comprising a 2100 mm long bar and 2000 long grou<sup>t</sup> section, both bar and grou<sup>t</sup> have an influence on the natural frequencies of the model. Therefore, these elements can be considered as parallel coupled resistors which provide a resistance to the motion of the wave through the model. In this case, the wave velocity can be calculated as follows:

$$v\_{f\text{g\\_mod}} = \frac{v\_{\text{steel}} \cdot v\_{\text{groot}}}{v\_{\text{steel}} + v\_{\text{groot}}} \tag{10}$$

The wave velocity for fully grouted model is 2176 m/s. The first three frequencies are then 544, 1088, and 1632 Hz, as shown in Table 1. Given the many factors that are the result of the expected error of the experimental model relative to the above calculations, it can be concluded that the matching of the first three frequencies is relatively good and a relevant frequency range could be established. The fact that the presence of grou<sup>t</sup> causes a decrease of the wave velocity in rock bolts was stated in the literature [24].

Further, calculation of natural frequencies of the partially grouted models (pg\_models) can be analytically calculated by considering the grouted sections as the parallel coupled resistors which are connected in series with the non-grouted sections. A coe fficient of grouting percentage (μ) has to be implemented into equation, which has a form:

$$
\upsilon\_{\text{pg\\_model}} = \left(\frac{\upsilon\_{\text{steel}} \upsilon\_{\text{groot}}}{\upsilon\_{\text{steel}} + \upsilon\_{\text{groot}}}\right) \mu + \upsilon\_{\text{steel}'} (1 - \mu) \tag{11}
$$

The properties of a grou<sup>t</sup> are implemented in Equation (11) through the velocity, based on Young's elasticity modulus and density obtained from laboratory tests conducted on grou<sup>t</sup> samples taken during construction of models. For example, in case of a 70% grouted rock bolt model (μ = 0.7), wave velocity is 3038 m/s, which gives the values of first three frequencies 723, 1447, and 2170 Hz. It is worth noting that Equation (11) determines the wave velocity based on grouting percentage and grou<sup>t</sup> properties. The position of the defects in grouting are not covered by this equation.
