## 3 Validation of the Proposed Method
### Test Results  
According to theory of designing the transducer, the relationship between the compressive stress applied on the sensor face and the output frequency can be expressed as follows:  
\[P=K(f^{2}-f_{0}^{2}) \tag{3}\]  
where \(P\) is the compressive stress,\(f\) is the output frequency and\(f_{0}\) is the initial frequency. The coefficient \(K\) is related to the properties of transducer and usually determined via calibrating test. It should be noted that the calibrating coefficient \(K\) generally is obtained in the condition that the compressive stress \(P\) is applied directly on the sensor face. When the transducers are buried into the rock and used to measure the rock stress, the calibrating coefficient \(K\) cannot be used directly because the compressive stress acting on the sensor face is very complicated and is not equal to the initial rock stress. As shown in Figure 11, to simulate the measuring performance of a transducer embedded in a rock mass, cement mortar body model tests are carried out in a true-triaxial loading system (Manufactured by the Institute of Rock and Soil Mechanics, Chinese Academy of Science, China). The relationship between the exterior loading stress and the output frequencies can be expressed as:  
\[P_{c}=LF^{-1}\cdot K(f^{2}-f_{0}^{2}) \tag{4}\]  
where \(P_{c}\) is the exterior loading acting on the sample face, \(LF\) is the loading factor and defined as the ratio of the measured compressive stress to the exterior loading. To obtain the LF, tests have been conducted on four cement mortar samples with elastic modulus varying from 0.86 GPa to 6.48 GPa. The test results show that the value of LF decreases as the elastic modulus increases. Figure 12 shows the test results, and it can be found that the value of LF decreases as the elastic modulus increases. Figure 13 shows the linear fitting relationship between \(LF\) and the elastic modulus of the cement mortar. According to the elastic modulus listed in Table 2, the values of \(LF\) in PDS No. 1 and No. 11 coal mines are 0.591 and 0.777, respectively. Figure 10: Test process for the RSR method in the PDS No. 1 coal mine: (**a**) Drilling the hole; (**b**) installing the column transducer; (**c**) connecting the pushrods; (**d**) injection pipe for grouting with the cement mortar. The information of transducers embedded in the PDS No. 1 coal mine is presented in Table 3, where the Azimuthal angle and dip angle are based on geodetic coordinates (x axis: north, y axis: west, z axis: vertical). Figure 14 shows the frequency evolution process of the six sensor elements with time during the measurement in PDS No. 1 coal mine. As shown in Figure 14, the testing time lasted 181 days. Figure 11: Model tests on a cement mortar body. ((**a**). steel support frame; (**b**). steel dowels; (**c**). hydraulic jack; (**d**). pressure gauge; (**e**). cement mortar body with a transducer embedded in the middle). Figure 12: Results of calibration tests. Figure 13: Relationship between elastic modulus and loading factor. According to Equation (4), the variation in the compressive stress output from the transducer with test time can be obtained (see Figure 15). From Figure 15, it can be observed that the measured stresses increase rapidly during the first few days and gradually become stable when the time of the test exceeded 140 days. The characteristics of stress curves show are similar to classic creep curves for rock masses where the transient creep stage and steady-state creep stage are contained. It was also found that the time for the stress become stable was much longer than the relaxation time obtained from the creep test, which generally is in hundreds of hours. This indicated that the rheological process of the rock mass is much slower than that of the rock specimens. Numerical simulations for transducers embedded in rock masses show recovery rate, which is defined as the ratio of the stress by a transducer to the initial stress of the rock mass before drilling, depends on the rheological properties of the rock mass [18].