Bridge vibration shape construction method based on energy balance

Abstract

This invention discloses a bridge vibration mode construction method based on energy balance, which relates to the field of bridge health monitoring technology. The method involves extracting the bridge contact point acceleration response using a central difference method combined with signal processing methods, based on the vibration control equation at the static equilibrium position of the vehicle-bridge system and utilizing the acquired vehicle acceleration response. An exponential term is introduced to adjust the energy distribution of the left and right parts of the signal. Based on the principle of energy balance, an analytical signal free of damping attenuation is obtained. Further signal processing is performed to obtain the instantaneous amplitude of the analytical signal, which is normalized to obtain the undistorted vibration mode of the bridge. Therefore, using this method, the modal shape of the bridge can be directly determined without precalculating the damping ratio or using a reference stationary vehicle, simplifying the bridge vibration mode identification process.

Description

Technical Field

[0001] The present 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 a vital role in the maintenance and assessment of transportation infrastructure. Traditional bridge mode shape identification methods require deploying a large number of sensors on bridges, which is cumbersome, labor-intensive, and cost-prohibitive. Indirect measurement methods are widely used in bridge mode shape identification due to their greater convenience and cost-effectiveness.

[0003] Currently, bridge modal identification methods based on vehicle response facilitate the collection of data on bridge dynamic characteristics. They do not require additional sensor installation or complex testing equipment, making them convenient, efficient, and traffic-friendly. However, existing methods for bridge modal identification based on vehicle response affect the results due to the influence of the bridge's damping ratio and frequency. This requires detailed dynamic testing and complex processing of the structure to obtain modal parameters. Therefore, a more efficient method is needed to simplify the process of obtaining modal parameters and improve the efficiency of bridge modal 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 objectives, 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, calculate or measure the vehicle acceleration response, and use the vehicle acceleration response to obtain the contact point acceleration response. Then, use signal processing methods to obtain the single-frequency contact point acceleration response of the bridge;

[0007] S2. Introduce the exponential term and construct the product signal. 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 then normalize it to obtain the distortion-free vibration mode of the bridge.

[0008] Preferably, in step S2, adjusting the energy distribution of the left and right parts of the product signal according to the energy balance principle 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 exponential term introduced, α represents the penalty term, represents the contact point acceleration response of the i-th sampling point in the signal;

[0012] Define energy asymmetry, the expression is:

[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 perform iterative optimization through energy asymmetry using the bisection method 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 ∈ represents 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 modal shape and eliminates 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 mode construction method based on energy balance;

[0023] Figure 2 2. Schematic diagram of a bridge modal identification process in an embodiment of a bridge vibration mode 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] Example 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:

[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 δ represents the Dirac function and g represents the acceleration due to gravity.

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

[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 example, variational modal decomposition (VMD) is used to obtain modal components. Insufficient data at the signal boundaries during VMD can lead to distortion in the edge regions, manifesting as endpoint effects. These distortions hinder accurate signal modeling, thereby reducing the accuracy of the overall decomposition. To mitigate these effects, this example symmetrically extends the original bridge contact response signal at both ends using odd-number extension, minimizing boundary distortion and ensuring a smooth transition.

[0036] S2. Based on the vibration control equation of the vehicle-bridge system at the static equilibrium position, 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] Where, It represents the simplified acceleration response of the corresponding contact point, 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 vehicle speed.

[0040] Low frequency signal sin(Ω n t) modulates the high frequency signal sin(ω Dn t). Due to β 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 them together, divide the signal into two parts, and calculate the average energy of the left and right parts of the signal. 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] Where, 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. It 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 The difference between (α)) changes from a positive value to a negative value, that is, the right end of the signal increases faster than the left end. Therefore, in this embodiment, a bisection 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 satisfied. R -α L |<∈, where ∈ defines the desired computational precision.

[0051] When the energy asymmetry η(α) reaches its minimum, 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, that is, 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 undistorted 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 from 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 solutions of the present invention rather than to limit the same. 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 solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions 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, calculate or measure the vehicle acceleration response, and use the vehicle acceleration response to obtain the contact point acceleration response. Then, use signal processing methods to obtain the single-frequency contact point acceleration response of the bridge; S2. Introduce the exponential term to construct the product signal. 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 undistorted vibration mode of the bridge. According to the energy balance principle, the energy distribution of the left and right parts of the product signal is adjusted, including: Define the energy of the left and right parts of the signal, the expression is: ； ； Where, 、 represent the average energy of the left and right parts respectively, represents the number of modal components, represents the introduced exponential term, represents the penalty term, Indicates the signal The contact point acceleration response of each sampling point; Define energy asymmetry, the expression is: ； Where, It indicates the energy asymmetry between the left and right parts; According to the damping ratio and frequency distribution of conventional small and medium span bridge structures, the penalty term is set The range of , and iteratively optimize through energy asymmetry using the bisection method as follows: like , then Updated to Otherwise, Updated to ; Iterate the optimization until the convergence criterion is met ,in Indicates the calculation accuracy.

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