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De-icing Method for Coupled Transmission Tower-Line System
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Energy Procedia 17 (2012) 1383 – 1389
2012 International Conference on Future Electrical Power and Energy Systems
De-icing Method for Coupled Transmission Tower-Line System Qu Cheng-Zhong, Liu Yue-Jun School of Civil Engineering , Northeast Dianli University, Jilin, China
Abstract For the disaster brought by the transmission line regelation, a new de-icing method is presented in this paper. It is supposed that the ice and transmission line are combined closely when there is no external loads, the composite structure of ice-transmission line is set up, then the coupled model of the composite structure-tower is set up. The equations of motion for the coupled system are derived. The composite structure can be applied to exciting loads, the frequency of which is close to the natural frequency of ice. The natural frequency of ice is far from that of the transmission line. Based on the resonance theory, the ice occurs to deform and crush, finally the ice divorces from the conductors under the function of the exciting load, but those transmission lines are still very well, the analytical solution can be obtained and the new de-icing method for transmission line based on the theory of resonance is got. It is suitable for the transmission line to de-ice through the computation of the concrete example.
© 2012 and/or peer-review under responsibility of Hainan University. 2011 Published Publishedby byElsevier ElsevierLtd. Ltd.Selection Selection and/or peer-review under responsibility of [name organizer] Keywords: the composite structure of ice-transmission line; resonance; de-icing
1. Introduction Ice accumulating on transmission lines can cause extensive damage to the lines and towers. At the beginning of 2005, Hunan province was affected the worst by ice storm, many transmission-towers happened to collapse[1]. Since mid-January 2008, most of districts were troubled in the catastrophe weather. Many transmission towers, conductors, insulators and electric power fittings were damaged seriously[2]. Over a million of consumers lost their electricity supply for some period. It deeply influenced the development of the social economy and people’s life. So it is the urgent question to seek the de-icing method for the transmission tower-line system, which is in order to guarantee the safety of the power network. The commonly used de-icing method at present is Āad hocā[3], it only can deal with a small amount of ice and the speed of de-icing is very slow. The large current ice-melting method has the high efficiency
1876-6102 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. doi:10.1016/j.egypro.2012.02.256
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
of de-icing, but it also consumes a lot of energy and operates very complex, it does a large harm to the transmission tower-line system. The chemistry method not only needs the suitable working environment and good traffic conditions but also needs to consider the corrosion to the tower. The mechanical de-icing method is difficult to meet the environment of transmission line. Ice-melting by short-circuit current, which needs to stop all the overhead transmission lines in the ice-melting circuit, and it is difficult to choose the ice-melting power supply[4~7]. The regelation becomes the biggest problem to the transmission tower-line system, it raises a hot discussion on how to de-ice fast and effectively. In order to avoid recurrence of such a situation, it is quite necessary to study and exploit a new de-icing method to reduce the disaster caused by transmission line regelation. Therefore, the composite structure of ice-transmission line is set up in this paper, the new de-icing method for transmission line based on the resonance theory is got. 2. The principle of the de-icing It makes the ice equivalent to annular section. It is supposed that the ice and transmission line is combined closely when there is no external loads, the composite structure of ice-transmission line is set up. Because the natural frequency of ice is far from that of the line, the exciting load that the frequency is close to the natural frequency of ice is applied on the composite structure. It makes the amplitude of forced vibration become bigger obviously. The ice belongs to the brittle material, whose strength is much less than that of line. Based on the theory of resonance, the tensile stress of the composite structure is larger than the allowable tensile stress of ice, but it is far less than the allowable tensile stress of transmission line, which make the composite structure generate the small rotational angle and deflection. However, the rotational angle and deflection of the transmission lines are both smaller than the allowable rotational angle and allowable deflection. The ice occurs to deform and crush because its strength and rigidity are not enough, but those of the transmission lines are satisfied. So the transmission lines are still very well. 2.1 The composite structure of ice-transmission line According to the designing standard of transmission line, the section of ice seems as the annular, it is supposed that the ice closely wrappes on the transmission line when there is no external loads. It seems the ice and line as a whole. So the composite structure of ice-transmission line is set up, shown in Fig.1 (d- the diameter of transmission line, b-the thickness of ice).
G Figure 1. The composite structure of ice-transmission line
E
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
2.2 The coupled model of the composite structure-tower and the equation of motion The coupled model of the composite structure-tower is set up, shown in Fig.2 (m- the lumped mass of the composite structure, mi-the lumped mass of transmission tower ).
P
P
PL P1 P1[
P1
[
P1
PL
P
P
P
P1
P
[
P1
\
[
\
Figure 2. Coupled model of the composite structure-tower
The equation of motion for the coupled model of the composite structure-tower:
>M @^u` >K @^u` ^p`
(1) The mass matrices of the composite structure are superimposed by the ice and transmission line, the line and ice are parallel connected, therefore, the rigidity matrices of ice and line are found to be :
>K @lineice >K @line >K @ice ˄2˅ 2.3 Analytical solution of the mathematical model The composite structure can be applied the exciting loads, the frequency of which is close to the natural frequency of ice. Based on the theory of resonance, the analytical solution can be obtained. Using the method of mode shape superposition to solve the equation, the mode shape of the system can be calculated, the equation can be transferred as:
M n q K n q
Pn ˄n=1,2 ! N˅
᷊᷉
the equation can be transferred to˖
qn Z 2 q n where H
1 , Pn Mn
1 Pn Mn
˄4˅
F cosQt ˈQ is the interference frequency .in which case,the Eq.(4) becomes:
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
qn Z 2 q n
HF cosQt
˄5˅
The resonance solutions can be solved using the asymptotic method under the function of the exciting loads when the ice happens to resonate[8~10] : q n a cos Qt T ;
a
T
D*
1 D * QF sin T ; 2
1 a' Q 2 F cos T 2 aQ
D [ a Z 2 Q 2 2 aQ Q 2 F cos T ]
The initial conditions are that:
x0 0 a0 , x 0 0 0 a a0 ˈ T 0 because of
T
0
D > a Z 2 Q 2 Q 2 F cos T @ 0 2aQ hence, Q
a 0Z 2 a0 F
Therefore, the solutions of equation are found to be
§ a Z2 · 0 q n t a 0 cos¨ t¸ ¨ a0 F ¸ ¹ ©
˄6˅
the displacement of the system at any time is N
^u t ` ¦ ^I `n q n t
˄7˅
n 1
dynamic coefficient:
1
P
§Q · 1 ¨ ¸ ©Z ¹
2
˄8˅
the composite structure of ice-transmission line’s tensile stress:
V
Ppl 4W
˄9˅
The deflection of composite structure:
y
Ppu
˄10˅ where p is the amplitude of the exciting loads; l is the length of the composite structure of icetransmission line; w represents the module of bending section.
Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
3. Numerical example Meteorological conditions are distinct from different areas, according to the designing standard of transmission line, the thickness of ice in our country is designed to 15mm, in this paper, 10mm can be chose to calculate in order to clear the ice from the transmission line. A 220kv model of coupled transmission tower-line system is chose in this numerical example[11], the height of tower is 59.59m, shown in Fig.3. The span of tower is 300m, the ice is seemed as cylinder which closely wrappes on the transmission line, the composite structure of ice-transmission line is set up, shown in Fig.4.
Figure 3. The computational model of tower
P
P
P
P
Figure 4. The coupled model of the composite structure-tower
The transmission tower is simplified to some lumped mass, each conductor is represented by 60 lumped masses with the same mass, M line 3.03kg , the diameter of conductor is D=10mm, the thickness of ice is d=10mm, the elastic modulus of the ice is E=10Gpa, the Poisson's ratio of ice is r=0.3. The first-order circular frequency of ice is¹1=8.88rad/s, the second-order circular frequency of ice is¹ 2=19.86 rad/s, according to Eqs.(10~13) , when the disturbing force is P=0.005N, the tensile stress of the composite structure is 7.65Mpa, it is larger than the allowable tensile stress of ice [³ice]=5.175 Mpaˈbut
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
it is far less than the allowable tensile stress of transmission line [³line]=1.70h102 Mpa. The deflection is 2.6mm, rotational angle is 0.16o, the deformation of the ice and transmission line are both small. Therefore it can be concluded that the ice happens to destruct because its strength is not enough, but strength and rigidity of conductor are both satisfied, so the ice occurs to deform and crush, finally the ice divorces from the conductors. The conductors have not been damaged. Through some computation, it can be concluded that the ice divorces from the conductors when the exciting load is very small, however, the transmission lines are still keeping well. Therefore, the new deicing method for transmission lines based on the theory of resonance can clear the ice from the lines. So this method can clear away the ice from the transmission lines effectively. 4. Conclusions 1) According to the designing standard of transmission line, the section of ice seems as the annular. It is supposed that the ice closely wrappes on the transmission line when there is no external loads, it seems the ice and line as a whole. So the composite structure of ice-transmission line is set up. 2) The equation of motion for the coupled model can be derived, the analytical solution of de-icing method can be obtained based on the resonance theory. 3) To the concrete example, the difference of natural oscillation frequency between the ice and conductor is very obvious, the small exciting load that the frequency is close to the natural frequency of ice is applied on the composite structure. It can make the strength and rigidity of the ice are both not enough, the ice divorces from the conductors, but those of conductors are enough, so the conductors are keeping well when the ice divorces from the conductors. 4) The new de-icing method based on the resonance theory has been obtained, which can enhance the efficiency of de-icing and prevent the deposition on the transmission lines. It can reduce the manpower and material resources consumed in the de-icing progress. Therefore, the prospect of new de-icing method based on the theory of resonance is broad. References [1]LIU Chun, LU Jia-zheng, CHEN Hong-dong. Cause analysis of tower falling down and ice accretion in Hunan 500KV power transmission line[J]. Hunan Electric Power,2005,25(5):1-3. [2]Zhang Xin. Fault Analysis on Collapse of Transmission Towers of Minhe Line No.I in 500KV Hunan Power Networks[J]. Power System Technology,2008;32(1):159~160. [3]LI Zai-hua, BAI Xiao-min, zhou Zi-guan, etal. Prevention and Treatment Methods of Ice Coating in Power Networks and Its Recent Study[J]. Power System Technology,2008;32(4):7. [4]HUANG Xin-bo, LIU Jia-bing, CAI Wei, etal, Present Research Situation of Icing and Snowing of Overhead Transmission Lines in China and Foreign Countries[J]. Power System Technology,2008;32(4):27~28. [5]Polhman J C, Landers P. Present stated-of-the –art of transmission line icing[J]. IEEE Trans on Power Apparatus and system, 1982, 101(8)˖2443~2450. [6]Ehbert R I, Schrag R L, Bernhart W D. An investigation of power line de-icing by electro-impulse methods[J]. IEEE Trans on Power Delivery, 1989, 4(3): 1855~1861. [7]Sullivan C R, Petrenko V F, Mccurdy J D. Breaking the ice transmission line icing[J]. IEEE Industry Applications Magazine, 2003, 9(5): 49~54. [8]Peng Xian, Sheng Guogang. Asymptotic method for solving resonance solutions of strongly nonlinear systems[J].Journal of vibration and shock, 2004, 23(1): 44~46.
Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389 [9]Tsuyoshi INOUE, Yukio ISHIDA, and Shintaro YAMADA. Vibration of the Translation and the Inclination Motions Coupled System Under the Periodic Base Motion(Auto-parametric Resonance and Influences of Height of Center of Mass, Unbalance of Mass and Difference between Support Stiffness) [J]. Journal of System Design and Dynamics, 2007, 4(1):736-741. [10]Gaetan Kerschen, D. Michael McFarland, Jeffrey J. Kowtko, Young S. Lee, Lawrence A. [11]Bergman, Alexander F. Vakakis. Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity[J]. Journal of Sound and Vibration 299 (2007):823-836. [12]Li Hong-nan, Wang Qian-xin. Dynamic charateristics of long-span transmission lines and their supporting towers[J].Journal of civil engineering,1997;30(5):33.
1389
Energy Procedia 17 (2012) 1383 – 1389
2012 International Conference on Future Electrical Power and Energy Systems
De-icing Method for Coupled Transmission Tower-Line System Qu Cheng-Zhong, Liu Yue-Jun School of Civil Engineering , Northeast Dianli University, Jilin, China
Abstract For the disaster brought by the transmission line regelation, a new de-icing method is presented in this paper. It is supposed that the ice and transmission line are combined closely when there is no external loads, the composite structure of ice-transmission line is set up, then the coupled model of the composite structure-tower is set up. The equations of motion for the coupled system are derived. The composite structure can be applied to exciting loads, the frequency of which is close to the natural frequency of ice. The natural frequency of ice is far from that of the transmission line. Based on the resonance theory, the ice occurs to deform and crush, finally the ice divorces from the conductors under the function of the exciting load, but those transmission lines are still very well, the analytical solution can be obtained and the new de-icing method for transmission line based on the theory of resonance is got. It is suitable for the transmission line to de-ice through the computation of the concrete example.
© 2012 and/or peer-review under responsibility of Hainan University. 2011 Published Publishedby byElsevier ElsevierLtd. Ltd.Selection Selection and/or peer-review under responsibility of [name organizer] Keywords: the composite structure of ice-transmission line; resonance; de-icing
1. Introduction Ice accumulating on transmission lines can cause extensive damage to the lines and towers. At the beginning of 2005, Hunan province was affected the worst by ice storm, many transmission-towers happened to collapse[1]. Since mid-January 2008, most of districts were troubled in the catastrophe weather. Many transmission towers, conductors, insulators and electric power fittings were damaged seriously[2]. Over a million of consumers lost their electricity supply for some period. It deeply influenced the development of the social economy and people’s life. So it is the urgent question to seek the de-icing method for the transmission tower-line system, which is in order to guarantee the safety of the power network. The commonly used de-icing method at present is Āad hocā[3], it only can deal with a small amount of ice and the speed of de-icing is very slow. The large current ice-melting method has the high efficiency
1876-6102 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. doi:10.1016/j.egypro.2012.02.256
1384
Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
of de-icing, but it also consumes a lot of energy and operates very complex, it does a large harm to the transmission tower-line system. The chemistry method not only needs the suitable working environment and good traffic conditions but also needs to consider the corrosion to the tower. The mechanical de-icing method is difficult to meet the environment of transmission line. Ice-melting by short-circuit current, which needs to stop all the overhead transmission lines in the ice-melting circuit, and it is difficult to choose the ice-melting power supply[4~7]. The regelation becomes the biggest problem to the transmission tower-line system, it raises a hot discussion on how to de-ice fast and effectively. In order to avoid recurrence of such a situation, it is quite necessary to study and exploit a new de-icing method to reduce the disaster caused by transmission line regelation. Therefore, the composite structure of ice-transmission line is set up in this paper, the new de-icing method for transmission line based on the resonance theory is got. 2. The principle of the de-icing It makes the ice equivalent to annular section. It is supposed that the ice and transmission line is combined closely when there is no external loads, the composite structure of ice-transmission line is set up. Because the natural frequency of ice is far from that of the line, the exciting load that the frequency is close to the natural frequency of ice is applied on the composite structure. It makes the amplitude of forced vibration become bigger obviously. The ice belongs to the brittle material, whose strength is much less than that of line. Based on the theory of resonance, the tensile stress of the composite structure is larger than the allowable tensile stress of ice, but it is far less than the allowable tensile stress of transmission line, which make the composite structure generate the small rotational angle and deflection. However, the rotational angle and deflection of the transmission lines are both smaller than the allowable rotational angle and allowable deflection. The ice occurs to deform and crush because its strength and rigidity are not enough, but those of the transmission lines are satisfied. So the transmission lines are still very well. 2.1 The composite structure of ice-transmission line According to the designing standard of transmission line, the section of ice seems as the annular, it is supposed that the ice closely wrappes on the transmission line when there is no external loads. It seems the ice and line as a whole. So the composite structure of ice-transmission line is set up, shown in Fig.1 (d- the diameter of transmission line, b-the thickness of ice).
G Figure 1. The composite structure of ice-transmission line
E
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
2.2 The coupled model of the composite structure-tower and the equation of motion The coupled model of the composite structure-tower is set up, shown in Fig.2 (m- the lumped mass of the composite structure, mi-the lumped mass of transmission tower ).
P
P
PL P1 P1[
P1
[
P1
PL
P
P
P
P1
P
[
P1
\
[
\
Figure 2. Coupled model of the composite structure-tower
The equation of motion for the coupled model of the composite structure-tower:
>M @^u` >K @^u` ^p`
(1) The mass matrices of the composite structure are superimposed by the ice and transmission line, the line and ice are parallel connected, therefore, the rigidity matrices of ice and line are found to be :
>K @lineice >K @line >K @ice ˄2˅ 2.3 Analytical solution of the mathematical model The composite structure can be applied the exciting loads, the frequency of which is close to the natural frequency of ice. Based on the theory of resonance, the analytical solution can be obtained. Using the method of mode shape superposition to solve the equation, the mode shape of the system can be calculated, the equation can be transferred as:
M n q K n q
Pn ˄n=1,2 ! N˅
᷊᷉
the equation can be transferred to˖
qn Z 2 q n where H
1 , Pn Mn
1 Pn Mn
˄4˅
F cosQt ˈQ is the interference frequency .in which case,the Eq.(4) becomes:
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
qn Z 2 q n
HF cosQt
˄5˅
The resonance solutions can be solved using the asymptotic method under the function of the exciting loads when the ice happens to resonate[8~10] : q n a cos Qt T ;
a
T
D*
1 D * QF sin T ; 2
1 a' Q 2 F cos T 2 aQ
D [ a Z 2 Q 2 2 aQ Q 2 F cos T ]
The initial conditions are that:
x0 0 a0 , x 0 0 0 a a0 ˈ T 0 because of
T
0
D > a Z 2 Q 2 Q 2 F cos T @ 0 2aQ hence, Q
a 0Z 2 a0 F
Therefore, the solutions of equation are found to be
§ a Z2 · 0 q n t a 0 cos¨ t¸ ¨ a0 F ¸ ¹ ©
˄6˅
the displacement of the system at any time is N
^u t ` ¦ ^I `n q n t
˄7˅
n 1
dynamic coefficient:
1
P
§Q · 1 ¨ ¸ ©Z ¹
2
˄8˅
the composite structure of ice-transmission line’s tensile stress:
V
Ppl 4W
˄9˅
The deflection of composite structure:
y
Ppu
˄10˅ where p is the amplitude of the exciting loads; l is the length of the composite structure of icetransmission line; w represents the module of bending section.
Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
3. Numerical example Meteorological conditions are distinct from different areas, according to the designing standard of transmission line, the thickness of ice in our country is designed to 15mm, in this paper, 10mm can be chose to calculate in order to clear the ice from the transmission line. A 220kv model of coupled transmission tower-line system is chose in this numerical example[11], the height of tower is 59.59m, shown in Fig.3. The span of tower is 300m, the ice is seemed as cylinder which closely wrappes on the transmission line, the composite structure of ice-transmission line is set up, shown in Fig.4.
Figure 3. The computational model of tower
P
P
P
P
Figure 4. The coupled model of the composite structure-tower
The transmission tower is simplified to some lumped mass, each conductor is represented by 60 lumped masses with the same mass, M line 3.03kg , the diameter of conductor is D=10mm, the thickness of ice is d=10mm, the elastic modulus of the ice is E=10Gpa, the Poisson's ratio of ice is r=0.3. The first-order circular frequency of ice is¹1=8.88rad/s, the second-order circular frequency of ice is¹ 2=19.86 rad/s, according to Eqs.(10~13) , when the disturbing force is P=0.005N, the tensile stress of the composite structure is 7.65Mpa, it is larger than the allowable tensile stress of ice [³ice]=5.175 Mpaˈbut
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Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389
it is far less than the allowable tensile stress of transmission line [³line]=1.70h102 Mpa. The deflection is 2.6mm, rotational angle is 0.16o, the deformation of the ice and transmission line are both small. Therefore it can be concluded that the ice happens to destruct because its strength is not enough, but strength and rigidity of conductor are both satisfied, so the ice occurs to deform and crush, finally the ice divorces from the conductors. The conductors have not been damaged. Through some computation, it can be concluded that the ice divorces from the conductors when the exciting load is very small, however, the transmission lines are still keeping well. Therefore, the new deicing method for transmission lines based on the theory of resonance can clear the ice from the lines. So this method can clear away the ice from the transmission lines effectively. 4. Conclusions 1) According to the designing standard of transmission line, the section of ice seems as the annular. It is supposed that the ice closely wrappes on the transmission line when there is no external loads, it seems the ice and line as a whole. So the composite structure of ice-transmission line is set up. 2) The equation of motion for the coupled model can be derived, the analytical solution of de-icing method can be obtained based on the resonance theory. 3) To the concrete example, the difference of natural oscillation frequency between the ice and conductor is very obvious, the small exciting load that the frequency is close to the natural frequency of ice is applied on the composite structure. It can make the strength and rigidity of the ice are both not enough, the ice divorces from the conductors, but those of conductors are enough, so the conductors are keeping well when the ice divorces from the conductors. 4) The new de-icing method based on the resonance theory has been obtained, which can enhance the efficiency of de-icing and prevent the deposition on the transmission lines. It can reduce the manpower and material resources consumed in the de-icing progress. Therefore, the prospect of new de-icing method based on the theory of resonance is broad. References [1]LIU Chun, LU Jia-zheng, CHEN Hong-dong. Cause analysis of tower falling down and ice accretion in Hunan 500KV power transmission line[J]. Hunan Electric Power,2005,25(5):1-3. [2]Zhang Xin. Fault Analysis on Collapse of Transmission Towers of Minhe Line No.I in 500KV Hunan Power Networks[J]. Power System Technology,2008;32(1):159~160. [3]LI Zai-hua, BAI Xiao-min, zhou Zi-guan, etal. Prevention and Treatment Methods of Ice Coating in Power Networks and Its Recent Study[J]. Power System Technology,2008;32(4):7. [4]HUANG Xin-bo, LIU Jia-bing, CAI Wei, etal, Present Research Situation of Icing and Snowing of Overhead Transmission Lines in China and Foreign Countries[J]. Power System Technology,2008;32(4):27~28. [5]Polhman J C, Landers P. Present stated-of-the –art of transmission line icing[J]. IEEE Trans on Power Apparatus and system, 1982, 101(8)˖2443~2450. [6]Ehbert R I, Schrag R L, Bernhart W D. An investigation of power line de-icing by electro-impulse methods[J]. IEEE Trans on Power Delivery, 1989, 4(3): 1855~1861. [7]Sullivan C R, Petrenko V F, Mccurdy J D. Breaking the ice transmission line icing[J]. IEEE Industry Applications Magazine, 2003, 9(5): 49~54. [8]Peng Xian, Sheng Guogang. Asymptotic method for solving resonance solutions of strongly nonlinear systems[J].Journal of vibration and shock, 2004, 23(1): 44~46.
Qu Cheng-Zhong and Liu Yue-Jun / Energy Procedia 17 (2012) 1383 – 1389 [9]Tsuyoshi INOUE, Yukio ISHIDA, and Shintaro YAMADA. Vibration of the Translation and the Inclination Motions Coupled System Under the Periodic Base Motion(Auto-parametric Resonance and Influences of Height of Center of Mass, Unbalance of Mass and Difference between Support Stiffness) [J]. Journal of System Design and Dynamics, 2007, 4(1):736-741. [10]Gaetan Kerschen, D. Michael McFarland, Jeffrey J. Kowtko, Young S. Lee, Lawrence A. [11]Bergman, Alexander F. Vakakis. Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity[J]. Journal of Sound and Vibration 299 (2007):823-836. [12]Li Hong-nan, Wang Qian-xin. Dynamic charateristics of long-span transmission lines and their supporting towers[J].Journal of civil engineering,1997;30(5):33.
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