Extraction of vertical and flexural–torsional frequencies of thin-walled box bridges with road roughness and damping from the residual contact response of a two-axle vehicle

Extraction of bridge mode shapes from the response of a two-axle passing vehicle using a two-peak spectrum idealized filter approach

As an essential reference to bridge dynamic characteristics, the identification of bridge frequencies has far-reaching consequences for the health monitoring and damage evaluation of bridges. This study proposes a uniform scheme to identify bridge frequencies with two different subspace-based methodologies, i.e., an improved Short-Time Stochastic Subspace Identification (ST-SSI) method and an improved Multivariable Output Error State Space (MOESP) method, by simply adjusting the signal inputs. One of the key features of the proposed scheme is the dimensionless description of the vehicle-bridge interaction system and the employment of the dimensionless response of a two-axle vehicle as the state input, which enhances the robustness of the vehicle properties and speed. Additionally, it establishes the equation of the vehicle biaxial response difference considering the time shift between the front and the rear wheels, theoretically eliminating the road roughness information in the state equation and output signal effectively. The numerical examples discuss the effects of vehicle speeds, road roughness conditions, and ongoing traffic on the bridge identification. According to the dimensionless speed parameter Sv1 of the vehicle, the ST-SSI (Sv1 < 0.1) or MOESP (Sv1 ≥ 0.1) algorithm is applied to extract the frequencies of a simply supported bridge from the dimensionless response of a two-axle vehicle on a single passage. In addition, the proposed methodology is applied to two types of long-span complex bridges. The results show that the proposed approaches exhibit good performance in identifying multi-order frequencies of the bridges, even considering high vehicle speeds, high levels of road surface roughness, and random traffic flows.

Extracting Bridge Frequencies from The Dynamic Responses of Moving and Non-moving Vehicles

A vibration amplifier is first proposed for adding to a test vehicle to enhance its capability to detect frequencies of the bridge under scanning. The test vehicle adopted is of single-axle and modeled as a single degree-of-freedom (DOF) system, which was proved to be successful in previous studies. The amplifier is also modeled as a single-DOF system, and the bridge as a simple beam of the Bernoulli–Euler type. To unveil the mechanism involved, closed-form solutions were first derived for the dynamic responses of each component, together with the transmissibility from the vehicle to amplifier. Also presented is a conceptual design for the amplifier. The approximations adopted in the theory were verified to be acceptable by the finite element simulation without such approximations. Since road roughness can never be avoided in practice and the test vehicle has to be towed by a tractor in the field test, both road roughness and the tractor are included in the numerical studies. For the general case, when the amplifier is not tuned to the vehicle frequency, the bridge frequencies can better be identified from the amplifier than vehicle response, and the tractor is helpful in enhancing the overall performance of the amplifier. Besides, the amplifier can be adaptively adjusted to target and detect the bridge frequency of concern. For the special case when the amplifier is tuned to the vehicle frequency, the amplifier can improve the vehicle performance by serving as a tuned mass damper, as conventionally known. This case is of limited use since it does not allow us to target the bridge frequencies. Both bridge damping and vehicle speed are also assessed with their effects addressed.

Scanning the bridge’s frequencies from the passing vehicle’s vibration data has been frequently investigated recently. However, in previous studies, vehicles were typically simplified to quarter- or half-car models, and apparent disparity could be observed between the models and real vehicles. To make the vehicle model more practical, in this study, a 3D vehicle model is built to extract the bridge’s frequencies from vehicle vibrations. For the first time, equations for calculating the contact-point (CP) response of the 3D vehicle model are derived with tire damping. Furthermore, residual CP responses between front and rear wheels are utilized to eliminate the inverse effects of road roughness, making the bridge frequencies outstanding in the frequency domain. The robustness of the proposed method is tested under different influence factors, and two possible measurement errors are as follows: the sensor position and axle distance when applying the proposed method in engineering. Results show that the proposed method performs stably under the influence of different road roughness classes and tire damping. Bridge frequencies can be identified when the vehicle is travelling at a highway speed (108 km/h in this study). Environmental noises can submerge the bridge’s high-order frequencies but have little influence on the low-frequency range. High bridge damping will restrain the transmission of bridge vibration to the vehicle, making high-order bridge frequencies less visible. In addition, the errors introduced by a vehicle body sensor position can be eliminated when calculating the CP responses for tires, thus will not influence bridge frequency identification. To avoid possible errors induced by manual measurement of the axle distance, a novel cross-correlation function-based method is employed, which is verified effective and practical for calculating residual CP responses.

Highways, urban roads, and bridges are the important transportation infrastructures for the economic development of modern society. The evaluation of bridge and road quality is crucial to the maintenance and management of the bridge and road industry. Road roughness is a widely accepted indicator in the evaluation of road quality and safety, which is a major input source for vehicles. The vehicle responses-based method of identifying road roughness is efficient and convenient. However, the dynamic characteristics of the vehicle have an important impact on the interaction between the vehicle and the road. When the vehicle parameters are not yet clear, the coupling of unknown parameters and unknown road roughness results in the need for mutual iteration when the existing methods simultaneously identify vehicle parameters and road roughness. To address this issue, this study proposes an effective method for the combined identification of vehicle parameters and road roughness using vehicle responses. The test vehicle is modeled as a four-degree-of-freedom half-vehicle model. In view of the coupling effect between tire stiffness and road roughness, the unknown vehicle physical parameters, except for tire stiffness, are first included in the extended state vector. Based on the extended Kalman filter for unknown excitation (EKF-UI), unknown vehicle physical parameters and unknown forces on the axle are identified. Subsequently, based on the property that the front and rear axles of the vehicle pass through the same road roughness area at a fixed time lag, the tire stiffness is identified by combining the identified unknown forces on the axle. Finally, the road roughness is obtained using the identified vehicle parameters and unknown forces. Numerical studies with different levels of roughness, different noise levels, and different vehicle speeds have verified the accuracy of this method in identifying vehicle parameters and road roughness.

Scanning and separating vertical and torsional–flexural frequencies of thin-walled girder bridges by a single-axle test vehicle

Fatigue life evaluation of bridge stay cables subject to monitoring traffic and considering road roughness

Measuring the bridge frequencies indirectly from an instrumented test vehicle is a potentially powerful technique for its mobility and economy, compared with the conventional direct technique that requires vibration sensors to be installed on the bridge. However, road surface roughness may pollute the vehicle spectrum and render the bridge frequencies unidentifiable. The objective of this paper is to study such an effect. First, a numerical simulation is conducted using the vehicle-bridge interaction element to demonstrate how the surface roughness affects the vehicle response. Then, an approximate theory in closed form is presented, for physically interpreting the role and range of influence of surface roughness on the identification of bridge frequencies. The latter is then expanded to include the action of an accompanying vehicle. Finally, measures are proposed for reducing the roughness effect, while enhancing the identifiability of bridge frequencies from the passing vehicle response.

Two factors are critical to the effectiveness of the vehicle scanning method for bridge frequencies. One is the frequency of the test vehicle itself. This can be eliminated by using the vehicle–bridge contact point response calculated from the vehicle response. The other is the surface roughness of the bridge, which can be removed by using the residual response of two connected vehicles. In this paper, it is demonstrated for the first time that both vehicle’s frequency and surface roughness can be simultaneously eliminated using the contact residue of two connected vehicles. Theoretically, a formulation is presented for both the contact response and residues. In the numerical study, the contact response is demonstrated to outperform the vehicle response as more bridge frequencies can be identified, while the contact residue is verified to work well for various surface roughnesses, vehicle spacings, and bridge damping ratios. For damped bridges with rough surfaces, the contact residue enables us to extract the first three bridge frequencies.

In recent years, a rapid bridge health monitoring technology has been developed using an instrumental moving vehicle. Using recorded vehicle vibration data, bridge frequencies are identified for bridge health monitoring or finite element model updating. Target bridge frequencies with significant amplitudes in the vehicle’s vibration frequency spectra are expected to be found. However, in the coupled vehicle–bridge interaction (VBI) system, bridge vibration-relevant vehicle dynamics might not be noticeable. The bridge frequency would be difficult to identify because of the potential influence of road roughness. To resolve this difficulty, a novel bridge frequency identification method is proposed to mitigate the negative effects of road roughness. First, theoretical derivations are done to ascertain the VBI system dynamic characteristics considering road surface roughness. Our findings showed that the road roughness-relevant vehicle dynamics are closely related with the traveling speed, whereas the bridge frequency remains approximately constant. Theoretical investigations indicated that cross-power spectra between vehicle dynamics at multiple moving speeds are effective to mitigate the negative effects of road roughness. Presumably, it is feasible to identify the target bridge frequency from the cross-power spectra. Both the dynamic characteristics of the VBI system and the effectiveness of the proposed method for bridge frequency identification were examined using finite element simulations and laboratory experiments. Compared to existing methods, the proposed method is widely applicable to real-world situations and difficulties.

Recently, drive-by-bridge inspection methods have attracted substantial scholarly interest; however, their practical implementation has encountered significant challenges. In engineering practice, even simply extracting bridge frequencies from ordinary or commercial vehicles appears to be difficult; components related to factors such as road roughness often dominate vehicle vibration responses. This study proposes a novel coherence-PPI (Prominent Peak Identification) algorithm based on the Bayesian framework and applies it to city bus bridge monitoring to extract bridge frequencies. The fundamental idea is to recognize the bridge frequency as a common vibration component across various vehicle runs. The algorithm comprises the following three steps: First, coherences were computed for all vehicle runs to interpret the signal relationships. Second, a Bayesian framework was established to statistically determine the threshold that can maximize the occurrence of bridge frequency. Third, the prominent peaks (PPs) were selected based on the threshold, and their distribution was counted to identify the fundamental frequency of the bridge. In contrast to the previous studies that focused on avoiding differences (e.g., by trying to reduce variation, driving in the same lane, and using the same speed), this methodology encourages the introduction of variability in drive-by measurements to filter bridge frequencies, rendering it particularly compelling for practical engineering applications. The proposed methodology was validated through numerical studies, including the Monte Carlo method, and field tests using city buses. The results demonstrated that the method can effectively eliminate the effects of road roughness, environmental noise, and vehicle parameter variations and accurately identify the bridge frequency.

Indirect measurements of bridge modal properties using instrumental passing vehicles offer a cost-effective technique for the health monitoring of numerous bridges without disrupting traffic. However, existing studies primarily rely on two-dimensional (2-D) test vehicles and bridges, failing to capture essential 3-D dynamics of real-world systems. This study addresses this gap by providing theoretical insights into the use of an instrumental 3-D two-axle passing vehicle to identify spatial mode shapes of 3-D bridges. Closed-form solutions are first derived for the dynamic responses of a 3-D two-axle vehicle interacting with a 3-D thin-walled simple beam, supported by custom-developed 3-D vehicle-bridge interaction (VBI) elements for finite element validation. A hybrid signal processing approach combining variational mode decomposition (VMD) and continuous wavelet transform (CWT) is then proposed to isolate bridge frequencies and reconstruct spatial mode shapes. Numerical examples are employed to validate the approach, examining the effects of vehicle speed, vehicle damping, and road roughness. Theoretical derivations and finite element simulations confirm that vehicle response spectra contain both vertical flexural and lateral flexural-torsional bridge frequencies. The VMD-CWT method successfully distinguishes these frequencies and retrieves spatial mode shapes, achieving modal assurance criterion values exceeding 0.994 and 0.971 for the first and second modes, respectively. Parametric studies indicate that vehicle speed minimally impacts accuracy, vehicle damping enhances frequency identification but slightly degrades mode shape reconstruction, and road roughness reduces overall precision. This study advances vehicle-based indirect measurement by enabling spatial mode identification of 3-D bridges using practical 3-D two-axle vehicles—an achievement beyond the reach of existing 2-D models.

Ride has always been an important aspect in vehicle design, driven by the customer’s increasing demand for vehicles with better comfort. The vibrational response of the vehicle is one of many factors contributing to the overall ride perception, with road inputs being the major excitation source. The improved capabilities of vehicle simulation models and virtual proving grounds have supplemented experimental prototype testing for tuning suspensions. Final tuning, and ride evaluation, is however still done through physical on-road testing. Four-post rig tests hold potential cost and time savings when used appropriately in the development process. The four-post rig imposes only vertical inputs at the tire contact patches and thus it is expected for the vertical response of the vehicle to be the dominant component. The objective of this paper is to determine whether ride comfort can sufficiently be evaluated on the four-post rig using only the vertical seat Component Ride Value (CRV). The response of an instrumented vehicle on the four-post rig was measured by a tri-axial seat pad accelerometer. Vertical, longitudinal and lateral seat CRVs as well as the seat Point Ride Value (PRV) were calculated from the measured seat acceleration in the three translational directions using the BS6841 (1987) standard. The PRV is the square root of the sums of squares of the three CRVs. The CRVs and PRV, obtained from tests at various speeds and road roughnesses, were analyzed to determine whether the vertical seat CRV is sufficient in capturing the perceived ride comfort. Results showed that the longitudinal and lateral CRVs are in excess of 26% and 63%, respectively of the dominant vertical seat CRV over the various tests. This implies that the vertical seat CRV underestimated the discomfort when compared to the seat PRV. It was also observed that the three CRVs had different sensitivities to test parameters and conditions. For example, at different speeds over the same road roughness the longitudinal CRV increased significantly more than the other two components. The differences in sensitivities may be due to the specific boundary conditions imposed on the vehicle by the four-post rig. It is concluded that the vertical seat CRV may not be sufficient to evaluate the ride comfort on a four-post rig.

Damped test vehicle for scanning bridge frequencies: Theory, simulation and experiment