- Email: [email protected]
Effect of a preload force on anchor system frequency
International Journal of Mining Science and Technology 23 (2013) 135–138
Contents lists available at SciVerse ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Effect of a preload force on anchor system frequency Lu Aihong a,b,⇑, Xu Jinhai a, Liu Haishun a a b
State Key Laboratory of Geomechanics & Deep Underground Engineering, China University of Mining & Technology, Xuzhou 221008, China School of Mechanics and Civil Engineering, China University of Mining & Technology, Xuzhou 221116, China
a r t i c l e
i n f o
Article history: Received 5 April 2012 Received in revised form 7 May 2012 Accepted 6 June 2012 Available online 18 April 2013 Keywords: Frequency Preload force Anchor detection Numerical simulation
a b s t r a c t The interrelationship between preload forces and natural frequencies of anchors was obtained from the structure of an anchor and its mechanical characteristics. We established a numerical model for the dynamic analysis of a bolt support system taking into consideration the working surroundings of the anchor. The natural frequency distribution of the system under various preload forces of the anchor was analyzed with ANSYS. Our results show that each order of the system frequency varied with an increase in preload forces. A single order frequency decreased with an increase in the preload force. A preload force affected low-order frequencies more than high-order frequencies. We obtained a functional relationship by fitting preload forces and fundamental frequencies, which was in agreement with our theoretical considerations. This study provides theoretical support for the detection of preload forces. Ó 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Coupled with the improvement in pre-stressed techniques, prestressed anchors are widely applied in mining engineering. The magnitude of pre-stress in practical work is a continuous concern to engineers. At present, detective measures for the quality of bolt anchoring apply a hydraulic jack to carry out destructive drawing tests and obtain the pre-stress value. But these kinds of measurements have two shortcomings. The first one is that it requires much time and labor to carry out the experiments, whereas the second one is that the experiments produce strong local disturbance on wall rocks and decrease the effectiveness of the anchor support as yet, which is especially disadvantageous to the soft rock or fractured strata. Since the 1990s, many investigations have been carried out on bolt anchoring [1–5]. However, there is not an accepted measurement to detect the service load of the anchor and anchoring force. It is therefore necessary to find a simple and effective method acceptable to detect the service load of an anchor and its anchoring force. According to the theory by Clough and Penzien, the natural vibration frequency of an anchor subjected to axial force is caused by the axial load, and this natural vibration frequency decreases with an increase in the axial force [6]. Theoretically, the changes in the axial force can be obtained from tests of natural frequency. In this paper, we investigate the pre-stress by testing for natural frequencies and much effort has been made by many researchers [7–10].
⇑ Corresponding author. Tel.: +86 516 83885205. E-mail address: [email protected] (A. Lu).
In our research, the vibration characteristics of an anchor under different pre-stresses were simulated and we obtained a functional relationship between the preload force and its fundamental frequency. 2. Analysis of vibration characteristics of pre-stressed anchor A fixed site was exposed to dual effects from both bonding resin and the surrounding rock. Given the co-operation of the bonding analysis of the anchor vibration, we posed the following hypothesis [9–11]. Anchors stimulate vibrations in the elastic range. When an anchor vibrates, the relationship among displacement, stress and strain of each part of the anchor body obeys the Hooker law. In low strain dynamic tests, with a small exciting force which can be controlled, the vibration of the anchor can meet the conditions for the hypothesis. When the anchor material is uniform or uniform in segments and isotropic, the anchor also approximately meets the conditions for the hypothesis in low strain tests. When the stimulated anchor vibrates, its cross section remains a plane, and the size of all particles and the direction of the displacement in the same section are similar. There is no phase difference or exceeding or lagging in vibration. For a distance x from the end of the anchor, we take an element of length dx, and u represents the displacement of the section at the distance x at time t, as shown in Fig. 1. Given that its cross section remains a plane during longitudinal vibration, the lateral deformation can be neglected. Then, u(x, t) is a binary function of section position x and time t; q is the mass per unit volume of anchor; E is the elastic modulus of the anchor.
2095-2686/$ - see front matter Ó 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2013.03.003
136
A. Lu et al. / International Journal of Mining Science and Technology 23 (2013) 135–138
u u+
∂u dx ∂x X(x)
X(x)
x x
σ+
σ
dx L1
∂σ dx ∂x
dx L Fig. 1. Longitudinal vibration of anchor.
According to Newton’s second law of movement:
@2u qAdx 2 ¼ @t
@r @r @ @u dx rx þ x rx A ¼ A x dx ¼ EA @x @x @x @x
F ¼ EAD
If we neglect the higher order terms, we have:
ð2Þ
Eq. (2) is the longitudinal one-dimensional wave equation, qffiffiffi showing that the longitudinal wave spreads at a speed of qE . Eq. (2) can be calculated by means of variable separation method. If we assume that the length of the non-anchored section of the anchor is l, when the two ends, x = 0 and x = l are subjected to certain restrictions, then Eq. (2) is a control equation of longitudinal vibration. The form of the solution is assumed as follows:
uðx; tÞ ¼ XðxÞTðtÞ
T€ X 00 ¼ 2 ¼ k2 X c0 T
ð4Þ
i.e.,
0
2pf c0
" 2 # 2p f 1 2pfl F ¼ EAD 1 c0 2! c0
ð15Þ
From Eq. (15), we can see that the working load of the anchor is proportional to some power of its natural frequency. But because realistic conditions of a working anchor are very complex, it is very difficult to express them as analytical formulas. We applied the software ANSYS to discuss the laws of change in the natural frequency of the anchor system under different pre-tightening forces. 3. Numerical model of bolt supporting system vibration
ð3Þ
where X(x) is the main or normal function defining the vibration mode, and T(t) determines the development of the vibration mode with time t. Substituting Eq. (3) into Eq. (2), we obtain:
X 00 þ k2 X ¼ 0 T€ þ c2 k2 T ¼ 0
ð14Þ
ð1Þ
i.e.,
@2u E @2u ¼ @t2 q @x2
" # 2 2p f 1 2pfl 1 þ c0 2! c0
ð5Þ ð6Þ
qffiffiffi E
If k ¼ ; c0 ¼ q are the propagation speeds of the wave and the natural frequency of the system respectively, then the solutions to Eq. (5) are:
XðxÞ ¼ C cos kx þ D sin kx
ð7Þ
TðtÞ ¼ A cos c0 kt þ B sin c0 kt
ð8Þ
The constants C and D in Eq. (7) can be determined from the constraints at the anchor ends. In order to simplify the calculations, one end of the anchor is fixed and the other is subject to an axial tensile force F. If F is the working load of the anchor, then the boundary conditions are:
uð0; tÞ ¼ 0 @u EA ¼ F @x
ð9Þ 4. Analysis of numerical simulation of pre-stress anchor system
ð10Þ
x¼l
And the simultaneous solutions to Eqs. (5), (9), and (10) can obtain:
Xð0Þ ¼ 0
ð11Þ
Setting C = 0
EAX 0 ðlÞ ¼ F
ð12Þ
then
F ¼ EADk cos kl ¼ EAD
When the end of the anchor was subjected to external loads, the resin medium, the surrounding rock, anchor and the support plate form one system by the action of resin medicine-one and the support plate. The system will vibrate together with the anchor. The resin anchor system commonly used in coal mines is shown in Fig. 1. The system can be regarded as an infinite half space problem. Considering the force and deformation symmetry of the bolting system and the boundary effect of the bolt action in numerical analysis, the numerical model can be taken in the analysis as a finite axial symmetric plane element model, as shown in Fig. 2. l is the anchor length and l1 is the anchoring length. It is assumed that l1 = 0.5 m. We restricted the symmetry of the border by displacement, i.e., the symmetry was placed on left side of the model, and the horizontal chain pole restrictions was set on the right side of the model in order to simulate infinite boundaries, as shown in Fig. 2. The mechanical parameters of the anchor system are listed in Table 1. We applied a preloaded force F to the anchor with an element of PRETS179. The anchor, the surrounding wall and the resin developed into an asymmetric type of plane element 42. The finite element girding the anchor system is shown in Fig. 3. The stress distribution of the surrounding wall along the axial direction under a pre-tightening force is shown in Fig. 4.
2p f 2pfl cos c0 c0
The Taylor series expansion of Eq. (13) is given by:
ð13Þ
4.1. Vibration frequency of anchor system under different preload forces The natural frequencies of different preload forces of an anchor system are listed in Table 2, where the anchor length l = 2.0 m and the anchoring length l1 = 0.5 m. fi (i = 1, 2, . . ., 5) is the ith order frequency in Table 2. When a preload force is applied to an anchor system, each frequency order to the system changes with an increase in the anchor preload force F; a single frequency order decreases with an increase in the anchor preload force F. When F increases from 10 to 110 kN, the decrement value of the primary frequency is 1130.9 Hz; and the decrement rate is 92.1%. The decrement value
137
A. Lu et al. / International Journal of Mining Science and Technology 23 (2013) 135–138
A L Surrounding wall
R
l
Support plate
l1 r
P (t) 20 anchor
Resin A-A
A Fig. 2. Dynamical model of bolt supporting system.
Table 1 Physical and mechanical properties of bolt supporting system materials.
Anchor Resin Support plate Surrounding wall
Density q (kg/ m3)
Elastic modulus E (GPa)
Poisson’s ratio
7840 1800 7840 2500
200 16 200 28
0.30 0.35 0.30 0.25
l
Fig. 4. Vertical stress distribution of anchorage system under the action of preload force.
Table 2 Relation between natural frequency and preload force of anchored.
of the third frequency is 968.7 Hz, with a decrement rate of 25.6%. The decrement value of the fifth frequency is 329.6 Hz and its decrement rate is 25.1%. A single frequency order of the system decreases with an increase in the anchor preload force F. The decrease at the higher frequency is slightly smaller than that at the lower frequencies. It shows that the anchor preload force F has a great effect at low frequencies of an anchor system. Low frequencies can reflect change in anchor preload forces better. 4.2. Relation between working load of anchor and fundamental frequency
f1
f2
f3
f4
f5
10 15 20 25 30 35 40 45 50 60 70 80 90
1223.0 1175.2 604.79 387.62 314.63 285.13 265.41 251.68 237.33 211.56 180.26 139.38 96.20
1816.6 1816.7 1816.8 1816.9 1810.6 1761.8 1709.9 1665.5 1620.5 1537.7 1464.6 1397.1 1334.5
3501.0 3377.2 2677.4 2587.6 2567.3 2560.4 2556.3 2553.8 2551.1 2546.3 2541.3 2536.8 2533.3
4639.3 4639.4 4639.5 4639.6 4635.1 4597.9 4545.8 4490.4 4423.7 4273.6 4116.2 3951.0 3783.0
5280.2 5273.5 5187.9 5156.7 5147.9 5144.2 5142.1 5142.5 5140.8 5133.2 4643.3 4017.3 4643.3
120
F (kN)
Fig. 3. Finite element girding of anchorage system.
F (kN)
y= 3×10-7 x3 +0.0006x 2 0.5002x+155.25 R2 = 0.9927
90 60 30
In the previous section we saw that the anchor preload force F has a great effect on the fundamental frequency of an anchor system. On the basis of vibration frequency analysis, we carried out a further study on the fundamental frequency of an anchor system under the action of preload force. Fig. 5 shows the change in
0
300
600
900
1200
f (Hz) Fig. 5. Fitting curve of relation between preload force and anchorage system.
138
A. Lu et al. / International Journal of Mining Science and Technology 23 (2013) 135–138
fundamental frequency from the preload force F. This fundamental frequency and preload force F can be fitted by a third order polynomial. Letting the preload force F = a0 + a1f + a2f2 + a3f3, where a0, a1, a2 and a3 are constants related to the anchor system. The constants a0 = 155.2, a1 = 0.5002, a2 = 0.0006 and a3 = 3 107 were obtained by fitting the polynomial to our frequency data. A correlation coefficient R2 is 0.99 implied a very good fitting of our simulation results and agrees closely with our theoretical considerations.
Acknowledgments The study was conducted with the financial support from the National Basic Research Program of China (No. 2013CB227900), the China Postdoctoral Science Foundation (No. 20110491483) and the State Key Laboratory of Coal Resources and Mine Safety (No. 10F08). References
5. Conclusions (1) Each frequency order of the system changes with an increase in the anchor preload force F; a single frequency order decreases with an increase in F. When F increased from 10 to 110 kN, the decrement values were 1130.9, 968.7 and 329.6 Hz, with decrement rates of 92.1%, 25.6% and 25.1% for the primary, the third and the fifth frequency orders respectively. A single frequency order of the system decreased with an increase in the anchor preload force F, where the decrement in the higher frequency orders is less than that at lower frequency orders. This showed that the anchor preload force F has a great effect at low frequency orders of the anchor system. The lower frequency order reflected the change of anchor preload force F better. (2) Our theoretical analysis and numerical simulation indicate that there was a typical power function relationship between the working load and fundamental frequencies of the anchor; its correlation coefficient was 0.9927.
[1] Xu M, Zhang YX. Numerical study of low strain dynamic test for bolt. Chin J Rock Mech Eng 2003;22(9):1538–41. [2] Yang H, Wang C. Mathematical models and responding analysis for bolt and surrounding rock system. J Test Meas Technol 2002;16(1):41–4. [3] Li Y, Xu M, Zhang YX. Analytic solution of anchor bolts with dynamic detection. J Chongqing Jianzhu Univ 2005;25(2):48–53. [4] Yang H, Wang C. Study on propagation law of elastic wave in bolt anchoring system. J Test Meas Technol 2003;17(2):145–9. [5] Tannant DD, Brummer Rk. Journal of Geotechnical and Geoviron: field tests and modeling. Int J Rock Mech Sci Geomech Abstr 1995;32(6):537–50. [6] Clough RW, Penzien J. Dynamic of structures. USA: McGraw Hill; 1982. [7] Li QF. Study on nondestructive testing mechanism and methods for rockbolt supporting based on stress wave. Xuzhou: China University of Mining and Technology; 2007. [8] Liu HF, Yang WW, Li Y. Research on real-time test technique of work in load of anchorage bolt. Min Res Dev 2008;28(2):54–7. [9] Zhang YT, Wang XL, Li RG. Experimental research on nature frequency of prestressed concrete beams. Huazhong Univ of Sci Tech 2007;35(2):12–5. [10] Wang MW, Wang HL. Technique and method field tests of non destructive inspection of anchoring quality of rock bolts. Hydrogeol Eng Geol 1998;1:56–8. [11] Ivanovic A, Neilson RD, Rodger AA. Lumped parameter modeling of singletendon ground anchorage systems. Geotech Eng 2001;149(2):103–13.
Contents lists available at SciVerse ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Effect of a preload force on anchor system frequency Lu Aihong a,b,⇑, Xu Jinhai a, Liu Haishun a a b
State Key Laboratory of Geomechanics & Deep Underground Engineering, China University of Mining & Technology, Xuzhou 221008, China School of Mechanics and Civil Engineering, China University of Mining & Technology, Xuzhou 221116, China
a r t i c l e
i n f o
Article history: Received 5 April 2012 Received in revised form 7 May 2012 Accepted 6 June 2012 Available online 18 April 2013 Keywords: Frequency Preload force Anchor detection Numerical simulation
a b s t r a c t The interrelationship between preload forces and natural frequencies of anchors was obtained from the structure of an anchor and its mechanical characteristics. We established a numerical model for the dynamic analysis of a bolt support system taking into consideration the working surroundings of the anchor. The natural frequency distribution of the system under various preload forces of the anchor was analyzed with ANSYS. Our results show that each order of the system frequency varied with an increase in preload forces. A single order frequency decreased with an increase in the preload force. A preload force affected low-order frequencies more than high-order frequencies. We obtained a functional relationship by fitting preload forces and fundamental frequencies, which was in agreement with our theoretical considerations. This study provides theoretical support for the detection of preload forces. Ó 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Coupled with the improvement in pre-stressed techniques, prestressed anchors are widely applied in mining engineering. The magnitude of pre-stress in practical work is a continuous concern to engineers. At present, detective measures for the quality of bolt anchoring apply a hydraulic jack to carry out destructive drawing tests and obtain the pre-stress value. But these kinds of measurements have two shortcomings. The first one is that it requires much time and labor to carry out the experiments, whereas the second one is that the experiments produce strong local disturbance on wall rocks and decrease the effectiveness of the anchor support as yet, which is especially disadvantageous to the soft rock or fractured strata. Since the 1990s, many investigations have been carried out on bolt anchoring [1–5]. However, there is not an accepted measurement to detect the service load of the anchor and anchoring force. It is therefore necessary to find a simple and effective method acceptable to detect the service load of an anchor and its anchoring force. According to the theory by Clough and Penzien, the natural vibration frequency of an anchor subjected to axial force is caused by the axial load, and this natural vibration frequency decreases with an increase in the axial force [6]. Theoretically, the changes in the axial force can be obtained from tests of natural frequency. In this paper, we investigate the pre-stress by testing for natural frequencies and much effort has been made by many researchers [7–10].
⇑ Corresponding author. Tel.: +86 516 83885205. E-mail address: [email protected] (A. Lu).
In our research, the vibration characteristics of an anchor under different pre-stresses were simulated and we obtained a functional relationship between the preload force and its fundamental frequency. 2. Analysis of vibration characteristics of pre-stressed anchor A fixed site was exposed to dual effects from both bonding resin and the surrounding rock. Given the co-operation of the bonding analysis of the anchor vibration, we posed the following hypothesis [9–11]. Anchors stimulate vibrations in the elastic range. When an anchor vibrates, the relationship among displacement, stress and strain of each part of the anchor body obeys the Hooker law. In low strain dynamic tests, with a small exciting force which can be controlled, the vibration of the anchor can meet the conditions for the hypothesis. When the anchor material is uniform or uniform in segments and isotropic, the anchor also approximately meets the conditions for the hypothesis in low strain tests. When the stimulated anchor vibrates, its cross section remains a plane, and the size of all particles and the direction of the displacement in the same section are similar. There is no phase difference or exceeding or lagging in vibration. For a distance x from the end of the anchor, we take an element of length dx, and u represents the displacement of the section at the distance x at time t, as shown in Fig. 1. Given that its cross section remains a plane during longitudinal vibration, the lateral deformation can be neglected. Then, u(x, t) is a binary function of section position x and time t; q is the mass per unit volume of anchor; E is the elastic modulus of the anchor.
2095-2686/$ - see front matter Ó 2013 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2013.03.003
136
A. Lu et al. / International Journal of Mining Science and Technology 23 (2013) 135–138
u u+
∂u dx ∂x X(x)
X(x)
x x
σ+
σ
dx L1
∂σ dx ∂x
dx L Fig. 1. Longitudinal vibration of anchor.
According to Newton’s second law of movement:
@2u qAdx 2 ¼ @t
@r @r @ @u dx rx þ x rx A ¼ A x dx ¼ EA @x @x @x @x
F ¼ EAD
If we neglect the higher order terms, we have:
ð2Þ
Eq. (2) is the longitudinal one-dimensional wave equation, qffiffiffi showing that the longitudinal wave spreads at a speed of qE . Eq. (2) can be calculated by means of variable separation method. If we assume that the length of the non-anchored section of the anchor is l, when the two ends, x = 0 and x = l are subjected to certain restrictions, then Eq. (2) is a control equation of longitudinal vibration. The form of the solution is assumed as follows:
uðx; tÞ ¼ XðxÞTðtÞ
T€ X 00 ¼ 2 ¼ k2 X c0 T
ð4Þ
i.e.,
0
2pf c0
" 2 # 2p f 1 2pfl F ¼ EAD 1 c0 2! c0
ð15Þ
From Eq. (15), we can see that the working load of the anchor is proportional to some power of its natural frequency. But because realistic conditions of a working anchor are very complex, it is very difficult to express them as analytical formulas. We applied the software ANSYS to discuss the laws of change in the natural frequency of the anchor system under different pre-tightening forces. 3. Numerical model of bolt supporting system vibration
ð3Þ
where X(x) is the main or normal function defining the vibration mode, and T(t) determines the development of the vibration mode with time t. Substituting Eq. (3) into Eq. (2), we obtain:
X 00 þ k2 X ¼ 0 T€ þ c2 k2 T ¼ 0
ð14Þ
ð1Þ
i.e.,
@2u E @2u ¼ @t2 q @x2
" # 2 2p f 1 2pfl 1 þ c0 2! c0
ð5Þ ð6Þ
qffiffiffi E
If k ¼ ; c0 ¼ q are the propagation speeds of the wave and the natural frequency of the system respectively, then the solutions to Eq. (5) are:
XðxÞ ¼ C cos kx þ D sin kx
ð7Þ
TðtÞ ¼ A cos c0 kt þ B sin c0 kt
ð8Þ
The constants C and D in Eq. (7) can be determined from the constraints at the anchor ends. In order to simplify the calculations, one end of the anchor is fixed and the other is subject to an axial tensile force F. If F is the working load of the anchor, then the boundary conditions are:
uð0; tÞ ¼ 0 @u EA ¼ F @x
ð9Þ 4. Analysis of numerical simulation of pre-stress anchor system
ð10Þ
x¼l
And the simultaneous solutions to Eqs. (5), (9), and (10) can obtain:
Xð0Þ ¼ 0
ð11Þ
Setting C = 0
EAX 0 ðlÞ ¼ F
ð12Þ
then
F ¼ EADk cos kl ¼ EAD
When the end of the anchor was subjected to external loads, the resin medium, the surrounding rock, anchor and the support plate form one system by the action of resin medicine-one and the support plate. The system will vibrate together with the anchor. The resin anchor system commonly used in coal mines is shown in Fig. 1. The system can be regarded as an infinite half space problem. Considering the force and deformation symmetry of the bolting system and the boundary effect of the bolt action in numerical analysis, the numerical model can be taken in the analysis as a finite axial symmetric plane element model, as shown in Fig. 2. l is the anchor length and l1 is the anchoring length. It is assumed that l1 = 0.5 m. We restricted the symmetry of the border by displacement, i.e., the symmetry was placed on left side of the model, and the horizontal chain pole restrictions was set on the right side of the model in order to simulate infinite boundaries, as shown in Fig. 2. The mechanical parameters of the anchor system are listed in Table 1. We applied a preloaded force F to the anchor with an element of PRETS179. The anchor, the surrounding wall and the resin developed into an asymmetric type of plane element 42. The finite element girding the anchor system is shown in Fig. 3. The stress distribution of the surrounding wall along the axial direction under a pre-tightening force is shown in Fig. 4.
2p f 2pfl cos c0 c0
The Taylor series expansion of Eq. (13) is given by:
ð13Þ
4.1. Vibration frequency of anchor system under different preload forces The natural frequencies of different preload forces of an anchor system are listed in Table 2, where the anchor length l = 2.0 m and the anchoring length l1 = 0.5 m. fi (i = 1, 2, . . ., 5) is the ith order frequency in Table 2. When a preload force is applied to an anchor system, each frequency order to the system changes with an increase in the anchor preload force F; a single frequency order decreases with an increase in the anchor preload force F. When F increases from 10 to 110 kN, the decrement value of the primary frequency is 1130.9 Hz; and the decrement rate is 92.1%. The decrement value
137
A. Lu et al. / International Journal of Mining Science and Technology 23 (2013) 135–138
A L Surrounding wall
R
l
Support plate
l1 r
P (t) 20 anchor
Resin A-A
A Fig. 2. Dynamical model of bolt supporting system.
Table 1 Physical and mechanical properties of bolt supporting system materials.
Anchor Resin Support plate Surrounding wall
Density q (kg/ m3)
Elastic modulus E (GPa)
Poisson’s ratio
7840 1800 7840 2500
200 16 200 28
0.30 0.35 0.30 0.25
l
Fig. 4. Vertical stress distribution of anchorage system under the action of preload force.
Table 2 Relation between natural frequency and preload force of anchored.
of the third frequency is 968.7 Hz, with a decrement rate of 25.6%. The decrement value of the fifth frequency is 329.6 Hz and its decrement rate is 25.1%. A single frequency order of the system decreases with an increase in the anchor preload force F. The decrease at the higher frequency is slightly smaller than that at the lower frequencies. It shows that the anchor preload force F has a great effect at low frequencies of an anchor system. Low frequencies can reflect change in anchor preload forces better. 4.2. Relation between working load of anchor and fundamental frequency
f1
f2
f3
f4
f5
10 15 20 25 30 35 40 45 50 60 70 80 90
1223.0 1175.2 604.79 387.62 314.63 285.13 265.41 251.68 237.33 211.56 180.26 139.38 96.20
1816.6 1816.7 1816.8 1816.9 1810.6 1761.8 1709.9 1665.5 1620.5 1537.7 1464.6 1397.1 1334.5
3501.0 3377.2 2677.4 2587.6 2567.3 2560.4 2556.3 2553.8 2551.1 2546.3 2541.3 2536.8 2533.3
4639.3 4639.4 4639.5 4639.6 4635.1 4597.9 4545.8 4490.4 4423.7 4273.6 4116.2 3951.0 3783.0
5280.2 5273.5 5187.9 5156.7 5147.9 5144.2 5142.1 5142.5 5140.8 5133.2 4643.3 4017.3 4643.3
120
F (kN)
Fig. 3. Finite element girding of anchorage system.
F (kN)
y= 3×10-7 x3 +0.0006x 2 0.5002x+155.25 R2 = 0.9927
90 60 30
In the previous section we saw that the anchor preload force F has a great effect on the fundamental frequency of an anchor system. On the basis of vibration frequency analysis, we carried out a further study on the fundamental frequency of an anchor system under the action of preload force. Fig. 5 shows the change in
0
300
600
900
1200
f (Hz) Fig. 5. Fitting curve of relation between preload force and anchorage system.
138
A. Lu et al. / International Journal of Mining Science and Technology 23 (2013) 135–138
fundamental frequency from the preload force F. This fundamental frequency and preload force F can be fitted by a third order polynomial. Letting the preload force F = a0 + a1f + a2f2 + a3f3, where a0, a1, a2 and a3 are constants related to the anchor system. The constants a0 = 155.2, a1 = 0.5002, a2 = 0.0006 and a3 = 3 107 were obtained by fitting the polynomial to our frequency data. A correlation coefficient R2 is 0.99 implied a very good fitting of our simulation results and agrees closely with our theoretical considerations.
Acknowledgments The study was conducted with the financial support from the National Basic Research Program of China (No. 2013CB227900), the China Postdoctoral Science Foundation (No. 20110491483) and the State Key Laboratory of Coal Resources and Mine Safety (No. 10F08). References
5. Conclusions (1) Each frequency order of the system changes with an increase in the anchor preload force F; a single frequency order decreases with an increase in F. When F increased from 10 to 110 kN, the decrement values were 1130.9, 968.7 and 329.6 Hz, with decrement rates of 92.1%, 25.6% and 25.1% for the primary, the third and the fifth frequency orders respectively. A single frequency order of the system decreased with an increase in the anchor preload force F, where the decrement in the higher frequency orders is less than that at lower frequency orders. This showed that the anchor preload force F has a great effect at low frequency orders of the anchor system. The lower frequency order reflected the change of anchor preload force F better. (2) Our theoretical analysis and numerical simulation indicate that there was a typical power function relationship between the working load and fundamental frequencies of the anchor; its correlation coefficient was 0.9927.
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