Vehicle-bridge Interaction Model Research Articles

A Frequency Identification Approach for Damaged Heavy-Haul Railway Bridge with Bogie Accelerations

Bridge structural health monitoring (BSHM) is extensively employed to assess whether bridge damage exceeds maintenance limits and helps maintenance decision-making. Frequency identification is an essential part of BSHM, which utilizes the first-order modal frequency as an evaluation indicator of bridge health status. This paper proposes a method for bridge damage identification based on extracting the first-order modal frequency of the bridge from dynamic responses of vehicle bogies. Firstly, the feasibility of extracting bridge modal frequency from bogie accelerations is theoretically deduced via a vehicle–bridge interaction model. The result indicates that the bogie accelerations encompass three frequency components, namely vehicle driving frequency, vehicle pitching frequency and bridge first-order modal frequency. Then, a multi-domain conjoint analysis method incorporating the time domain, frequency domain and time–frequency domain is proposed to extract the first-order frequency of damaged bridge from bogie accelerations. By defining a damage sensitivity index of the sliding window, the time-domain sensitive boundaries of bogie acceleration to bridge damage are pinpointed. Leveraging prior knowledge of the bridge modal frequency range, the frequency-domain sensitive boundaries bogie acceleration to bridge frequency are determined. On this basis, the sensitive region of bogie acceleration to bridge frequency can be precisely localized in the Hilbert Huang transform spectrum, ultimately identifying bridge frequency through searching local maximum instantaneous energy points within this region. Finally, comprehensive experiments, considering different vehicle speeds and track spectra scenarios, are carried out to investigate the effectiveness and robustness of the proposed method. The results affirm the performance of the method to accurately identify the first-order modal frequency of the damaged bridge, further underscoring the promising prospect of on-board BSHM.

A vehicle-bridge interaction model considering contact patch size and vehicle self-generated excitation – A theoretical study

This paper presents an improved theoretical damped single-axle vehicle-bridge dynamic interaction model to consider the effect of the contact patch size and motor-induced vehicle excitation. The contact patch issue is critical as it determines the minimum time step for simulation and maximum identifiable frequency, while the inclusion of the motor-induced vehicle excitation benefits the design of autonomous self-driven rather than towed vehicles. Estimations of the contact patch size for both the pneumatic tire and solid wheel scenarios are discussed. The contact patch responses, which degenerate into contact point responses when the contact patch size is assumed to be infinitely small, were derived for the first time both from the vehicle and bridge responses to confirm their equivalence. The minimum time step, which determines maximum identifiable frequency but is arbitrarily chosen in literature, is proposed to be determined by the vehicle speed and contact patch length. The procedures to extract multiple bridge triad information including natural frequencies, mode shapes, and damping ratios from the vehicle responses are also presented. Based on extensive parametric analyses, the sinusoidal vehicle excitation becomes more prominent as its amplitude and/or frequency increase and may overshadow the analysis of bridge frequencies of interest. The vehicle acceleration leads to a more accurate extraction of bridge mode shapes and damping ratios than the vehicle displacement since the displacement is dominated by the fundamental mode of bridge vibration. The damping ratio extraction shows an average error of 0.28% from the instantaneous amplitude of the vehicle acceleration signal.

A deep‐learning framework for classifying the type, location, and severity of bridge damage using drive‐by measurements

AbstractThis paper proposes a new deep‐learning framework for drive‐by bridge condition monitoring. The proposed approach represents a bridge monitoring regime that enables the presence, type, location, and severity of bridge damage to be identified purely from measurements taken on a passing vehicle, without needing any pre‐measured training data. The computational framework adopts a numerical vehicle–bridge interaction (VBI) model, which is automatically calibrated using only the vehicle vibration measurements. The calibrated model is used to generate labeled training data, eliminating the practical difficulties associated with collecting training data for damaged bridge scenarios. The numerically simulated data are used to train a convolutional neural network that can then classify the actual damage characteristics of a bridge. The method is tested using a laboratory‐scale VBI model, and results show that the algorithm can accurately identify the presence of seized bearings and cracking in the bridge beams. The algorithm shows good accuracy when identifying the type and extent of damage. The location of bearing damage can generally be identified; however, the location of cracking is less accurate at lower damage levels. The proposed framework represents a significant improvement on existing techniques for indirect bridge monitoring.

An improved local maximum synchrosqueezing transform with adaptive window width for instantaneous frequency identification of time-varying structures

Civil engineering structures usually exhibit time-varying characteristics under operational conditions due to changes in environment and loads. The identification of instantaneous frequencies (IFs) from measured responses of time-varying structures is a challenge when using the local maximum synchrosqueezing transform (LMSST) due to the difficulty in selecting the appropriate window width. In this study, an improved LMSST with adaptive window width (ALMSST) is proposed to circumvent the limitations of LMSST. The window width of ALMSST can be adaptively determined by combining the autoregressive power spectrum-based variational modal decomposition (AR-VMD) and a window width optimization algorithm. The AR-VMD is used to decompose the multi-component signal into mono-component signals. The Rényi entropy is adopted as an evaluation index in the window width optimization algorithm for selecting the optimal window width for each mono-component signal. Therefore, ALMSST can provide a highly concentrated time–frequency (TF) representation for all mono-component signals. Two simulated signals demonstrate that the ALMSST improves the accuracy of identified IFs compared with LMSST. Numerical simulation of a three-story shear building model with time-varying stiffness shows that ALMSST can accurately identify the IFs of time-varying structures under heavy noise. A cable test with linear and sinusoidal varying tension forces and a vehicle-bridge interaction (VBI) model system with different vehicle weights are investigated to verify the applicability of ALMSST to track the IFs of time-varying structures. Numerical simulation and experimental results illustrate that ALMSST performs well in identifying the IFs of time-varying structures.

Dynamic amplification of multi-span simply-supported prestressed concrete girder viaducts subjected to multi-body heavy vehicles

High productivity freight vehicles (HPFVs) are used in many countries to address increasing freight demand. However, it is then necessary to assess the safety of existing bridge structures subjected to HPFVs because of their increasing mass and dimension. A critical part of this work is to assess the dynamic amplification of such vehicles, without undue conservatism, often based on vehicle–bridge interaction (VBI) modelling. In the VBI problem, the vehicle model is usually derived using the Principle of Virtual Work or Euler–Lagrange methods. However, HPFVs are characterized by multiple bodies and articulation points and these techniques quickly become intractable. Alongside the vehicle model, the bridge is often assumed to be a single simply-supported span, or a few continuous spans, whereas many urban bridge viaducts, in particular, have multiple prestressed concrete (PSC) girder simply-supported spans. Furthermore, these bridges typically have an upwards hog as a result of the prestress and it is hypothesized that certain combinations of vehicle, hog, span length, and number of spans, could produce a large dynamic amplification of static load effects. In this work, Kane’s Method is applied to heavy vehicle with multi-trailers combinations which provides a relatively easy, systematic, and numerical manipulation means of determining the vehicle dynamic equations. The 9-axle B-Double vehicle, common in Australia, is used as an example to illustrate the procedure of deriving the vehicle dynamic equations. We apply the resulting vehicle models in a comprehensive VBI model to consider the dynamic amplification of PSC girder viaducts. The results indicate that the studied PSC girder viaducts experience higher amplification than the equivalent single-span simply-supported bridges. Recommendations for future studies and practice, such as the number of spans in the viaduct to obtain the ‘convergence’ of dynamic amplification, are given.

Investigation of Traffic Induced Dynamic Response of a Bridge

Abstract Bridges are special structures that provide a transport connection between areas divided by a physical barrier. Vehicles passing over bridges cause vibrations in the structural elements of the bridge, which can damage these structural elements over time. The aim of our research was to investigate the vibrations generated by vehicles on Hungary’s first steel girder welded bridge, which was subjected to the maximum allowable load. In this paper, we describe the measurements we have carried out, analyse the results, present the vehicle-bridge interaction model we have constructed and the computer simulation we have carried out by using it. Our research results are intended to contribute to accident prevention and the safe operation of bridges.

Post-fire nonlinear vehicle–bridge interaction analysis considering hysteretic material behavior

Deteriorating bridges and potential vulnerabilities are crucial in the structural assessment of railroad bridges. Among various hazards that a bridge may encounter, evaluating the bridge after a fire is important because it may change the behavior of the bridge. More specifically, the post-fire bridge exhibits structural stiffness degradation, residual stress accumulation, and residual deflection. Due to those residual responses further affecting the dynamics of the bridge, analyzing the post-fire behavior of a bridge subject to normal traffic usage becomes important. Thus, this paper proposes a framework that assesses the dynamic behavior of a post-fire bridge considering the material hysteresis due to thermal loads and vehicle-bridge interaction (VBI). The proposed model is different from previous approaches in that the nonlinear VBI analysis is analyzed in an integrated system equation. So far, most related works exchange the data between separate finite element analysis software and in-house codes, intrinsically facing convergence problems. Instead, by adopting the augmented representation approach, the proposed framework becomes computationally efficient. Herein, nonlinear structural fire analysis is performed to obtain post-fire responses of the bridge. Then, the dynamic analysis is realized by incorporating material hysteretic behavior and VBI systems. The proposed framework is validated against the linear VBI model under gravity load. Subsequently, parametric studies are presented to represent various fire scenarios and long-term serviceability. The results demonstrate that the proposed model provides an effective tool for evaluating post-fire bridge dynamics by including material hysteresis under cyclic traffic loads. Moreover, the proposed framework can easily accommodate various types of hazards as well that cause the nonlinear behavior of the system.

Detect the stiffness transition in beam structures by using the passive tap-scan method

Damage detection of bridges through the response of a passing vehicle has drawn much attention in recent years. Most related researches focus on the impact of the bridge stiffness on vehicle responses. However, this paper demonstrates that the vehicle response is more sensitive to the stiffness transition in bridge girders. The theoretical basis is established on a vehicle-bridge interaction model, in which the change of an effective vehicle stiffness is introduced to represent the change of beam stiffness. The solution to this problem clearly shows that the acceleration of vehicle has an abrupt change at the stiffness transition point due to the discontinuity of the stiffness gradient. This effect will be enhanced when applying a tapping force near the vehicle resonance frequency. This tapping force can be economically generated by coordinating the vehicle velocity and the geometric pattern of toothed wheels, which was called the “passive tap-scan method” in a recent paper. This theory also reveals that the amplitude of the acceleration of vehicle is a quadratic function of the stiffness change ratio. Hence, a novel damage detection method is proposed to give a quantitative evaluation of the damage. These theoretical findings are quantitatively verified by numerical simulations and experimental tests on a standard pre-stressed concrete T-beam. Then, the practical capability of this damage detection method is further demonstrated through a field test on a real bridge.

Investigation of Spatial-Varying Frequencies Concerning Effects of Moving Mass on a Beam

The spatial-varying frequency of a vehicle-bridge interaction (VBI) system subjected to a moving mass is theoretically derived and numerically investigated through a three-dimensional VBI model, in which the effects of moving mass are introduced through the inertial force and centrifugal force in the equation of motion of the bridge. For a large vehicle-to-bridge mass ratio, it has been known that the frequency of a VBI system could change with respect to the location of a moving vehicle. As such, this study derives the analytical solution based on a moving mass-beam system to account for frequency variation and further builds the numerical model with detailed implementation for practical applications. The numerical results show the following findings: (1) The frequency of a VBI system is a function of velocity and location of a moving vehicle. (2) The reduction of spatial-varying frequency ratio for a particular mode decreases with respect to the mode of higher order. (3) The maximum reduction of spatial-varying frequency ratio of the first mode in a moving mass-beam system occurs in the location where the bridge has the maximum deflection as a result of local mode excitation. (4) For the VBI system with high suspension stiffness and large vehicle-to-bridge mass ratio, the absolute variation of spatial-varying frequency ratio of the first mode can be up to 30–40%.

Improving running safety of railway bridges during earthquakes by equipping vehicle with lateral fuse stopper

Ensuring the safety of the railway vehicles over bridges shaken by earthquakes is a practical issue in the seismic design of railway bridges. To improve the vehicle’s running safety on railway bridges during earthquakes, the stopper (or bump-stop) between the bogie and the car-body of the vehicle was modified as a fuse-type element, i.e., it was designed to break under large impact due to earthquake occurrence. The efficacy of the fuse stopper was illustrated by a full-scale vehicle-track shake table test. The fuse stopper element was introduced to the vehicle-bridge interaction model, whose accuracy was also demonstrated by comparison with the test results. The fuse stopper was designed to remain intact under service level earthquakes (SLE) to avoid frequent replacement or repair, while it should fuse to improve vehicle safety on a group of bridges under design basis earthquakes (DBE). The bridge group includes 24 simply-supported bridges with pier heights ranging from 6m to 60m, covering the most simply-supported bridges in the China railway network. A total of 10 ground motion records were used in the analysis to account for the uncertainty of ground motions. Through vehicle-bridge incremental dynamic analysis (IDA) simulation, the optimal fusion strength of 188 kN and the extended gap of 0.04 m were proposed for the fuse stopper at DBE intensity, in which the bridge group can achieve derailment-free. The design procedure of the fuse stopper may provide a cost-effective way to improve the vehicle’s running safety on a group of bridges during earthquakes.

Bridge Static Influence Line Identification Based on Structural Dynamic Responses Under High-Speed Trains

Bridge influence lines can reflect the performance status of bridge structures and are an important tool to evaluate high-speed railway bridges. Because a high-speed train can cause large dynamic response on a bridge, it is difficult to identify the static influence line of the bridge. To extract the static bridge influence lines for bridge evaluation, this paper proposes a low-pass filter design method for static influence line identification. By maximizing the difference in bridge static and dynamic energy, the relationship between the frequency domain characteristics of the bridge dynamic response and the filter parameters is established. Based on this relationship, reasonable values of the low-pass filter parameters can be determined. After filtering the dynamic response, the quasi-static response can be effectively obtained, and the bridge static influence line can be identified through the least square regularization method. Finally, a numerical example of the vehicle–bridge interaction model is used to verify the effectiveness of the proposed method. The results show that the low-pass filter designed through the proposed principle can accurately remove the dynamic effects, and its stability and robustness are good. The research results provide a reliable method for the static influence line identification of high-speed railway bridges.

Dynamic performance-based assessment for tied-arch bridges subjected to heavy multi-axial tractor-trailers

With rapid industrial development, the overloading of bridges caused by heavy multi-axle vehicles has become a critical problem worldwide. This work conducts a performance-based evaluation for tied-arch bridges subjected to heavy multi-axle vehicles, and a prediction function of the dynamic impact factor (DIF) for tied-arch bridges is proposed considering the bridge frequency, vehicle speeds, and road quality level. Specifically, the main parameters affecting the bridge response were first identified theoretically using a simplified multi-axle vehicle-bridge model. A tractor-trailer mechanical model and a tied-arch bridge model were subsequently constructed and coupled using displacement coordination conditions to explore the characteristics of the bridge response under heavy multi-axle tractor-trailers. Additionally, based on the identified factors, a detailed investigation of DIF was performed using the developed special vehicle-bridge interaction model. Finally, a suggested computational method for predicting DIF of tied-arch bridges was proposed. The results obtained indicate the impact effect of special tractor-trailers on bridges should not be ignored, even at low operating speeds and in good road conditions. Moreover, additional monitoring measures should be placed on the structural components that are sensitive to dynamic responses to regulate responses within an acceptable range.

Towards probabilistic seismic performance of vehicle‐bridge interaction systems: From stochastic dynamic model to fragility analysis

AbstractThis paper aims to assess the seismic fragility of vehicle‐bridge‐interaction (VBI) systems considering the effects of vehicle types, traffic conditions, and road surface qualities. A stochastic nonlinear mechanical model for the earthquake‐VBI system is developed, and the fragility functions for the proposed VBI model are derived by considering the relevant probabilistic seismic demand parameters. On the basis of a typical four‐span continuous prestressed concrete highway bridge in China, a complete numerical model for the VBI system is built considering multiple uncertainties from bridge and vehicle parameters, as well as the road surface qualities. A total of 120 real ground motion records with different combinations of magnitude‐source‐to‐site distance (M‐R) and earthquake intensity characteristics are selected. Meanwhile, 80 scenarios in terms of different combinations of vehicle types, vehicle speeds, and road surface irregularities are defined. In this context, 96,000 nonlinear time‐history analyses are performed, and the developed fragility models are applied to the VBI system at both component and system levels. Results indicate that the fragilities of pier drift, bearing shear strain, and the overall VBI system increase with the increase of the vehicle weight or the decrease of the vehicle speed, while the vertical deck displacement is dominated by the vehicle weight. It is also found that the road surface quality has a negligible effect on both component and system fragilities.

Fragility analysis for vehicle derailment on railway bridges under earthquakes

With the rapid development of high-speed railways around the globe, the safety of vehicles running on bridges during earthquakes has been paid more attention to. In the design of railway bridges, in addition to ensuring the safety of the bridge structure in earthquake, the vehicle safety should also be ensured. Previous studies have focused on the detailed analysis of vehicle derailment on bridges, proposing complex numerical algorithms for wheel–rail contact analysis as well as for parametric analysis, but they are inconvenient for designers. Intensity measure (IM) used in performance-based earthquake engineering is introduced in this study. A method to evaluate the vehicle safety on bridges under earthquakes is proposed with respect to the optimal IM. Then, the vehicle derailment case of the Kumamoto earthquake in Japan verifies the decoupling method of vehicle–bridge interaction model. In the assessment of vehicle derailments, eight IMs are systematically compared: the IMs of bridge deck motion are generally better than those of ground motion; the variation coefficient of spectral intensity of the bridge deck is the smallest at different frequencies. Finally, the derailment fragility cloud map is presented to evaluate the vehicle safety on bridges during earthquakes.

Improving vehicle’s seismic safety by equipping railway bridges with FPB and misalignment control device

Running safety of the railway vehicles on the bridge during earthquakes is a major concern for railway engineering. To reduce the derailment risk of railway vehicles on bridges, friction pendulum bearings (FPB) are proposed to be equipped on simply supported bridges in this study. The full nonlinear behavior of the FPB is introduced into the vehicle-bridge interaction model. The effect of FPB’s manufacturing variations, including the shear pin’s strength and friction coefficient, on the misalignment was investigated. The manufacturing variations of the FPB were found to produce large lateral misalignment, which further contributes to large wheel-rail forces when the vehicle passes over the girder ends. It diminishes the improvement in the vehicle’s seismic safety provided by FPBs. Thus, a misalignment control device is proposed to limit the misalignment of the railway bridges equipped with FPBs. The vehicle-bridge interaction analysis results show that no wheel uplift occurred on the bridge equipped with FPBs and misalignment control devices during an earthquake. It indicates that the FPB significantly reduces the vehicle’s derailment risk on bridges compared with the non-isolated bearings.

A Bayesian deep learning approach for random vibration analysis of bridges subjected to vehicle dynamic interaction

Vehicle actions represent the main operational loading for various types of bridges. It is essential to conduct random vibration analysis due to the unavoidable uncertainties arising from both the vehicle and bridge structure. This paper proposes a novel approach for the vehicle-induced random vibration analysis of bridges integrating Bayesian deep learning. The dynamic equation of the stochastic vehicle-bridge interaction system in state space form is deduced, based on which an ensemble of deep neural network is proposed to construct the surrogate model consisting of two designed functional modules, i.e., convolutional layers for excitation input feature extraction and long short-term memory (LSTM) layers for bridge response time series prediction. According to the deduced state space equation, the conventional LSTM cell is modified by introducing randomness to a portion of cell parameters. Probability distributions of the selected network parameters are then estimated by Bayesian inference, enabling the surrogate model to convey the uncertainties of the vehicle-bridge interaction system and rapidly estimate random vibration responses of the bridge. The proposed approach is applied on a railway bridge under high-speed train loading to demonstrate its efficacy. A deep Bayesian neural network is tailored and developed using the present methodology for the studied train-bridge coupling dynamic system. Time-domain statistics and frequency-domain responses of the bridge are acquired through the Bayesian deep learning model and compared with the results from a validated vehicle-bridge interaction model. Robustness of the Bayesian deep learning approach is further examined by investigating the influence of training dataset size, vehicle speed, and model input noise.

Dynamic Behavior of a Historic Metallic Bridge under Metro Vehicles Based on Advanced Interaction Models

The present article focuses on the evaluation of the dynamic behaviour of a centenary steel arch bridge, located in Portugal, under light railway traffic loads. The work aims to assess the dynamic behaviour of the bridge subjected to an alternative type of railway vehicle, more specifically, a typical underground vehicle that is currently in service in the Lisbon Metro. The dynamic response of the system has been evaluated using two distinct methodologies, namely a moving loads model and a vehicle-bridge interaction model. To achieve this goal, finite element (FE) models from both the bridge and the vehicle have been developed and a comprehensive study has been carried to evaluate the influence of distinct factors in the dynamic response of the bridge-train system, namely the methodology used to assess the dynamic response, the location of the response reference point in the deck, the train speed and the vehicle configuration (single or double vehicle). Moreover, both the traffic safety, passenger comfort and pedestrian comfort have also been evaluated using normative criteria based on acceleration responses. The results shown that the normative limits related to traffic safety and passenger comfort were never exceeded in any condition analysed in the study. However, the pedestrian comfort was jeopardized when the train speed exceeded 20 km/h.

Bridge Damping Extraction Method from Vehicle–Bridge Interaction System Using Double-Beam Model

When vehicles interact with a bridge, a vehicle–bridge interaction (VBI) system is created. The frequency and modal shape of VBI systems have been widely studied, but the damping of VBI systems has not been adequately investigated. In recent years, several incidents of abnormal bridge vibration due to changes in bridge damping have occurred and aroused widespread concern in society. Damping is an important evaluation index of structural dynamic performance. Knowing the damping ratio of a VBI system is useful for analyzing the damping changes while a bridge is in service. This paper presents a method to extract bridge damping values from a VBI system, which can serve as a guide for bridge damping evaluation. First, a double-beam theoretical model was used to simplify the VBI system for cases involving uniform traffic flow. The damping ratio equation for the simplified VBI system was obtained using the extended dynamic stiffness method (EDSM). A double-beam finite element model and a VBI finite element model were established. The damping ratios of the two models were separately calculated and then compared with the simplified VBI model results. The results verified the accuracy of the simplified method. This paper then explains that bridge damping values can be extracted by estimating the equivalent traffic flow parameters and using the damping formula for the simplified VBI system. The bridge damping ratios extracted using this method in an engineering case ranged from 0.75% to 0.78%, which is smaller than the range that was directly identified using monitoring data (0.83–1.19%). The results show that the method can effectively extract bridge damping ratios and improve damping ratio identification.

An indirect bridge frequency identification method using dynamic responses of high-speed railway vehicles

The identification of bridge frequencies using dynamic responses of moving vehicles has been studied by many researchers, however, most of them assume that the vehicle moves at a relatively low speed. In the application of high-speed railway bridges, the bridge span is typically short and the train always moves at a very high speed, which implies that the duration of the vehicle traveling on the bridge is very short and the signal recorded from the vehicle is not long enough for signal processing to extract the bridge frequency. The present study proposes a new method to identify the bridge frequency from vehicles moving at a high speed by combining the responses of multiple vehicles. The theoretical analysis is conducted using the simple vehicle-bridge interaction (VBI) model. By subtracting the acceleration responses of adjacent vehicles at the same position on the bridge, the residual vehicle responses were obtained and combined, which eliminated the additional driving frequencies and avoided its interference on the bridge frequency identification. Numerical simulations are then conducted to verify the proposed method, which shows that by combining the acceleration of several vehicles to extend the overall duration of the signal, the frequency resolution can be improved and the bridge frequency can be successfully identified using vehicles travelling at a high speed. The influence of track irregularity, vehicle speed, bridge stiffness, and bridge damping on the identification results has also been studied. In addition, a multi-degree-of-freedom vehicle model is used to further verify the proposed method.

Research on Simplified Calculation Method of Coupled Vibration of Vehicle-Bridge System

A modified moving loads’ model is proposed for the vehicle-bridge coupling vibration simulation. Taking the vehicle-bridge interaction model (VBI) as the reference, the accuracy and applicability of the three calculation models, namely, moving loads’ model, moving mass model, and spring-damper-mass model, are compared using the frequently-used railway simply-supported beam with a span of 32 meters as the research object. Influencing factors such as vehicle speed, mass ratio of vehicle and beam, and primary spring stiffness on the dynamic response of the vehicle-bridge system are discussed in detail. The results show that the moving load model has the best performance on the stability of the deviation rate, but its calculation results are smaller than the other two methods as well as the VBI. The values of the deviation rate for the moving mass model and the spring-damper-mass model are large, and the stability of those are insufficient in the range of 80%∼120% of the first resonance velocity. Except for that, the results of the two models are in good agreement with the VBI model. According to above analysis, a modified moving loads’ model with two amplification coefficients, namely, 1.10 for the range of 90%∼105% of the first resonance velocity and 1.05 for other velocities, are proposed, which has higher calculation efficiency and accuracy.