A method for identifying modal shapes of a curved bridge based on contact responses of vehicle bridges

Abstract

The present application belongs to the technical field of bridge health monitoring and detection, and proposes a curved bridge modal vibration mode identification method based on vehicle-bridge contact response, which comprises data acquisition and processing, inverse calculation of mobile vehicle-curved bridge contact response, and construction of curved bridge modal vibration mode. The method can be applied to bridge health state evaluation by identifying modal vibration mode. The present application fills the gap of curved bridge modal vibration mode identification technology based on mobile vehicle-curved bridge contact response. It can efficiently identify vertical and radial modal vibration modes of curved bridges and eliminate the influence of vehicle body vertical, swing and lateral responses on bridge modal vibration mode identification, which is of great significance for bridge health detection.

Description

Technical Field

[0001] This invention belongs to the field of bridge health monitoring and detection technology, specifically relating to a method for identifying vertical and radial mode vibration patterns of curved bridges based on the contact response between a moving vehicle and a curved bridge. Background Technology

[0002] Bridges, as vital structures that break down natural or man-made geographical barriers, play a crucial role in people's transportation, communication, and urban economic development. However, during their service life, bridges inevitably suffer from various factors such as vehicle loads and environmental erosion, making their health a constant concern. At the end of 2021, my country had 961,100 highway bridges, totaling 73.8021 million meters in length. For these nearly one million bridges in my country, developing rapid and efficient bridge health monitoring technologies is essential to ensuring their safe operation.

[0003] Vibration-based methods are widely used to monitor the health of bridges. Traditional monitoring methods involve directly installing numerous vibration sensors on the bridge to continuously collect data on its vibration response. However, due to high installation and maintenance costs, these methods are typically limited to monitoring the health of long-span, extra-long-span bridges, and bridges with special structures. For the small- and medium-span bridges that account for over 90% of the total number of bridges in my country, there is an urgent need to research an economical and efficient bridge health monitoring technology.

[0004] When a vehicle crosses a bridge, the moving vehicle and the bridge form a coupled system. Within this system, the movement of the vehicle causes the bridge to vibrate, and simultaneously, the bridge transmits its own vibrations to the moving vehicle. Based on this characteristic, bridge health monitoring methods that acquire bridge vibration data based on the response of moving vehicles have attracted widespread attention. This method deploys a small number of sensors on the moving vehicle and is characterized by mobility, economy, and versatility.

[0005] Furthermore, current research focuses on straight bridges, where the vertical response reflects the bridge's health condition. There is a lack of methods for identifying modal vibration modes of curved bridges based on the contact response between a moving vehicle and the curved bridge. For curved bridges, both vertical and radial (horizontal) modal vibration modes reflect the bridge's health condition. Effectively identifying the vertical and radial modal vibration modes of curved bridges and eliminating the influence of the vehicle's own vertical and lateral (horizontal) responses on modal vibration identification is crucial for bridge health monitoring. Summary of the Invention

[0006] In this field, frequency, as an important modal parameter, is one of the key indicators for assessing the health status of bridges. The Chinese invention patent application "A Method for Identifying Vertical and Radial Modal Parameters of Curved Bridges Based on Moving Vehicle Response" (application number CN202310076351X) designs a model by treating the vehicle as a mass block M. During the data acquisition phase, accelerometers are used to collect the vertical and lateral vibration responses of the vehicle body. In the data processing phase, interference from the vertical and lateral frequencies of the vehicle body is eliminated to achieve visibility of bridge frequency identification, thus enabling its application in bridge health status assessment. However, as this is an early stage of cutting-edge research, the Chinese invention patent application "A Method for Identifying Vertical and Radial Modal Parameters of Curved Bridges Based on Moving Vehicle Response" (application number CN202310076351X) only treats the model as a mass block and does not yet consider the dynamic parameter characteristics of swaying.

[0007] In this field, mode shape is another important modal parameter and one of the important indicators for assessing the health status of bridges.

[0008] The technical problem to be solved by this invention is to address the shortcomings of existing technologies by providing a method for identifying the vertical and radial mode shapes of curved bridges based on the contact response between a moving vehicle and a curved bridge. The wisdom and strategy embodied in the inventive method are as follows:

[0009] (1) Taking into account the vertical, sway, and lateral dynamic parameters of the vehicle body, the vehicle body is treated as a steel axle, which improves the model and makes it more consistent with the motion state of everyday vehicles. Sensors for collecting vertical and lateral vibration data are installed on the vehicle body axle near the left and right wheels and at the center of the axle, respectively. Based on this, the vertical, sway, and lateral responses of the vehicle body are calculated. Based on the vehicle-bridge coupling effect in the vertical and horizontal directions, the vertical and radial modal parameters of the curved bridge are indirectly identified from the vehicle body response.

[0010] (2) Based on the vertical, swaying and lateral responses of the vehicle body, a method for back-calculating the contact response between the moving vehicle and the curved bridge is established, thereby eliminating the influence of the vehicle body's vertical, swaying and lateral natural vibration on the identification of the modal parameters of the curved bridge and enhancing the identification effect of the vertical and radial modal parameters of the curved bridge.

[0011] (3) Based on the contact response of the moving vehicle-curved bridge, combined with variational mode decomposition and synchronous squeezing wavelet transform, the vertical and radial mode vibration modes of the curved bridge can be efficiently identified, and thus applied to the health status assessment of the bridge.

[0012] A method for identifying vertical and radial mode shapes of curved bridges based on the contact response between a moving vehicle and a curved bridge includes data acquisition and processing, inverse calculation of the contact response between the moving vehicle and the curved bridge, construction of the mode shapes of the curved bridge, and identification of the mode shapes, which can then be applied to bridge health status assessment.

[0013] Specifically, the steps include the following:

[0014] Data collection and processing:

[0015] Step 1: Place the accelerometer S vl S vr (l represents the left wheel, r represents the right wheel) Mounted on the vehicle body axle near the left and right wheels, acceleration sensor S r Installed at the center of the vehicle body axle, such as Figure 2 As shown. Based on the vehicle-bridge coupling principle, the vertical and radial vibration responses of a curved bridge will be transmitted to the vehicle, causing the vehicle body to experience vertical, swaying, and lateral vibrations.

[0016] Step 2: Measure the distance the vehicle travels across the curved bridge, sensor S vl S vr Vertical vibration responses at the left and right wheels of the vehicle can be collected. and Sensor S r The lateral vibration response of the vehicle body can be collected. The vertical and sway vibration responses of the vehicle body are calculated using formula (26). and

[0017] Inverse calculation of the contact response between the two moving vehicles and the curved bridge:

[0018] Step 3: Analyze the vertical vibration response of the vehicle body using formula (25). swaying vibration response and lateral vibration response Take the derivative with respect to time t;

[0019] Step 4: Substitute formula (25) into formula (24) to solve for G. vl G vr and G r ;

[0020] Step 5: Calculate the vertical and radial response of the contact acceleration between the moving vehicle and the curved bridge using formula (23). and It also identifies the vertical and radial frequencies of curved bridges.

[0021] Construction of the mode shapes of a triboidal bridge:

[0022] Step 6: In order to identify the mode shape, variational mode decomposition technology is used to separate the single relevant components containing the vertical and radial responses of the curved bridge from the vertical and radial acceleration responses of the moving vehicle-curved bridge contact obtained in Step 5 (i.e., formulas (23a, 23b)), as shown in formulas (27a, 27b).

[0023] Step 7: Apply continuous wavelet transform to the vertical acceleration component response of the moving vehicle-curved bridge contact (Equation 27a) containing the vertical response of the curved bridge, and the radial acceleration component response of the moving vehicle-curved bridge contact (Equation 27b) containing the radial response of the curved bridge, respectively, to obtain the continuous wavelet coefficients W of the vertical and radial component responses of the curved bridge. uv (a,b) and W ur (a,b) is represented by formulas (28a, 28b);

[0024] Step 8: For a fixed wavelet center frequency a bv,n =ω0 / ω bv,n and a br,1 =ω0 / ω br,1 Continuous wavelet coefficients |W uv (a bv,n b)| and |W ur (a br,1 b)| respectively reach the maximum value in the region, as shown in formulas (29a, 29b), where This reflects the mode shape corresponding to the component response of the curved bridge;

[0025] Step 9: Combine synchronous squeezing wavelet transform technique, that is, for continuous wavelet coefficients W uv (a,b) and W uv Redistribute (a, b) respectively to obtain the synchronous squeezed wavelet transform coefficients T uv (ω l b) and T ur (ω l ,b), as shown in formulas (30a, 30b), further yields the time-frequency diagrams of the curved bridge in both the vertical and radial directions;

[0026] Step 10: Extract the ridge lines related to the component responses of the curved bridge from the time-frequency diagram and perform normalization processing to finally obtain the vertical and radial mode shapes of the curved bridge.

[0027] Formulas (23)-(30) are respectively:

[0028]

[0029]

[0030]

[0031]

[0032]

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] IV. In application, the health status of bridges is assessed by identifying the modal vibration modes of the structure curves.

[0046] This invention fills a gap in the field of modal mode identification technology for curved bridges based on the contact response between a moving vehicle and a curved bridge. It can efficiently identify the vertical and radial modal modes of curved bridges and eliminate the influence of the vehicle's own vertical, swaying, and lateral (horizontal) responses on bridge modal mode identification, which is of great significance for bridge health monitoring. Attached Figure Description

[0047] Figure 1 Flowchart of the method of the present invention

[0048] Figure 2 Sensor layout scheme for the measuring vehicle system (lateral: measuring lateral vibration; vertical: measuring vertical vibration and swaying vibration)

[0049] Figure 3 The mechanical model of this invention: moving vehicle-curved bridge system

[0050] Figure 4 The embodiment is based on the vehicle body vertical response obtained by sensor arrangement according to the method of the present invention.

[0051] Figure 5 The embodiment is based on the vehicle body sway response obtained by sensor arrangement according to the method of the present invention.

[0052] Figure 6 The embodiment is based on the radial response of the vehicle body obtained by the sensor arrangement method of the present invention.

[0053] Figure 7 The embodiment is based on the vehicle-curved bridge contact response calculation method of the present invention to obtain the vertical contact acceleration response of the vehicle-curved bridge.

[0054] Figure 8 The embodiment is based on the radial contact acceleration response of the vehicle-curved bridge obtained by the vehicle-curved bridge contact response calculation method of the present invention.

[0055] Figure 9 The embodiments are based on the vertical and radial component responses of curved bridges obtained by variational mode decomposition techniques according to the present invention.

[0056] Figure 10 The embodiments are based on the time-frequency diagrams of the vertical and radial component responses of curved bridges obtained by the synchronous squeezing wavelet transform technique of the present invention.

[0057] Figure 11 The embodiments are based on the vertical and radial mode shape identification results of curved bridges obtained by the method of the present invention. Detailed Implementation

[0058] The technical solutions provided in this application will be further described below with reference to specific embodiments and accompanying drawings. The advantages and features of this application will become clearer from the following description.

[0059] A method for identifying vertical and radial mode shapes of curved bridges based on the contact response between a moving vehicle and a curved bridge is presented as an application method. The process is as follows: Figure 1 As shown):

[0060] Data collection and processing:

[0061] Step 1: Place the accelerometer S vl S vr (l represents the left wheel, r represents the right wheel) Mounted on the vehicle body axle near the left and right wheels, acceleration sensor S r Installed at the center of the vehicle body axle, such as Figure 2 As shown. Based on the vehicle-bridge coupling principle, the vertical and radial vibration responses of a curved bridge will be transmitted to the vehicle, causing the vehicle body to experience vertical, swaying, and lateral vibrations.

[0062] Step 2: Measure the distance the vehicle travels across the curved bridge, sensor S vl S vr Vertical vibration responses at the left and right wheels of the vehicle can be collected. and Sensor S r The lateral vibration response of the vehicle body can be collected. The vertical and sway vibration responses of the vehicle body are calculated using formula (26). and

[0063] Inverse calculation of the contact response between the moving vehicle and the curved bridge:

[0064] Step 3: Analyze the vertical vibration response of the vehicle body using formula (25). swaying vibration response and lateral vibration response Take the derivative with respect to time t;

[0065] Step 4: Substitute formula (25) into formula (24) to solve for G. vr G vr and G r ;

[0066] Step 5: Calculate the vertical and radial response of the contact acceleration between the moving vehicle and the curved bridge using formula (23). and It also identifies the vertical and radial frequencies of curved bridges.

[0067] Construction of modal vibration modes of curved bridges:

[0068] Step 6: In order to identify the mode shape, variational mode decomposition technology is used to separate the single relevant components containing the vertical and radial responses of the curved bridge from the vertical and radial acceleration responses of the moving vehicle-curved bridge contact obtained in Step 5 (i.e., formulas (23a, 23b)), as shown in formulas (27a, 27b).

[0069] Step 7: Apply continuous wavelet transform to the vertical acceleration component response of the moving vehicle-curved bridge contact (Equation 27a) containing the vertical response of the curved bridge, and the radial acceleration component response of the moving vehicle-curved bridge contact (Equation 27b) containing the radial response of the curved bridge, respectively, to obtain the continuous wavelet coefficients W of the vertical and radial component responses of the curved bridge. uv (a,b) and W ur (a,b) is represented by formulas (28a, 28b);

[0070] Step 8: For a fixed wavelet center frequency a bv,n =ω0 / ω bv,n and a br,1 =ω0 / ω br,1 Continuous wavelet coefficients |W uv (a bv,n b)| and |W ur (a br,1 b)| respectively reach the maximum value in the region, as shown in formulas (29a, 29b), where This reflects the mode shape corresponding to the component response of the curved bridge;

[0071] Step 9: Combine synchronous squeezing wavelet transform technique, that is, for continuous wavelet coefficients W uv (a,b) and W ur Redistribute (a, b) respectively to obtain the synchronous squeezed wavelet transform coefficients T uv (ω l b) and T ur (ω l ,b), as shown in formulas (30a, 30b), further yields the time-frequency diagrams of the curved bridge in both the vertical and radial directions;

[0072] Step 10: Extract the ridge lines related to the component responses of the curved bridge from the time-frequency diagram and perform normalization processing to finally obtain the vertical and radial mode shapes of the curved bridge.

[0073] As can be seen from the above technical solution process, by measuring the vertical acceleration response of the vehicle body axle near the left and right wheels... Lateral acceleration response of the vehicle body (like Figure 2 As shown, the vertical, swaying, and lateral vibration responses of the vehicle body can be calculated, thereby inversely calculating the vertical and radial responses of the moving vehicle-curved bridge contact. Based on the vertical and lateral responses of the moving vehicle-curved bridge contact, combined with variational mode decomposition and synchronous squeezing wavelet transform techniques, efficient identification of the vertical and radial mode shapes of the curved bridge is achieved. This is the core innovation and technical contribution of the application method of this invention.

[0074] Part One: The Construction Process of the Theoretical Foundation of the Technical Solution

[0075] The moving vehicle-curved bridge system is a coupled system, involving the coupling of vertical and radial (horizontal) responses. When the vehicle travels across the curved bridge, the bridge's vertical and radial vibration responses are transmitted to the vehicle body through the wheels, causing vertical, swaying, and lateral vibrations. Therefore, accelerometers are installed on the vehicle axles near the left and right wheels and at the axle center to collect vertical and radial vibration data. Figure 2 As shown, the vertical, sway, and lateral responses of the vehicle body can be calculated, and then the vertical and radial responses of the moving vehicle-curved bridge contact can be further calculated. This allows for the indirect identification of the vertical and radial mode shapes of the curved bridge, while eliminating the influence of the vehicle body's own vertical, sway, and lateral vibrations on the identification of the bridge's mode shapes.

[0076] like Figure 3 The model shown in the figure has the following meanings: x, y, and z represent the three-dimensional coordinate system of the mechanical model; the curved bridge is simplified to an Euler-Bernoulli simply supported beam model, u r u a u vθ and θ represent the radial, axial, vertical, and torsional displacement responses of the curved beam, respectively; O is the center of the curved beam; R is the radius of curvature of the curved beam; β is the central angle of the curved beam; m is the mass per unit length of the curved beam; E and G are the elastic and shear moduli, respectively; I y Let I be the moment of inertia about the y-axis. z Let J be the moment of inertia about the z-axis, J be the torsional constant, and A be the cross-sectional area of ​​the curved beam; m v and J v These represent the vehicle's mass and moment of inertia, respectively. v θ v and y r These represent the vertical, swaying, and lateral displacements of the vehicle body, respectively, k v c v k represents the vertical stiffness and vertical damping of the left and right wheels of the vehicle body. r c r For the lateral stiffness and lateral damping of the vehicle body, u cvl u cvr For the vertical displacement of the left and right wheels of the moving vehicle at the contact point with the curved bridge, u cr The radial displacement of the contact between the moving vehicle and the curved bridge.

[0077] When the measuring vehicle crosses the bridge, the control equations for the vertical, rotational, axial, and radial vibrations of the curved bridge are as follows:

[0078]

[0079]

[0080]

[0081]

[0082] In the formula, δ is the Dixlade function, and f is the vertical contact force. cvs (s = l, r represent the left and right wheels of the vehicle body, respectively), radial contact force f cr and torque T(t) are respectively

[0083]

[0084]

[0085] T(t) = e clfcvl (t)+e cr f cvr (t) (7)

[0086] In the formula, g is the acceleration due to gravity, and e cr d is the distance by which the contact point of the right wheel of the vehicle body deviates laterally from the center axis of the curved bridge, and d is the distance between the centers of the left and right wheels of the vehicle body.

[0087] Based on the modal superposition method, analytical expressions for the vertical, rotational, axial, and radial displacement responses of curved bridges can be derived.

[0088]

[0089]

[0090]

[0091]

[0092] The coefficients in the formula are:

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[0097]

[0098]

[0099]

[0100]

[0101] Based on the contact relationship between the moving vehicle and the curved bridge, and letting x = vt, the theoretical expressions for the vertical and radial responses of the moving vehicle-curved bridge contact are as follows:

[0102]

[0103]

[0104] The coefficients in the formula are:

[0105] p cvs,n =p v1,n -e cs p θ1,n (twenty two)

[0106] From equations (20) and (21) above, it can be seen that the theoretical expressions for the vertical and radial responses of the moving vehicle-curved bridge contact include the vertical and radial vibration frequencies of the curved bridge, thus theoretically proving the feasibility of identifying the modal parameters of the curved bridge from the contact response of the moving vehicle-curved bridge. Furthermore, the theoretical expressions for the vertical and radial responses of the moving vehicle-curved bridge contact do not include the vertical, swaying, and lateral frequencies of the vehicle body, eliminating the influence of the vehicle body's own frequencies on the identification of the curved bridge modal parameters. Therefore, the application method of this invention inevitably improves the identification effect and accuracy of the bridge modal parameters.

[0107] Part Two: Data Acquisition and Processing, Inverse Calculation of Contact Response between Moving Vehicle and Curved Bridge:

[0108] Step 1: Place the accelerometer S vl S vr (l represents the left wheel, r represents the right wheel) Mounted on the vehicle body axle near the left and right wheels, acceleration sensor S r Installed at the center of the vehicle body axle, such as Figure 2 As shown. Based on the vehicle-bridge coupling principle, the vertical and radial vibration responses of a curved bridge will be transmitted to the vehicle, causing the vehicle body to experience vertical, swaying, and lateral vibrations.

[0109] Step 2: Measure the distance the vehicle travels across the curved bridge, sensor S vl S vr Vertical vibration responses at the left and right wheels of the vehicle can be collected. and Sensor S r The lateral vibration response of the vehicle body can be collected. The vertical and sway vibration responses of the vehicle body are calculated using formula (26). and

[0110] Step 3: Analyze the vertical vibration response of the vehicle body using formula (25). swaying vibration response and lateral vibration response Take the derivative with respect to time t;

[0111] Step 4: Substitute formula (25) into formulas (24a, 24b, 24c) to solve for G. vl G vr and G r ;

[0112] Step 5: Calculate the vertical and radial response of the contact acceleration between the moving vehicle and the curved bridge using formula (23). cvs and ü cr This is used to provide information to the third part; thereby, the vertical and radial frequencies of the curved bridge are identified simultaneously.

[0113] The above steps 1-5 involve formulas (23)-(26), specifically:

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[0115]

[0116]

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[0118]

[0119]

[0120]

[0121]

[0122]

[0123] From the above equations (23) and (24), it can be seen that the present invention comprehensively considers the vertical, swaying and lateral vibrations generated by the vehicle body during driving, which is more in line with the actual vehicle body motion state. Thus, the vertical and radial responses of the moving vehicle-curved bridge contact are further calculated. Based on the identification of the curved bridge mode vibration based on the moving vehicle-curved bridge contact response, the influence of the vehicle body's own vertical, swaying and lateral vibrations on the bridge mode vibration identification is eliminated, thereby effectively enhancing the identification effect of the curved bridge mode vibration.

[0124] Part Three: Construction of Mode Shapes of Curved Bridges: Based on the contact response between the moving vehicle and the curved bridge, and combining variational mode decomposition and synchronous squeeze wavelet transform techniques, the vertical and radial mode shapes of the curved bridge are constructed.

[0125] Step 6: In order to identify the mode shape, variational mode decomposition technology is used to separate the single relevant components containing the vertical and radial responses of the curved bridge from the vertical and radial acceleration responses of the moving vehicle-curved bridge contact obtained in Step 5 (i.e., formulas (23a, 23b)), as shown in formulas (27a, 27b).

[0126] Step 7: Apply continuous wavelet transform to the vertical acceleration component response of the moving vehicle-curved bridge contact (Equation 27a) containing the vertical response of the curved bridge, and the radial acceleration component response of the moving vehicle-curved bridge contact (Equation 27b) containing the radial response of the curved bridge, respectively, to obtain the continuous wavelet coefficients W of the vertical and radial component responses of the curved bridge. uv (a,b) and W ur(a,b) is represented by formulas (28a, 28b);

[0127] Step 8: For a fixed wavelet center frequency a bv,n =ω0 / ω bv,n and a br,1 =ω0 / ω br,1 Continuous wavelet coefficients |W uv (a bv,n b)| and |W ur (a br,1 b)| respectively reach the maximum value in the region, as shown in formulas (29a, 29b), where This reflects the mode shape corresponding to the component response of the curved bridge;

[0128] Step 9: Combine synchronous squeezing wavelet transform technique, that is, for continuous wavelet coefficients W uv (a,b) and W ur Redistribute (a, b) respectively to obtain the synchronous squeezed wavelet transform coefficients T uv (ω l b) and T ur (ω l ,b), as shown in formulas (30a, 30b), further yields the time-frequency diagrams of the curved bridge in both the vertical and radial directions;

[0129] Step 10: Extract the ridge lines related to the component responses of the curved bridge from the time-frequency diagram and perform normalization processing to finally obtain the vertical and radial mode shapes of the curved bridge.

[0130] Steps 6-10 above involve formulas (27)-(30), specifically:

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[0132]

[0133]

[0134]

[0135]

[0136]

[0137]

[0138]

[0139] In formula (27a), n is a natural number, n = 1, 2, 3…, representing the nth curved bridge response component in the contact response between the moving vehicle and the curved bridge. In formulas (28a) and (28b), ψ… * Let Ψ denote the complex conjugate of the mother wavelet ψ, and in formulas (29a) and (29b) Ψ denotes the Fourier transform of the mother wavelet ψ.

[0140] From the above process of constructing the vertical and radial mode shapes of curved bridges, it can be seen that applying variational mode decomposition (VMD) to the contact response of the moving vehicle-curved bridge can effectively decompose the vertical and radial component responses of the curved bridge. Applying synchronous squeezing wavelet transform (MST), which has high time-frequency resolution and energy concentration, to the component responses of the curved bridge can yield clear and stable time-frequency diagrams, thereby extracting ridge lines to construct mode shapes. This invention, based on the contact response of the moving vehicle-curved bridge, creatively utilizes a combined technique of variational mode decomposition and synchronous squeezing wavelet transform to achieve efficient identification of the vertical and radial mode shapes of curved bridges, representing the core innovation and technological contribution of this invention's application method.

[0141] The following examples provide further numerical verification of the technical solution of the present invention.

[0142] Example parameters:

[0143] The curved bridge has a span L = 30m, a radius of curvature R = 100m, an elastic modulus E = 33.2 GPa, a Poisson's ratio v = 0.2, and a torsional constant J = 21.18m. 4 Moment of inertia I y =18.75m 4 I z =2.43m 4 The cross-sectional area A = 9m² 2 The density of the curved bridge is ρ = 2400 kg / m³. 3 Vehicle mass m v =1000kg, moment of inertia J v =240kg·m 2 Vertical stiffness k of the left and right wheels of the vehicle body v =800kN / m, vehicle body lateral stiffness k r =1,000kN / m, the distance d between the left and right wheel contact points of the vehicle body is 1.4m, and the eccentricity e between the left wheel contact point of the vehicle body and the center of gravity axis of the curved bridge is... cl = -1.4m, the contact point of the right wheel of the vehicle body is along the centroidal axis of the curved bridge, i.e., e cr =0m, the vehicle's running speed is v=5m / s.

[0144] Summary 1

[0145] To verify that the method of the present invention can identify the vertical and radial modal parameters of a curved bridge from the vehicle body response, numerical simulations were performed in this embodiment: Figure 4 The vertical response of the vehicle body is obtained based on the sensor arrangement method of this invention. Figure 5 The vehicle body sway response is obtained based on the sensor arrangement method of this invention. Figure 6 This refers to the vehicle body lateral response obtained based on the sensor arrangement method of this invention. From... Figure 4 , Figure 5 and Figure 6 As can be seen from the present invention, the sensor arrangement method can be used to successfully identify the vertical and radial frequencies of the curved bridge from the vehicle body response.

[0146] Summary 2

[0147] To verify the calculation effect of the method of the present invention on the vertical and radial contact acceleration response of the moving vehicle-curved bridge, numerical simulation was performed in this embodiment: Figure 7 The vertical contact acceleration response of the vehicle-curved bridge is obtained based on the vehicle-curved bridge contact response calculation method of this invention. Figure 8 This is the radial contact acceleration response of the vehicle-curved bridge obtained based on the vehicle-curved bridge contact response calculation method of this invention. Figure 7 , Figure 8 It can be seen that the proposed method for calculating the contact response between the moving vehicle and the curved bridge yields vertical and radial contact acceleration responses that are almost identical to those obtained directly from the curved bridge.

[0148] And will Figure 7 and Figure 8 and Figure 4 , Figure 5 and Figure 6 The comparison reveals that the vehicle's own frequencies are filtered out in the vehicle-curved bridge contact response, while the visibility of higher-order vertical and radial frequencies of the curved bridge is improved.

[0149] As can be seen from the inverse calculation method, the entire vehicle-curved bridge contact response identification process only utilizes the vehicle's own response. Therefore, the vehicle-curved bridge contact response calculation method proposed in this invention is not limited to the structural form of the curved bridge.

[0150] Summary 3

[0151] To verify that the method of the present invention can identify the vertical and radial mode shapes of a curved bridge from the contact response of a moving vehicle and a curved bridge, numerical simulations were performed: Figure 9 The vertical and radial component responses of the curved bridge are obtained based on the variational mode decomposition technique of this invention. Figure 10 The above are time-frequency diagrams of the vertical and radial component responses of a curved bridge obtained using the synchronous squeezing wavelet transform technique of this invention. Figure 11The results show the vertical and radial mode shape identification of the curved bridge obtained based on the method of this invention. Figure 9 , Figure 10 and Figure 11 It can be seen that the proposed variational mode decomposition and synchronous squeezing wavelet transform joint technology can be successfully applied to the identification of vertical and radial mode shapes of curved bridges, and the identified vertical and radial mode shapes of curved bridges have high identification accuracy.

[0152] The above description is merely a description of preferred embodiments of this application and is not intended to limit the scope of this application in any way. Any changes or modifications made by those skilled in the art based on the above-disclosed technical content should be considered as equivalent and valid embodiments and fall within the scope of protection of the technical solution of this application.

Claims

1. A method for modal shape identification of a curved bridge based on axle contact response, characterized in that, The application relates to a method for identifying modal shapes of a curved bridge, comprising data acquisition and processing, inverse calculation of moving vehicle-curved bridge contact response, construction of modal shapes of the curved bridge, and application of the modal shapes to bridge health state evaluation. I. Data acquisition and processing: Step 1: Acceleration sensor S vl , S vr is installed on the vehicle body axle near the left and right wheels, wherein l represents the left wheel, r represents the right wheel, and acceleration sensor S r is installed at the center of the vehicle body axle. Based on the principle of bridge coupling, the vertical and radial vibration responses of the curved bridge will be transmitted to the vehicle, causing the vehicle body to produce vertical, swing and lateral vibrations. Step 2: Measure the vehicle driving over the curve bridge, sensor S vl , S vr Collect the vertical vibration response of the left and right wheels of the vehicle body and , sensor S r Collect the lateral vibration response of the vehicle body , calculate the vertical and swing vibration response of the vehicle body using formula (26) and ; II. Inverse calculation of moving vehicle-curved bridge contact response: Step 3: Derivation of the vertical vibration response of the vehicle body with respect to time t using equation (25) , the roll vibration response , and the lateral vibration response of the vehicle body with respect to time t using equation (25) t . Step 4: Substitute equation (25) into equation (24) and solve for , and ; Step 5: Calculate the vertical and radial response of the moving car-curve bridge contact acceleration using equation (23) and and identify the vertical and radial frequencies of the curve bridge. III. Construction of modal shapes of the curved bridge: Step 6: In order to identify the modal shapes, a variational modal decomposition technique is used to separate single related components containing vertical and radial responses of the curved bridge from the moving vehicle-curved bridge contact vertical and radial acceleration responses calculated in step 5, i.e. formulae (23a, 23b), as shown in formulae (27a, 27b); Step 7: Apply the continuous wavelet transform to the moving vehicle-curved bridge contact vertical acceleration component response, i.e., equation (27a) containing the curved bridge vertical response, and the moving vehicle-curved bridge contact radial acceleration component response, i.e., equation (27b) containing the curved bridge radial response, respectively, to obtain the continuous wavelet coefficients of the curved bridge vertical and radial component responses and are shown in equations (28a, 28b); Step 8: For a fixed wavelet center frequency and the successive wavelet coefficients and respectively reach a local maximum, as shown in equations (29a, 29b), where reflects the mode shape corresponding to the component response of the curved bridge. Step 9: Reallocating the continuous wavelet coefficients and respectively, to obtain the synchronous extrusion wavelet transform coefficients and , as shown in formulas (30a, 30b), to further obtain the time-frequency diagram of the curved bridge in the vertical and radial directions. Step 10: Ridge lines related to the curved bridge component responses are extracted in a time-frequency diagram, and normalized processing is carried out, so that the vertical and radial modal shapes of the curved bridge are identified; The formulae (23)-(30) are respectively:

2. The method of claim 1, wherein, Further comprising steps: IV. Application: The modal shapes of the curved bridge are identified and constructed to evaluate the bridge health state.

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