A calculation method for spatial coupling dynamics of long-span arch bridges and vehicles
By establishing a large span arch bridge-vehicle space coupling dynamic model, the problem of vibration coupling between large span arch bridge and vehicle is solved, efficient and accurate calculation methods are achieved, and the effect of bridge design and vehicle riding comfort is improved.
Patent Information
- Application Number
- CN202211660574.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-12-23
- Publication Date
- 2025-08-22
- Estimated Expiration
- 2042-12-23
AI Technical Summary
The existing technology is difficult to effectively solve the problem of space coupling vibration between large span arch bridges and vehicles, resulting in a decrease in bridge fatigue durability and vehicle ride comfort. The commercial software modeling efficiency is low and the parameterization is complex, and there is a lack of theoretical models that consider bridge torsional vibration.
The spatial beam unit is used to define the local and global coordinate system transformation matrix, and the spatial dynamic model of large span arch bridge and vehicle is constructed. Combined with the road surface unevenness and vehicle vibration model, a large span arch bridge-vehicle space coupling dynamic model is established, and the programmatic calculation is performed in MATLAB, and the solution is solved using the Newmark-β method.
It improves computing efficiency and accuracy, can consider bridge torsional vibration, provides fast and efficient theoretical calculation methods for axle coupling dynamics, fills the relevant technology gaps, and improves the reliability of design optimization and state evaluation.
Smart Images
Figure CN115719009B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of vehicle-bridge coupling dynamics, and in particular to a method for calculating long-span arch bridge-vehicle spatial coupling dynamics. Background Art
[0002] Long-span arch bridges are widely used worldwide for their reasonable cost, high bearing capacity, and excellent seismic performance. However, the light weight and low stiffness of the bridge deck structure make it prone to vibration under external excitation, especially for mid-through and through-type long-span arch bridges. Furthermore, vehicles traveling on the bridge deck, affected by deck deformation and road surface unevenness, will generate coupled vibrations with the bridge. This dynamic variation in vehicle loads increases the dynamic complexity of the vehicle-bridge system, impacting bridge fatigue durability, vehicle ride comfort, and driving safety, posing challenges to the design, optimization, and condition assessment of long-span arch bridges. Therefore, studying the dynamics of long-span arch bridge-vehicle coupling systems is of great significance. Due to the structural complexity of long-span arch bridges, the mainstream research approach is to use commercial software to develop numerical models of vehicle-bridge coupling systems. However, these approaches suffer from complex model parameterization, difficulty integrating with intelligent optimization algorithms, and time-consuming solution times. Furthermore, existing theoretical models simplify spatial structures into planar structures and lack theoretical models for long-span arch bridge-vehicle coupling dynamics that address the spatial characteristics of vehicle loads and consider the torsional vibrations of the bridge. This hinders the study of the coupling mechanism of long-span arch bridges with vehicles.
[0003] In order to solve the above problems, the present invention proposes a method for calculating the spatial coupling dynamics of large-span arch bridges and vehicles. This method will promote the development of vehicle-bridge coupling vibration control technology, improve the efficiency and reliability of large-span arch bridge-vehicle spatial coupling dynamics calculations, provide theoretical support for the design optimization of large-span arch bridges, fill the gaps in related technologies, promote the development of coupling dynamics theory, and at the same time have great social and economic benefits. Summary of the Invention
[0004] To overcome the shortcomings of existing technologies and fill the gaps in related technologies, the present invention provides a method for calculating the spatial coupling dynamics of a long-span arch bridge and a vehicle. The technical solutions adopted by the present invention to solve the technical problems are as follows:
[0005] A method for calculating spatial coupling dynamics of a long-span arch bridge and a vehicle, characterized by comprising the following steps:
[0006] Step (1): Define the local coordinate system Oxyz, the reference coordinate system Ox′-y′-z′ and the global coordinate system of the spatial beam element The transformation matrix between the local coordinate system and the global coordinate system is defined as:
[0007]
[0008] In formula (1), t1 is the transformation matrix between the local coordinate system and the reference coordinate system, which is defined as formula (2);
[0009]
[0010] In formula (2),
[0011]
[0012] In formula (3), are the x, y, and z coordinate values of the spatial beam element node i in the global coordinate system;
[0013] t2 is the transformation matrix between the reference coordinate system and the global coordinate system. When the x-axis of the local coordinate system of the spatial beam element is parallel to the z-axis of the global coordinate system, it is defined as Equation (4a), otherwise it is defined as Equation (4b);
[0014]
[0015] In formula (4), α is the angle between the y-axis of the local coordinate system and the y′-axis of the reference coordinate system;
[0016] Step (2): Construct a spatial dynamic model of a long-span arch bridge. Use spatial beam elements to mesh the long-span arch bridge and establish a spatial dynamic model of the long-span arch bridge. Its dynamic equation is defined as:
[0017]
[0018] In formula (5), M b 、C b and K b is the mass matrix, damping matrix and stiffness matrix of the long-span arch bridge, x b and F b represents the node displacement and node load of the long-span arch bridge, ∪ is the standard finite element assembly method, n is the number of spatial beam elements, β1 and β2 are the Ruili damping coefficients, and are the unit displacement vector, mass matrix, stiffness matrix and load vector of the spatial beam element in the global coordinate system, respectively, and are defined as:
[0019]
[0020] In formula (6), is the node displacement vector of the spatial beam element in the local coordinate system, defined as formula (7); and are the mass matrix, stiffness matrix and unit load vector of the spatial beam element in the local coordinate system, respectively, and are defined as Equation (8), Equation (9) and Equation (10);
[0021]
[0022] In formula (8), ρ, A, and J are the density, cross-sectional area, and polar moment of inertia of the spatial beam element, respectively;
[0023]
[0024] In formula (9), E, I y and I z are the elastic modulus of the spatial beam element, the section moment of inertia in the xz coordinate plane, and the section moment of inertia in the xy coordinate plane;
[0025]
[0026] In formula (10), R w is the reaction force of the road on the wheel, N e is the shape function of the spatial beam element under the wheel load, defined as formula (11);
[0027]
[0028] In formula (11), d is the distance from the load position to the node 1 of the spatial beam element;
[0029] Step (3): Construct a vehicle vibration model and simplify the vehicle model using the lumped mass method. The vehicle vibration includes the vertical vibration of the sprung mass x v and pitch vibration θ v , front unsprung mass vertical vibration x1, rear unsprung mass vertical vibration x2, front tire mass vertical vibration x3, rear tire mass vertical vibration x4, then the vehicle dynamics model is calculated by the following formula:
[0030]
[0031] In formula (12),
[0032]
[0033] In formula (13), m v , m1, m2, and m w are sprung mass, front unsprung mass, rear unsprung mass and tire mass, J v is the sprung pitching moment of inertia, k s1 、k s2 、k w1 and k w2 are the front suspension stiffness, rear suspension stiffness, front wheel stiffness and rear wheel stiffness respectively, c s1 、c s2 、c w1 and c w2are the front suspension damping, rear suspension damping, front tire damping and rear tire damping respectively. l1 and l2 are the distances from the front suspension and rear suspension to the sprung mass center respectively. g1 and F g2 are the forces exerted by the vehicle’s own weight on the front and rear wheels, R w1 and R w2 They are the forces acting on the front and rear wheels by the bridge surface;
[0034] Step (4): Construct a road roughness model, which is defined as:
[0035]
[0036] In formula (14), r1 and r2 are the time-domain correlated random road excitations on the front and rear wheels, respectively; v is the vehicle speed; n q and n0 are the lower cutoff frequency and spatial reference frequency respectively; G q (n0) is the road roughness coefficient; ω(t) is the white noise with a power intensity of 0.5; t d is the ratio of the distance from the front wheel to the rear wheel to the vehicle speed;
[0037] Step (5): Construct a spatial coupling dynamic model of a long-span arch bridge and a vehicle. When a wheel travels on the bridge deck, the vertical displacement of the wheel y w Vertical displacement y b The relationship can be expressed as:
[0038]
[0039] Considering the effects of multiple wheels, the following wheel-bridge displacement relationship is obtained:
[0040]
[0041] In formula (16), a is the vehicle acceleration, N b is the shape function matrix of all nodes on the bridge formed by the wheel load, which is defined as formula (17):
[0042]
[0043] In formula (17), k is the number of wheel loads;
[0044] r is the road excitation applied to all wheels, which is defined as formula (18):
[0045]
[0046] Further substituting the above equations (5) and (16) into the above equation (12), we can obtain the long-span arch bridge-vehicle spatial coupling dynamic model:
[0047]
[0048] In formula (19),
[0049]
[0050] Step (6): Calculation of the spatial coupling dynamics of the long-span arch bridge and the vehicle. The spatial coupling dynamics model of the long-span arch bridge and the vehicle described in step (4) is programmed in MATLAB software and solved using the Newmark-β method to output the motion displacement, velocity and acceleration of all bridge nodes and vehicle concentrated masses.
[0051] Compared with the existing technology, the beneficial effects of the present invention are as follows: this method has accurate calculation and high calculation efficiency, can avoid the problem of inaccurate calculation of vehicle-bridge coupling dynamics caused by the plane simply supported beam theory ignoring actual torsional vibration and the differences in arch bridge structures, and provides a fast and efficient large-span arch bridge-vehicle spatial coupling dynamics theoretical calculation method, which has very broad engineering application prospects, can not only fill the gaps in related technologies, but also generate greater social and economic benefits. BRIEF DESCRIPTION OF THE DRAWINGS
[0052] Figure 1 This is a flow chart for the implementation of the long-span arch bridge-vehicle spatial coupling dynamics calculation method;
[0053] Figure 2 It is the global coordinate system, local coordinate system and reference coordinate system of the spatial beam element;
[0054] Figure 3 It is a model of a long-span arch bridge;
[0055] Figure 4 It is the long-span arch bridge model after mesh division;
[0056] Figure 5 is the vehicle vibration model;
[0057] Figure 6 It is a schematic diagram of the vehicle-bridge coupling working condition under different lanes;
[0058] Figure 7 (a) is the mid-span displacement response of the bridge under different lane conditions;
[0059] Figure 7 (b) is the vehicle sprung vibration acceleration response under different lane conditions;
[0060] Figure 8 (a) is the maximum mid-span displacement of the bridge under different vehicle speed conditions;
[0061] Figure 8 (b) is the root mean square value of the sprung vibration acceleration of the vehicle under different vehicle speed conditions;
[0062] Figure 9 This is a schematic diagram of the long-span arch bridge-multi-vehicle axle coupling working condition;
[0063] Figure 10 It is the displacement response of the bridge deck centerline along the bridge length under the long-span arch bridge-multi-vehicle bridge coupling working condition. DETAILED DESCRIPTION
[0064] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1 — Figure 10 The specific embodiments of the present invention are described in detail.
[0065] Reference Figure 1 A method for calculating the spatial coupling dynamics of a long-span arch bridge and a vehicle comprises the following steps:
[0066] Step (1): Definition Figure 2 The local coordinate system Oxyz, the reference coordinate system Ox′-y′-z′ and the global coordinate system of the spatial beam element are shown The transformation matrix between the local coordinate system and the global coordinate system is defined as:
[0067]
[0068] In formula (1), t1 is the transformation matrix between the local coordinate system and the reference coordinate system, which is defined as formula (2). t2 is the transformation matrix between the reference coordinate system and the global coordinate system. When the x-axis of the local coordinate system of the spatial beam element is parallel to the z-axis of the global coordinate system, it is defined as formula (4a). Otherwise, it is defined as (4b).
[0069]
[0070] In formula (2),
[0071]
[0072] In formula (3), are the x, y, and z coordinate values of the spatial beam element node i in the global coordinate system;
[0073]
[0074] In formula (4), α is the angle between the y-axis of the local coordinate system and the y′-axis of the reference coordinate system;
[0075] Step (2): Construct a spatial dynamic model of a long-span arch bridge. Figure 3 For a long-span arch bridge model, the spatial beam unit is used to divide the grid, and then Figure 4The meshed long-span arch bridge model shown in the figure can be used to establish a spatial coupled dynamic model of the long-span arch bridge. Its dynamic equation is defined as:
[0076]
[0077] In formula (5), M b 、C b and K b is the mass matrix, damping matrix and stiffness matrix of the long-span arch bridge, x b and F b represents the node displacement and node load of the long-span arch bridge, ∪ is the standard finite element assembly method, n is the number of spatial beam elements, β1 and β2 are the Ruili damping coefficients, and are the unit displacement vector, mass matrix, stiffness matrix and load vector of the spatial beam element in the global coordinate system, respectively, and are defined as:
[0078]
[0079] In formula (6), is the node displacement vector of the spatial beam element in the local coordinate system, which is defined as Equation (7). and are the mass matrix, stiffness matrix and unit load vector of the spatial beam element in the local coordinate system, respectively, and are defined as Equations (8), (9) and (10);
[0080]
[0081] In formula (8), ρ, A, and J are the density, cross-sectional area, and polar moment of inertia of the spatial beam element, respectively;
[0082]
[0083] In formula (9), E, I y and I z are the elastic modulus of the spatial beam element, the section moment of inertia in the xz coordinate plane, and the section moment of inertia in the xy coordinate plane;
[0084]
[0085] In formula (10), R w is the reaction force of the road on the wheel, N e is the element shape function of the spatial beam element under the wheel load;
[0086]
[0087] In formula (11), d is the distance from the load position to the node 1 of the spatial beam element;
[0088] Step (3): Construct a vehicle vibration model, such as Figure 5 As shown, the lumped mass method is used to simplify the vehicle model. After simplification, the vehicle vibration includes the vertical vibration of the sprung mass x v and pitch vibration θ v , front suspension unsprung mass vertical vibration x1, rear suspension unsprung mass vertical vibration x2, front tire mass vertical vibration x3, rear tire mass vertical vibration x4, then the vehicle dynamics model can be expressed as:
[0089]
[0090] In formula (12),
[0091]
[0092] In formula (13), m v , m1, m2, and m w are sprung mass, front unsprung mass, rear unsprung mass and tire mass, J v is the sprung pitching moment of inertia, k s1 、k s2 、k w1 and k w2 are the front suspension stiffness, rear suspension stiffness, front wheel stiffness and rear wheel stiffness respectively, c s1 、c s2 、c w1 and c w2 are the front suspension damping, rear suspension damping, front tire damping and rear tire damping respectively. l1 and l2 are the distances from the front suspension and rear suspension to the sprung mass center respectively. g1 and F g2 are the forces exerted by the vehicle's own weight on the front and rear tires, R w1 and R w2 They are the forces acting on the front and rear tires by the bridge surface;
[0093] Step (4): Construct a road roughness model, which is defined as:
[0094]
[0095] In formula (14), r1 and r2 are the time-domain correlated random road excitations on the front and rear wheels, respectively; v is the vehicle speed; n q and n0 are the lower cutoff frequency and spatial reference frequency respectively; G q (n0) is the road roughness coefficient; ω(t) is the white noise with a power intensity of 0.5; t d is the ratio of the distance from the front wheel to the rear wheel to the vehicle speed;
[0096] Step (5): Construct a spatial coupling dynamic model of a long-span arch bridge and a vehicle. When a wheel travels on the bridge deck, the vertical displacement of the wheel y w Vertical displacement y b The relationship can be expressed as:
[0097]
[0098] Considering the effects of multiple wheels, the following wheel-bridge displacement relationship is obtained:
[0099]
[0100] In formula (16), a is the vehicle acceleration, N b is the shape function matrix formed by the wheel load on all nodes on the bridge, defined as formula (17);
[0101]
[0102] In formula (17), k is the number of wheel loads;
[0103] r is the road excitation applied to all wheels, which is defined as formula (18):
[0104]
[0105] Further substituting the above equations (5) and (16) into the above equation (12), we can obtain the long-span arch bridge-vehicle spatial coupling dynamic model:
[0106]
[0107] In formula (19),
[0108]
[0109] Step (6): Calculate the spatial coupling dynamics of the long-span arch bridge and vehicle. The spatial coupling dynamics model of the long-span arch bridge and vehicle described in step (4) is programmed in MATLAB software. The Newmark-β method is used to solve the problem and output the motion displacement, velocity and acceleration of all bridge nodes and vehicle concentrated masses. The calculation proposed by the present invention can quickly and efficiently obtain the dynamic response of the vehicle-bridge coupling system. In order to analyze the influence of driving in different lanes on the dynamics of the vehicle-bridge coupling system, a design is made. Figure 6 The vehicle-bridge coupling working conditions under different lanes shown in the figure correspond to the bridge mid-span displacement and vehicle sprung vibration acceleration respectively. Figure 7 (a) and Figure 7 (b) Different driving speeds also have a significant impact on the dynamic response of the vehicle-bridge coupling system. Figure 8 (a) and Figure 8(b) The maximum displacement of the bridge mid-span and the root mean square value of the vehicle sprung vibration acceleration under different vehicle speed conditions; Figure 9 This is a schematic diagram of the long-span arch bridge-multi-vehicle bridge coupling working condition, and the corresponding bridge deck centerline displacement response along the bridge length direction is as follows: Figure 10 shown.
[0110] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.
Claims
1. A method for calculating the spatial coupling dynamics of a long-span arch bridge and a vehicle, characterized in that The following steps are involved: Step (1): Define the local coordinate system Oxyz, the reference coordinate system Ox′-y′-z′ and the global coordinate system of the spatial beam element The transformation matrix between the local coordinate system and the global coordinate system is defined as: In formula (1), t1 is the transformation matrix between the local coordinate system and the reference coordinate system, which is defined as formula (2); In formula (2), In formula (3), are the x, y, and z coordinate values of the spatial beam element node i in the global coordinate system; t2 is the transformation matrix between the reference coordinate system and the global coordinate system. When the x-axis of the local coordinate system of the spatial beam element is parallel to the z-axis of the global coordinate system, it is defined as Equation (4a), otherwise it is defined as Equation (4b); In formula (4), α is the angle between the y-axis of the local coordinate system and the y′-axis of the reference coordinate system; Step (2): Construct a spatial dynamic model of a long-span arch bridge. Use spatial beam elements to mesh the long-span arch bridge and establish a spatial dynamic model of the long-span arch bridge. Its dynamic equation is defined as: In formula (5), M b 、C b and K b is the mass matrix, damping matrix and stiffness matrix of the long-span arch bridge, x b and F b represents the node displacement and node load of the long-span arch bridge, ∪ is the standard finite element assembly method, n is the number of spatial beam elements, β1 and β2 are the Ruili damping coefficients, and are the unit displacement vector, mass matrix, stiffness matrix and load vector of the spatial beam element in the global coordinate system, respectively, and are defined as: In formula (6), is the node displacement vector of the spatial beam element in the local coordinate system, defined as formula (7); and are the mass matrix, stiffness matrix and unit load vector of the spatial beam element in the local coordinate system, respectively, and are defined as Equation (8), Equation (9) and Equation (10); In formula (8), ρ, A, and J are the density, cross-sectional area, and polar moment of inertia of the spatial beam element, respectively; In formula (9), E, I y and I z are the elastic modulus of the spatial beam element, the section moment of inertia in the xz coordinate plane, and the section moment of inertia in the xy coordinate plane; In formula (10), R w is the reaction force of the road on the wheel, N e is the shape function of the spatial beam element under the wheel load, defined as formula (11); In formula (11), d is the distance from the load position to the node 1 of the spatial beam element; Step (3): Construct a vehicle vibration model and simplify the vehicle model using the lumped mass method. The vehicle vibration includes the vertical vibration of the sprung mass x v and pitch vibration θ v , front unsprung mass vertical vibration x1, rear unsprung mass vertical vibration x2, front tire mass vertical vibration x3, rear tire mass vertical vibration x4, then the vehicle dynamics model is calculated by the following formula: In formula (12), In formula (13), m v , m1, m2, and m w are sprung mass, front unsprung mass, rear unsprung mass and tire mass, J v is the sprung pitching moment of inertia, k s1 、k s2 、k w1 and k w2 are the front suspension stiffness, rear suspension stiffness, front wheel stiffness and rear wheel stiffness respectively, c s1 、c s2 、c w1 and c w2 are the front suspension damping, rear suspension damping, front tire damping and rear tire damping respectively. l1 and l2 are the distances from the front suspension and rear suspension to the sprung mass center respectively. g1 and F g2 are the forces exerted by the vehicle’s own weight on the front and rear wheels, R w1 and R w2 They are the forces acting on the front and rear wheels by the bridge surface; Step (4): Construct a road roughness model, which is defined as: In formula (14), r1 and r2 are the time-domain correlated random road excitations on the front and rear wheels, respectively; v is the vehicle speed; n q and n0 are the lower cutoff frequency and spatial reference frequency respectively; G q (n0) is the road roughness coefficient; ω(t) is the white noise with a power intensity of 0.5; t d is the ratio of the distance from the front wheel to the rear wheel to the vehicle speed; Step (5): Construct a spatial coupling dynamic model of a long-span arch bridge and a vehicle. When a wheel travels on the bridge deck, the vertical displacement of the wheel y w Vertical displacement y b The relationship can be expressed as: Considering the effects of multiple wheels, the following wheel-bridge displacement relationship is obtained: In formula (16), a is the vehicle acceleration, N b is the shape function matrix of all nodes on the bridge formed by the wheel load, which is defined as formula (17): In formula (17), k is the number of wheel loads; r is the road excitation applied to all wheels, which is defined as formula (18): Further substituting the above equations (5) and (16) into the above equation (12), we can obtain the long-span arch bridge-vehicle spatial coupling dynamic model: In formula (20), Step (6): Calculation of the spatial coupling dynamics of the long-span arch bridge and the vehicle. The spatial coupling dynamics model of the long-span arch bridge and the vehicle described in step (4) is programmed in MATLAB software and solved using the Newmark-β method to output the motion displacement, velocity and acceleration of all bridge nodes and vehicle concentrated masses.