CN115659455A - Theoretical detection method for determining local void of monolithic roadbed - Google Patents
Theoretical detection method for determining local void of monolithic roadbed Download PDFInfo
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Abstract
The invention relates to a theoretical detection method for determining local void of an integral ballast bed, which comprises the following steps: obtaining a train-integral track bed-lining-foundation coupling vibration equation according to a control equation of a steel rail, an integral track bed and lining and a dynamic balance equation of the train, and calculating the vertical acceleration of the train body by utilizing a Newmark algorithm; obtaining a Hilbert-Huang transform spectrogram of the vertical acceleration of the vehicle body; and judging the position of the local void area of the whole track bed according to the energy change of the Hilbert-Huang transform spectrum of the imf2 decomposition curve in the void state. The beneficial effects of the invention are: the position of a local void area of the whole track bed can be judged according to the change of the Hilbert-Huang transform spectrum energy value of the vertical acceleration imf2 of the train body; whether the ballast bed is empty and the initial position of the empty ballast bed can be judged, the severity of the empty length can be qualitatively estimated according to the variation length of the energy value, and the subway track operation and maintenance cost can be reduced.
Description
Technical Field
The invention belongs to the technical field of underground engineering, and particularly relates to a theoretical detection method for determining local void of an integral ballast bed.
Background
In big cities with small and precious land, the traffic space underground becomes the inevitable trend of planning and developing cities in the future, and subways play a role in keeping the daily operation of the cities and realizing the expansion of the city scale. However, in the case of long-term operation of a subway, the rail structure may be damaged by, for example, air separation. When the subway track is poured, the surface of the duct piece is directly poured on the whole track bed, and the surface bonding capability of the duct piece are weaker. The subway shield tunnel in the soft soil area is easy to deform greatly due to the influence of factors such as soil body properties, peripheral construction, tunnel burial depth and the like, and the local stripping between the integral ballast bed and the duct pieces is caused due to the inconsistent deformation between the duct pieces and the integral ballast bed, so that the separation is caused. The phenomena of falling off, disengaging and the like of a track bed and duct pieces occur in the operation of the subways in a plurality of medium and large cities, the structural life of the subway tunnel is influenced, the riding comfort is also influenced, and meanwhile, huge potential safety hazards are brought to the operation of the subways in the cities. Therefore, the local empty of the track bed is detected and identified in time so as to prevent early diseases from being treated slightly and gradually, and the method has important significance for guaranteeing the integrity and the operation safety of the subway track system.
At present, the method mainly utilizes modern geophysical detection technology to identify the track bed separation, and specifically mainly comprises a ground penetrating radar method, an impact echo method, an ultrasonic method and the like.
Ground penetrating radar method: and detecting the diseases of the whole track bed of the railway tunnel by utilizing a ground penetrating radar, such as Lizayun and the like, comprehensively drilling and sampling actual measurement data, performing mathematical simulation analysis, and determining the specific position of the whole track bed void by adopting nondestructive detection of the ground penetrating radar.
Impact echo method: lixing and the like judge the defects of the track bed such as stripping, void and the like by using an impact echo method.
An ultrasonic method: the Zhao-arm uses nonlinear ultrasonic frequency mixing technique to excite different frequency mixing signals before and after the void, and uses FFT transform to obtain signal frequency component, and the void position of the track bed is identified according to the frequency component.
However, these methods for detecting the ballast bed void have obvious disadvantages: firstly, the test detection method has long consumed time and high cost; and secondly, the detection has certain range limitation, and the whole operation road section is difficult to monitor in real time.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a theoretical detection method for determining the local void of the integral ballast bed.
The theoretical detection method for determining the local void of the monolithic roadbed comprises the following steps:
s1, obtaining a train-integral track bed-lining-foundation coupling vibration equation according to a control equation of a steel rail, an integral track bed and a lining and a dynamic balance equation of a train, and calculating the vertical acceleration of a train body by utilizing a Newmark algorithm;
s2, performing Hilbert-Huang transformation on the vertical acceleration of the vehicle body to obtain Hilbert-Huang transformation spectrograms of decomposition curves imf1 to imf6, and expressing the magnitude of the energy value through the color scale numerical value of the curves;
and S3, comparing the Hilbert-Huang transform spectrograms of the vertical acceleration of the vehicle body under the condition of no void, and judging the position of the local void area of the whole track bed according to the energy change of the Hilbert-Huang transform spectrogram of the imf2 decomposition curve under the void state.
Preferably, step S1 is specifically: establishing a power balance equation of the train according to the Dalnbel principle
M, C and K are respectively a mass matrix, a damping matrix and a rigidity matrix of the train; v is a displacement vector of the train, and F is an external force matrix borne by each part of the train;
the steel rail is simulated by Euler beams simply supported at two ends, fasteners below the steel rail are distributed in a scattered mode at equal intervals, and a spring-damping unit is adopted for simulation; the method comprises the following steps that (1) an integral ballast bed and a lining are simulated by using Timoshenko beams simply supported at two ends respectively, and interface bonding between the integral ballast bed and the lining is simulated by using discrete spring-damping units distributed at equal intervals; the foundation is regarded as the spring damping unit of the uniform distribution and is directly connected with the lining;
obtaining an ordinary differential vibration equation of the steel rail, the integral ballast bed and the lining through orthogonal decomposition by adopting a modal superposition method, combining the ordinary differential vibration equation of the steel rail, the ballast bed and the lining with a dynamic balance equation of the train to obtain a train-integral ballast bed-lining-foundation coupling vibration equation, and finally carrying out numerical calculation on the equation set by utilizing a Newmark algorithm to obtain the vertical acceleration of the train body.
Preferably, in step S1: the steel rail is simulated by Euler beams simply supported at two ends, and the control equation is
In the formula:
E r I r the unit symbol is N.m for the bending rigidity of the rail 2 ；v r The unit symbol is m for the vertical displacement of the steel rail; x is the number of rs,j The position of the jth fastener is shown, and the unit symbol is m; ρ is a unit of a gradient r A r The unit symbol is kg/m for the distribution mass of the steel rail; n is rs The number of the fasteners is; x is the number of w,i (t) is the position of the ith wheel pair at the moment t; n is a radical of an alkyl radical c Number of trains marshalling; p is rs,j For the fastening force of the jth fastener, the formula is
Wherein, K rs The unit symbol is N/m for the rigidity of the fastener; c rs The damping is the fastener damping, and the unit symbol is N.s/m; p is a,i (x w,i (t)) is the wheel-rail contact force of the ith wheel pair of the a-th vehicle, and the calculation formula is
P a,i (x w,i (t))＝K wr (z w,a,i (t)-ε(x w,a,i )v r (x w,a,i ,t))
Wherein K wr The unit symbol is N/m for the contact rigidity of the wheel and the rail; z is a radical of w,a,i The unit symbol is m for the vertical displacement of the ith wheel set of the a-th vehicle;
preferably, in step S1: the track bed is simulated by simply supporting Timoshenko beams at two ends, and the control equation is as follows
In the formula:
κ h A h G h the unit symbol is N for the shearing rigidity of the whole ballast bed;
F h (x, t) is the vertical external force applied to the integral ballast bed, and the unit symbol is N;
m h (x, t) is an external bending moment applied to the integral track bed, and the unit symbol is N.m;
ρ h the unit symbol is kg/m for the density of the monolithic track bed 3 ；ρ h A h The unit symbol is kg/m which is the distribution mass of the whole track bed;
I h is the section moment of inertia of the monolithic ballast bed, with unit symbol m 4 ；E h I h The unit symbol is N.m for the flexural rigidity of the monolithic track bed 2 ；
v h The unit symbol is m for the vertical displacement of the integral track bed;
Preferably, in step S1: the lining is also simulated by simply supported Timoshenko beams at two ends, the control equation of the lining is the same as that of the integral ballast bed, and the subscripts in the formula and the variables are changed from h to t.
Preferably, in step S2: in a Hilbert-Huang transform spectrogram of the vertical acceleration of the vehicle body, hilbert-Huang transform spectrums of imf1-imf6 decomposition curves are sequentially arranged from top to bottom, wherein each curve color scale numerical value is a negative number, the absolute value of the color scale numerical value is in negative correlation with an energy value, and the smaller the absolute value of the color scale numerical value is, the larger the energy value is.
Preferably, in step S3: compared with vertical acceleration imf1 to imf6 Hilbert-Huang transform spectrograms of the vehicle body under the condition of no emptying, a Hilbert-Huang transform spectrum frequency value curve of an imf2 decomposition curve under the condition of emptying changes along with time, and specifically, before reaching an emptying area, the frequency in the IMf2 curve oscillates periodically; when the empty area is approached, the imf2 frequency curve fluctuates obviously, specifically, the frequency value is a minimum value point with the reduction amplitude exceeding 15% after being increased for a short time, the time corresponding to the minimum value point is the time when the center of the front bogie of the first vehicle reaches the initial point of the empty area, and the central position of the front bogie of the first vehicle is calculated through the time, so that the coordinates of the initial point of the empty area of the whole track bed can be obtained.
Preferably, in step S3: the color scale value of the curve imf2 is reduced along with the increase of the length of the void area, the reduction trend of the color scale value of the imf2 in the non-void area is greater than that of the void area, according to the change of the color scale value of the curve imf2 in different time periods, if the absolute value reduction amplitude of the color scale value outside a certain time period exceeds 15%, the condition that the train passes through the whole track bed section in the time period is judged to have a void situation, and according to a minimum value point with the first reduction amplitude nearest to the time period exceeding 15% and the train speed, the starting point position of the whole track bed void area is calculated.
The invention has the beneficial effects that:
1) The method comprises the steps that through the influence of the integral track bed void on the dynamic characteristics of a track, the vertical acceleration of a vehicle body is used as a sensitive index for void identification, and on the basis, a Hilbert-Huang transform method is adopted to identify a void section of the integral track bed; and judging the position of the local void area of the whole track bed according to the change of the Hilbert-Huang transform spectrum energy value of the vertical acceleration imf2 of the vehicle body.
2) The vibration signal in the running process of the vehicle is utilized to realize real-time identification of the whole running road section, whether the ballast bed is empty or not and the initial position of the empty ballast bed can be judged by combining the energy value change of the imf2 and the train speed, the severity of the empty ballast bed length is qualitatively estimated according to the change length of the energy value change, a new simple method is provided for track inspection, the operation and maintenance cost of the subway track is reduced, and a theoretical basis is provided for the research on the local empty ballast bed in the whole future.
Drawings
FIG. 1 is a partial void section of a subway monolithic track bed track according to the present invention;
FIG. 2 is a subway rail coupling vibration calculation model according to the present invention;
FIG. 3 is a Hilbert-Huang transform spectrum of the vertical acceleration of the vehicle body without the invention being vacated;
FIG. 4 is a Hilbert-Huang transform spectrum of the vertical acceleration of the vehicle body at 0.4m of the inventive ride-through;
FIG. 5 is a Hilbert-Huang transform spectrum of the vertical acceleration of the vehicle body at 0.8m of the inventive vehicle body clearance;
FIG. 6 is a Hilbert-Huang transform spectrum of vertical acceleration of a vehicle body under a condition of a clearance of 1.2m in accordance with the present invention;
FIG. 7 is a Hilbert-Huang transform spectrum of the vertical acceleration of the vehicle body at 1.6m of the void according to the present invention.
Detailed Description
The present invention will be further described with reference to the following examples. The following examples are set forth merely to aid in the understanding of the invention. It should be noted that modifications can be made to the invention by a person skilled in the art without departing from the principle of the invention, and these modifications and modifications also fall within the scope of the claims of the invention.
Example one
As an embodiment, a theoretical detection method for determining local void of a monolithic track bed is provided, which specifically includes the following steps:
s1, obtaining a train-integral track bed-lining-foundation coupling vibration equation according to a control equation of a steel rail, an integral track bed and lining and a dynamic balance equation of the train, and calculating the vertical acceleration of a train body by utilizing a Newmark algorithm;
the simulation conditions of the car body, the steel rail, the integral ballast bed and the lining are as follows: the vehicle body adopts a 20-degree-of-freedom rigid model of two carriages, namely the vehicle body and the bogie consider vertical and nodding displacement, the wheel set only considers vertical displacement, and the bogie and the wheel set, and the carriage and the bogie are respectively connected by primary suspension and secondary suspension; the steel rail is simulated by Euler beams simply supported at two ends, fasteners below the steel rail are distributed in a scattered mode at equal intervals, and a spring-damping unit is adopted for simulation; the integral ballast bed and the lining are respectively simulated by using Timoshenko beams simply supported at two ends, and the interface bonding between the integral ballast bed and the lining is simulated by using discrete spring-damping units distributed at equal intervals; the foundation is regarded as the spring damping unit of the uniform distribution and is directly connected with the lining; the steel rail is in an ideal state, and the irregularity of the rail is not considered; in the void area, the stiffness coefficient and the damping coefficient of the discrete spring-damping unit between the integral ballast bed and the lining interface are set to be 0, and the nonlinear contact is not considered.
And the following parameters are defined:
v c the unit symbol is m for the vertical displacement of the vehicle body;
ψ c the vehicle body nodding displacement is represented by rad;
m c the unit symbol is kg for the vehicle body mass;
J c is the moment of inertia of the vehicle body and has a unit symbol of kg.m 2 ；
k 2 The secondary suspension stiffness is N/m;
c 2 the damping is secondary suspension damping, and the unit symbol is N.s/m;
m b the unit symbol is kg for the bogie mass;
J b the unit symbol is kg.m for the moment of inertia of the bogie 2 ；
v b Is the vertical displacement of the bogie, the unit symbol is m;
ψ b the unit symbol is rad for the nodding displacement of the bogie;
k 1 the unit symbol is N/m, and the unit symbol is a series of suspension stiffness;
c 1 is a series of suspension resistorsNi, unit symbol is N · s/m;
z wi (i =1, \8230;, 8) is the vertical displacement of eight wheel pairs, in the unit symbol m;
E r I r the unit symbol is N.m for the bending rigidity of the rail 2 ；
ρ r A r The unit symbol is kg/m for the distribution mass of the steel rail;
ρ h A h the unit symbol is kg/m which is the distribution mass of the whole ballast bed;
κ h A h G h the unit symbol is N for the shear stiffness of the integral ballast bed;
E h I h the unit symbol is N.m for the flexural rigidity of the monolithic track bed 2 ；
ρ t A t The unit symbol is kg/m for the distribution mass of the tunnel lining;
κ t A t G t the unit symbol is N for the shearing rigidity of the tunnel lining;
E t I t bending stiffness for tunnel lining, unit symbol is N.m 2 ；
K hs The unit symbol is N/m, and is the bonding rigidity between the lining of the ballast bed;
C hs the unit symbol is N.s/m for the bonding damping between the road bed linings;
K ts the unit symbol is N/m for equivalent rigidity of the foundation;
C ts the unit symbol is N.s/m for equivalent damping of the foundation;
based on the Daronbel principle, the power balance equation of the train is established as follows:
in the formula: m, C and K are respectively a mass matrix, a damping matrix and a rigidity matrix of the train; v is the displacement vector of the train, and F is the external force matrix borne by each part of the train.
The steel rail is simulated by an Euler beam simply supported at two ends, and the control equation is as follows:
in the formula:
E r I r the unit symbol is N.m for the bending rigidity of the rail 2 ；v r The unit symbol is m for the vertical displacement of the steel rail; x is the number of rs,j The position of the jth fastener is shown, and the unit symbol is m; rho r A r The unit symbol is kg/m for the distribution mass of the steel rail; n is rs The number of the fasteners is; x is the number of w,i (t) is the position of the ith wheel pair at the moment t; n is c The number of trains to be marshalled; p rs,j For the fastening force of the jth fastener, the calculation formula is
Wherein, K rs The unit symbol is N/m for the rigidity of the fastener; c rs The damping is the fastener damping, and the unit symbol is N.s/m; p a,i (x w,i (t)) is the wheel-rail contact force of the ith wheel pair of the a-th vehicle, and the calculation formula is
P a,i (x w,i (t))＝K wr (z w,a,i (t)-ε(x w,a,i )v r (x w,a,i ,t))
Wherein K wr The unit symbol is N/m for the contact rigidity of the wheel and the rail; z is a radical of w,a,i The unit symbol is m for the vertical displacement of the ith wheel set of the a-th vehicle;
the track bed is simulated by simply supporting Timoshenko beams at two ends, and the control equation is as follows
In the formula:
κ h A h G h the unit symbol is N for the shear stiffness of the integral ballast bed;
F h (x, t) is the vertical external force applied to the integral ballast bed, and the unit symbol is N;
m h (x, t) is an external bending moment applied to the integral track bed, and the unit symbol is N.m;
ρ h the unit symbol is kg/m for the density of the monolithic track bed 3 ；ρ h A h The unit symbol is kg/m which is the distribution mass of the whole ballast bed;
I h is the section moment of inertia of the monolithic ballast bed, with unit symbol m 4 ；E h I h The unit symbol is N.m for the flexural rigidity of the monolithic track bed 2 ；
v h The unit symbol is m for the vertical displacement of the integral track bed;
The lining is also simulated by simply supported Timoshenko beams at two ends, the control equation of the lining is the same as that of the integral ballast bed, and only subscripts in formulas and variables are changed from h to t.
And then obtaining an ordinary differential vibration equation of the steel rail, the integral ballast bed and the lining through orthogonal decomposition by adopting a modal superposition method, combining the ordinary differential vibration equation with a subway train power equation to obtain a train-integral ballast bed-lining-foundation coupling vibration equation, and finally carrying out numerical calculation on the equation set by utilizing a Newmark algorithm to obtain the vertical acceleration of the train body.
S2, performing Hilbert-Huang transformation on the vertical acceleration of the vehicle body to obtain Hilbert-Huang transformation spectrograms of decomposition curves imf1 to imf6, and expressing the energy value through the color scale numerical value of the curves; the Hilbert-Huang transform spectrogram sequentially comprises Hilbert-Huang transform spectrums of imf1-imf6 decomposition curves from top to bottom, wherein the color scale numerical value of each curve is a negative number, the absolute value of the color scale numerical value is in negative correlation with the energy value, and the smaller the absolute value of the color scale numerical value is, the larger the energy value is.
S3, judging the position of the local void area of the whole track bed according to the energy change of the Hilbert-Huang transform spectrum of the imf2, wherein the specific method comprises the following steps: compared with vertical acceleration imf1 to imf6 Hilbert-Huang transform spectrograms of the vehicle body under the condition of no emptying, a Hilbert-Huang transform spectrum frequency value curve of an imf2 decomposition curve under the condition of emptying is remarkably changed along with time, specifically, before reaching an emptying area, the frequency in the imf2 curve oscillates periodically, and the oscillation center value is consistent with the imf2 frequency center value under the condition of no emptying; when the empty area is approached, the imf2 frequency curve fluctuates obviously, the frequency value is increased for a short time, the frequency value which exceeds 15 percent is reduced obviously, the time corresponding to the minimum value point of the obvious reduction is the time when the center of the front bogie of the first vehicle reaches the initial point of the empty area, and the central position of the front bogie of the first vehicle is calculated through the time, so that the coordinates of the initial point of the empty area of the whole track bed can be obtained.
The color scale value of the curve imf2 is reduced along with the increase of the length of the void area, the reduction trend of the color scale value of the imf2 in the non-void area is greater than that of the void area, according to the change of the color scale value of the curve imf2 in different time periods, if the absolute value reduction amplitude of the color scale value outside a certain time period exceeds 15%, the condition that the train passes through the whole track bed section in the time period is judged to have a void situation, and according to a minimum value point with the first reduction amplitude nearest to the time period exceeding 15% and the train speed, the starting point position of the whole track bed void area is calculated.
Example two
The embodiment provides a use example of the theoretical detection method for determining the local void of the monolithic ballast bed, which is provided in the first embodiment:
under the assumption that the stiffness coefficient and the damping coefficient of a discrete spring-damping unit between a track bed and a lining of a void area are set to be 0 and nonlinear contact is not considered, numerical simulation that a train drives through the void area at the speed of 72km/h is carried out, wherein the void lengths are respectively 0.4m, 0.8m, 1.2m and 1.6m and correspond to void area numbers 1622 to 1625, 1622 to 1629, void area numbers 1622 to 1633 and void area numbers 1622 to 1637 of a spring-damping unit section of the figure 1, so that Hilbert-Huang transform spectra of vertical acceleration of the train body are analyzed.
The hubert-yellow transform spectra of the vehicle body acceleration at different void lengths are shown in fig. 4-7: the hilbert-yellow transform spectrum frequency value of the vehicle body vertical acceleration imf1 (the first curve from top to bottom in fig. 4 to 7) does not change significantly, but the color of the curve gradually darkens, indicating that the energy value decreases with the increase of the void length, and particularly, the change of the void length from 1.2m to 1.6m is most significant.
In the non-void region, the frequency value of the hilbert-yellow transform spectrum of imf2 (the second curve from top to bottom in fig. 4 to 7) remains stable, but the energy value decreases as the void length increases, and the scale value of imf2 decreases more in the non-void region than in the void region. When the train runs to the vicinity of the vacated position and the whole train runs out of the vacated position, the energy value changes significantly, so that the time point of the significant change of the energy is related to the position of the vacated section. Therefore, the position of the void region can be judged by segmenting the Hilbert-Huang transform spectrum energy change region of the imf 2.
Meanwhile, the hilbert-yellow transform spectrum frequency values of the imf2 (the second curves from top to bottom in fig. 4 to 7) all show the phenomenon that the frequency value oscillates periodically, then the maximum value of the frequency is obviously increased at a certain moment, and then the minimum value of the frequency is obviously reduced, and the time period of the phenomenon has certain coincidence with the time period of the hilbert-yellow transform spectrum energy change of the imf 2.
In conclusion, the position of the whole track bed local void starting point is roughly judged according to the time point when the Hilbert-Huang transform spectrum energy value of the imf2 starts to change remarkably, and the coordinate of the whole track bed void area starting point can be obtained according to the frequency value change of the imf 2; in addition, the severity of the whole track bed void length can be qualitatively judged according to the energy value change of the Hilbert-Huang transform spectrums of the vertical accelerations imf1 and imf2 of the vehicle body and the length of the significant change section of the imf 2;
the result shows that the method can judge whether the local void of the whole ballast bed exists and the position of the void area on the theoretical level. The method provides a new simple method for track inspection, can utilize vibration signals in the running process of vehicles to realize real-time identification of the whole running road section, provides a theoretical basis for the research on local emptying of the whole track bed in the future, and is beneficial to reducing the operation and maintenance cost of the subway track.
Claims (8)
1. A theoretical detection method for determining local void of a monolithic roadbed is characterized by comprising the following steps:
s1, obtaining a train-integral track bed-lining-foundation coupling vibration equation according to a control equation of a steel rail, an integral track bed and lining and a dynamic balance equation of the train, and obtaining a vertical acceleration of a train body by utilizing a Newmark algorithm;
s2, performing Hilbert-Huang transformation on the vertical acceleration of the vehicle body to obtain Hilbert-Huang transformation spectrograms of decomposition curves imf1 to imf6, and expressing the energy value through the color scale numerical value of the curves;
and S3, judging whether the whole track bed is partially empty or not and judging the position of an empty area according to the energy change of the Hilbert-Huang transform spectrum of the imf 2.
2. The theoretical detection method for determining the local void of the monolithic roadbed according to claim 1, wherein the step S1 specifically comprises: establishing a power balance equation of the train according to the Dalabel principle
Wherein M, C and K are respectively a mass matrix, a damping matrix and a rigidity matrix of the train; v is a displacement vector of the train, and F is an external force matrix borne by each part of the train;
the steel rail is simulated by Euler beams simply supported at two ends, fasteners below the steel rail are distributed in a scattered mode at equal intervals, and a spring-damping unit is adopted for simulation; the integral ballast bed and the lining are respectively simulated by using Timoshenko beams simply supported at two ends, and the interface bonding between the integral ballast bed and the lining is simulated by using discrete spring-damping units distributed at equal intervals; the foundation is regarded as the spring damping unit of the uniform distribution and is directly connected with the lining;
obtaining the ordinary differential vibration equation of the steel rail, the integral ballast bed and the lining through orthogonal decomposition by adopting a modal superposition method, combining the ordinary differential vibration equation of the steel rail, the ballast bed and the lining with the dynamic balance equation of the train to obtain a train-integral ballast bed-lining-foundation coupling vibration equation, and finally carrying out numerical calculation on the equation set by utilizing a Newmark algorithm to obtain the vertical acceleration of the train body.
3. The theoretical detection method for determining the local void of the monolithic roadbed according to claim 2, wherein in step S1: the steel rail is simulated by Euler beams simply supported at two ends, and the control equation is
In the formula:
E r I r the unit symbol is N.m for the bending rigidity of the rail 2 ；v r The unit symbol is m for the vertical displacement of the steel rail; x is the number of rs,j The position of the jth fastener is shown, and the unit symbol is m; rho r A r The unit symbol is kg/m for the distribution mass of the steel rail; n is rs The number of fasteners; x is the number of w,i (t) is the position of the ith wheel pair at the moment t; n is c Number of trains marshalling; p rs,j For the fastening force of the jth fastener, the calculation formula is
Wherein, K rs The unit symbol is N/m for the rigidity of the fastener; c rs The unit symbol is N.s/m for fastener damping; p a,i (x w,i (t)) is the wheel-rail contact force of the ith wheel pair of the a-th vehicle, and the calculation formula is
P a,i (x w,i (t))＝K wr (z w,a,i (t)-ε(x w,a,i )v r (x w,a,i ,t))
Wherein K wr The unit symbol is N/m for the contact rigidity of the wheel and the rail; z is a radical of w,a,i The unit symbol is m for the vertical displacement of the ith wheel pair of the a-th vehicle;
4. the theoretical detection method for determining the local void of the monolithic roadbed according to claim 2, wherein in the step S1: the track bed is simulated by simply supporting Timoshenko beams at two ends, and the control equation is as follows
In the formula:
κ h A h G h the unit symbol is N for the shearing rigidity of the whole ballast bed;
F h (x, t) is vertical external force applied to the whole track bed, and the unit symbol is N;
m h (x, t) is an external bending moment applied to the whole track bed, and the unit symbol is N.m;
ρ h the unit symbol is kg/m for the density of the monolithic track bed 3 ；ρ h A h The unit symbol is kg/m which is the distribution mass of the whole ballast bed;
I h is the section moment of inertia of the monolithic ballast bed, with unit symbol m 4 ；E h I h The unit symbol is N.m for the flexural rigidity of the monolithic track bed 2 ；
v h The unit symbol is m for the vertical displacement of the integral track bed;
5. The theoretical detection method for determining the local void of the monolithic roadbed according to claim 4, wherein in step S1: the lining is also simulated by simply supported Timoshenko beams at two ends, the control equation of the lining is the same as that of the integral ballast bed, and the subscripts in the formula and the variables are changed from h to t.
6. The theoretical detection method for determining the local void of the monolithic roadbed according to claim 1, wherein in step S2: in a Hilbert-Huang transform spectrogram of the vertical acceleration of the vehicle body, hilbert-Huang transform spectrums of imf1-imf6 decomposition curves are sequentially arranged from top to bottom, wherein the color scale numerical value of each curve is a negative number, the absolute value of the color scale numerical value is in negative correlation with the energy value, and the smaller the absolute value of the color scale numerical value is, the larger the energy value is.
7. The theoretical detection method for determining the local void of the monolithic roadbed according to claim 1, wherein in the step S3: compared with vertical acceleration imf1 to imf6 Hilbert-Huang transform spectrograms of the vehicle body under the condition of no emptying, a Hilbert-Huang transform spectrum frequency value curve of an imf2 decomposition curve under the condition of emptying changes along with time, and specifically, before reaching an emptying area, the frequency in the IMf2 curve oscillates periodically; when the empty area is approached, the imf2 frequency curve fluctuates, which is specifically represented as that a minimum value point with the frequency value reduced by more than 15% appears after the frequency value is increased for a short time, the time corresponding to the minimum value point is the time when the center of the front bogie of the first vehicle reaches the initial point of the empty area, and the central position of the front bogie of the first vehicle is calculated through the time, so that the coordinates of the initial point of the empty area of the whole track bed can be obtained.
8. The theoretical detection method for determining the local void of the monolithic roadbed of claim 7, wherein in the step S3: the color scale value of the curve imf2 is reduced along with the increase of the length of the void area, the reduction trend of the color scale value of the imf2 in the non-void area is greater than that of the void area, according to the change of the color scale value of the curve imf2 in different time periods, if the absolute value reduction amplitude of the color scale value existing outside a certain time period exceeds 15%, the condition that the train passes through the whole track bed section in the time period is judged to be void, and according to the minimum value point with the nearest first reduction amplitude exceeding 15% in the time period and the train speed, the starting point position of the whole track bed void area is calculated.
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