Abstract
Modal analysis theory of rock is proposed and the modeling of vibration characteristics of fractured rock is undertaken in this study. The modeling includes two aspects, namely, the natural frequency of rock with a single fracture and the crack expansion energy of rock with multiple fractures. Also, the results of numerical analysis are presented. Four main control parameters are considered, including the material properties, crack size, crack trend and the number of cracks. It is confirmed that the natural frequency of rock will be reduced by the cracks in it. The expansion energy of a single crack is inversely proportional to the number of such cracks in the rock. Namely, the more the similar cracks are, the less the energy required for a single crack expansion is and the smaller the excitation frequency needed for rock resonance is. The vibration characteristics of fractured rock are validated by numerical analysis. The natural frequency of rock increases with the increase of elastic modulus, and decreases with the increase of the angle, the length, the width and the number of cracks.
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Abbreviations
- \({\mathbf{M}}\) :
-
Mass matrix of rock
- \({\mathbf{K}}\) :
-
Stiffness matrix of rock
- \({\mathbf{x}}\) :
-
Displacement array of rock
- \({{\ddot{\mathbf{x}}}}\) :
-
Acceleration array of rock
- \({\mathbf{F}}({\mathbf{t}})\) :
-
Excitation force array on the rock
- \({\varvec{\Phi}}\) :
-
Free response amplitude array of rock
- \(\omega_{i}\) :
-
The i order natural frequency
- \(k\) :
-
Stiffness of rock
- \(m\) :
-
Mass of rock
- \(L\) :
-
Length of rock
- \(E\) :
-
Elastic modulus of rock
- \(A\) :
-
Cross-sectional area of rock
- \(\sigma\) :
-
Fracture strength
- \(a\) :
-
Half length of fracture
- \(\gamma_{s}\) :
-
Density of surface energy
- \(p_{i}\) :
-
The number of rock unit which contains i class crack
- \(E_{io}\) :
-
Expansion energy of crack
- \(\dot{x}\) :
-
Vibration velocity of rock
- \(c\) :
-
Damping coefficient
- \(\left| {H(\omega_{i} )} \right|\) :
-
Amplitude–frequency characteristics of rock
- \(n\) :
-
The number of times of vibration
- \(T\) :
-
Harmonic force on the surface of rock
- \(f(t)\) :
-
Excitation force
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Acknowledgments
The support of National Natural Science Foundation Major Project of China (No. 51490650) and Scientific Research and Technology Development Project of CNPC (No. 2014A-4211) are gratefully acknowledged. The authors would also like to thank the support of Graduate Student Innovation Research Projects of Northeast Petroleum University (No. YJSCX2014-011NEPU).
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An erratum to this article can be found at http://dx.doi.org/10.1007/s00603-016-1072-3.
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Li, S., Yan, T., Li, W. et al. Simulation on Vibration Characteristics of Fractured Rock. Rock Mech Rock Eng 49, 515–521 (2016). https://doi.org/10.1007/s00603-015-0762-6
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DOI: https://doi.org/10.1007/s00603-015-0762-6