CN114186362A - Analysis method for predicting track irregularity and optimizing configuration of sensors thereof based on vibration response of operation train - Google Patents

Analysis method for predicting track irregularity and optimizing configuration of sensors thereof based on vibration response of operation train Download PDF

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CN114186362A
CN114186362A CN202111442801.XA CN202111442801A CN114186362A CN 114186362 A CN114186362 A CN 114186362A CN 202111442801 A CN202111442801 A CN 202111442801A CN 114186362 A CN114186362 A CN 114186362A
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train
displacement
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史康
胡孝霜
杨永斌
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Chongqing University
China State Railway Group Co Ltd
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China State Railway Group Co Ltd
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Abstract

The invention discloses an analysis method for predicting track irregularity and sensor optimal configuration based on vibration response of an operating train, relates to the field of railway track engineering, and has strong applicability to design, construction and maintenance of tracks. The invention utilizes sensor technology and designs a new algorithm to separate train response signals collected by a sensor to realize the identification of track irregularity, and comprises the steps of establishing an axle coupling system, obtaining a train body response input expression by utilizing a Newmark-beta method, and establishing an output response expression by utilizing train response measured by the sensor, thereby obtaining a load vector and finally realizing the identification of the track irregularity.

Description

Analysis method for predicting track irregularity and optimizing configuration of sensors thereof based on vibration response of operation train
Technical Field
The invention relates to the field of railway track engineering, in particular to an analysis method for predicting track irregularity and optimal configuration of a sensor thereof based on vibration response of an operating train.
Background
The rail irregularity is an important interference source of train vibration, directly influences the running stability, comfort and safety of the train, is also an important factor for controlling the maximum speed of the train, and whether good rail irregularity can be obtained is one of the key problems for determining the success or failure of the high-speed railway.
In recent decades, rail surfaces have been deteriorating at faster rates due to increased train axle weights, increased traffic volumes and operating speeds, and climate change, thus requiring faster and frequent detection of rail health.
Disclosure of Invention
The invention aims to provide an analysis method for predicting track irregularity and sensor optimal configuration based on vibration response of an operating train so as to solve the problem of detection of the track irregularity.
The technical scheme adopted for achieving the aim of the invention is that the analysis method for predicting the track irregularity and the sensor optimal configuration based on the vibration response of the operating train comprises the following steps:
1) constructing a vehicle track bridge coupling system; the vehicle track bridge coupling system comprises a detection train and a bridge, wherein a sensor is arranged on the detection train;
2) establishing a vehicle body motion equation, and calculating a mass matrix M, a rigidity matrix K and a damping matrix C of a vehicle model;
3) solving a displacement response expression, a speed response expression and an acceleration response expression of train running by using a Newmark-beta method, wherein the displacement response expression, the speed response expression and the acceleration response expression are used as vibration input responses;
4) collecting and detecting the vibration response of a train running on a bridge as a vibration output response, and establishing a system output response equation; wherein the vibrational response comprises a displacement response, a velocity response, and an acceleration response;
5) constructing a relation between an input response equation and an output response equation, and obtaining a theoretical expression of the relation between a load matrix P and an output response Y of the vehicle body system;
6) solving the ill-conditioned problem of the output response Y by using a Tikhonov regularization method, introducing the expression obtained in the step 5) into a regularization term, and converting the ill-conditioned problem into a proper problem;
7) and obtaining a regularization solution of the load matrix, and calculating the track irregularity by utilizing the relation between the load and the track irregularity.
Further, in the vehicle track bridge coupling system in the step 1), the detection train is simplified into a 4-degree-of-freedom vehicle model consisting of a vehicle body and two wheel pairs, the vehicle body comprises all vehicle body parts above a bogie and wheels, and the parameter of the detection train comprises the mass m of the vehicle bodyvVehicle body moment of inertia IvVehicle body rotation angle thetavVertical displacement y of vehicle bodyvFront wheel mass m1Rigidity K for connecting vehicle body and front wheels1Damping Cs1Front wheel vertical displacement y1Front wheel and track contact point abscissa x1Front wheel stiffness Kt1Rear wheel mass m2Connection rigidity K of vehicle body and rear wheels2Damping Cs2Vertical displacement of rear wheel y2X abscissa of contact point between rear wheel and rail2Rear wheel stiffness Kt2Train running speed v, wheel set spacing S and a describing position of mass center of train body1And a2。
Further, the vehicle body motion equation established in the step 2) is as follows:
Figure BDA0003383941090000021
in the formula: m, C, K are the mass matrix, damping matrix and stiffness matrix of the vehicle structural system, respectively, P (t) is the external force vector on the structure, L is the mapping matrix of the input,
Figure BDA0003383941090000022
Figure BDA0003383941090000023
x (t) is the acceleration, speed and displacement vector of the vehicle body system respectively;
wherein:
M＝diag[mv Iv m1 m2] (2)
Figure BDA0003383941090000024
Figure BDA0003383941090000025
Figure BDA0003383941090000026
further, the step 3) comprises the following sub-steps:
3-1) introducing a coefficient gamma and a coefficient beta, the coefficient gamma representing the weight of the initial acceleration and the final acceleration contributing to the speed change in the time interval at; β represents the weight of the initial acceleration and the final acceleration contribution to the displacement change over the Δ t time interval, interpreted in mathematical terms as:
Figure BDA0003383941090000031
Figure BDA0003383941090000032
3-2) to tiAnd ti+1The acceleration between is integrated to obtain ti+1The displacement and velocity at time are:
Figure BDA0003383941090000033
Figure BDA0003383941090000034
in the formula: Δ t is tiTime and ti+1A time step between moments;
3-3) substituting equations (6) and (7) into equations (8) and (9), respectively, yields:
Figure BDA0003383941090000035
Figure BDA0003383941090000036
3-4) carrying out item shifting on the formula (11) to obtain an acceleration expression at the end of the delta t time period:
Figure BDA0003383941090000037
3-5) substituting equation (12) into equation (10) to obtain the velocity expression at the end of the Δ t period:
Figure BDA0003383941090000038
3-6) listed at ti+1Equation of dynamic balance of vehicle at moment:
Figure BDA0003383941090000039
3-7) substituting equations (12) and (13) into equation (14) to obtain the time-step-only end position displacement Xi+1By appropriate merging of similar terms, this formula is written as:
Figure BDA00033839410900000310
the formula (15) is a form of a static equilibrium equation including equivalent stiffness
Figure BDA00033839410900000311
And equivalent load
Figure BDA00033839410900000312
Wherein:
Figure BDA0003383941090000041
Figure BDA0003383941090000042
wherein
Figure BDA0003383941090000043
Represented by the currently applied force and the restoring force of the previous time step, the end of time step displacement Xi+1Expressed as:
Figure BDA0003383941090000044
wherein:
Figure BDA0003383941090000045
Figure BDA0003383941090000046
Figure BDA0003383941090000047
Figure BDA0003383941090000048
3-8) calculating the displacement of the end point of the time step from the formula (18), and calculating the velocity at that time from the formula (13)
Figure BDA0003383941090000049
Converting constant coefficient of matrix in equation into unit matrix
Figure BDA00033839410900000410
Then, the terms are combined and unified into a form as shown in formula (18):
Figure BDA00033839410900000411
wherein:
Figure BDA00033839410900000412
Figure BDA00033839410900000413
Figure BDA00033839410900000414
Figure BDA0003383941090000051
3-9) substituting equation (18) into equation (12) for the acceleration at the end of the time step
Figure BDA0003383941090000052
The method is simplified as follows:
Figure BDA0003383941090000053
wherein:
Figure BDA0003383941090000054
Figure BDA0003383941090000055
Figure BDA0003383941090000056
Figure BDA0003383941090000057
3-10) in combination with equations (18), (23), (28):
Figure BDA0003383941090000058
3-11) equation (33) is expressed as the initial response, resulting in the train at tiThe displacement response, the speed response and the acceleration response at the moment are as follows:
Figure BDA0003383941090000059
further, the step 5) comprises the following sub-steps:
5-1) obtaining the displacement, speed and acceleration vector of the detection train according to the acquisition signal of the sensor on the detection train, and establishing an output response equation:
Figure BDA00033839410900000510
wherein: setting R ═ diag [ R ]d Rv Ra]
5-2) substitution of formula (34) for formula (35) gives:
Figure BDA0003383941090000061
assuming that the initial response of the structure is zero, then there are:
Figure BDA0003383941090000062
5-3) placing formula (36) at t1To tnWriting in a matrix convolution form in a time range:
Y＝HLP (38)
wherein:
Figure BDA0003383941090000063
solving the load vector P directly using equation (38) is:
Figure BDA0003383941090000064
further, in step 6), introducing the formula (38) in step 5-3) into a regularization term, and identifying the force P by using a Tikhonov regularization method, wherein the formula is as follows:
Figure BDA0003383941090000065
wherein: λ is a regularization parameter, λ > 0.
Further, when the track irregularity is calculated by using the relationship between the load and the track irregularity in the step 7), the following formula is adopted:
P(t)＝Kt(yw(t)-yc(t))＝Kt(yw(t)-yb(t)-r(t)) (42)
wherein: ktAs wheel stiffness, ywFor vertical wheel displacement, w ═ 1 denotes the front wheel, w ═ 2 denotes the rear wheel, y denotes the rear wheelcDisplacement of the contact point of the vehicle body with the rail, ybAnd r (t) is the rail irregularity.
The technical effect of the invention is undoubtedly that the invention utilizes the sensor technology and adopts a novel algorithm to estimate the irregularity of the railway track, compared with the prior art, the invention opens up a new road to identify the irregularity of the track and provides a new idea for track monitoring work.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a simplified physical model of a vehicle track bridge coupling system;
FIG. 3 is a time discretization graph of acceleration response;
FIG. 4 is a random irregular profile of a simulated orbit using a German high-low interference spectrum (PSD);
FIG. 5 is a comparison graph of the real track irregularity value and the train front wheel identification estimated value;
fig. 6 is a comparison graph of the real track irregularity value and the train rear wheel identification estimated value.
Detailed Description
The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.
Example 1:
referring to fig. 1, the embodiment discloses an analysis method for predicting track irregularity and sensor optimal configuration thereof based on vibration response of an operating train, which comprises the following steps:
1) constructing a vehicle track bridge coupling system; the vehicle track bridge coupling system comprises a detection train and a bridge, wherein a sensor is arranged on the detection train; specifically, in the vehicle track bridge coupling system, assuming that the track is tightly attached to the bridge, the track is uniformly simplified into an euler simple beam, the detection train is simplified into a 4-degree-of-freedom vehicle model consisting of a vehicle body and two wheel sets, the vehicle body comprises a bogie and all vehicle body parts above the wheels, see fig. 2, and the parameters of the detection train comprise vehicle body mass mvVehicle body moment of inertia IvVehicle body rotation angle thetavVertical displacement y of vehicle bodyvFront wheel mass m1Rigidity K for connecting vehicle body and front wheels1Damping Cs1Front wheel vertical displacement y1Front wheel and track contact point abscissa x1Front wheel stiffnessKt1Rear wheel mass m2Connection rigidity K of vehicle body and rear wheels2Damping Cs2Vertical displacement of rear wheel y2X abscissa of contact point between rear wheel and rail2Rear wheel stiffness Kt2Train running speed v, wheel set spacing S and a describing position of mass center of train body1And a2。
2) Establishing a vehicle body motion equation, and calculating a mass matrix M, a rigidity matrix K and a damping matrix C of a vehicle model; specifically, the established vehicle body motion equation is as follows:
Figure BDA0003383941090000071
in the formula: m, C, K are the mass matrix, damping matrix and stiffness matrix of the vehicle structural system, respectively, P (t) is the external force vector on the structure, L is the mapping matrix of the input,
Figure BDA0003383941090000072
Figure BDA0003383941090000081
and X (t) are respectively an acceleration vector, a speed vector and a displacement vector of the vehicle body system.
Wherein:
M＝diag[mv Iv m1 m2] (2)
Figure BDA0003383941090000082
Figure BDA0003383941090000083
Figure BDA0003383941090000084
3) solving a displacement response expression, a speed response expression and an acceleration response expression of train running by using a Newmark-beta method as vibration input responses, wherein the Newmark-beta method is a Newmark-beta method; specifically, the method comprises the following steps:
3-1) the Newmark-beta method is based on the assumed acceleration change (the discretization of time is shown in figure 3), and a coefficient gamma and a coefficient beta are introduced to improve the accuracy and stability of the calculation of the acceleration, wherein the coefficient gamma represents the weight of the contribution of the initial acceleration and the final acceleration to the speed change in the delta t time interval; β represents the weight of the initial acceleration and the final acceleration contribution to the displacement change over the Δ t time interval, interpreted in mathematical terms as:
Figure BDA0003383941090000085
Figure BDA0003383941090000086
3-2) to tiAnd ti+1The acceleration between is integrated to obtain ti+1The displacement and velocity at time are:
Figure BDA0003383941090000087
Figure BDA0003383941090000088
in the formula: Δ t is tiTime and ti+1A time step between moments;
3-3) substituting equations (6) and (7) into equations (8) and (9), respectively, yields:
Figure BDA0003383941090000091
Figure BDA0003383941090000092
3-4) carrying out item shifting on the formula (11) to obtain an acceleration expression at the end of the delta t time period:
Figure BDA0003383941090000093
3-5) substituting equation (12) into equation (10) to obtain the velocity expression at the end of the Δ t period:
Figure BDA0003383941090000094
3-6) listed at ti+1Equation of dynamic balance of vehicle at moment:
Figure BDA0003383941090000095
3-7) substituting equations (12) and (13) into equation (14) to obtain the time-step-only end position displacement Xi+1By appropriate merging of similar terms, this formula is written as:
Figure BDA0003383941090000096
the formula (15) is a form of a static equilibrium equation including equivalent stiffness
Figure BDA0003383941090000097
And equivalent load
Figure BDA0003383941090000098
Wherein:
Figure BDA0003383941090000099
Figure BDA00033839410900000910
wherein
Figure BDA00033839410900000911
Represented by the currently applied force and the restoring force of the previous time step, the end of time step displacement Xi+1Expressed as:
Figure BDA00033839410900000912
wherein:
Figure BDA0003383941090000101
Figure BDA0003383941090000102
Figure BDA0003383941090000103
Figure BDA0003383941090000104
3-8) calculating the displacement of the end point of the time step from the formula (18), and calculating the velocity at that time from the formula (13)
Figure BDA0003383941090000105
Converting constant coefficient of matrix in equation into unit matrix
Figure BDA0003383941090000106
Then, the terms are combined and unified into a form as shown in formula (18):
Figure BDA0003383941090000107
wherein:
Figure BDA0003383941090000108
Figure BDA0003383941090000109
Figure BDA00033839410900001010
Figure BDA00033839410900001011
3-9) substituting equation (18) into equation (12) for the acceleration at the end of the time step
Figure BDA00033839410900001012
The method is simplified as follows:
Figure BDA00033839410900001013
wherein:
Figure BDA00033839410900001014
Figure BDA00033839410900001015
Figure BDA0003383941090000111
Figure BDA0003383941090000112
3-10) in combination with equations (18), (23), (28):
Figure BDA0003383941090000113
3-11) equation (33) is expressed as the initial response, resulting in the train at tiThe displacement response, the speed response and the acceleration response at the moment are as follows:
Figure BDA0003383941090000114
4) collecting and detecting the vibration response of a train running on a bridge as a vibration output response, and establishing a system output response equation; wherein the vibrational response comprises a displacement response, a velocity response, and an acceleration response;
5) constructing a relation between an input response equation and an output response equation, and obtaining a theoretical expression of the relation between a load matrix P and an output response Y of the vehicle body system; specifically, the method comprises the following steps:
5-1) obtaining the displacement, speed and acceleration vector of the detection train according to the acquisition signal of the sensor on the detection train, and establishing an output response equation:
Figure BDA0003383941090000115
wherein: setting R ═ diag [ R ]d Rv Ra]
5-2) substitution of formula (34) for formula (35) gives:
Figure BDA0003383941090000116
assuming that the initial response of the structure is zero, then there are:
Figure BDA0003383941090000121
5-3) placing formula (36) at t1To tnWriting in a matrix convolution form in a time range:
Y＝HLP (38)
wherein:
Figure BDA0003383941090000122
solving the load vector P directly using equation (38) is:
Figure BDA0003383941090000123
due to the matrix HLThe linear dependence between the columns is large,
Figure BDA0003383941090000124
has a determinant close to zero, i.e.
Figure BDA0003383941090000125
Close to singularity, directly solving the formula (40) becomes an ill-posed problem;
6) solving the ill-conditioned problem of the output response Y by using a Tikhonov regularization method, introducing the expression obtained in the step 5) into a regularization item, and converting the ill-conditioned problem into a proper problem; specifically, a regularization term is introduced into the formula (38) in the step 5-3), and the Tikhonov regularization method identifies the force P, wherein the formula is as follows:
Figure BDA0003383941090000126
wherein: λ is a regularization parameter, λ > 0;
7) obtaining a regularization solution of the load matrix, and calculating the track irregularity by using the relation between the load and the track irregularity, wherein the following formula is specifically adopted:
P(t)＝Kt(yw(t)-yc(t))＝Kt(yw(t)-yb(t)-r(t)) (42)
wherein: ktAs wheel stiffness, ywFor vertical wheel displacement, w ═ 1 denotes the front wheel, w ═ 2 denotes the rear wheel, y denotes the rear wheelcDisplacement of the contact point of the vehicle body with the rail, ybAnd r (t) is the rail irregularity.
Based on the theoretical derivation, MATLAB is used for finite element modeling and calculation to obtain the imaging of a theoretical formula, and the theoretical correctness of the method is numerically verified. Specifically, finite element modeling is carried out in MATLAB to obtain a simple supported track bridge with the span of 30m, and the condition that a four-degree-of-freedom train provided with a sensor passes through the simple supported beam is simulated, so that the mass m of a train bodyv17735kg, moment of inertia of vehicle body Jv＝1.47×105kg·m2Mass m of front and rear wheels1＝m21000kg, rigidity K of connection between vehicle body and wheels1＝Ks2＝2.47×106N/m, damping Cs1＝Cs2＝3×104N/m, wheel stiffness Kt1＝Kt2＝3.74×106N/m, wheel damping Ct1＝Ct2When the train running speed v is 20m/S and the wheel set spacing S is 4.2m, and the train body is a homogeneous rectangle, a1＝a20.5; the relevant parameters of the bridge are as follows: mass per unit cross section mb6000kg/m, and a section bending rigidity EI of 2.5 × 1010N·m2. The time step is taken to be 0.001.
Through numerical calculation, the track irregularity identified by the front wheel is shown in fig. 5, wherein the real value of the track irregularity (red line in the figure) adopts a random irregular section of the track simulated by a german high-low interference spectrum (PSD), as shown in fig. 4; the rail irregularity estimate (blue line in the figure) is back-calculated from the established output response equation (40) for the vehicle body response measured by the sensors on the vehicle body. The identified track irregularity of the rear wheels is shown in fig. 6. The estimated value and the real value of the track irregularity in the two figures are compared, and the track irregularity estimated value obtained by the method is found to be well matched with the real track irregularity condition, and the track irregularity identification result of the front wheel is consistent with the track irregularity identification result of the rear wheel, so that the correctness of the method is proved.
Example 2:
the embodiment discloses an analysis method for predicting track irregularity and sensor optimal configuration based on vibration response of an operating train, which comprises the following steps of:
1) constructing a vehicle track bridge coupling system; the vehicle track bridge coupling system comprises a detection train and a bridge, wherein a sensor is arranged on the detection train;
2) establishing a vehicle body motion equation, and calculating a mass matrix M, a rigidity matrix K and a damping matrix C of a vehicle model;
3) solving a displacement response expression, a speed response expression and an acceleration response expression of train running by using a Newmark-beta method, wherein the displacement response expression, the speed response expression and the acceleration response expression are used as vibration input responses;
4) collecting and detecting the vibration response of a train running on a bridge as a vibration output response, and establishing a system output response equation; wherein the vibrational response comprises a displacement response, a velocity response, and an acceleration response;
5) constructing a relation between an input response equation and an output response equation, and obtaining a theoretical expression of the relation between a load matrix P and an output response Y of the vehicle body system;
6) solving the ill-conditioned problem of the output response Y by using a Tikhonov regularization method, introducing the expression obtained in the step 5) into a regularization term, and converting the ill-conditioned problem into a proper problem;
7) and obtaining a regularization solution of the load matrix, and calculating the track irregularity by utilizing the relation between the load and the track irregularity.
Example 3:
the main steps of this embodiment are the same as those of embodiment 2, and further, in the vehicle track bridge coupling system in step 1), the detection train is simplified into a 4-degree-of-freedom vehicle model consisting of a vehicle body and two wheel sets, the vehicle body comprises all vehicle body parts above a bogie and wheels, and the parameters of the detection train comprise vehicle body mass mvVehicle body moment of inertia IvVehicle body rotation angle thetavVertical displacement y of vehicle bodyvFront wheel mass m1Rigidity K for connecting vehicle body and front wheels1Damping Cs1Front wheel vertical displacement y1Front wheel and track contact point abscissa x1Front wheel stiffness Kt1Rear wheel mass m2Connection rigidity K of vehicle body and rear wheels2Damping Cs2Vertical displacement of rear wheel y2X abscissa of contact point between rear wheel and rail2Rear wheel stiffness Kt2Train running speed v, wheel set spacing S and a describing position of mass center of train body1And a2。
Example 4:
the main steps of this embodiment are the same as those of embodiment 3, and further, the vehicle body motion equation established in step 2) is:
Figure BDA0003383941090000141
in the formula: m, C, K are the mass matrix, damping matrix and stiffness matrix of the vehicle structural system, respectively, P (t) is the external force vector on the structure, L is the mapping matrix of the input,
Figure BDA0003383941090000142
Figure BDA0003383941090000143
and X (t) are respectively an acceleration vector, a speed vector and a displacement vector of the vehicle body system.
Wherein:
M＝diag[mv Iv m1 m2] (2)
Figure BDA0003383941090000144
Figure BDA0003383941090000145
Figure BDA0003383941090000151
example 5:
the main steps of this embodiment are the same as embodiment 4, and further, step 3) includes the following sub-steps:
3-1) introducing a coefficient gamma and a coefficient beta, the coefficient gamma representing the weight of the initial acceleration and the final acceleration contributing to the speed change in the time interval at; β represents the weight of the initial acceleration and the final acceleration contribution to the displacement change over the Δ t time interval, interpreted in mathematical terms as:
Figure BDA0003383941090000152
Figure BDA0003383941090000153
3-2) to tiAnd ti+1The acceleration between is integrated to obtain ti+1The displacement and velocity at time are:
Figure BDA0003383941090000154
Figure BDA0003383941090000155
in the formula: Δ t is tiTime and ti+1A time step between moments;
3-3) substituting equations (6) and (7) into equations (8) and (9), respectively, yields:
Figure BDA0003383941090000156
Figure BDA0003383941090000157
3-4) carrying out item shifting on the formula (11) to obtain an acceleration expression at the end of the delta t time period:
Figure BDA0003383941090000158
3-5) substituting equation (12) into equation (10) to obtain the velocity expression at the end of the Δ t period:
Figure BDA0003383941090000161
3-6) listed at ti+1Equation of dynamic balance of vehicle at moment:
Figure BDA0003383941090000162
3-7) substituting equations (12) and (13) into equation (14) to obtain the time-step-only end position displacement Xi+1By appropriate merging of similar terms, this formula is written as:
Figure BDA0003383941090000163
the formula (15) is a form of a static equilibrium equation including equivalent stiffness
Figure BDA0003383941090000164
And equivalent load
Figure BDA0003383941090000165
Wherein:
Figure BDA0003383941090000166
Figure BDA0003383941090000167
wherein
Figure BDA0003383941090000168
Represented by the currently applied force and the restoring force of the previous time step, the end of time step displacement Xi+1Expressed as:
Figure BDA0003383941090000169
wherein:
Figure BDA00033839410900001610
Figure BDA00033839410900001611
Figure BDA00033839410900001612
Figure BDA00033839410900001613
3-8) calculating the displacement of the end point of the time step from the formula (18), and calculating the velocity at that time from the formula (13)
Figure BDA00033839410900001614
Converting constant coefficient of matrix in equation into unit matrix
Figure BDA00033839410900001615
Then, the terms are combined and unified into a form as shown in formula (18):
Figure BDA0003383941090000171
wherein:
Figure BDA0003383941090000172
Figure BDA0003383941090000173
Figure BDA0003383941090000174
Figure BDA0003383941090000175
3-9) substituting equation (18) into equation (12) for the acceleration at the end of the time step
Figure BDA0003383941090000176
The method is simplified as follows:
Figure BDA0003383941090000177
wherein:
Figure BDA0003383941090000178
Figure BDA0003383941090000179
Figure BDA00033839410900001710
Figure BDA00033839410900001711
3-10) in combination with equations (18), (23), (28):
Figure BDA00033839410900001712
3-11) equation (33) is expressed as the initial response, resulting in the train at tiThe displacement response, the speed response and the acceleration response at the moment are as follows:
Figure BDA0003383941090000181
example 6:
the main steps of this embodiment are the same as those of embodiment 5, and further, step 5) includes the following sub-steps:
5-1) obtaining the displacement, speed and acceleration vector of the detection train according to the acquisition signal of the sensor on the detection train, and establishing an output response equation:
Figure BDA0003383941090000182
wherein: setting R ═ diag [ R ]d Rv Ra]
5-2) substitution of formula (34) for formula (35) gives:
Figure BDA0003383941090000183
assuming that the initial response of the structure is zero, then there are:
Figure BDA0003383941090000184
5-3) placing formula (36) at t1To tnWriting in a matrix convolution form in a time range:
Y＝HLP (38)
wherein:
Figure BDA0003383941090000185
solving the load vector P directly using equation (38) is:
Figure BDA0003383941090000186
example 7:
the main steps of this embodiment are the same as those of embodiment 6, and further, in step 6), the regularization term is introduced into the formula (38) in step 5-3), and the Tikhonov regularization method identifies the force P, and the formula is as follows:
Figure BDA0003383941090000191
wherein: λ is a regularization parameter, λ > 0.
Example 8:
the main steps of this embodiment are the same as those of embodiment 7, and further, when the track irregularity is calculated by using the relationship between the load and the track irregularity in step 7), the following formula is adopted:
P(t)＝Kt(yw(t)-yc(t))＝Kt(yw(t)-yb(t)-r(t)) (42)
wherein: ktAs wheel stiffness, ywFor vertical wheel displacement, w ═ 1 denotes the front wheel, w ═ 2 denotes the rear wheel, y denotes the rear wheelcDisplacement of the contact point of the vehicle body with the rail, ybAnd r (t) is the rail irregularity.

Claims (7)

1. An analysis method for predicting track irregularity and sensor optimal configuration based on vibration response of an operating train is characterized by comprising the following steps of: the method comprises the following steps:
1) constructing a vehicle track bridge coupling system; the vehicle track bridge coupling system comprises a detection train and a bridge, wherein a sensor is arranged on the detection train;
2) establishing a vehicle body motion equation, and calculating a mass matrix M, a rigidity matrix K and a damping matrix C of a vehicle model;
3) solving a displacement response expression, a speed response expression and an acceleration response expression of train running by using a Newmark-beta method, wherein the displacement response expression, the speed response expression and the acceleration response expression are used as vibration input responses;
4) collecting and detecting the vibration response of a train running on a bridge as a vibration output response, and establishing a system output response equation; wherein the vibrational response comprises a displacement response, a velocity response, and an acceleration response;
5) constructing a relation between an input response equation and an output response equation, and obtaining a theoretical expression of the relation between a load matrix P and an output response Y of the vehicle body system;
6) solving the ill-conditioned problem of the output response Y by using a Tikhonov regularization method, introducing the expression obtained in the step 5) into a regularization term, and converting the ill-conditioned problem into a proper problem;
7) and obtaining a regularization solution of the load matrix, and calculating the track irregularity by utilizing the relation between the load and the track irregularity.
2. The method of claim 1 for analyzing track irregularity prediction and sensor optimization configuration thereof based on operational train vibration response, wherein the method comprises the following steps: in the vehicle track bridge coupling system in the step 1), the detection train is simplified into a 4-freedom-degree vehicle model consisting of a vehicle body and two wheel pairs, the vehicle body comprises all vehicle body parts above a bogie and wheels, and the parameter of the detection train comprises the mass m of the vehicle bodyvVehicle body moment of inertia IvVehicle body rotation angle thetavVertical displacement y of vehicle bodyvFront wheel mass m1Rigidity K for connecting vehicle body and front wheels1Damping Cs1Front wheel vertical displacement y1Front wheel and track contact point abscissa x1Front wheel stiffness Kt1Rear wheel mass m2Connection rigidity K of vehicle body and rear wheels2Damping Cs2Vertical displacement of rear wheel y2X abscissa of contact point between rear wheel and rail2Rear wheel stiffness Kt2Train running speed v, wheel set spacing S and a describing position of mass center of train body1And a2。
3. The method of claim 1 for analyzing track irregularity prediction and sensor optimization configuration thereof based on operational train vibration response, wherein the method comprises the following steps: the vehicle body motion equation established in the step 2) is as follows:
Figure FDA0003383941080000011
in the formula: m, C, K are the mass matrix, damping matrix and stiffness matrix of the vehicle structural system, respectively, P (t) is the external force vector on the structure, L is the mapping matrix of the input,
Figure FDA0003383941080000021
Figure FDA0003383941080000022
x (t) is the acceleration, speed and displacement vector of the vehicle body system respectively;
wherein:
M＝diag[mv Iv m1 m2] (2)
Figure FDA0003383941080000023
Figure FDA0003383941080000024
Figure FDA0003383941080000025
4. the method of claim 3 for analyzing track irregularity prediction and sensor optimization configuration thereof based on operational train vibration response, wherein the method comprises the following steps: the step 3) comprises the following sub-steps:
3-1) introducing a coefficient gamma and a coefficient beta, the coefficient gamma representing the weight of the initial acceleration and the final acceleration contributing to the speed change in the time interval at; β represents the weight of the initial acceleration and the final acceleration contribution to the displacement change over the Δ t time interval, interpreted in mathematical terms as:
Figure FDA0003383941080000026
Figure FDA0003383941080000027
3-2) to tiAnd ti+1The acceleration between is integrated to obtain ti+1The displacement and velocity at time are:
Figure FDA0003383941080000028
Figure FDA0003383941080000031
in the formula: Δ t is tiTime and ti+1A time step between moments;
3-3) substituting equations (6) and (7) into equations (8) and (9), respectively, yields:
Figure FDA0003383941080000032
Figure FDA0003383941080000033
3-4) obtaining an acceleration expression at the final end of the time period of the delta t by arranging the equation (11) in a term shifting way:
Figure FDA0003383941080000034
3-5) substituting equation (12) into equation (10) to obtain a speed expression at the end of the Δ t period:
Figure FDA0003383941080000035
3-6) listed at ti+1Equation of dynamic balance of vehicle at moment:
Figure FDA0003383941080000036
3-7) substituting equations (12) and (13) into equation (14) to obtain the time-step-only end position displacement Xi+1By appropriate merging of similar terms, this formula is written as:
Figure FDA0003383941080000037
the formula (15) is a form of a static equilibrium equation including equivalent stiffness
Figure FDA0003383941080000038
And equivalent load
Figure FDA0003383941080000039
Wherein:
Figure FDA00033839410800000310
Figure FDA00033839410800000311
wherein
Figure FDA00033839410800000312
Represented by the currently applied force and the restoring force of the previous time step, the end of time step displacement Xi+1Expressed as:
Figure FDA0003383941080000041
wherein:
Figure FDA0003383941080000042
Figure FDA0003383941080000043
Figure FDA0003383941080000044
Figure FDA0003383941080000045
3-8) calculating the displacement of the end point of the time step from the formula (18), and calculating the velocity at that time from the formula (13)
Figure FDA0003383941080000046
Converting constant coefficient of matrix in equation into unit matrix
Figure FDA0003383941080000047
Then combined and unified into a form as in formula (18):
Figure FDA0003383941080000048
wherein:
Figure FDA0003383941080000049
Figure FDA00033839410800000410
Figure FDA00033839410800000411
Figure FDA00033839410800000412
3-9) substituting equation (18) into equation (12) for the acceleration at the end of the time step
Figure FDA00033839410800000413
The method is simplified as follows:
Figure FDA00033839410800000414
wherein:
Figure FDA00033839410800000415
Figure FDA0003383941080000051
Figure FDA0003383941080000052
Figure FDA0003383941080000053
3-10) in combination with equations (18), (23), (28):
Figure FDA0003383941080000054
3-11) equation (33) is expressed as the initial response, resulting in the train at tiThe displacement response, the speed response and the acceleration response at the moment are as follows:
Figure FDA0003383941080000055
5. the method of claim 4 for analyzing track irregularity prediction and sensor optimization configuration thereof based on operational train vibration response, wherein the method comprises the following steps: step 5) comprises the following sub-steps:
5-1) obtaining the displacement, speed and acceleration vector of the detection train according to the acquisition signal of the sensor on the detection train, and establishing an output response equation:
Figure FDA0003383941080000056
wherein: setting R ═ diag [ R ]d Rv Ra]
5-2) substitution of formula (34) for formula (35) gives:
Figure FDA0003383941080000057
assuming that the initial response of the structure is zero, then there are:
Figure FDA0003383941080000058
5-3) placing formula (36) at t1To tnWriting in a matrix convolution form in a time range:
Y＝HLP (38)
wherein:
Figure FDA0003383941080000061
solving the load vector P directly using equation (38) is:
Figure FDA0003383941080000062
6. the method of claim 5 for analyzing track irregularity prediction and sensor optimization configuration thereof based on operational train vibration response, wherein the method comprises the steps of: in step 6), introducing the formula (38) in step 5-3) into a regularization term, and identifying the force P by using a Tikhonov regularization method, wherein the formula is as follows:
Figure FDA0003383941080000063
wherein: λ is a regularization parameter, λ > 0.
7. The method of claim 6, wherein the method comprises the steps of: in the step 7), when the track irregularity is calculated by using the relationship between the load and the track irregularity, the following formula is adopted:
P(t)＝Kt(yw(t)-yc(t))＝Kt(yw(t)-yb(t)-r(t)) (42)
wherein: ktIn order to achieve the rigidity of the wheel,ywfor vertical wheel displacement, w ═ 1 denotes the front wheel, w ═ 2 denotes the rear wheel, y denotes the rear wheelcDisplacement of the contact point of the vehicle body with the rail, ybAnd r (t) is the rail irregularity.
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CN116258040A (en) * 2022-12-30 2023-06-13 武汉理工大学 Track irregularity detection method
CN116305456A (en) * 2023-03-09 2023-06-23 武汉理工大学 Method and device for simultaneously estimating bridge frequency and track irregularity and electronic equipment

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Publication number Priority date Publication date Assignee Title
CN116258040A (en) * 2022-12-30 2023-06-13 武汉理工大学 Track irregularity detection method
CN116258040B (en) * 2022-12-30 2024-01-23 武汉理工大学 Track irregularity detection method
CN116305456A (en) * 2023-03-09 2023-06-23 武汉理工大学 Method and device for simultaneously estimating bridge frequency and track irregularity and electronic equipment
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