Bridge vibration mode construction method based on energy balance

Abstract

The invention discloses a bridge vibration mode construction method based on energy balance, which relates to the technical field of bridge health monitoring, and comprises the following steps: based on a vibration control equation at a static balance position of an axle system, extracting a contact point acceleration response of a bridge by utilizing an obtained vehicle acceleration response and combining a central difference method with a signal processing method; an exponential term is introduced, energy distribution of the left part and the right part of a signal is adjusted, an analytic signal without a damping attenuation effect is obtained according to the energy balance principle, the instantaneous amplitude of the analytic signal is further obtained through signal processing, the instantaneous amplitude is normalized, and the distortionless vibration mode of the bridge is obtained. Therefore, by the adoption of the method, the modal shape of the bridge can be directly determined, the damping ratio does not need to be calculated in advance or a static vehicle does not need to be referred, and the bridge vibration mode recognition process is simplified.

Description

Technical Field

[0001] The invention relates to the technical field of bridge health monitoring, and in particular to a bridge vibration mode construction method based on energy balance. Background Art

[0002] Bridge health monitoring plays an important role in the maintenance and evaluation of transportation infrastructure. Traditional bridge vibration mode identification methods require the deployment of a large number of sensors on the bridge, which is cumbersome, laborious and costly. Indirect measurement methods are widely used in bridge vibration mode identification methods due to their greater convenience and economy.

[0003] At present, the bridge vibration mode identification method based on vehicle response is helpful to collect data on the dynamic characteristics of the bridge. It does not require additional sensor installation or complex test equipment, and is convenient, efficient, and does not affect traffic. However, in the existing bridge vibration mode identification method based on vehicle response, the damping ratio and frequency of the bridge will affect the results, which requires detailed dynamic testing and complex processing of its structure to obtain modal parameters. Therefore, it is necessary to provide a more efficient method that can simplify the process of obtaining modal parameters and improve the efficiency of bridge vibration mode identification. Summary of the invention

[0004] The purpose of the present invention is to provide a bridge vibration mode construction method based on energy balance, which can directly use vehicle response to determine the undistorted mode shape of the bridge and simplify the identification process.

[0005] To achieve the above object, the present invention provides a method for constructing a distortion-free vibration mode of a bridge based on energy balance, comprising the following steps:

[0006] S1. Based on the vibration control equation at the static equilibrium position of the vehicle-bridge system, the vehicle acceleration response is calculated or measured, and the contact point acceleration response is obtained using the vehicle acceleration response, and then the single-frequency contact point acceleration response of the bridge is obtained through a signal processing method;

[0007] S2. Introduce exponential terms and construct product signals. According to the energy balance principle, adjust the energy distribution of the left and right parts of the product signal to obtain an analytical signal without damping attenuation effect. Use signal processing to obtain the instantaneous amplitude of the analytical signal, and normalize it to obtain the distortion-free vibration mode of the bridge.

[0008] Preferably, in step S2, according to the energy balance principle, adjusting the energy distribution of the left and right signals of the product signal includes:

[0009] Define the energy of the left and right parts of the signal, the expression is:

[0010]

[0011] Where PL (α), P R (α) represents the average energy of the left and right parts respectively, N represents the number of modal components, e αi represents the introduced exponential term, α represents the penalty term, represents the contact point acceleration response of the i-th sampling point in the signal;

[0012] Energy asymmetry is defined as:

[0013]

[0014] Where η(α) represents the energy asymmetry between the left and right parts.

[0015] Preferably, according to the damping ratio and frequency distribution of conventional small and medium span bridge structures, the interval range of the penalty term α is set [α L , α R ] and use the bisection method to iteratively optimize through energy asymmetry as follows:

[0016] If η((α L +α R ) / 2)>0, then α L Update to (α L +α R ) / 2; otherwise, update α to (α L +α R ) / 2;

[0017] Iterate optimization until the convergence criterion |α is met R -α L |<∈, where ∈ indicates the calculation accuracy.

[0018] Therefore, the present invention adopts the above-mentioned bridge vibration mode construction method based on energy balance, which has the following technical effects:

[0019] (1) Based on the characteristics of energy balance, the analytical signal of the contact acceleration response is obtained, and the distortion-free bridge vibration mode is obtained by normalizing the instantaneous amplitude of the analytical signal, which simplifies the identification process of the bridge mode shape without the need to precalculate the damping ratio or refer to a stationary vehicle.

[0020] (2) When using signal processing, the odd number extension method is used to symmetrically extend the data at both ends of the signal to reduce the endpoint effect.

[0021] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1is a schematic diagram of a simplified model of a vehicle moving at a speed v in an embodiment of a bridge vibration shape construction method based on energy balance;

[0023] Figure 2 Schematic diagram of the bridge modal identification process in an embodiment of the bridge vibration shape construction method based on energy balance. DETAILED DESCRIPTION

[0024] The present invention can be explained in more detail by the following examples. The purpose of disclosing the present invention is to protect all changes and improvements within the scope of the present invention. The present invention is not limited to the following examples.

[0025] Embodiment 1

[0026] like Figure 1 As shown in the simplified model of a vehicle moving at a speed v, the vehicle is modeled as a mass m v The car body relies on the stiffness k v The vehicle is supported by springs; accelerometers are mounted on the vehicle body to measure its vertical motion; the bridge is modeled as a Bernoulli-Euler beam with a span length L, mass per unit length m, stiffness EI, and damping coefficient c.

[0027] Based on the simplified vehicle model, the vibration control equation of the vehicle-bridge system at the static equilibrium position is as follows:

[0028] The equation of motion for the vehicle is:

[0029]

[0030] The dynamic equation of the bridge is:

[0031]

[0032] Among them, u(x, t), y v (t) represent the vertical displacement of the bridge and the vehicle, u c (t) represents the bridge contact response, They represent the first and second derivatives of u(x, t) with respect to time t, respectively, and u″″(x, t) represents the fourth derivative with respect to the longitudinal coordinate x. Represents y v (t) is the second-order derivative with respect to time t, where the symbol δ represents the Dirac function and g represents the acceleration due to gravity.

[0033] Acceleration response at the contact point In this embodiment, the influence of vehicle frequency has been completely eliminated, which has obvious advantages in identifying the modal parameters (such as frequency, damping ratio and modal shape) of the bridge. Therefore, the contact point acceleration response is used to extract the modal information of the bridge in this embodiment.

[0034] S1. In the field test, due to the acceleration response of the contact point It is time-dependent and difficult to measure directly. In practical applications, the measured vehicle response is generally used to accurately extract the single-frequency contact acceleration response of the bridge through the central difference method combined with signal processing.

[0035] In this embodiment, variational mode decomposition (VMD) is used to obtain modal components. In the application process of VMD, insufficient signal boundary data will lead to distortion in the edge area, which is manifested as endpoint effect. These distortions will hinder the accurate modeling of the signal, thereby reducing the accuracy of the overall decomposition. To mitigate these effects, this embodiment symmetrically extends the two ends of the original signal of the bridge contact response through odd extension to minimize boundary distortion and ensure smooth transition.

[0036] S2. According to the vibration control equation at the static equilibrium position of the vehicle-bridge system, the simplified contact point acceleration response expression is obtained as follows:

[0037]

[0038] E n (t) = B Dn sin(ω Dn t)sin(Ω n t);

[0039] In the formula, represents the simplified corresponding contact point acceleration response, E n (t) represents the analytical signal without damping effect in the n-th order contact point acceleration response. This component is not affected by exponential decay. Dn represents the signal amplitude, ω bn represents the nth-order undamped frequency of the bridge, Ω n Indicates the driving frequency caused by the vehicle speed.

[0040] Low frequency signal sin(Ω n t) modulates the high frequency signal sin(ω Dn t). n =Ω n / ω bn ≈0(ω bn ≈ω Dn ), indicating that ω Dn Significantly greater than Ω n , so the envelope period contains multiple high-frequency oscillation periods. In this case, the high-frequency oscillations are densely distributed throughout the envelope curve, resulting in the overall energy distribution of the signal being dominated by the low-frequency component sin(Ω n t) envelope control, and the high frequency component sin(ω Dnt) contributes to the details of the oscillation. In addition, the low-frequency signal sin(Ω n t)(Ω n =nπvt / L) presents a symmetrical distribution in the time interval [0, L / v], resulting in the signal E n (t) The energy distribution along the time axis also shows symmetry.

[0041] It can be seen that the signal E n (t) The left and right parts have the characteristics of energy balance. Based on this, an exponential term e is constructed in this embodiment αt (α>0), and compare it with the acceleration response Multiply, divide the signal into two parts, and calculate the average energy of the left and right parts. When the average energy of the left and right parts is equal, the output signal is the analytical signal E without damping attenuation effect. n (t), as follows:

[0042] The average energy P of the left and right parts L (α) and P R (α) is defined as:

[0043]

[0044] In the formula, The single-frequency contact acceleration response of the bridge at the i-th sampling point, e αi represents the exponential term introduced by the i-th sampling point, α represents the penalty term, and N represents the total number of samples.

[0045] Energy asymmetry η(α) is defined as the ratio of the energy difference between the left and right parts to the total energy, which is used to adjust the energy of the left and right parts of the signal. The expression is as follows:

[0046]

[0047] Considering the damping ratio and frequency distribution characteristics of conventional small and medium span bridge structures, the limit of the exponential coefficient α can be selected within [α L , α R ] range, generally take α L =0,α R =10, the specific value can be adjusted according to actual conditions.

[0048] As α increases, (P L (α)-P R (α)) changes from a positive value to a negative value, that is, the increase rate of the right end of the signal is greater than the increase rate of the left end. Therefore, in this embodiment, the binary method is used to iteratively optimize α, as follows:

[0049] If η((α L +α R) / 2)>0, then α L Update to (α L +α R ) / 2; otherwise, α R Update to (α L +α R ) / 2.

[0050] The iterative process continues until the convergence criterion |α is met. R -α L |<∈, where ∈ defines the desired computational precision.

[0051] When the energy asymmetry η(α) reaches its minimum value, it indicates that the energy distribution of the left and right parts has achieved symmetry. At this time, the output signal E n (t) can be used to accurately characterize the dynamic behavior of the system, i.e., an analytical signal without damping attenuation effects.

[0052] Through E n (t) Perform Hilbert transform (HT) to extract the analytical signal E n The instantaneous amplitude of (t) is normalized to obtain the distortion-free vibration mode of the bridge. In addition, before extracting the instantaneous amplitude of the analytical signal, the odd number extension method is used to symmetrically extend the data at both ends of the signal to reduce the endpoint effect.

[0053] In other embodiments, variational mode decomposition (VMD) and Hilbert transform (HT) may also be replaced by other signal processing methods to extract corresponding modal components in the signal.

[0054] Therefore, the present invention adopts the above-mentioned bridge vibration mode construction method based on energy balance, which can directly determine the distortion-free modal shape of the bridge without pre-calculating the damping ratio or referring to a stationary vehicle, thereby simplifying the bridge vibration mode identification process.

[0055] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A bridge vibration mode construction method based on energy balance, characterized in that: The following steps are involved: S1. Based on the vibration control equation at the static equilibrium position of the vehicle-bridge system, the vehicle acceleration response is calculated or measured, and the contact point acceleration response is obtained using the vehicle acceleration response, and then the single-frequency contact point acceleration response of the bridge is obtained through a signal processing method; S2. Introduce exponential terms and construct product signals. According to the energy balance principle, adjust the energy distribution of the left and right parts of the product signal to obtain an analytical signal without damping attenuation effect. Use signal processing to obtain the instantaneous amplitude of the analytical signal, and normalize it to obtain the distortion-free vibration mode of the bridge.

2. The method for constructing a bridge vibration mode based on energy balance according to claim 1, characterized in that: In step S2, according to the energy balance principle, the energy distribution of the left and right signals of the product signal is adjusted, including: Define the energy of the left and right parts of the signal, the expression is: Where P L (α), P R (α) represents the average energy of the left and right parts respectively, N represents the number of modal components, e αi represents the introduced exponential term, α represents the penalty term, represents the contact point acceleration response of the i-th sampling point in the signal; Energy asymmetry is defined as: Where η(α) represents the energy asymmetry between the left and right parts.

3. The method for constructing a bridge vibration mode based on energy balance according to claim 1, characterized in that: According to the damping ratio and frequency distribution of conventional small and medium span bridge structures, the interval range of the penalty term α is set [α L , α R ] and use the bisection method to iteratively optimize through energy asymmetry as follows: If η((α L +α R ) / 2)>0, then α L Update to (α L +α R ) / 2; otherwise, update α to (α L +α R ) / 2; Iterate optimization until the convergence criterion |α is met R -α L |<∈, where ∈ indicates the calculation accuracy.

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