CN111027213A - Frequency domain method-based transverse vibration reaction calculation method for vehicle-induced bridge - Google Patents

Frequency domain method-based transverse vibration reaction calculation method for vehicle-induced bridge Download PDF

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CN111027213A
CN111027213A CN201911264049.7A CN201911264049A CN111027213A CN 111027213 A CN111027213 A CN 111027213A CN 201911264049 A CN201911264049 A CN 201911264049A CN 111027213 A CN111027213 A CN 111027213A
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bridge
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spectrum
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晋智斌
任娟娟
袁李刚
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Southwest Jiaotong University
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Abstract

The invention discloses a method for calculating the transverse vibration reaction of a vehicle-induced bridge based on a frequency domain method, and provides a novel method for determining the transverse vibration reaction of a railway bridge, compared with the traditional time domain analysis method, the method subdivides a train load frequency spectrum into a load sequence spectrum and a bridge harmonic mode spectrum, researches the influence relationship of transverse simple harmonic frequency, vehicle speed, load sequence and different load amplitude values on the transverse simple harmonic frequency, the vehicle speed, the load sequence and the different load amplitude values, and theoretically simplifies calculation; the transverse vibration of the axle is directly decoupled, the transverse force of the wheel set is simplified to the force of the bogie, and random vibration analysis is facilitated by adopting a frequency domain algorithm through the power spectrum density of the transverse force of the wheel set; therefore, the method provided by the invention has the advantages of small calculation amount and high calculation efficiency; and based on the determined number of modes, the calculation accuracy is also better.

Description

Frequency domain method-based transverse vibration reaction calculation method for vehicle-induced bridge
Technical Field
The invention belongs to the technical field of railway bridge transverse vibration calculation, and particularly relates to a frequency domain method-based method for calculating a transverse vibration reaction of a vehicle-induced bridge.
Background
Conventionally, the transverse vibration of a railway bridge caused by moving vehicles is evaluated by a time domain simulation method, a detailed vehicle-rail-bridge model needs to be established, the wheel-rail relationship is complex, and the method has large calculation amount and low calculation efficiency. Especially, it is difficult to use the time domain simulation analysis method when a large number of examples are required for correlation analysis. Such as bridge vibration assessment on a whole high-speed railway line, probability analysis of railway bridge vibration. In addition, the track irregularity is random, and the result of performing time domain simulation by using a certain specified track irregularity is not representative. The vibration decoupling of the vehicle-induced bridge is realized, the wheel set transverse force is considered in the form of power spectral density, the random reaction of the bridge is calculated, and the method is high in calculation efficiency.
The method is characterized in that a time domain simulation method is used for calculating the transverse vibration of the railway bridge, the time domain simulation method is complex in establishing a simulation model, large in calculation amount and low in calculation efficiency, rail irregularity is a typical random process, an integration method is generally adopted in consideration of time domain solving of axle coupling vibration analysis, such as a Newmark- β method, a central difference method and a Wilson-theta method, and the like, a certain requirement is imposed on an integration step length by partial algorithms, and even if a modal superposition method is adopted to simplify calculation of the bridge, the calculation efficiency is still low compared with a frequency domain method, and the calculation efficiency is undoubtedly an important influence factor for research and calculation requiring a large number of examples.
Disclosure of Invention
Aiming at the defects in the prior art, the method for calculating the transverse vibration of the vehicle-induced bridge based on the frequency domain method solves the problems of large calculation amount and low calculation efficiency when the transverse vibration is determined by the traditional time domain analysis method.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that: a method for calculating the transverse vibration reaction of a vehicle-induced bridge based on a frequency domain method comprises the following steps:
s1, simplifying the transverse force of the train wheel set into the force of the bogie acting on the bridge structure, and determining the transverse force F of the kth bogiek(t)；
S2, constructing a bridge dynamic equation through a finite element method, and according to the bridge displacement Xb(t) decomposing the motion into a plurality of orders of bridge modal motion, and establishing a bridge modal motion equation;
s3, determining the transverse force F of the bogiek(t) induced ith order modal force p of bridgei(t)；
S4, according to the unit load sequence spectrum C1(omega) determining train load at different speedsSequence-carrying spectrum Cv(ω)；
S5, i-th order modal spectrum of bridge
Figure BDA0002312369900000021
Determining ith harmonic modal spectrum of bridge
Figure BDA0002312369900000022
S6, load sequence spectrum C of trainv(omega) and i-th harmonic modal spectra of bridges
Figure BDA0002312369900000023
Multiplying to obtain the ith order modal force p of the bridgei(t) a corresponding modal force spectrum P (ω);
s7, multiplying the ith-order modal force spectrum P (omega) of the bridge by the ith-order modal frequency response function H (omega) of the bridge to obtain an ith-order modal reaction spectrum Q (omega) of the bridge;
s8, carrying out inverse Fourier transform on the ith order modal response frequency spectrum Q (omega) of the bridge and combining the transient response to obtain the complete ith order modal response Q (omega) of the bridgehT) and obtaining the transverse vibration reaction of the vehicle-caused bridge based on the determined mode number.
Further, in the step S1, the lateral force F of the kth bogiek(t) is:
Fk(t)＝F0(t-tk)
wherein, the subscript k is the number of bogies included in the train, and k is 0,1, …,2Nc-1，NcThe number of vehicles in the train;
t is the time;
tkis the kth bogie lateral force lag time, and tk＝xk/v，xkIs the distance between the kth bogie and the first bogie and v is the running speed of the train.
Further, the bridge dynamic equation constructed in step S2 is:
Figure BDA0002312369900000031
in the formula, MbA mass matrix of the bridge structure;
Cba damping matrix of the bridge structure;
Kba stiffness matrix for a bridge structure;
Xb(t) is the displacement of the bridge at time t;
Figure BDA0002312369900000032
is the acceleration of the bridge, i.e. the second derivative of displacement with respect to time;
Figure BDA0002312369900000033
is the speed of the bridge, i.e. the first derivative of displacement with respect to time;
Pb(t) is an external force acting on the bridge;
in step S2, the method for decomposing the bridge dynamic equation into the bridge modal equation of motion specifically includes:
a1 displacement of bridge X by mode superposition methodb(t) decomposition into a plurality of modal shapes φiIs superposed of Xb(t) is expressed as:
Figure BDA0002312369900000034
in the formula, qi(t) is the ith order modal displacement;
a2 bridge displacement X based on mode shape superpositionb(t) decomposing the bridge dynamic equation to obtain an ith-order bridge modal motion equation:
Figure BDA0002312369900000035
in the formula (I), the compound is shown in the specification,
Figure BDA0002312369900000036
the i-th order modal acceleration of the bridge is obtained;
Figure BDA0002312369900000037
the i-th order modal speed of the bridge;
qithe displacement is the ith order modal displacement of the bridge;
ζithe damping ratio of the ith order bridge mode vibration is obtained;
ωb,ithe circular frequency of the ith order bridge mode vibration;
Pi(t) is the i-th order bridge modal force, and
Figure BDA0002312369900000038
Figure BDA0002312369900000039
is the ith order bridge mode vector phiiThe transposing of (1).
Further, in step S3, the i-th order modal force pi (t) of the bridge caused by the bogie lateral force is:
Figure BDA0002312369900000041
in the formula, L is a bridge span;
pi [. cndot. ] is a control variable of the load acting on the bridge structure, and pi (u) ═ 1, | u ∈ [0,1 ]; 0| other };
when the transverse force of the bogie is simple harmonic load, the bridge modal force spectrum can be decomposed into a product of a train load sequence spectrum and a bridge harmonic modal spectrum;
the simple harmonic load is as follows:
Figure BDA0002312369900000042
in the formula, ωhThe circular frequency of the simple harmonic.
Further, the unit load sequence spectrum C in the step S41(ω)；
Figure BDA0002312369900000043
Wherein exp (. cndot.) is an exponential function;
i is an imaginary number;
omega is the circular frequency;
lathe distance between the front wheel and the rear wheel of the same bogie is the same;
lcthe distance between the centers of front and rear bogies of the same vehicle;
the load sequence spectrum C of the train in the step S4v(ω) is:
Cv(ω)＝C1(ω/v)。
further, the ith harmonic mode spectrum of the bridge in the step S5
Figure BDA0002312369900000044
Comprises the following steps:
Figure BDA0002312369900000045
in the formula, v ≠ 1, ωh≠0
Figure BDA0002312369900000046
Is v ═ 1, ω h0, and
Figure BDA0002312369900000047
Figure BDA0002312369900000051
further, the i-th order modal force spectrum P (ω) of the bridge in the step S6 is:
Figure BDA0002312369900000052
further, in step S7, the i-th order modal response spectrum Q (ω) of the bridge is:
Q(ω)＝H(ω)P(ω)
wherein H (omega) is the ith order modal frequency response function of the bridge, and
Figure BDA0002312369900000053
further, in the step S8, the i-th order modal response q (ω) of the complete bridgehAnd t) is:
Figure BDA0002312369900000054
wherein, pi is a circumference ratio;
A. b is a transient solution constant term;
Figure BDA0002312369900000055
Figure BDA0002312369900000056
the invention has the beneficial effects that:
compared with the traditional time domain analysis method, the train load frequency spectrum is subdivided into a load sequence spectrum and a bridge harmonic mode spectrum, the influence relation of transverse simple harmonic frequency, vehicle speed, load sequence and different load amplitude values is researched, and the calculation is simplified theoretically; the transverse vibration of the axle is directly decoupled, the transverse force of the wheel set is simplified to the force of the bogie, and random vibration analysis is facilitated by adopting a frequency domain algorithm through the power spectrum density of the transverse force of the wheel set; therefore, the method provided by the invention has the advantages of small calculation amount and high calculation efficiency; and based on the determined number of the modes, the calculation precision is better.
Drawings
FIG. 1 is a flow chart of a method for calculating a lateral vibration response of a vehicle induced bridge based on a frequency domain method.
Fig. 2 is a schematic diagram of a general railway train bogie arrangement in an embodiment of the present invention.
FIG. 3 is a graph illustrating a comparison of a Newmark- β bridge response with a spectral method in accordance with an embodiment of the present invention.
FIG. 4 is a power spectrum diagram of the lateral force of the wheel set at different vehicle speeds according to an embodiment of the present invention.
Fig. 5 is a schematic diagram of a standard deviation of a displacement response of a simply supported beam according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
As shown in fig. 1, a method for calculating a lateral vibration reaction of a vehicle-induced bridge based on a frequency domain method includes the following steps:
s1, simplifying the transverse force of the train wheel set into the force of the bogie acting on the bridge structure, and determining the transverse force F of the kth bogiek(t)；
S2, constructing a bridge dynamic equation through a finite element method, and according to the bridge displacement Xb(t) decomposing the motion into a plurality of orders of bridge modal motion, and establishing a bridge modal motion equation;
s3, determining the transverse force F of the bogiek(t) induced ith order modal force p of bridgei(t)；
S4, according to the unit load sequence frequency spectrum C1(omega), determining train load sequence spectrum C under different speedsv(ω)；
S5, i-th order modal spectrum of bridge
Figure BDA0002312369900000061
Determining ith harmonic modal spectrum of bridge
Figure BDA0002312369900000062
S6, load sequence spectrum C of trainv(omega) and i-th harmonic modal spectra of bridges
Figure BDA0002312369900000063
Multiplying to obtain the ith order modal force p of the bridgei(t) a corresponding modal force spectrum P (ω);
s7, multiplying the ith-order modal force spectrum P (omega) of the bridge by the ith-order modal frequency response function H (omega) of the bridge to obtain an ith-order modal reaction spectrum Q (omega) of the bridge;
s8, carrying out inverse Fourier transform on the ith order modal response frequency spectrum Q (omega) of the bridge and combining the transient response to obtain the complete ith order modal response Q (omega) of the bridgehT) and obtaining the transverse vibration reaction of the vehicle-caused bridge based on the determined mode number.
For the above step S1, it has been found that the influence of the vibration of the bridge on the wheel-rail force is limited, the response of the train and the bridge can be solved in a decoupled manner, and since the lateral forces of the first wheel pair and the third wheel pair of the train are almost the same and the lateral forces of the second wheel pair and the fourth wheel pair are also similar, the lateral forces under the two bogies are almost the same by summing up the lateral forces of the two wheel pairs under each bogie into one bogie force. Since the distance between the front and rear wheelsets under the same bogie is usually 1.5-2.5m, which is much smaller than the span of the bridge, the train wheelset transverse force can be simplified to bogie force acting on the bridge structure. As shown in FIG. 2, the lateral force of each bogie can be approximately considered to be the same, the lateral force of different bogies has a time lag, and the lateral force of the k-th bogie is denoted as Fk(t), which is expressed as:
Fk(t)＝F0(t-tk) (1)
wherein, the subscript k is the number of bogies included in the train, and k is 0,1, …,2Nc-1，NcThe number of vehicles in the train;
t is the time;
tkis the kth bogie lateral force lag time, and tk＝xk/v，xkIs the kth bogie andthe distance between the first bogies, v, is the running speed of the train.
The bridge dynamic equation constructed in the step S2 is:
Figure BDA0002312369900000071
in the formula, MbA mass matrix of the bridge structure;
Cba damping matrix of the bridge structure;
Kba stiffness matrix for a bridge structure;
Xb(t) is the displacement of the bridge at time t;
Figure BDA0002312369900000081
is the acceleration of the bridge, i.e. the second derivative of displacement with respect to time;
Figure BDA0002312369900000082
is the speed of the bridge, i.e. the first derivative of displacement with respect to time;
Pb(t) is an external force acting on the bridge due to the moving force;
the method for decomposing the bridge dynamic equation into the bridge modal motion equation specifically comprises the following steps:
a1 displacement of bridge X by mode superposition methodb(t) decomposition into a plurality of modal shapes φiIs superposed of Xb(t) is expressed as:
Xb(t)＝∑iqi(t)φi(3)
in the formula, qi(t) is the ith order modal displacement;
a2 bridge displacement X based on mode shape superpositionb(t) decomposing the bridge dynamic equation;
specifically, formula (3) is substituted for formula (2), and left-multiplied on both sides
Figure BDA0002312369900000083
And performing normalization processing of modal quality
Figure BDA0002312369900000084
The dynamic equation of the bridge is decomposed into n single-degree-of-freedom (SD0F) bridge modal motion equations as follows:
Figure BDA0002312369900000085
in the formula (I), the compound is shown in the specification,
Figure BDA0002312369900000086
the i-th order modal acceleration of the bridge is obtained;
Figure BDA0002312369900000087
the i-th order modal speed of the bridge;
qithe displacement is the ith order modal displacement of the bridge;
ζithe damping ratio of the ith order bridge mode vibration is obtained;
ωb,ithe circular frequency of the ith order bridge mode vibration;
Pi(t) is the i-th order bridge modal force, and
Figure BDA0002312369900000088
Figure BDA0002312369900000089
is the ith order bridge mode vector phiiThe transposing of (1).
In step S3, the i-th order modal force p of the bridge due to the lateral force of the bogiei(t) is:
Figure BDA0002312369900000091
where δ (·) is the dirac function;
l is the bridge span, and x represents the position on the bridge;
pi [. cndot. ] is a control variable of the load acting on the bridge structure, and pi (u) ═ 1, | u ∈ [0,1 ]; 0| other }.
In the above steps S4 to S6, the transverse force applied to the bridge by the train is generated by the wheel set snaking motion caused by the track irregularity, and the track irregularity is random, so the Power Spectral Density (PSD) is often used to generate the track irregularity, and the transverse force of the wheel set is further described. Therefore, the transverse vibration reaction of the bridge under the action of the simple harmonic load is particularly important, any function can be decomposed into superposition of the simple harmonic load according to the Fourier transform principle, and if the transverse force of the wheel pair is the simple harmonic load, the circular frequency is omegahSimple harmonic load
Figure BDA0002312369900000092
Comprises the following steps:
Figure BDA0002312369900000093
wherein k is 0,1,2, …,2Nc-1；
ωhCircular frequency which is a simple harmonic;
i is an imaginary number;
at this time, the ith order modal force p of the bridgei(ωhAnd t) is:
Figure BDA0002312369900000094
the frequency spectrum of the ith order modal force of the bridge is as follows:
Figure BDA0002312369900000095
in the formula, ω is a circle frequency;
let t-tkT', the spectrum P (ω) of the i-th order modal force of the bridge can be written as:
Figure BDA0002312369900000101
wherein, Cv(omega) is the load sequence spectrum of the train;
Figure BDA0002312369900000102
Figure BDA0002312369900000103
is the ith harmonic mode spectrum of the bridge;
Figure BDA0002312369900000104
for a typical railroad train as shown in fig. 2, the unit load sequence spectrum can be simplified as:
Figure BDA0002312369900000105
in the formula, x2k＝klc，x2k+1＝klc+la；
lcIs the distance between the centers of the front and rear bogies of the same vehicle;
lais the distance between the centers of the front and rear wheel pairs under the same bogie;
order to
Figure BDA0002312369900000106
It can be deduced that:
Figure BDA0002312369900000107
Z＝[exp(IωNclc)-1]/[exp(Iωlc)-1](14) the unit load sequence spectrum C in step S41(ω) is:
Figure BDA0002312369900000108
for exp (I ω l)c) In the special case of 0, formula (15) obviously does not apply and can be obtained according to formula (12):
C1(ω)＝Nc[1+exp(Iωla)](16) display deviceHowever, the load sequence spectra at different speeds satisfy the following relations:
Cv(ω)＝C1(ω/v) (17)
in the formula, C1(ω) is a unit load sequence spectrum at a speed of 1m/S, thereby obtaining a load sequence spectrum C of the train in step S4v(ω)；
For v ═ 1, ωhWhen the value is 0, the i-th harmonic mode spectrum of the bridge is as follows:
Figure BDA0002312369900000111
for v ≠ 1, ωhNot equal to 0, according to equation (11), the ith harmonic mode spectrum of the bridge obtained in step S5 is:
Figure BDA0002312369900000112
therefore, based on equation (9), the spectrum P (ω) of the i-th order modal force of the bridge in step S6 is:
Figure BDA0002312369900000113
in step S7, the ith-order bridge motion equation of equation (4) is solved by a frequency domain method, so as to obtain a spectrum Q (ω) of the ith-order modal response of the bridge. I.e. the frequency spectrum P (ω) equal to the i-th order modal force of the bridge multiplied by the i-th order modal frequency response function H (ω) of the bridge, i.e.:
Q(ω)＝H(ω)P(ω) (21)
Figure BDA0002312369900000114
for the step S8, under the condition of considering the transient reaction, the spectrum Q (ω) of the i-th order modal reaction of the bridge is subjected to inverse fourier transform to obtain the complete i-th order modal reaction Q (ω) of the bridgehAnd t) is:
Figure BDA0002312369900000115
wherein, pi is a circumference ratio;
A. b is a transient solution constant term;
Figure BDA0002312369900000121
in an embodiment of the present invention, a fast fourier transform implementation method for determining the lateral vibration of the vehicle-induced bridge based on the frequency domain method is provided, where the frequency domain solution of the modal vibration of a certain order of the bridge is shown below, and phi in the following refers to a modal shape of a certain order of the bridge.
In step S2, the bridge displacement X is calculated by a modal superposition methodb(t) decomposition into a plurality of modal shapes φiAnd (3) superposition. For a certain-order modal motion of the bridge, discretizing a vibration mode of the bridge, wherein the shape of the mode of the bridge is represented as:
φ[n]＝φ(nΔx) (25)
wherein N is 0,1,.., N-1;
Δ x ═ L/(N-1), is the dimension of the bridge discrete length. L is the bridge span. N is the number of the bridge modal discrete points;
FFT (fast fourier transform) is performed on the discrete bridge mode shape according to equation (18) to obtain:
Figure BDA0002312369900000122
wherein i ═ 0, 1.., N-1;
φ1，0[i]the corresponding spatial frequencies are:
Ω(i)＝iΔΩ (27)
where Δ Ω ═ 2 pi/N is the fundamental spatial frequency of the FFT conversion.
When calculating the bridge harmonic mode frequency spectrum, the spatial frequency is omega according to the equation (19) due to the frequency shift relation of the front Fourier transformhThe bridge harmonic mode spectrum of j delta omega is
Figure BDA0002312369900000123
The second formula of equation (28) is due to1，0Is a function of the period N, i.e.
φ1，0[i]＝φ1，0[N+i](29)
Considering the characteristics of the actual FFT calculation, the frequency range ω (i) (i ═ 0, …, N-1) should become
Figure BDA0002312369900000131
At this time, the harmonic mode spectrum Φ 'of the bridge is obtained in step S5'v,j[i]Namely:
Figure BDA0002312369900000132
the frequency response function of the bridge mode in the step S6 is:
Figure BDA0002312369900000133
the sequence spectrum of the load is according to equation (21)
Figure BDA0002312369900000134
The bridge mode response spectrum Q' i in the step S7 is thus obtained as:
Q′[il＝H[i]Cv[i]φ′j，v[i](33)
wherein i ═ N-1)/2, …, (N-1)/2;
according to equation 24, the transient solution constant term a, B of the bridge modal response is:
Figure BDA0002312369900000135
for the
Figure BDA0002312369900000136
Inverse Fourier transform of (1), frequencyThe range should be changed to:
Figure BDA0002312369900000137
in step S8, the modal response of the bridge time domain may be obtained by performing inverse fourier transform on Q [ i ], and then superimposing the initial conditions. Namely:
Figure BDA0002312369900000138
wherein N is 0,1, …, N-1;
ωh＝jvΔΩ,t[n]nL/(N-1) represents discrete time points.
It should be noted that, in order to make the time span cover the whole process of train passing through the bridge, the length L should be greater than the sum of the bridge span and the train length. In the extended length range, the modal shape outside the bridge span should be assigned a value of zero.
In one embodiment of the invention, by specific calculation, the bridge response of a series of simple harmonic moving forces analyzed by a spectrum method through a simple beam is compared with the bridge response calculated by a Newmark- β time domain integration method:
calculating parameters: bridge span L is 32m, i-th order frequency omega of bridgeb,i/(2π)＝13.57i2Hz, the mass m of the bridge per meter is 28125kg/m, and the modal damping ratio is zetai0.05, load amplitude F017.5kN, and the vehicle speed v 350km/h, lc＝26m，la17.5m, number of vehicles NcFrequency omega of simple harmonic force 8hAnd/(2 pi) ═ 10.80Hz, i refers to the ith order bridge mode.
As shown in FIG. 3, when the frequency domain method of the present invention is compared with the bridge response obtained by Newmark- β, it can be seen that the bridge response obtained by the two methods is consistent, wherein (a) is the frequency domain method, and (b) is the time domain analysis method;
the bridge response in fig. 3 is due to the moving wheel pair lateral force with only one frequency component. In fact, the frequency range of the transverse force of the wheel set is wide due to the fact that the track irregularity process is random. As shown in fig. 4, a typical lateral force power spectrum is used as the random input to the bridge.
And analyzing and counting the response of the simply supported beam bridge under the mobile random transverse force by using a frequency spectrum method. The maximum standard deviation of the bridge displacement at the midspan and 1/4 midspan at the speed of 100-. The bridge displacements obtained using the first order modes are very close to those obtained including the first seven order modes. Obviously, in the real train speed range, the contribution of the high-order modes other than the first-order mode to the bridge displacement is very limited.
The maximum standard deviation of the bridge acceleration at the midspan and 1/4 midspan at the speed of 100-350km/h is shown in FIG. 5. The mid-span acceleration of the bridge obtained by only considering the first-order mode is very close to the result obtained by the first seven-order mode. However, for an acceleration of 1/4 bridge span, the vehicle speed is 250-350km/h, the result calculated by the first-order mode is much smaller than that of the first seven-order mode.
The above calculation example shows that when the appropriate modal order is adopted, the precision of the frequency domain algorithm is basically consistent with that of the Newmark- β method.
The invention has the beneficial effects that:
compared with the traditional time domain analysis method, the train load frequency spectrum is subdivided into a load sequence spectrum and a bridge harmonic mode spectrum, the influence relation of transverse simple harmonic frequency, vehicle speed, load sequence and different load amplitude values on the train load frequency spectrum is researched, and the calculation is simplified theoretically; the transverse vibration of the axle is directly decoupled, the transverse force of the wheel set is simplified to the force of the bogie, and random vibration analysis is facilitated by adopting a frequency domain algorithm through the power spectrum density of the transverse force of the wheel set; therefore, the method provided by the invention has the advantages of small calculation amount and high calculation efficiency; and based on the determined number of the modes, the calculation precision is better.

Claims (9)

1. A method for calculating the transverse vibration reaction of a vehicle-induced bridge based on a frequency domain method is characterized by comprising the following steps:
s1, simplifying the transverse force of the train wheel set into the force of the bogie acting on the bridge structure, and determining the transverse force F of the kth bogiek(t)；
S2, constructing a bridge dynamic equation through a finite element method, and according to the bridge displacement Xb(t) decomposing the motion into a plurality of orders of bridge modal motion, and establishing a bridge modal motion equation;
s3, determining the transverse force F of the bogiek(t) induced ith order modal force p of bridgei(t)；
S4, according to the unit load sequence spectrum C1(omega), determining train load sequence spectrum C under different speedsv(ω)；
S5, i-th order modal spectrum of bridge
Figure FDA0002312369890000011
Determining ith harmonic modal spectrum of bridge
Figure FDA0002312369890000012
S6, load sequence spectrum C of trainv(omega) and i-th harmonic modal spectra of bridges
Figure FDA0002312369890000013
Multiplying to obtain the ith order modal force p of the bridgei(t) a corresponding modal force spectrum P (ω);
s7, multiplying the ith-order modal force spectrum P (omega) of the bridge by the ith-order modal frequency response function H (omega) of the bridge to obtain an ith-order modal reaction spectrum Q (omega) of the bridge;
s8, carrying out inverse Fourier transform on the ith order modal response frequency spectrum Q (omega) of the bridge and combining the transient response to obtain the complete ith order modal response Q (omega) of the bridgehT) and obtaining the transverse vibration reaction of the vehicle-caused bridge based on the determined mode number.
2. The method for calculating the lateral vibration reaction of the vehicle-induced bridge based on the frequency domain method as claimed in claim 1, wherein the method is characterized in thatIn the step S1, the lateral force F of the kth bogiek(t) is:
Fk(t)＝F0(t-tk)
wherein, subscript k is the number of bogies included in the train, and k is 0,1c-1，NcThe number of vehicles in the train;
t is the time;
tkis the kth bogie lateral force lag time, and tk＝xk/v，xkIs the distance between the kth bogie and the first bogie and v is the running speed of the train.
3. The method for calculating the transverse vibration reaction of the vehicle-induced bridge based on the frequency domain method as claimed in claim 2, wherein the bridge dynamic equation constructed in the step S2 is:
Figure FDA0002312369890000021
in the formula, MbA mass matrix of the bridge structure;
Cba damping matrix of the bridge structure;
Kba stiffness matrix for a bridge structure;
Xb(t) is the displacement of the bridge at time t;
Figure FDA0002312369890000022
is the acceleration of the bridge, i.e. the second derivative of displacement with respect to time;
Figure FDA0002312369890000023
is the speed of the bridge, i.e. the first derivative of displacement with respect to time;
Pb(t) is an external force acting on the bridge;
in step S2, the method for decomposing the bridge dynamic equation into the bridge modal equation of motion specifically includes:
a1 displacement of bridge X by mode superposition methodb(t) decomposition into a plurality of modal shapes φiIs superposed of Xb(t) is expressed as:
Figure FDA0002312369890000024
in the formula, qi(t) is the ith order modal displacement;
a2 bridge displacement X based on mode shape superpositionb(t) decomposing the bridge dynamic equation to obtain an ith-order bridge modal motion equation:
Figure FDA0002312369890000025
in the formula (I), the compound is shown in the specification,
Figure FDA0002312369890000026
the i-th order modal acceleration of the bridge is obtained;
Figure FDA0002312369890000027
the i-th order modal speed of the bridge;
qithe displacement is the ith order modal displacement of the bridge;
ζithe damping ratio of the ith order bridge mode vibration is obtained;
ωb，ithe circular frequency of the ith order bridge mode vibration;
Pi(t) is the i-th order bridge modal force, and
Figure FDA0002312369890000031
Figure FDA0002312369890000032
is the ith order bridge mode vector phiiThe transposing of (1).
4. The frequency domain method-based transverse vibration reactor for vehicle-induced bridges of claim 3The calculation method is characterized in that in the step S3, the ith order modal force p of the bridge caused by the transverse force of the bogiei(t) is:
Figure FDA0002312369890000033
in the formula, L is a bridge span;
pi [. cndot. ] is a control variable of the load acting on the bridge structure, and pi (u) ═ 1, | u ∈ [0,1 ]; 0| other };
when the transverse force of the bogie is simple harmonic load, the bridge modal force spectrum can be decomposed into a product of a train load sequence spectrum and a bridge harmonic modal spectrum;
the simple harmonic load is as follows:
Figure FDA0002312369890000034
in the formula, ωhThe circular frequency of the simple harmonic.
5. The method for calculating the transverse vibration reaction of the vehicle-induced bridge based on the frequency domain method as claimed in claim 4, wherein the unit load sequence spectrum C in the step S41(ω)；
Figure FDA0002312369890000035
Wherein exp (. cndot.) is an exponential function;
i is an imaginary number;
omega is the circular frequency;
lathe distance between the front wheel and the rear wheel of the same bogie is the same;
lcthe distance between the centers of front and rear bogies of the same vehicle;
the load sequence spectrum C of the train in the step S4v(ω) is:
Cv(ω)＝C1(ω/v)。
6. root of herbaceous plantThe method for calculating the transverse vibration response of the vehicular bridge based on the frequency domain method as claimed in claim 5, wherein the i-th harmonic mode spectrum of the bridge in the step S5
Figure FDA0002312369890000041
Comprises the following steps:
Figure FDA0002312369890000042
in the formula, v ≠ 1, ωh≠0
Figure FDA00023123698900000410
Is v ═ 1, ωh0, and
Figure FDA0002312369890000043
Figure FDA0002312369890000044
7. the method for calculating the transverse vibration reaction of the vehicle-induced bridge based on the frequency domain method as claimed in claim 6, wherein the i-th order modal force spectrum P (ω) of the bridge in the step S6 is:
Figure FDA0002312369890000045
8. the method for calculating the transverse vibration response of the vehicle-induced bridge based on the frequency domain method as claimed in claim 7, wherein the i-th order modal response spectrum Q (ω) of the bridge in the step S7 is:
Q(ω)＝H(ω)P(ω)
wherein H (omega) is the ith order modal frequency response function of the bridge, and
Figure FDA0002312369890000046
9. the method for calculating the transverse vibration response of the vehicle-induced bridge based on the frequency domain method as claimed in claim 8, wherein in the step S8, the ith order modal response q (ω) of the complete bridge is obtainedhAnd t) is:
Figure FDA0002312369890000047
wherein, pi is a circumference ratio;
A. b is a transient solution constant term;
Figure FDA0002312369890000048
Figure FDA0002312369890000049
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