Bridge damping ratio calculation method based on logarithmic differentiation

Abstract

The invention discloses a bridge damping ratio calculation method based on logarithmic differentiation, and belongs to the technical field of bridge health monitoring, and the method comprises the steps: constructing a dynamic equation of a vehicle-bridge system at a static balance position, and obtaining the acceleration response of a contact point of a vehicle through the inverse calculation of the acceleration response of the vehicle; a first-order acceleration response of the bridge is obtained through variational mode decomposition; extracting a ridge amplitude and a dominant frequency of the modal response by using a wavelet; and then logarithm and differential are carried out on the ridge amplitude, and finally an average value is taken to obtain a damping ratio. By adopting the bridge damping ratio calculation method based on logarithmic differentiation, the damping ratio of the bridge can be identified with high precision by combining variational mode decomposition, wavelet ridge amplitude extraction and logarithmic differentiation technologies, the method has relatively high noise interference resistance, can also avoid the requirement on additional excitation equipment, and is suitable for bridge health monitoring.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge health monitoring, and particularly to a method for calculating the damping ratio of a bridge based on logarithmic differentiation. Background Art

[0002] Bridges are an important part of modern transportation infrastructure, and their safety and durability are crucial for the stable operation of transportation. However, during long-term service, bridges are affected by vehicle loads, environmental factors (such as wind, temperature, humidity, etc.), and material aging, resulting in changes in structural stiffness and damping characteristics. The damping ratio is one of the important parameters of the dynamic characteristics of a bridge, which can reflect the energy dissipation capacity of the bridge structure and is of great significance for evaluating the health status of the bridge, predicting fatigue damage, and optimizing structural design. Therefore, accurately and efficiently identifying the damping characteristics of a bridge is crucial for bridge health monitoring (BHM).

[0003] Traditional bridge damping identification methods mainly include the free vibration method, the decaying signal fitting method, the time-frequency analysis method, and the system identification method based on stochastic subspace, etc. Among them, the free vibration method extracts damping information by analyzing the free vibration response of the bridge after transient external force action, but it is difficult to obtain pure free vibration signals in actual engineering; the decaying signal fitting method fits based on the exponential decay characteristics of the vibration response signal, and the calculation accuracy is easily affected by noise; time-frequency analysis methods (such as wavelet transform, Hilbert-Huang transform, etc.) can extract local instantaneous damping characteristics, but the calculation complexity is relatively high; the stochastic subspace method uses the output signal to construct a state space model for damping identification, which is applicable to large-scale bridge structures, but has high requirements for computing resources and is difficult to ensure accuracy in a low signal-to-noise ratio environment.

[0004] In recent years, bridge parameter identification methods based on vehicle-bridge coupling vibration analysis have received extensive attention. These methods use moving loads (such as moving vehicles) to excite the bridge and analyze the dynamic responses of the vehicle or the bridge to invert the bridge structure parameters. Compared with traditional methods, the damping identification method based on the vehicle-bridge system can realize online monitoring by using the natural excitation of the moving vehicle without additional excitation equipment. However, existing methods still face challenges in aspects such as bridge response signal decomposition and damping extraction, such as insufficient signal decomposition accuracy and easy interference of damping calculation by noise.

[0005] Therefore, there is an urgent need to propose a bridge damping identification method based on logarithmic differentiation to provide a more reliable technical means for bridge health monitoring. Summary of the Invention

[0006] The object of the present invention is to provide a method for calculating the damping ratio of a bridge based on logarithmic differentiation. By combining variational mode decomposition, wavelet ridge amplitude extraction, and logarithmic differentiation technology, it can accurately identify the damping ratio of the bridge, has strong anti-noise interference ability, and can also avoid the need for additional excitation equipment, and is applicable to bridge health monitoring.

[0007] To achieve the above object, the present invention provides a method for calculating the damping ratio of a bridge based on logarithmic differentiation, including the following steps:

[0008] S1. Construct the dynamic equation of the vehicle-bridge system at the static equilibrium position, and calculate or measure the vehicle acceleration response;

[0009] S2. Use the vehicle acceleration response to inversely calculate the contact point acceleration response of the vehicle in a recursive manner;

[0010] S3. Use variational mode decomposition to obtain the first-order acceleration response of the bridge, and then use wavelet transform to extract the ridge amplitude and frequency of the acceleration response;

[0011] S4. Take the logarithm of the ridge amplitude and perform differentiation, and obtain the bridge damping ratio by taking the average value.

[0012] Preferably, in the above S1, constructing the dynamic equation of the vehicle-bridge system at the static equilibrium position includes:

[0013] The motion equation of the vehicle:

[0014]

[0015] The dynamic equation of the bridge:

[0016]

[0017] Among them, m v represents the vehicle body mass, c v represents the vehicle damping, k v represents the stiffness of the support spring, m represents the mass per unit length of the bridge, c represents the damping coefficient, x represents the position of the vehicle on the bridge, EI represents the stiffness of the bridge, u(x,t), y v (t) respectively represent the vertical displacements of the bridge and the vehicle, u c (t) represents the vertical displacement of the vehicle-bridge contact point, represents the first derivative of the vehicle-bridge contact displacement u c (t), ü(x,t) respectively represent the first derivative and the second derivative of u(x,t) with respect to time t, u””(x,t) represents the fourth derivative with respect to the longitudinal coordinate x, and respectively represent y v(t) The first and second derivatives with respect to time t, the symbol δ represents the Dirac function, g represents the acceleration due to gravity, and v represents the vehicle moving speed.

[0018] Preferably, the vibration equation of the simply supported beam is simulated as a series of sine functions and modal generalized coordinates q n (t) in superposition form:

[0019]

[0020] where L represents the length of the bridge and n represents the modal order;

[0021] Substitute Equation (3) into Equation (1), and since After appropriate transformation, the approximate equation for q n (t) is:

[0022]

[0023] where ω n and ξ n represent the natural frequency and damping ratio of the bridge, respectively, represent the first derivative and second derivative of q n (t) with respect to time t, respectively, ξ n = c / (2mω n ), Ω n = nπv / L represents the driving frequency;

[0024] Solving the above formula, the bridge vibration is finally obtained as follows:

[0025]

[0026] where, and represent the steady-state vibration components caused by external force excitation; and represent the free vibration components determined by the initial conditions of the bridge, ω Dn represents the damping frequency of the nth mode of the bridge,

[0027] Substitute Equation (5) into Equation (3) and let x = vt; then take the second derivative to obtain the contact point acceleration response ü c (t) as:

[0028]

[0029] where, and are determined by the external excitation, corresponding to the steady-state vibration components of the frequency 2Ω n ; and is determined by the initial conditions of the free vibration of the bridge and the vehicle-bridge coupling effect, corresponding to the left-shift frequency component (ω Dn -Ω n ); and corresponds to the right-shift frequency component (ω Dn +Ω n ).

[0030] Preferably, in S2, the acceleration response of the contact point of the vehicle is obtained by back-calculation in a recursive manner using the vehicle acceleration response, including;

[0031] First, rewrite the vehicle vibration formula (1) as follows:

[0032]

[0033] where and respectively represent the first derivative and the second derivative of y v (t) with respect to time t, represents the first derivative of u c (t) with respect to time t; take the second derivative of both sides of the above equation and rewrite it in the following form:

[0034]

[0035] where the second derivative is calculated using the central difference method, and its discrete expression is:

[0036]

[0037] where τ represents the sampling point and Δt is the sampling interval;

[0038] Integrate over the time interval [0, t]. Assuming that the accelerations of the vehicle and the contact point at the initial time (t = 0) are zero, the back-calculation formula for the contact point acceleration is obtained:

[0039]

[0040] Considering the acceleration value ü c (t + Δt) after the time step Δt, the recurrence relationship between the acceleration values ü c (t + Δt) and ü c (t) is obtained:

[0041]

[0042] where the integral term in the above equation is calculated by trapezoidal integration:

[0043]

[0044] Substitute it into the recurrence relation and simplify to obtain the final form:

[0045]

[0046] Thus, the derivation of the recurrence formula for the contact point acceleration is completed.

[0047] Preferably, in step S3, after obtaining the single-frequency response, the ridge amplitude and frequency ω are extracted by wavelet transform D1 ;

[0048] The single-frequency response formula is:

[0049]

[0050] Research shows that when the vehicle speed is low, it satisfies The ridge amplitude H of the above equation is obtained by wavelet transform n (t):

[0051]

[0052] where

[0053] Preferably, in step S4, after obtaining the ridge amplitude of the first-order single-frequency response when n = 1, take the logarithm of the ridge amplitude first and then differentiate it:

[0054]

[0055] where, -ξ 1 ω 1 is the DC component, and cot(πvt / L) is symmetric about the midpoint [L / (2v), 0] of the time interval [0, L / (v)], so the effective interval length In the time interval the damping ratio is calculated by the following equation:

[0056]

[0057] where i represents the sampling points within the selected interval, N represents the total number of sampling points within the interval, ω 1 satisfies the formula Since the damping ratio ξ in the bridge 1 is small, generally take ω 1 = ω D1 .

[0058] Therefore, the present invention adopts the above-mentioned method for calculating the damping ratio of a bridge based on logarithmic differentiation. By combining variational mode decomposition, wavelet ridge amplitude extraction, and logarithmic differentiation technology, it can accurately identify the damping ratio of the bridge, has strong anti-noise interference ability, and can also avoid the need for additional excitation equipment, and is applicable to bridge health monitoring.

[0059] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 is a schematic diagram of bridge calculation for an embodiment of the method for calculating the damping ratio of a bridge based on logarithmic differentiation according to the present invention;

[0061] Figure 2 is a vehicle acceleration response diagram for an embodiment of the method for calculating the damping ratio of a bridge based on logarithmic differentiation according to the present invention;

[0062] Figure 3 is a vehicle contact point response diagram for an embodiment of the method for calculating the damping ratio of a bridge based on logarithmic differentiation according to the present invention;

[0063] Figure 4 is a first-order bridge response diagram of contact point extraction for an embodiment of the method for calculating the damping ratio of a bridge based on logarithmic differentiation according to the present invention;

[0064] Figure 5 is a ridge amplitude diagram for an embodiment of the method for calculating the damping ratio of a bridge based on logarithmic differentiation according to the present invention;

[0065] Figure 6 is a logarithmic differentiation curve diagram within a selected interval for an embodiment of the method for calculating the damping ratio of a bridge based on logarithmic differentiation according to the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0066] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0067] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meanings understood by those of ordinary skill in the field to which the present invention belongs.

[0068] Embodiment 1

[0069] The present invention provides a method for calculating the damping ratio of a bridge based on logarithmic differentiation, including the following steps:

[0070] S1. Construct a dynamic equation of the vehicle-bridge system at the static equilibrium position, calculate or measure the vehicle acceleration response, and its curve diagram is as Figure 2 shown. It includes:

[0071] The motion equation of the vehicle:

[0072]

[0073] Dynamic equation of the bridge:

[0074]

[0075] Wherein, m v represents the mass of the car body, c v represents the vehicle damping, k v represents the stiffness of the support spring, m represents the mass per unit length of the bridge, c represents the damping coefficient, x represents the position of the vehicle on the bridge, EI represents the stiffness of the bridge, u(x, t), y v (t) represent the vertical displacements of the bridge and the vehicle respectively, u c (t) represents the vertical displacement of the vehicle-bridge contact point, represents the vehicle-bridge contact displacement u c (t) of the first derivative, ü(x, t) respectively represent the first derivative and the second derivative of u(x, t) with respect to time t, u””(x, t) represents the fourth derivative with respect to the longitudinal coordinate x, and respectively represent the first and second derivatives of y v (t) with respect to time t, the symbol δ represents the Dirac function, g represents the acceleration due to gravity, and v represents the vehicle moving speed.

[0076] As Figure 1 shown, the vibration equation of the simply supported beam of the bridge can be simulated as a series of sine functions and the modal generalized coordinate q n (t) superposition form:

[0077]

[0078] Wherein, L represents the length of the bridge, and n represents the modal order;

[0079] Substitute formula (3) into formula (1), and since After appropriate transformation, the approximate equation for q n (t) is:

[0080]

[0081] Where ω n and ξ n respectively represent the natural frequency and damping ratio of the bridge, respectively represent the first derivative and the second derivative of q n (t) with respect to time t, ξ n = c / (2mω n ), Ωn = nπv / L represents the driving frequency;

[0082] Solving the above formula, the bridge vibration can be finally obtained as follows:

[0083]

[0084] where, and represent the steady-state vibration components caused by external excitation (vehicle load, driving frequency); and represent the free vibration components determined by the initial conditions of the bridge, ω Dn represents the damping frequency of the nth mode of the bridge,

[0085] Substitute formula (5) into formula (3) and let x = vt; then take the second derivative to obtain the contact point acceleration response ü c (t) as:

[0086]

[0087] where, and are determined by the external excitation (such as vehicle load), corresponding to the steady-state vibration component with frequency 2Ω n ; and are determined by the initial conditions of the free vibration of the bridge and the vehicle-bridge coupling effect, corresponding to the left-shift frequency component (ω Dn -Ω n ); and then correspond to the right-shift frequency component (ω Dn +Ω n ).

[0088] S2. Using the vehicle acceleration response, the contact point acceleration response of the vehicle is obtained by back-calculation in a recursive manner, and its curve is as shown in Figure 3 ;

[0089] First, rewrite the vehicle vibration formula (1) as follows:

[0090]

[0091] where, and represent the first derivative and the second derivative of y v (t) with respect to time t respectively, represents the first derivative of u c (t) with respect to time t; take the second derivative of both sides of the above equation and rewrite it in the following form:

[0092]

[0093] Among them, the second derivative is calculated by the central difference method, and its discrete expression is:

[0094]

[0095] where τ represents the sampling point and Δt is the sampling interval;

[0096] Integrating over the time interval [0, t], assuming that the accelerations of the vehicle and the contact point at the initial time (t = 0) are zero, the inverse calculation formula for the contact point acceleration is obtained:

[0097]

[0098] Considering the acceleration value ü c (t + Δt) after the time step Δt, the acceleration value ü c (t + Δt) and ü c (t) recurrence relation:

[0099]

[0100] Among them, the integral term in the above equation can be calculated by various integration methods. In this invention, the trapezoidal integral is taken as an example for calculation:

[0101]

[0102] Substituting it into the recurrence relation and arranging to obtain the final form:

[0103]

[0104] So far, the derivation of the recurrence formula for the contact point acceleration is completed.

[0105] S3. Use variational mode decomposition to obtain the first-order acceleration response of the bridge, and its curve graph is as Figure 4 shown, and then use wavelet transform to extract the ridge line amplitude and frequency of the acceleration response.

[0106] After obtaining the single-frequency response, extract the ridge line amplitude and frequency ω D1 ;

[0107] The single-frequency response formula is:

[0108]

[0109] Research shows that due to the low vehicle speed, satisfying The ridge line amplitude H n (t) of the above equation can be obtained by wavelet transform, and its curve graph is asFigure 5 as shown

[0110]

[0111] Among them,

[0112] S4. Take the logarithm of the ridge line amplitude and perform differentiation, and obtain the bridge damping ratio by taking the average value.

[0113] After obtaining the first-order (n = 1) single-frequency response ridge line amplitude, take the logarithm of the ridge line amplitude first and then differentiate:

[0114]

[0115] Among them, -ξ 1 ω 1 is the DC component, ξ 1 represents the damping ratio in the bridge, and cot(πvt / L) is symmetric about the midpoint [L / (2v), 0] of the time interval [0, L / (v)], so the effective interval length is selected Select the logarithmic differentiation curve within the interval as shown in Figure 6 as shown, within the time interval the damping ratio can be calculated by the following equation:

[0116]

[0117] Among them, i represents the sampling points within the selected interval, N represents the total number of sampling points within the interval, ω 1 satisfies the formula Since the damping ratio ξ 1 in the bridge is small, generally ω 1 = ω D1 .

[0118] According to the average value solution result of formula (17), where the theoretical setting of the bridge damping ratio is 0.03, the obtained result is 0.0298, and the accuracy is 99.3%.

[0119] Therefore, the present invention adopts the above-mentioned method for calculating the bridge damping ratio based on logarithmic differentiation. By combining variational mode decomposition, wavelet ridge line amplitude extraction and logarithmic differentiation technology, it can accurately identify the damping ratio of the bridge, has strong anti-noise interference ability, and can also avoid the need for additional excitation equipment, and is applicable to bridge health monitoring.

[0120] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions of the present invention or make equivalent replacements, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A bridge damping ratio calculation method based on logarithmic differentiation, characterized in that: The following steps are involved: S1. Construct the dynamic equations of the vehicle-bridge system in the static equilibrium position and calculate or measure the vehicle acceleration response; S2, using the vehicle acceleration response, back-calculating the contact point acceleration response of the vehicle by recursive method; S3, using variational mode decomposition to obtain the first-order acceleration response of the bridge, and then using wavelet transform to extract the ridge amplitude and frequency of the acceleration response; S4. Take the logarithm of the ridge amplitude and differentiate it, and obtain the bridge damping ratio by taking the average value.

2. The bridge damping ratio calculation method based on logarithmic differentiation according to claim 1 is characterized in that: In S1, the dynamic equation of the vehicle-bridge system at the static equilibrium position is constructed, including: The equation of motion for the vehicle is: The dynamic equation of the bridge: Among them, m v represents the vehicle mass, c v represents the vehicle damping, k v represents the stiffness of the supporting spring, m represents the mass per unit length of the bridge, c represents the damping coefficient, x represents the position of the vehicle on the bridge, EI represents the stiffness of the bridge, u(x,t), y v (t) represent the vertical displacement of the bridge and the vehicle, u c (t) represents the vertical displacement of the vehicle-bridge contact point, represents the vehicle-bridge contact displacement u c The first derivative of (t), 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. and Respectively represent y v (t) The first and second order derivatives with respect to time t, the symbol δ represents the Dirac function, g represents the gravitational acceleration, and v represents the vehicle moving speed.

3. The bridge damping ratio calculation method based on logarithmic differentiation according to claim 2 is characterized in that: The vibration equation of a simply supported beam is simulated as a series of sinusoidal functions and modal generalized coordinates q n (t) Superposition form: Where L represents the length of the bridge and n represents the modal order; Substituting formula (3) into formula (1), and due to After appropriate transformation, we can get n The approximate equation for (t) is: where ω n and n denote the natural frequency and damping ratio of the bridge, respectively. Respectively represent q n (t) The first and second derivatives of time t, ξ n =c / (2mω n ),Ω n =nπv / L represents the driving frequency; Solving the above formula, the bridge vibration is finally obtained as follows: in, and Represents the steady-state vibration component caused by external excitation; and represents the free vibration component determined by the initial condition of the bridge, ω Dn represents the damping frequency of the nth mode of the bridge, Substitute formula (5) into formula (3) and let x = vt; then take the second-order derivative to obtain the contact point acceleration response: for: in, and Determined by external excitation, corresponding frequency 2Ω n The steady-state vibration component of and Determined by the initial condition of the free vibration of the bridge and the vehicle-bridge coupling effect, the corresponding left-shifted frequency component (ω Dn -Ω n ); and The corresponding right-shifted frequency component (ω Dn +Ω n ).

4. The bridge damping ratio calculation method based on logarithmic differentiation according to claim 1 is characterized in that: In S2, using the vehicle acceleration response to obtain the contact point acceleration response of the vehicle by recursive back calculation includes: First, rewrite the vehicle vibration formula (1) as follows: in, and Respectively represent y v (t) The first and second derivatives of time t, Indicates u c (t) The first derivative with respect to time t; Take the second derivative of both ends of the above equation and rewrite it into the following form: Among them, the second-order derivative The central difference method is used for calculation, and its discrete expression is: Among them, τ represents the sampling point, Δt is the sampling interval; Integrate over the time interval [0, t], assuming that the acceleration of the vehicle and the contact point at the initial time (t = 0) is zero, and obtain the inverse calculation formula of the contact point acceleration: Acceleration value after considering time step Δt Get the acceleration value and The recursive relationship is: The integral term in the above equation is calculated using trapezoidal integration: Substituting it into the recursive relation, we can get the final form: At this point, the recursive formula for the contact point acceleration is deduced.

5. The bridge damping ratio calculation method based on logarithmic differentiation according to claim 1 is characterized in that: In S3, after obtaining the single frequency response, the ridge amplitude and frequency ω are extracted by wavelet transform. D1 ; The single frequency response formula is: Studies have shown that when the vehicle speed is low, The ridge amplitude H of the above equation is obtained by wavelet transform n (t): in, 6. The bridge damping ratio calculation method based on logarithmic differentiation according to claim 1 is characterized in that: In S4, after obtaining the first-order single-frequency response ridge amplitude when n=1, the ridge amplitude is first logarithmically taken and then differentiated: Among them, -ξ1ω1 is the DC component, and cot(πvt / L) is symmetrical about the midpoint [L / (2v),0] of the time interval [0,L / v)], so the effective interval length is selected In the time interval The upper damping ratio is calculated by the following equation: Among them, i represents the sampling point in the selected interval, N represents the total number of sampling points in the interval, and ω1 satisfies the formula Since the damping ratio ξ1 in the bridge is small, ω1 = ω D1 .

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