JP2022026327A - Displacement estimation method for railroad bridge - Google Patents

Abstract

To provide a displacement estimation method for railroad bridge capable of targeting a railroad bridge such that it is easy to specify its external force caused by a train, and easily finding a high-precision displacement waveform at low cost from acceleration data.SOLUTION: A displacement estimation method for railroad bridge for estimating displacement occurred on the basis of acceleration data obtained from the railroad bridge on which a train travels comprises: a step S1 of acquiring an acceleration waveform when the train passes through the railroad bridge; steps S11, S12 of acquiring data such as a span length of the railroad bridge, kind information on the train, etc.; steps S13, S14 of using a natural vibration frequency and an attenuation ratio of the railroad bridge to calculate a frequency response function and a dynamic displacement waveform; steps S15, S16 of setting an evaluation section of a frequency region to identify static maximum displacement; and steps S31-S40 of performing replacement with a static displacement waveform based upon linear vibration theory in a correction section to generate a displacement waveform in train passage.SELECTED DRAWING: Figure 1

Description

The present invention relates to a displacement estimation method for a railway bridge that estimates the displacement generated based on acceleration data obtained from the railway bridge on which a train travels.

Acceleration monitoring is being promoted for the purpose of condition monitoring in each field. As disclosed in Patent Documents 1 and 2, the bridge is also equipped with an acceleration sensor, and the change in the state is evaluated by constant monitoring and analysis of the measurement result.

Displacement measurements for constant monitoring are generally more costly than acceleration and velocity measurements. Therefore, if the displacement waveform can be obtained from the acceleration waveform that can be measured easily and inexpensively, the practical benefit is great.

On the other hand, the displacement is understood to be obtained by time-integrating the measured acceleration to the second order, but the operation of time-integrating the acceleration to calculate the displacement is in the frequency domain lower than 1 / 2π (≈0.16) Hz. The absolute value of the displacement is significantly amplified.

In order to eliminate such an integration error, in Patent Document 1, a wavelet transform is performed on the measured value of the acceleration sensor, and the wavelet coefficient obtained by the previous wavelet transform is calculated within a frequency range suitable for the displacement analysis of the object. The displacement of the object is calculated by performing the wavelet inverse transform on the result of the second-order integration.

On the other hand, in Patent Document 2, after the measured value of the acceleration sensor is second-order integrated to calculate the displacement amount, the gravity component included in the measured value and a part of the integral error caused by the measurement error are removed by the first calculation unit. The displacement amount calculated in this manner is smoothed by the second calculation unit, and a spline curve whose control point is the bottom that appears before and after the timing at which the smoothed displacement amount shows a peak is specified. Further, the value indicated by the spline curve is subtracted from the displacement amount calculated by the first calculation unit to remove the integration error remaining in the displacement amount calculated by the first calculation unit.

Japanese Unexamined Patent Publication No. 2016-148549 Japanese Unexamined Patent Publication No. 2018-204952

However, in the method disclosed in Patent Document 1, it is necessary to appropriately set the frequency range for performing the wavelet inverse transform for each object. Further, in the method disclosed in Patent Document 2, the calculation cost is high because the process for removing the integration error is repeated after performing the second-order integration of the acceleration data.

Therefore, an object of the present invention is to provide a displacement estimation method for a railway bridge, which can easily and inexpensively obtain a highly accurate displacement waveform from acceleration data, for a railway bridge such as a train where an external force can be easily specified. There is.

In order to achieve the above object, the displacement estimation method of the railway bridge of the present invention is a displacement estimation method of the railway bridge that estimates the displacement generated based on the acceleration data obtained from the railway bridge on which the train travels. The step of acquiring the acceleration waveform when the train passes through the bridge, the step of acquiring the span length of the railway bridge, the type information regarding the length of the train, and the data regarding the train speed, the natural frequency of the railway bridge, and the natural frequency of the railway bridge. Using the frequency response function, the step of obtaining the frequency response function for the displacement of the railway bridge using the attenuation ratio, the step of calculating the dynamic displacement waveform based on the acceleration waveform, and the evaluation section of the frequency region are set. The step of identifying the static maximum displacement, the step of generating a static displacement waveform based on the linear vibration theory of the correction section set as the frequency region to be corrected, and the correction section of the dynamic displacement waveform are statically described. By replacing it with a displacement waveform, it is characterized by having a step of generating a displacement waveform when the train passes.

In the displacement estimation method of the railway bridge of the present invention configured as described above, the type information regarding the span length and the train length of the railway bridge and the data regarding the train speed are used. Further, the dynamic displacement waveform based on the acceleration waveform is replaced with the static displacement waveform based on the linear theoretical solution in the frequency domain set as the correction section to generate the displacement waveform when the train passes.

By targeting a railway bridge such as a train where it is easy to identify the external force in this way, it becomes possible to easily obtain the displacement waveform from the acceleration data. In addition, for sections where noise is large, such as in the low frequency region, high-precision displacement waveforms can be obtained at low calculation cost by using static displacement waveforms based on linear theoretical solutions instead of actually measured acceleration data. Can be done.

It is a flowchart explaining the process flow of the displacement estimation method of a railway bridge of this embodiment. It is explanatory drawing which illustrated the frequency characteristic of the external force by a train load train. It is explanatory drawing of the frequency response function of one degree of freedom system. It is explanatory drawing which illustrates the method of cutting out the residual waveform. It is explanatory drawing for demonstrating the concept of an evaluation section and a correction section. It is explanatory drawing which showed the verification result of the maximum acceleration at the time of passing a train. It is explanatory drawing which illustrates the correction waveform of the displacement when the span length is relatively short. It is explanatory drawing which illustrates the correction waveform of the displacement when the span length is relatively long. It is explanatory drawing of the result of having verified the estimation accuracy of static maximum displacement y _s ^max . It is explanatory drawing of another result which verified the estimation accuracy of static maximum displacement y _s ^max . It is explanatory drawing of the result of having verified the estimation accuracy of the maximum value of displacement. 11A is a continuation of FIG. 11A. It is explanatory drawing of the estimation accuracy of the maximum value of the displacement shown for explaining the vibration characteristic of a railway bridge. It is the continuation explanatory drawing of FIG. 12A.

Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a flowchart illustrating a processing flow of a displacement estimation method for a railway bridge according to the present embodiment.
The railway bridge described in this embodiment is a simple span bridge.

The railway bridge, which is a simple span bridge, is repeatedly subjected to external force as the train travels. The displacement generated in the railway bridge when an external force is applied changes depending on the state of the structure. That is, by monitoring the displacement and displacement waveform generated on the railway bridge, it becomes possible to grasp the state such as the degree of deterioration of the railway bridge.

Here, in the railway bridge, the type of the traveling train and the time when the railway bridge is affected by the traveling of the train can be clarified. Therefore, in the railway bridge displacement estimation method of the present embodiment, such information is used. By effectively utilizing it, a highly accurate displacement waveform can be obtained easily and inexpensively.

In the displacement estimation method of the railway bridge of the present embodiment, factors such as non-linearity such as cracks of the railway bridge to be applied and the influence of the interaction between the train and the railway bridge are not modeled without modeling. In order to make the method as non-parametric as possible, we decided to utilize the linear vibration theory to restore the displacement waveform based on the acceleration waveform when the train passes.

First, the outline of the linear vibration theory of the dynamic response of a simple span bridge will be described.
In the following, the dynamic response of a railway bridge expressed by a simple span when passing a train is formulated based on the linear beam theory. Specifically, the static response of the simple beam, the dynamic response of the simple beam, the frequency characteristic of the external force due to the train load train, and the frequency characteristic of the simple beam will be described. Then, the displacement estimation method of the railway bridge of the present embodiment will be described by using the equation developed based on these basic theories.

ここで、車両の各輪軸jの初期位置をx₀(j)として、一定速度vにて鉄道橋の上を通過しているとした単純梁の応答が、１次モードのみで表されると仮定したときに、モーダルリダクションにより以下の展開が成立する。 First, the static response of the simple beam will be described.
The dynamic response when a load train acts on a simple beam can be described by the following equation.

Here, assuming that the initial position of each wheel set j of the vehicle is x ₀ (j), the response of the simple beam assuming that it is passing over the railway bridge at a constant speed v is expressed only in the primary mode. Assuming, the following development is established by modal reduction.

ここで、yはモード変位であり着目点の変位（鉄道橋のスパン中央変位）、m_eqは等価質量、C_eqは等価減衰、k_eqは等価剛性である。

Here, y is the mode displacement, the displacement of the point of interest (center displacement of the span of the railway bridge), m _eq is the equivalent mass, C _eq is the equivalent damping, and k _eq is the equivalent stiffness.

Then, by expanding the expression, it can be described as follows.

λ (t) is standardized with the maximum value of λ (t) as the effective axle load factor λ _p , and P _eq = λ _p P _o is defined. Furthermore, κ is defined as a dimensionless quantity of the ratio of the external force and its own weight as the following equation.

上記λ(t)（λの上バー省略）の最大値が1であることから、y_s(t)の時刻歴上の最大値y_s ^maxは、以下のように記述できる。

Of the equations shown in [Equation 3] rewritten focusing on the external force term, the static response y _s (t), which is the displacement when the mass term and the damping term are ignored, can be described as follows. ..

Since the maximum value of λ (t) (the upper bar of λ is omitted) is 1, the maximum value y _s ^max in the time history of y _s (t) can be described as follows.

From the relationship between the effective axle load factor λ _p and the span length L _b , the effective axle load multiplier λ _p increases as the span length L _b of the simple beam increases. In addition, due to the shaft arrangement, the number of axes that effectively affect displacement decreases relative to the span length L _b , so the increase in the effective axle load factor λ _p at the span length L _b , which is the antiresonance point, increases. It becomes gentle.

ここで、zを等価モード座標上の応答値yと静的応答y_sの比として定義すると、以下の式に書き換えられる。

Next, the dynamic response of the simple beam will be described.
Using the relation of the above equation [Equation 3] and the relation of ω ² _eq = k _eq / m _eq and C _eq = 2ξ _eq ω _eq , the following equation can be described.

Here, if z is defined as the ratio of the response value y on the equivalent mode coordinates to the static response y _s , it can be rewritten as the following equation.

Further, the external force term and the velocity parameter α are described as follows.

Using the relationship between the dimensionless time t of the above equation [Equation 8] and the equation of [Equation 9], z, which is the ratio of the dynamic response y and the static response y _s , has the following relationship. Obtained by solving the equation.

ここで、y_dは動的応答の最大値である。 That is, z can be obtained by solving the above equation of motion with α, the damping ratio ξ _eq , and the wheel axis arrangement λ (the upper bar of λ is omitted) as parameters. In particular, the maximum value y _d of the dynamic response is an important evaluation index. The maximum value of z (τ) (the upper bar of τ is omitted) is called the impact coefficient and is defined as i _α by the following equation.

Where y _d is the maximum value of the dynamic response.

Next, the frequency characteristics of the external force due to the train load train will be described.
Focusing on P ₀ λ (t) of the external force term in the above equation [Equation 3], the only component that fluctuates with time is λ (t). Hereafter, the characteristics of the frequency domain of the mode external force λ (t) in the time domain will be considered. If the Fourier transform F (λ) of λ (t) is described as F _λ (ω), the following equation is obtained.

In addition, the above equation can be described as the sum of each wheel weight j as follows.

Then, as the expansion is continued, the following equation is obtained.

Here, F _{λ 0} (ω) is a vibration spectrum with a single axle, and the repetitive effect of multiple wheels is expressed by Σ _j e ^{-i ω t} ₀ ^(j) . For this Σ _j e ^-iωt ₀ ^(j) , the wheelset arrangement of each vehicle k can be described by [x ₀ + kL _v x ₀ + kL _v + ax ₀ + kL _v + bx ₀ + kL _v + a + b]. Continue to deploy using.

Summarizing the above equations of [Equation 13], [Equation 14] and [Equation 15], F _λ (ω) is expanded as follows.

Then, dimensionlessization is performed to organize the parameters. Several methods can be considered as the method of dimensionlessization. For example, if ω _v (= v π / L _b ) is used as a reference and described as t _b = π / ω _v , Ω = ω / ω _v , the following relationship is established for each component of the above equation [Equation 16]. can get.

On the other hand, if ω _vLv (= vπ / L _v ) is used as a reference and described as t _b = π / ω _v , Ω = ω / ω _vLv , the following relationship is obtained for each component of the above equation [Equation 16]. Is obtained.

ここで、F_wv(ω)は先述したように単一車軸による加振スペクトルであり、梁長さであるスパン長L_bを速度vで通過する加振周波数ω_vに依存する成分である。また、F_t0(ω)は、先頭輪軸が梁に進入する時刻t₀による位相差を表す成分であり、F_Lv(ω)，F_a(ω)，F_b(ω)はそれぞれ複数輪軸の繰り返し効果を表す成分である。 In the displacement estimation method of the railway bridge of the present embodiment, the following equation can be obtained by summarizing the equation of [Equation 16] by using the relation of the above [Equation 18].

Here, F _wv (ω) is a vibration spectrum with a single axle as described above, and is a component depending on the vibration frequency ω _v that passes through the span length L _b , which is the beam length, at a velocity v. F _t 0 (ω) is a component representing the phase difference depending on the time t ₀ when the leading wheel set enters the beam, and _FLv (ω), _{Fa (ω), and F b (} ω) are each of multiple wheel sets. It is a component that exhibits a repetitive effect.

FIG. 2 shows an example of the frequency characteristics of the external force due to the train load train. The vertical axis of FIG. 2 is the absolute value of each frequency function of each equation shown in the above [Equation 18], and the horizontal axis is Ω (= ω / ω _vLv ). Since | F _t0j (ω) | has an absolute value of 1, the description in FIG. 2 is omitted. Here, each parameter was set as L _b = 50, V = 285, t ₀ = 0.5, n _v = 16, L _v = 25, a = 2.5, b = 17.5.

| F _a (ω) | is a component due to the axle spacing a, and is a function that periodically becomes 0 when Ω = 1 + 2kL _v / a (k is a non-negative positive number) is satisfied.
| F _b (ω) | is a component due to the bogie center spacing b, has the same shape as | F _a (ω) |, and satisfies Ω = 1 + 2kL _v / b (k is a non-negative positive number). It is a function that periodically becomes 0 in some cases.
| F _Lv (ω) | is a component depending on the vehicle length L _v and the number of vehicles n _v , and the characteristic shape can be confirmed. The numerator of F _Lv (ω) becomes 0 when Ω = (1 + 2k) / n _v (k is a non-negative positive number) is satisfied, but the denominator becomes 0 periodically Ω = 1 + When it becomes 2k (k is a non-negative positive number), it does not become 0, and it shows a locally large value compared to before and after.

| F _wv (ω) | is a component that depends on the span length L _b , and tends to decrease as a whole as Ω increases. It also has the characteristic that it periodically becomes 0 when Ω = 1 + 2kL _v / L _b (k is a non-negative positive number) is satisfied.
| F _λ (ω) | (the upper bar of λ is omitted) is the product of each of the above components, but it seems that the influence of the component of | F _Lv (ω) | remains strong in terms of shape. .. When Ω = 2k (k = 1,2,3,4,6,7,8,9,11,12,13,14,16, ...), it shows a maximum value that increases periodically. ing.

Next, the frequency characteristics of the simple beam will be described.
Since the displacement response of the simple beam when the train passes can be generally reproduced by considering only the primary mode, the frequency characteristics of the displacement response of the one-degree-of-freedom system will be considered. In general, the transfer function (frequency response function) S _d for the displacement of a one-degree-of-freedom system with an intrinsic angular frequency ω _s and a mode damping ratio ξ _s is calculated by the following equation using the relationship Ω _s = ω / ω _s . expressed.

The transfer function (frequency response function) S _a for acceleration is expressed by the following equation.

FIG. 3 shows the transfer function S _d of the one-degree-of-freedom system shown by the above equation [Equation 20]. From this figure, | S _d | changes depending on the damping ratio ξ _s in the range of about 0.8 <Ω _s <1.2, and the influence of the damping ratio ξ _s is small in other regions.

In the region of Ω _s <0.5, | S _d | <1.3 is satisfied, and when Ω _s approaches 0, | S _d | gradually approaches 1. This is because the frequency components less than half of each natural angular frequency ω _s of the one-degree-of-freedom system are less than about 30% even if amplified, and if the frequency characteristics of the external force are known, the frequency response of the displacement in this region is the external force. It means that it is 1 to 1.3 times the characteristic.

Based on the concept of the linear vibration theory of the dynamic response of the simple span bridge explained above, the method of restoring the displacement waveform by the acceleration integration of the simple span bridge applied in the displacement estimation method of the railway bridge of the present embodiment will be described. ..

If the displacement waveform y (t) of the railway bridge can be calculated by linear theory, it can be obtained by solving the above equation [Equation 10], but the static displacement waveform y _s , the natural frequency f _eq , and the damping ratio ξ It is necessary to identify each parameter such as _eq . However, since the method based on parameter identification is limited in the case where the response of the actual bridge can assume a linear system, the influence of modeling error becomes large, the calculation load is large, and it can be obtained by constant monitoring. There are problems such as being unsuitable for processing large amounts of data. In particular, the natural frequency f _eq and damping ratio ξ _eq expressed by linear theory are parameters related to the vibration characteristics of bridges, but in the case of concrete bridges, the material is non-linear and the dynamics of trains and bridges. Strictly speaking, it changes from moment to moment due to the influence of interaction, so this error may not be negligible when assuming a stationary one-degree-of-freedom system.

FIG. 1 shows a schematic processing flow of the displacement estimation method for a railway bridge according to the present embodiment. In the displacement estimation method of the railway bridge of the present embodiment, processing is performed so that the displacement waveform can be obtained without assuming a model as much as possible.

First, the following is assumed as a premise for applying the displacement estimation method of the railway bridge of the present embodiment.
(Assumption 1) Only the acceleration waveform at a single point in the center of the span just below the track surface of the railway bridge is used as the measured data. In short, only the acceleration waveform obtained from the acceleration sensor installed in the center of the span of the railway bridge is used as the actually measured acceleration data.

(Assumption 2) The span length L _b of the bridge and the type of train passed are known information. The train type is type information related to the length of the train such as vehicle length L _v , axle spacing a, and bogie center spacing b.
(Assumption 3) The wheel load P ₀ of the train is unknown, but it is the same value for all vehicles.
(Assumption 4) The train speed v (m / s) (V (km / h)) is constant and does not change while passing through the railway bridge.

In the displacement estimation method of the railway bridge of the present embodiment, as shown in steps S31 to S34 of FIG. 1, only a small part of the low frequency of the acceleration waveform is corrected, but some corrections are made. The parameters need to be identified.

On the other hand, regarding the external force characteristics, when targeting railway vehicles, the type is limited to a certain range and the shaft arrangement is clearly known in advance, so a theoretical model is assumed using this. do. That is, the external force can be expressed by the right side of the above equation [Equation 1].

The parameters identified by the acceleration waveform are the number of vehicles n _v , the train passing speed (train speed V = 3.6v, which may change during passing but is treated as a constant average value), and the first axle is on the railway bridge. At the time of entry t ₀ , static maximum displacement (static deflection) y _s ^max . As can be seen from the equation [Equation 5] above, the static maximum displacement y _s ^max depends not only on the wheel load P ₀ but also on the unknown equivalent rigidity k _eq of the railway bridge, but here it is related to the external force characteristics. Treat as a parameter.

Further, since it is empirically known that the number of vehicles n _v , the train speed V, and the time t ₀ when the train enters can be obtained from many methods, detailed explanations are omitted and treated as known. Regarding the characteristics of the railway bridge, if the number of vehicles n _v , the train speed V, and the time when the train enters t ₀ are obtained in advance, the time region of the residual waveform will be clarified, so this is used. The natural frequency f _eq and the damping ratio ξ _eq are identified.

Next, a method for identifying the natural frequency f _eq and the damping ratio ξ _eq will be described.
In the displacement estimation method of the railway bridge of the present embodiment, the static maximum displacement y _s ^max is identified, and the frequency response function S _d of the railway bridge is used in this identification. Focusing on 0.5 f _eq or less in the frequency domain of the frequency response function S _d , the absolute value of the frequency response function S _d does not change significantly, so the natural frequency f _eq and attenuation ratio ξ, which are the parameters of the frequency response function S _d . It may not be necessary to determine the _eq exactly. However, if the vibration characteristics (f _eq , ξ _eq ) of the railway bridge are completely unknown, it is necessary to identify them by some method from the acceleration waveform. Here, the natural frequency f _eq and the damping ratio ξ _eq are identified by using the residual waveform after the train passes through the railway bridge.

FIG. 4 shows a diagram for explaining a method of cutting out the residual waveform. As a method of cutting out the residual waveform after passing the train from the measured waveform, there is a method of manually determining the time zone at the time of passing the train while checking the shape of the waveform one by one. It is difficult to process a large amount of monitoring data with this method.

On the other hand, since the identification of the natural frequency f _eq and the damping ratio ξ _eq is performed after identifying the number of vehicles n _v , the train speed v, and the time t ₀ when the train enters, these information and the vehicle length L _v Using the information, it is also possible to discriminate the residual waveform after t ₀ = n _v L _v / v.

Then, the residual waveform of the cut out acceleration is Fourier transformed into the frequency domain. Furthermore, focusing on the frequency domain of 120 L _b ^-0.8 or less based on the span length L _b of the railway bridge, the frequency showing the maximum value among the maximum points is defined as the natural frequency f _eq . In addition, four maximum values in the time domain of the residual waveform are extracted in descending order, and the logarithmic decrement rate is calculated from the four points in three ways to obtain the average value, and the attenuation ratio is ξ _eq .

ここで、フーリエ変換子Fを用いると、Y=F(y)、F_λ＝F(λ)であるため、上式は次のように記述できる。

Next, a method for identifying the static maximum displacement y _s ^max will be described. Here again, considering the forced vibration of a one-degree-of-freedom system, the Fourier transform of the above [Equation 9] gives the following equation.

Here, when the Fourier transformer F is used, Y = F (y) and F _λ = F (λ), so the above equation can be described as follows.

ここで、上式の右辺は、変位の周波数成分Y(ω)と、列車通過による外力の周波数成分F(ω)及び鉄道橋の周波数応答関数S_d(ω)であり、周波数領域で定義された関数である。ここで、上述したように周波数応答関数S_dの計算仮定である橋梁系が線形1自由度でノイズが全く無い状態であれば、上記[数２４]は全周波数領域で成立する。 Further, using the relation of the above equation [Equation 5], the static maximum displacement y _s ^max is defined by the following equation.

Here, the right side of the above equation is the frequency component Y (ω) of the displacement, the frequency component F (ω) of the external force due to the passage of the train, and the frequency response function S _d (ω) of the railway bridge, which are defined in the frequency domain. It is a function. Here, as described above, if the bridge system, which is the calculation assumption of the frequency response function S _d , has one linear degree of freedom and no noise, the above [Equation 24] holds in the entire frequency domain.

However, the low frequency region of the measured acceleration is often inaccurate due to the response characteristics of the accelerometer, and when simply integrated, the low frequency region of the displacement is susceptible to noise amplification. .. Furthermore, considering that the response of an actual bridge has a higher-order mode of second order or higher and non-linearity, the frequency domain in which the above [Equation 24] holds is limited under the conditions of an actual railway bridge. Will be.

In the identification of the static maximum displacement y _s ^max , the displacement estimation of the railway bridge of the present embodiment is performed by using a specific frequency region that is not easily affected by noise, high-order mode of the bridge, and uncertainty of vibration characteristics. There is a characteristic of the method. In particular, since the dynamic interaction between the vehicle and the structure and the non-linear response of the bridge affect the vibration characteristics of the railway bridge in the vicinity of the natural frequency f _eq , do not use the frequency component around the natural frequency f _eq . Therefore, the identification accuracy of the static maximum displacement y _s ^max can be improved.

ここで、ω_e1，ω_e2は、|Y/F_λS_d|の平均値を算出する評価区間となる周波数の下限及び上限であり、以下のように定義する。 Specifically, the static maximum displacement y _s ^max is identified by the following equation.

Here, ω _e1 and ω _e2 are the lower and upper limits of the frequency that is the evaluation interval for calculating the average value of | Y / F _λ S _d |, and are defined as follows.

<Applicable range f _eq ≧ 2v / L _v >
ω _e1 = 0.5 × πv / b, ω _e2 = 2 × πv / b
Here, L _v is the vehicle length and b is the bogie center spacing.
<Applicable range f _eq <2v / L _v >
ω _e1 = 0.1 × 2πf _eq , ω _e2 = 0.5 × 2πf _eq

Subsequently, the correction method will be described.
As shown in the following equation, the inverse Fourier transform F ^-1 of the displacement D defined in the frequency domain gives the displacement waveform d in the time domain.

Here, the measured acceleration waveform data is defined as a ^measure for the time domain and A ^measure for the frequency domain. Then, it is also possible to obtain the displacement waveform D ^measure when directly integrated in the frequency domain by the following equation.

With the direct integration method, the low-frequency noise contained in the a ^measure is amplified by the integration operation, so it is virtually impossible to restore data that can withstand the evaluation. The error when the acceleration is directly integrated is mainly due to the fact that the noise mixed in the low frequency region is amplified in the process of time integration.

Therefore, in the displacement estimation method of the railway bridge of the present embodiment, the measured acceleration is decomposed into the quasi-static component and the dynamic component of the railway bridge, and only the quasi-static component strongly affected by noise is theoretically solved. to correct.

As described above, in the linear theory, the static component and the dynamic component (impact coefficient) can be separated in the bridge response, and the response waveform of each can be calculated, but the quasi-static component referred to here. Does not mean the displacement response obtained by considering only the restoring force ignoring the inertial force and the damping force in the forced vibration of the bridge.

Then, it is impossible to divide the measured waveform of acceleration into a low frequency region having a large contribution to a static response and a high frequency region having a large contribution to a dynamic response by frequency analysis or the like. Therefore, a certain boundary is simply set in the frequency domain, a waveform having a low frequency component lower than that is referred to as a quasi-static component, and a waveform having a higher frequency component is referred to as a dynamic component.

ここで、ω_mは補正境界角周波数であり、以下に示す値を用いた。
＜適用範囲f_eq≧2v/L_v＞
ω_m=πv/b
＜適用範囲f_eq＜2v/L_v＞
ω_m=0.5f_eq In the displacement estimation method of the railway bridge of the present embodiment, D _modified in which only the low frequency region is corrected by D _theory assuming linear vibration is defined by the following equation.

Here, ω _m is the corrected boundary angular frequency, and the values shown below are used.
<Applicable range f _eq ≧ 2v / L _v >
ω _m = πv / b
<Applicable range f _eq <2v / L _v >
ω _m = 0.5f _eq

Then, the restored waveform d _modified of the displacement can be obtained by performing the inverse Fourier transform of the D _modified defined by the above [Equation 28] as shown by the above equation [Equation 26].
FIG. 5 shows a conceptual diagram of an evaluation section and a correction section.

Next, the result of verifying the validity of the displacement estimation method (this method) of the railway bridge of the present embodiment will be described.
First, the generation of the actually measured acceleration and displacement waveform will be described.

ここで、F^-1は逆フーリエ変換子、A(ω)は周波数領域上の加速度、D(ω)は周波数領域上の変位である。 The acceleration a (t) and displacement d (t) at the center of the span of the railway bridge are theoretically calculated by the following equations as the dynamic response when the train passes.

Here, F ^-1 is the inverse Fourier transformor, A (ω) is the acceleration in the frequency domain, and D (ω) is the displacement in the frequency domain.

The acceleration A (ω) and the displacement D (ω) on the frequency domain can be calculated by the following equations using the equations [Equation 20] and [Equation 21] described above.

By the processing on the frequency domain shown in the above [Equation 29] and [Equation 30], when the train passes at a much higher speed than the numerical integration of the equation of motion on the time axis as described above in the equation of [Equation 3]. It is possible to calculate the dynamic response of.

When considering restoring the displacement response waveform from the measured acceleration, self-noise is mixed in the measured acceleration waveform in addition to the pure response of the simple beam, and it is added to the absolute value on the time axis. There is. This noise includes those caused by disturbances such as wind and ground vibration other than the train load to be measured, and those caused by the characteristics of the power supply and the measuring equipment. Originally, self-noise is an internal factor generated from the electric circuit of the sensor, but it is difficult to separate such an internal factor from an external factor such as constant tremor and temperature characteristics. be. Therefore, in this verification, noise as an accelerometer system (accelerometer) including external factors included in the stationary state and A / D conversion performance is considered as self-noise. That is, the vibration actually measured without the train passing with the accelerometer installed becomes self-noise.

ここで、εは平均が0、分散がσ_ε ²で定義される正規分布に従う乱数によりモデル化する。 Therefore, noise ε is added to the theoretical solution calculated by the above equation [Equation 29].

Here, ε is modeled by a random number that follows a normal distribution with a mean of 0 and a variance of σ _ε ² .

The operation of integrating the acceleration over time to calculate the displacement is synonymous with multiplying each frequency component of the acceleration by 1 / (iω) ² , so in the frequency domain lower than 1 / 2π (≈0.16) Hz, The absolute value of the displacement will be significantly amplified. There are few general accelerometers that have this low frequency range as the frequency range that can be measured with high accuracy. Further, even if it is possible to measure by the performance of the sensor, noise is also amplified by the above-mentioned low frequency amplification when the integration operation is applied. That is, for frequency components of about 0.2 Hz or less, the reliability of the obtained displacement data becomes extremely low due to the influence of error amplification due to the integration operation. When the velocity is time-integrated to calculate the displacement, the noise amplification becomes smaller because the integration operation is only required once.

In the verification case, the parameter related to the train was set to V = 10,60, ..., 410,460km / h, assuming that the train speed was only V. The number of vehicles n _v was 12 cars, the axle load P ₀ was 120 kN, and the train approach time t ₀ was a fixed value of 2 seconds.

ここで、鉄道橋の単位長さあたりの質量m(上バーは省略)は、単線のコンクリート橋梁を想定し、固有振動数f_eqは比較的に柔構造の橋梁を想定した値である。また、減衰比ξ_eqは2％の固定値とした。 The parameters related to the railway bridge were L _b = 10,20, ..., 90,100 m, with only the span length L _b . The following relationship is assumed between the mass m per unit length of the railway bridge (upper bar omitted) and the natural frequency f _eq based on the span length L _b .

Here, the mass m per unit length of the railway bridge (the upper bar is omitted) is assumed to be a single-track concrete bridge, and the natural frequency f _eq is a value assuming a relatively flexible bridge. The attenuation ratio ξ _eq was set to a fixed value of 2%.

On the other hand, the standard deviation σε of noise was 0.001,0.005,0.010,0.015,0.020. The standard deviation σε of noise largely depends on the type of sensor and the installation conditions, but it is often about 0.001 to 0.005.

FIG. 6 shows the maximum acceleration a ^max when the train passes. For the maximum acceleration a ^max , the maximum absolute value was extracted from the time history of the acceleration a (t) shown by the above equation [Equation 29]. From FIG. 6, the maximum acceleration a ^max decreases as the train speed V becomes slower and the span length L _b becomes longer, and changes between 0.01 and 8 depending on the span length L _b and the train speed V. You can see that it does. Here, the standard deviation σε of the error is from 0.001 m / s ² to -0.020 m / s ² . When the span length L _b = 50m and the train speed V = 260km / h, the maximum acceleration a ^max is 0.244, and when σ _ε = 0.005, the 95% reliability region is about 0.01m / s ² , so SN. The ratio is about 25.

Next, the verification results of this method will be described.
7 and 8 show examples of displacement correction waveforms as verification results, with the horizontal axis representing the elapsed time (Time (s)) and the vertical axis representing the displacement (Disp (mm)). FIG. 7 is a verification result when the span length L _b is relatively short, and FIG. 8 is a verification result when the span length L _b is relatively long.

These verification results are for a noise standard deviation σε of 0.001 m / s ² and a train speed V of 260 km / h. The legend "Exact" in the figure shows the correct waveform of the displacement, and the legend "Estimated" shows the correction waveform by this method. Further, the legend "Modified part" shows the waveform in the low frequency region before the correction by this method. These results are under the condition that the number of vehicles n _v and the train speed V are known, that is, accurately identified.

From these figures, the correct waveform of the displacement (Exact) and the corrected waveform (Estimated) of this method are shown overlapping with almost the same waveform, and the validity of this method can be confirmed. On the other hand, the waveform before correction by this method (Modified part) is significantly different from the correct waveform in the range of 2s to 6s in each case, and it can be said that the correction of this method improved the estimation accuracy of the displacement. As described above, since this method can reproduce not only the maximum value of the waveform but also the shape, it can be seen that it is effective for a problem such as fatigue in which the amplitude and the number of repetitions are used as evaluation indexes.

Next, the estimation accuracy of the vibration characteristic f _eq of the railway bridge will be described.
This method uses an algorithm based on frequency analysis of the residual waveform, and it seems that the magnitude of noise has little effect on the estimation results. On the other hand, it was confirmed that the estimation accuracy tends to decrease as the span length L _b becomes longer and the train speed V becomes higher in the high-speed region. In addition, when the train speed V is low at 10 km / h and 60 km / h, or when the train speed is close to the antiresonance speed, the dynamic component contained in the waveform is small and the absolute value of the residual waveform is small, so the estimation error. Was found to tend to increase.

Next, the estimation accuracy of the static maximum displacement y _s ^max will be described.
FIG. 9 shows the estimation accuracy of the static maximum displacement y _s ^max when the standard deviation σε of noise is 0.001 m / s ² , and FIG. 10 shows the static maximum displacement y s max when the standard deviation σε of noise is 0.005 m / s ² . The estimation accuracy of the maximum target displacement y _s ^max is shown.

These results identify the number of vehicles n _v and the train speed V as known information in the flowchart shown in FIG. 1 (step S11), and the vibration characteristics of the railway bridge (natural frequency f _eq , damping ratio ξ _eq ). Is the result under known, i.e., accurately identified conditions (step S22).

The estimation error shown in these charts was calculated by the estimated value / correct answer value -1. From these results, it can be confirmed that the estimation error tends to increase as the train speed V is low, the span length L _b is long, and the noise is large. This is because when the speed is low, the span is long, and the noise is large, the absolute value of acceleration becomes small and the SN ratio becomes small. In addition, the estimation error shows a positive value in most areas, and it can be seen that the evaluation is on the safe side.

It is considered that the reason why the estimation error shows a positive value is that the absolute value of noise has an influence in the evaluation section of the frequency domain as described above. This suggests that the accuracy will be improved if the frequency distribution of the noise obtained in advance is taken into consideration in the correction process. Practically, we should focus on the region where the standard deviation σε of noise is 0.005 m / s ² or less and the train speed V is 110 km / h or more, and the static maximum displacement y _s ^max with an estimation error accuracy of about 5% or less. It turns out that it is possible to estimate.

Next, the estimation accuracy of the maximum displacement will be described.
11A and 11B are explanatory views of the result of verifying the estimation accuracy of the maximum value of the displacement. This verification result exemplifies the maximum displacement estimation accuracy when the standard deviation σε of noise is 0.005 m / s ² .

These results also identify the number of vehicles n _v and the train speed V as known information in the flowchart shown in FIG. 1 (step S11), and the vibration characteristics of the railway bridge (natural frequency f _eq , damping ratio ξ _eq ). Is the result under known, i.e., accurately identified conditions (step S22).

The legend "exact" in the upper figure of FIG. 11A shows the correct answer value of the maximum deflection and the uplift, and the legends "Estimates deflection" and "Estimated uplift" show the estimated value by this method. Further, the chart of FIG. 11B shows the accuracy of the maximum deflection, and is calculated by the estimated value / correct answer value -1.

From these figures, the maximum displacement on the positive side (uplift) and the negative side (maximum deflection) can be accurately evaluated in most cases, and the displacement waveform can be accurately restored as described above. It was also confirmed that the estimation error tends to increase as the train speed V is low, the span length L _b is long, and the standard deviation σε of noise increases. That is, when the standard deviation σε of noise is 0.001 m / s ² , the accuracy of estimation error is approximately 2% or less in the region where the train speed V is 110 km / h or more, and the standard deviation σε of noise is 0.005 m / s ² . In this case, the estimation error accuracy of about 2% or less was obtained in the region where the train speed V was 160 km / h or more.

Practically, it is sufficient to focus on the region where the standard deviation σε of noise is 0.005 m / s ² or less and the train speed V is 110 km / h or more, and the maximum displacement can be estimated with an estimation error accuracy of about 5% or less. It turns out that it is possible.

Here, FIGS. 12A and 12B show the estimation accuracy of the maximum value of the displacement in order to explain the vibration characteristics of the railway bridge. This figure shows the maximum displacement estimation accuracy when the standard deviation σε of noise is 0.001 m / s ² .

This result is the result when the vibration characteristics of the railway bridge (natural frequency f _eq , damping ratio ξ _eq ) are given as approximate values in the calculation of the frequency response function S _d shown in the above equation [Equation 20]. .. That is, this figure shows the effect of the accuracy of the frequency response function S _d on the waveform restoration result.

Specifically, the frequency response function S is set to 70L _b ^-0.8 when the natural frequency f _eq is actually given at 50L _b ^-0.8 , and 0.01 when the attenuation ratio ξ _eq is actually given at 0.02. _d was calculated. Looking at this figure, the error is increasing as a whole, and the tendency is remarkable in the long span and high speed region. This is because | S _d | was overcalculated because the attenuation ratio ξ _eq was set smaller than it actually is. However, the estimation error is generally within 10% in the region where the span length L _b is 30 m or less, and is approximately 20% or less even when the span length L _b is 100 m. Even when given, it can be seen that it has a certain estimation accuracy.

The identification algorithm of the vibration characteristics (natural frequency f _eq , damping ratio ξ _eq ) used in this method is very simple, and the estimation accuracy is greatly reduced at a specific train speed V, but in general, the bridge response. In the case of constant monitoring, it is unlikely that these values will change suddenly in multiple measurement data over a wide train speed V range, so use the information obtained in advance. It is desirable to calculate the frequency response function S _d , taking into account the plausibility of the identified values.

It is meaningful to identify the natural frequency f _eq and the damping ratio ξ _eq to confirm the change in the performance of the vibration characteristics of the railway bridge, even under adverse conditions such as ultra-low speed and antiresonance speed. Although it is possible to precisely identify the vibration characteristics by using a more advanced method, the calculation load increases significantly, but the ratio that contributes to the improvement of the estimation accuracy of the displacement waveform, which is the final evaluation purpose, is small.

Next, the flow and operation of the processing of the displacement estimation method of the railway bridge of the present embodiment will be described.
In the displacement estimation method of the railway bridge of the present embodiment, as shown in the flowchart of FIG. 1, the acceleration measured at the time of passing the train by the acceleration sensor installed in the center of the span of the railway bridge which is a simple span bridge in step S1. Acquire the measured value (acceleration data) of the waveform.

Subsequently, in steps S11 and S12, the number of vehicles n _v and the train speed V, which is the passing speed of the train, are identified, and the time t ₀ when the train enters the railway bridge is identified. As described above, these are used. A known value can be used for the value of.

Further, when it is necessary to identify the bridge characteristics, in step S21, the residual waveform is extracted from the acceleration waveforms (see FIG. 4), and the vibration characteristics (natural frequency f _eq , damping ratio ξ _eq ) are identified. (Step S22).

If the vibration characteristics of the railway bridge (natural frequency f _eq , damping ratio ξ _eq ) are obtained, the frequency response function S _d can be calculated (step S13). Further, in step S14, a dynamic displacement waveform (dynamic waveform) D ^measure is calculated using the measured acceleration waveform (see [Equation 27]).

Further, in steps S15 and S16, the evaluation interval of the frequency domain is set (see FIG. 5), and the static maximum displacement y _s ^max is identified (see [Equation 25]). This is the method for identifying the static maximum displacement y _s ^max .

Then, in steps S31 to S34, the dynamic displacement waveform D ^measure obtained from the acceleration waveform is corrected. First, in step S31, a frequency region having a frequency lower than the correction boundary angle frequency ω _m , which is the angular frequency used as the boundary, is set in the correction section (see FIG. 5).

Subsequently, in step S32, a static displacement waveform (quasi-static waveform D _theory ) based on the linear vibration theory is generated for the correction section. On the other hand, for areas other than the correction section, the dynamic displacement waveform D ^measure is stored as it is as the dynamic waveform D _mesure (step S33).

Then, in step S34, a D _modified is generated by synthesizing the quasi-static waveform D _theory of the correction section and the dynamic waveform D _mesure in the frequency region having the correction boundary angle frequency ω _m or more (see [Equation 28]). Further, by performing an inverse Fourier transform (see [Equation 26]) on this D _modified , the corrected displacement waveform d _modified when the train passes is restored (step S40).

In the railway bridge displacement estimation method of the present embodiment configured as described above, type information regarding the span length L _b of the railway bridge and the length of the train (vehicle length L _v , number of vehicles n _v , axle spacing a, trolley). Effective use of known data on center spacing b) and train speed V. Then, the dynamic displacement waveform (dynamic waveform D _mesure ) based on the acceleration waveform is replaced with the static displacement waveform (quasi-static waveform D _theory ) based on the linear theoretical solution in the frequency domain set as the correction interval. Generates a displacement waveform when the train passes.

By targeting a railway bridge such as a train where it is easy to identify the external force in this way, it becomes possible to easily obtain the displacement waveform from the acceleration data. In addition, for sections where noise is large, such as in the low frequency region, high-precision displacement waveforms can be obtained at low calculation cost by using static displacement waveforms based on linear theoretical solutions instead of actually measured acceleration data. Can be done.

Although the embodiment of the present invention has been described in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and design changes to the extent that the gist of the present invention is not deviated are made in the present invention. Included in the invention.

Claims (6)

It is a displacement estimation method for railway bridges that estimates the displacement generated based on the acceleration data obtained from the railway bridge on which the train runs.
The step to acquire the acceleration waveform when the train passes through the railway bridge, and
Steps to acquire data on the span length of the railway bridge, type information on the length of the train, and train speed, and
The step of obtaining the frequency response function to the displacement of the railway bridge using the natural frequency and the damping ratio of the railway bridge, and
The step of calculating the dynamic displacement waveform based on the acceleration waveform and
The step of setting the evaluation interval of the frequency domain and identifying the static maximum displacement using the frequency response function,
A step to generate a static displacement waveform based on the linear vibration theory of the correction section set as the frequency domain to be corrected, and
A method for estimating displacement of a railway bridge, comprising: a step of generating a displacement waveform when a train passes by replacing the correction section of the dynamic displacement waveform with the static displacement waveform. The displacement estimation method for a railway bridge according to claim 1, wherein the type information regarding the length of the train is a vehicle length, an axle spacing, a trolley center spacing, and a number of vehicles. The displacement estimation method for a railway bridge according to claim 1 or 2, wherein the frequency region to be corrected is a frequency region having a frequency lower than the boundary set based on the angular frequency. The displacement estimation method for a railway bridge according to claim 3, wherein the angular frequency set as a boundary differs depending on the relationship between the ratio of the train speed and the vehicle length and the natural frequency. The displacement of a railway bridge according to any one of claims 1 to 4, wherein the natural frequency and the damping ratio are identified by extracting a residual waveform from the acceleration waveform after the train has passed. Estimating method. The displacement estimation of the railway bridge according to any one of claims 1 to 5, wherein the acceleration waveform is a measured value of an acceleration sensor installed at the center of the span of the railway bridge, which is a simple span bridge. Method.

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