**2. Materials and methods**

### **2.1 Method and critical parameters**

Unlike previously mentioned approaches, the proposed method uses modal analysis procedures and excitation forces are generated by an impact hammer. To perform such a quality assessment of grouting rock bolts several conditions must be fulfilled. One of the primary conditions of realization of the method is identification of a modal model of a tested structure. The modal model of a mechanical system basically consists of two matrices [22–25]:


The modal model may be constructed starting from identification of a single modal eigenvector, and a more sophisticated model (not necessarily complete) would be a set of modal eigenvectors coordinates together with their natural frequencies and damping factors. From an individual characteristic of frequency response function Hjk(ω), where *j, k* stand for excitation and response points, evaluation of a natural frequency, a damping factor and a residue for an *r*-mode is possible, Eq. (**1**).

$$H\_{jk}(o) \to o\_r, \eta\_r, \_r A\_{jk}; \quad r = \mathbf{1}, m \tag{1}$$

In order to calculate foregoing elements of the modal matrix [Φ], as coordinates of modal vectors ϕjr, it is necessary to conduct a series of measurements of frequency response functions in different points of a tested mechanical system. The measurement of a frequency response function at excitation point is very

important. The coordinates of an r-mode may be calculated knowing a residue rAkk at this point using formula (2):

$$\left|\phi\_{kr}\right|^2 = {}\_rA\_{kk} \tag{2}$$

The rest of modal vector coordinates may be calculated using Eq. (3):

$$
\phi\_{\dot{r}r} = \frac{\mathcal{A}\_{jk}}{\phi\_{kr}} \tag{3}
$$

where: *ϕkr*, *ϕjr* - vector coordinates after the process of normalization.

So, for complete presentation of vibration motion of the tested structure with *n* degree of freedom, it is necessary to measure frequency response function at *n* different points of the structure, including a measurement at the excitation point [23, 24, 26, 27]. That is equivalent to the measurement of frequency response functions for a column or vector of a matrix [H]. In practice however, it is quite often appropriate to increase the number of measurement points and perform measurements of additional matrix elements, additional column or row of the matrix, for example hitting the rock bolt in an additional perpendicular direction to an axis of symmetry.

The measurement setup is shown in **Figure 1**. For realization of the method a response transducer was localized at a visible part of the rock bolt, attached using steel ring and stud [26] and a force transducer was localized at an impact hammer head. The direction of excitation was perpendicular to the symmetry axis of the rock bolt as well as is the main axis of the response transducer. After excitation of the rock bolt to transverse vibration, the signals from both the force transducer and the accelerometer (response transducer) were recorded and frequency response function (FRF) was calculated. The excitation was repeated at several points positioned along the outer part of the bar when the accelerometer remained at the same place. The subsequent frequency response functions were stored in universal file format in the computer memory. In the next step data were exported to a workstation where modal parameter extraction methods were realized. Since the installed rock bolt acts as an oscillator, its modal parameters are changed by different lengths and positions of grouting discontinuity. By proper extraction of those parameters, the intended identification was possible.

In this research the natural (resonant) frequency was the main modal parameter taken into account to differentiate foregoing cases of grouting discontinuity. To increase the accuracy of the method frequency response measurements were performed at 5-7 points positioned on the outer part of a rock bolt. It enabled calculating natural frequencies with the use of a larger number of equations and averaging obtained results in the least square sense. Additionally, it was possible to avoid a casual excitation at a nod point of a mode shape [27]. Mode shapes, which is self-understandable, could not be measured on the whole length of a rock bolt (the grouted part of a bar inaccessible). It is only possible in laboratory conditions. **Figures 2** and **3** present the example results of research on known cases of grouting discontinuities in real working conditions where a rock bolt support system was used. The amplitude of FRF function depends on the location of measured points and may differ for each pair of response and reference points, shown on **Figure 3**.

It was also necessary to have a reference point to compare our results with. With this aim the theoretical modal analysis was introduced [23–26] and a base of Finite Element (FE) models were built, encompassing different types of discontinuities (different boundary conditions).

### **Figure 1.**

*Measurement setup: (1) rock bolt, typical length 1.5 m-2.5 m; (2) the accelerometer; (3) the impact hammer; (4) portable measurement system; (5) workstation for modal analysis; (6) the surface of the upper roof section; (7) the grout. L is a grouted length.*

### **Figure 2.**

*An example of FRF functions (waterfall curve) for a known case of grouting discontinuities in real working conditions.*

**Figure 3.**

*An example of stabilization diagram for a known case of grouting discontinuities in real working conditions, the FRF curves for all measured points are shown (the order number of the model was set as 40).*

### **2.2 Reference model**

The lengths on which a rock bolt (a steel bar) is grouted into a roof section form defined border conditions. Different cases of grouting discontinuity can be modeled in theoretical models. To be used as a reference the theoretical model had to be reconciled to the experimental one taking into account a wide range of cases with controlled, known discontinuities of grouting.

In the presented research ANSYS program was used to build the finite element model of a grouted rock bolt, shown in **Figure 4**. The program enables modeling of finite elements with the help of advanced programming tools, so even very complicated geometry shapes can be developed. In the first phase geometry of the examined structure was involved. Then meshing process was realized and particular

**Figure 4.** *An example of a finite element model for an analyzed case study.*


### **Table 1.**

*The properties of components for FE modeling of the investigated rock bolt.*

### **Figure 5.**

*An example of an analyzed case study and a specific mode of natural frequency 232,5 Hz (the amplitude of the mode is scaled for presentation purposes).*

physical properties of materials were introduced including density, Young Modulus, Poisson Ratio etc., shown in **Table 1**. These parameters are crucial since mass and stiffness matrices are built in relation to them. Except steel these parameters were evaluated experimentally. Afterwards adequate physical parameters as well as loads and boundary conditions were attributed to groups of elements. The rock component was ascribed fixed support at the side and back faces and frictional support was attributed to connections between a nut, a plate and roof strata surface. Also, for modeling of torsion force applied to screw the nut and plate to a rock bolt, a bolt pretension feature was utilized (results for different torsion forces were calculated, but that value is controlled by an operator of a bolter and is given). After the process of reconciliation, which comprised all the above mentioned parameters, the validated FE model could be used as a reference base for unknown experimental cases. An example of an analyzed case study and a specific mode of natural frequency is presented in **Figure 5**. The influence of rock strata in which rock bolts are installed was also taken into account but the results obtained both in the experimental and working coal mines (sand stone and mud stone rock strata) showed that the type of rock strata has only a slight impact in comparison with grout discontinuity. As an example, for the particular case, lack of grout at the half length of a rock bolt, the difference ranged from 0.7% to 2.4% for 8 calculated natural frequencies.

As a result of theoretical modal analysis frequency response functions and natural frequencies were calculated (for excitation at the outer part of a rock bolt). Then sets of natural frequencies characteristic of different types of grouting discontinuities were collected and a large data base was set up. **Table 2** presents the sample of data sheets for the carried out calculations and obtained convergence charts of the analyzed cases.

### **2.3 Utilizing of regression methods**

In order to enable fast and effective comparison of theoretical (FE) and experimental models regression methods were reviewed. Cluster analysis seemed to be adequate for that purpose. It is one of the statistical methods to adjust parameters (in the presented research: natural frequencies) measured experimentally with


### **Table 2.**

*The calculated natural frequencies of the finite element models of investigated rock bolts.*

calculated data. The concept of cluster analysis, a term introduced in the work of Tryon [28] actually includes several different classification algorithms. For researchers of many disciplines it poses a major problem to organize the observed data in a sensible structure, or data grouping. In other words, cluster analysis is a tool for exploratory data analysis, whose aim is to arrange objects in a group, in such a way that the degree of binding properties of objects belonging to the same group is the largest, and with objects from other groups is as small as possible. Analysis of the cluster can be used to detect data structures without deriving interpretation/ explanation. In short: cluster analysis only detects structure in the data without explaining why it occurs. The general types of methods of cluster analysis are: agglomeration, grouping of objects and characteristics, and k-means clustering. Analysis of the cluster is not a statistical test, but a collection of different algorithms that group objects with specific features. Unlike many other statistical procedures, methods of cluster analysis are used mostly when we do not have any a priori hypotheses, while we are still in the exploratory phase of our research. Therefore, testing the statistical significance in the traditional sense of the term actually is not applicable. Instead measurement discrepancies or the distances between objects are used. The most direct way to calculate the distance between objects in multidimensional space is Euclidean distance calculation. If we have a two-or threedimensional space, this measure is the actual geometric distance between objects in space. From the point of view of a matching algorithm the actual distances or other derivatives of the distance may be used. The following are the types used.

The first is Euclidean distance. This is a geometric distance in multidimensional space. It should be calculated as follows: distance (x,y) = {Σ<sup>i</sup> (xi - yi) 2 } ½.

Euclidean distance (and squared Euclidean distance) are calculated based on the raw data, and not on the basis of the standardized data. This method has some

advantages (for example, the distance between any two objects is not affected by adding new objects that can be dispersed). However, the differences of units of dimensions may have a big impact on the way distances are calculated. In general, it is appropriate to standardize them in order to have a comparable data scale.

For squared Euclidean distance the distance is raised to a square, to assign more weight to objects that are more remote. It should be calculated as follows: distance (x,y) = {Σ<sup>i</sup> (xi - yi) 2 } ½.

Another type of distance is City distance (Manhattan, City block). This distance is simply the sum of the differences measured along the dimensions. In most cases, this distance measure yields similar results to the ordinary Euclidean distance. In the case of this measure, the impact of single large differences (outliers) is suppressed (because they are not raised to the square). City distance is calculated as follows: distance (x,y) = Σ<sup>i</sup> |xi - yi|.

We use the distance power when we want to increase or decrease the importance that is assigned to the dimensions for which the relevant properties are quite / completely different. This can be achieved using just the power. It is counted as follows: distance (x,y) = (Σ<sup>i</sup> |xi - yi| p ) 1/r, where *r* and *p* are parameters defined by the user. The parameter *p* increases / decreases the weight that is assigned to the differences in the individual dimensions, the parameter *r* increases / decreases the weight that is assigned to differences between objects. If *r* and *p* are equal to 2, then the distance is equal to the Euclidean distance.

In the realized research the application was developed to assign experimentally measured natural frequencies to the appropriate corresponding classes of cases of discontinuity calculated on the base of finite element models. The algorithm uses City distance (x,y) = Σ<sup>i</sup> |xi - yi|. The match procedure was realized in STATISTICA environment. So, for unknown cases we seek for the lowest value of that distance which represents the best fit to the theoretical model (an estimated grout length and position). **Figure 6** presents the characteristics of the transfer function for the analyzed case, **Table 3** the identified natural frequencies and **Figure 7** the scatter plot of the differences between FE model and data evaluated experimentally.

### **Figure 6.**

*The characteristics of the transfer function for the analyzed case (a), and the estimated grout length (b), the length of the rock bolt is equal to 1.8 m, the chart axis are: the vertical axis—inertance, in (m/s<sup>2</sup> )/N, the horizontal axis the frequency, in Hz.*

### *Quality Assessment of Installed Rock Bolts DOI: http://dx.doi.org/10.5772/intechopen.101125*


**Table 3.**

*The identified natural frequencies of the investigated rock bolt.*

### **Figure 7.**

*Scatter plot of the differences between FE model and data evaluated experimentally. The values of the lengths of discontinuity are: (a) 30 cm and (b) 90 cm. These values are specified for the first minimum differences of models and are clearly outside the values of the random scatter.*
