I have paper written in english, not sure its correct or not.
My english is kinda bad. grammar and spelling mistakes.
If someone can help me out with this one. the paper is not long 2 pages. so i think it a couple minutes work.
You can mark the mistakes with CAPS. Of course it would be more great if you can correct it, but that would be kinda lazy. Just need someone who can tell me whether there are mistake.
need to know whether the grammar spelling is good.
thanks in advance
THIS IS MY PAPER BELOW
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VVV
V
HIGH-FREQUENCY VIBRATION CHARACTERISTIC ANALYSIS FOR TRACK STRUCTURE
Abstract: A track structure calculation model is established with the track periodically supported characteristics; 3-dimension solid element in finite element method is employed to describe the rail. This model can image the response of track structure to high frequency excitation exactly. It excels the traditional beam model in calculation precision and applicable range. Finite element method is adopted to analyze track structure based on the model mentioned above. Various parts of track vibration response in high-frequency excitation and the vibration attenuation along rail are studied, and track structure high-frequency vibration impedance characteristic is determined. Some rules of track structure high frequency vibration are obtained.
Keywords: Wheel/Track, noise, High-Frequency Vibration, Mechanic Impedance, High-Speed Railway
Introduction
As large-scale railway speed-raising projects are implemented in China, a large number of high-speed trains have been put into operation. Environmental noise pollution induced by high-speed railway has become a prominent problem in the development of railway. Wheel-rail noise is one of the most important parts of railway noise. High-frequency vibration is the main source of wheel-rail noise. So it is the main approach to reduce the wheel-rail noise to study tracks high-frequency vibration characteristics and control that [P. J. Remington. (1987)].
3 The models of track structure
The rail is effectively an infinite structure. Its motion is therefore not modal but consists of travelling waves. The research before are limited to track models 1 and 2:
1. Continuously supported beam. [D. J. Thompson. (1993); this is a Timoshenko beam mounted on a two-layer continuous support.
2. Periodically supported beam [17]. This is similar to model 1, with the exception that the periodicity of the support is included.
Two degrees of freedom (DOF) vibration system with sleeper quality model is established [LIU linya. (2004)]. Rail vibration mechanical analysis model are shown in Figure 1. For further optimization of the model, the rail is simulated with three-dimensional solid element in this study, and takes periodically into consideration. The model is shown in figure 2.
3 The analysis method of vibration characteristics
Any sustained cyclic load will produce a sustained cyclic response (a harmonic response) in a structural system.
Through the harmonic (or sinusoidal) excitation, the amplitude ratio and phase difference between incentive and response can be employed directly to determine the system impedance:
Where: Zj represents the vibration impedance of structure. Fj and Vj represent the excitation and response respectively.
Sinusoidal force can be expressed as a plural:
Where: F represents the amplitude of excitation force; ω denote frequency; t is time; φ1：phase angle of force
The corresponding displacement, velocity, acceleration can also be expressed as plural too:
Where: F, X represent for the amplitude of sinusoidal excitation and displacement response amplitude respectively；φ2：phase angle of displacement velocity and acceleration.
So equation (1) can be written as:
Where: V denote the amplitude of response; φ denote phase angle of response.
Harmonic response analysis gives the ability to predict the sustained dynamic behavior of structures in Finial Element Method. The idea is to calculate the structure's response at several frequencies and to obtain some response quantity (usually displacements) versus frequency [ANSYS, Inc. (2005)].
Consider the equation of motion for a structural system:
here: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
Abstract: A track structure calculation model is established with the track periodically supported characteristics<REWORD, VERY AWKWARD>; 3-dimension solid element in finite<INFINITE?> element method is employed to describe the rail. This model can image the response of track structure to high frequency excitation exactly. It excels the traditional beam model in calculation precision and applicable range. Finite element method is adopted to analyze track structure based on the model mentioned above. Various parts of track vibration response in high-frequency excitation and the vibration attenuation along rail are studied, and track structure high-frequency vibration impedance characteristic is determined. Some rules of track structure high frequency vibration are obtained.
Keywords: Wheel/Track, noise, High-Frequency Vibration, Mechanic Impedance, High-Speed Railway
Introduction
As large-scale railway speed-raising projects are implemented in China, a large number of high-speed trains have been put into operation. Environmental noise pollution induced <CAUSED?>by high-speed railway has become a prominent problem in the development of railway. Wheel-rail noise is one of the most important parts of railway noise. High-frequency<NO HYPHEN, TWO WORDS> vibration is the main source of wheel-rail noise. So it is the main approach<SO THE MAIN APPROACH> to reduce the<DELETE THE> wheel-rail noise <IS>to study tracks<DELETE TRACKS> high-frequency<NO HYPHEN> vibration characteristics and control that <IT>[P. J. Remington. (1987)].
3 The models of track structure
The rail is effectively an infinite structure. Its motion is therefore not modal but consists of travelling<TRAVELING> waves. The research before are limited to track models 1 and 2:
<MODEL>1. <A> Continuously supported beam. [D. J. Thompson. (1993); this is a Timoshenko beam mounted on a two-layer continuous support.
<MODEL>2. <A>Periodically supported beam [17]. This is similar to model 1, with the exception that the periodicity of the support is included.
Two degrees of freedom (DOF) vibration system with sleeper quality model is established [LIU linya. (2004)]. Rail vibration mechanical<DELETE MECHANICAL> analysis model are<IS> shown in Figure 1. For further optimization of the model, the rail is simulated with <A>three-dimensional solid element in this study, and takes periodically into consideration. The model is shown in figure 2.
3 The analysis method of vibration characteristics
Any sustained cyclic load will produce a sustained cyclic response (a harmonic response) in a structural system.
Through the harmonic (or sinusoidal) excitation, the amplitude ratio and phase difference between incentive and response can be employed directly to determine the system impedance:
Where: Zj represents the vibration impedance of structure. Fj and Vj represent the excitation and response respectively.
Sinusoidal force can be expressed as a plural:
Where: F represents the amplitude of excitation force; ω denote frequency; t is time; φ1：phase angle of force
The corresponding displacement, velocity, acceleration can also be expressed as plural too:
Where: F, X represent for the amplitude of sinusoidal excitation and displacement response amplitude respectively；φ2：phase angle of displacement velocity and acceleration.
So equation (1) can be written as:
Where: V denote the amplitude of response; φ denote phase angle of response.
Harmonic response analysis gives the ability to predict the sustained dynamic behavior of structures in Finial Element Method. The idea is to calculate the structure's response at several frequencies and to obtain some response quantity (usually displacements) versus frequency [ANSYS, Inc. (2005)].
Consider the equation of motion for a structural system:
PART 2 didn't copy good
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Consider the equation of motion for a structural system:
(5)
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
（6）
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
(8)
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
(9)
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
PART TWO
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Consider the equation of motion for a structural system:
(5)
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
（6）
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
(8)
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
(9)
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
Consider the equation of motion for a structural system:
(5)
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
（6）
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
(8)
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
(9)
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
（6）
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
(8)
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
(9)
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
part 2
-----
-----
Consider the equation of motion for a structural system:
(5)
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
（6）
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
(8)
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
(9)
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
part 2
-----
-----
Consider the equation of motion for a structural system:
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
（6）
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
(7)
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
(8)
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
(9)
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.
PART 2
---------
---------
where: [M] represent structural mass matrix; [C]: structural damping matrix; [K]: structural stiffness matrix; {x} : nodal displacement vector; {Fa}: applied load vector As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:
where: xmax : maximum displacement; Ω:imposed circular frequency (radians/time); φ: displacement phase shift (radians)
Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation (6) can be rewritten as:
Where: {x1}: real displacement vector; {x2}: imaginary displacement vector.
The force vector can be specified analogously to the displacement:
where: Fmax : force amplitude; ψ = force phase shift; {F1}:real force vector; {F1}: imaginary force vector.
Substituting equation(7) and (8)into equation(5), The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed, then the follow equation can be get:
The response can be obtained by the solution to the equation.
4 Validation of the of method and model
An example of predicted and measured accelerance is given in Figure 3 for track structure.[experimental validation].
Figure 3. Track vertical accelerance:
Figure 4. Track vertical accelerance predicted by 3D model
Figure 4 gives the accelerance calculated from by using of 3D model and method established this paper.
Compare figure 4 with figure 3, the value predicted by tradition beam model lost the ‘peak’ especially for high frequency. But curve of figure 4 has good consistency to the measured result.
As shown in figure 4, the track structure is unanimous in vibration characteristics when excited above sleeper and between sleeper in low-frequency part. But it is inconsistent at high-frequency. Especially, there is a resonance point about 1250Hz when excited at mid-span. But the point is also an anti resonance point when motivated at sleeper supporting point.
Figure 5 represents the comparison of the vibrations picked up at rail head and rail foot respectively.
Figure 5. Comparison of the vibrations picked up at rail head and rail foot respectively.
As illustrated in figure 5, vibration of rail head and rail foot are in line below 1000Hz, the accordance indicate rail vibrated mainly as a whole. In the high-frequency part, rail impedance amplitude value separated. It illustrates that rail cross-section produces deformation vibration.
The traditional model can not reflect the rail cross-section deformation at high-frequency effectively. Thus it can not simulate the track structure vibration at high-frequency precisely. The models established this paper can express the deformation well, so it excels the tradition beam model in calculation precision.
5 Vibration distribution and decay rate
5.1 Vibration distribution and mode
Track vibration distribution in the track direction is varied with frequency. Figure 6 gives the representative vibration mode curves for different frequency band.
Figure 6 Vibration distribution along the track direction
Below 125Hz, vibration distribution with half-wavelength around 4 sleeper bays. And attenuation in the track length direction.
Between 160 and 500Hz, track vibration shown linear with track length.
At range of 630 to 1600Hz, track vibrated as pinned-pinned mode ( where sleeper separation equals half a bending wavelength ) especially for 1000 Hz.
At frequency band higher than 2000Hz, track structure vibrated as a more complex mode. There are more than one distribution distances for the vibration ‘peak’.
5.2 Vibration propagation decay rate
Just as illustrated in the curves above, the vibration attenuation along the track direction is directly related to frequency. The propagating wave decay rate is obtained [C.J.C. Jones. (2006)] by analysis the slope of the curves, shown in Figure 8.
Fig 7 Vibration propagation decay rate
At low frequency the decay rates are high. Just above the resonance frequency of the rail on the pad (about 300Hz), the rail becomes decoupled from the sleepers and, as the propagating wave cuts on, the decay rates become low. In this set of results a significant peak is seen around the pinned–pinned frequency at 1 kHz.
5.2 The influence of structure parameter on decay rate
Figure 8 gives the curves of decay rate for sleeper separation varied from 0.568 to 0.60, and the value shown in figure 7 are calculated by using of sleeper distance is 0.55m.
Figure 8 Influence of sleeper separation on decay rate
Compare figure 8 and figure 7, it can be extracted. It is useless on the decay rate to change the sleeper supporting distance for low frequency especially bellows 100Hz. The most obvious effect produced at the frequency band between 100Hz-1000Hz, bigger sleeper bay means high decay rate for 100Hz~200Hz, on the contrast, for 200~1000Hz, less sleeper separation increase decay rate, thereby reduce the vibration propagation, consequently control the noise radiation.
6 Conclusion
By means of establish three-dimensional vibration model of track structure, harmonic response analysis is employed to study the vibration characteristics of track structure. This model can reflect the rail cross-section deformation at high-frequency effectively, so its calculation precision in high-frequency is superior to the traditional beam model. Response of different exciting points and pick-up point are obtained by calculating the track structure. Vibration propagation decay rate are obtained. Meanwhile, some rules of track structure high frequency vibration are obtained:
Rail cross-section is vibrated mainly as a whole below 1000Hz; In high-frequency part, rail cross-section produced deformative vibration.
Track vibration propagation are relatives to the excited frequency. Less sleeper supporting distance are helpful for vibration propagation control.