Simplified calculation method for dynamic characteristics of suspension bridge

Abstract

The invention relates to the field of dynamic characteristic calculation of suspension bridges, and discloses a simplified calculation method for dynamic characteristics of a suspension bridge, which comprises the following steps: respectively acquiring vertical oscillatory differential equations associated with amplitude functions of a main beam unit and a main cable unit, and respectively deducing the amplitude functions of the main beam unit and the main cable unit based on the vertical oscillatory differential equations; respectively acquiring a node force matrix and a node displacement matrix of the main beam unit and the main cable unit based on the amplitude function, and acquiring a unit dynamic stiffness matrix of the main beam and the main cable based on an association relationship between the node force matrix and the node displacement matrix; the unit dynamic stiffness matrix is assembled to obtain an overall dynamic stiffness matrix, the nonlinear characteristic value is solved to obtain dynamic characteristic parameters of the suspension bridge, and the dynamic characteristic parameters of the suspension bridge comprise natural vibration frequency and vibration mode. Form finding is not needed, the dynamic characteristics of the suspension bridge can be optimized and analyzed from the unit scale, and meanwhile the analysis precision is improved.

Description

Technical Field

[0001] The invention relates to the field of calculation of dynamic characteristics of suspension bridges, and in particular to a simplified calculation method of the dynamic characteristics of suspension bridges. Background Art

[0002] Suspension bridges have become one of the preferred options for large bridge construction due to their excellent spanning capacity, stable structure, beautiful appearance, strong adaptability and economy. Understanding and analyzing the dynamic characteristics of suspension bridges is essential to ensure the safety, durability and comfort of bridges. The dynamic characteristics of suspension bridges refer to the response characteristics of bridges under dynamic loads, which determine the behavior of bridges under dynamic loads such as wind, earthquakes, and vehicle driving.

[0003] The parameter selection of suspension bridges often involves an iterative process of solving the "initial equilibrium state" of the structure, relying on the finite element method of "main cable form finding". In this process, a large amount of modeling work is required, which is particularly inconvenient for "dynamic characteristic design". Although the current analytical method based on deflection theory does not require form finding, its simplified analysis on the span scale is difficult to reflect the impact of changes in structural stiffness on the unit scale, and its accuracy is relatively limited. Summary of the invention

[0004] The present invention aims to provide a simplified calculation method for the dynamic characteristics of a suspension bridge, which does not require form-finding and optimizes the dynamic characteristics of the suspension bridge from a unit scale to improve the analysis accuracy.

[0005] In order to achieve the above object, the present invention adopts the following technical scheme: A simplified calculation method for the dynamic characteristics of a suspension bridge includes: S1, respectively obtaining the vertical vibration differential equation of the main beam unit according to the bending stiffness, the mass per unit length, the vertical displacement and the axial force, and obtaining the vertical vibration differential equation of the main cable unit according to the bending stiffness, the horizontal force, the mass per unit length and the vertical displacement, wherein the vertical displacement is associated with the amplitude function, and the amplitude functions of the main beam unit and the main cable unit are respectively derived based on the vertical vibration differential equation; S2, based on the amplitude function, the node force matrix and the node displacement matrix of the main beam unit and the main cable unit are obtained respectively, and the unit dynamic stiffness matrix of the main beam and the main cable is obtained based on the correlation between the node force matrix and the node displacement matrix; S3, assembling unit dynamic stiffness matrices to obtain an overall dynamic stiffness matrix, solving nonlinear eigenvalues ​​to obtain dynamic characteristic parameters of the suspension bridge, wherein the dynamic characteristic parameters of the suspension bridge include natural frequency and vibration mode.

[0006] The principle and advantages of this scheme are: in actual application, the dynamic stiffness matrix of the main beam unit is introduced, the amplitude function of the main cable unit and the dynamic stiffness matrix are derived successively, and finally the influence matrix of the assembly stiffness is established, thereby forming the dynamic stiffness matrix and equation of the three-span suspension bridge. The suspension bridge dynamic characteristics optimization analysis method provided by this scheme can significantly improve the efficiency of the suspension bridge dynamic characteristics analysis while ensuring the analysis accuracy through accurate vibration modeling, comprehensive dynamic stiffness matrix construction and efficient nonlinear eigenvalue solution, which not only provides strong technical support for the design and maintenance of the bridge, but also provides reliable guarantee for the safe operation and long-term stability of the suspension bridge. Through this scheme, the dynamic characteristics of the suspension bridge can be calculated without the need for suspension bridge form finding and complex finite element modeling analysis. The method is simple and convenient, and can calculate the dynamic characteristics of the suspension bridge from the unit scale with higher accuracy; compared with the traditional analytical method that can only achieve high accuracy in structures with uniform stiffness distribution along the span, this scheme absorbs the advantages of the finite element method and the analytical method: it not only has higher accuracy and is suitable for programming, but also has a wider application prospect, so it is more suitable for the preliminary design of any suspension bridge than the analytical method.

[0007] Preferably, as an improvement, the vertical vibration differential equation of the main beam unit is:

[0008] in, is the bending stiffness of the main beam, is the mass per unit length of the main beam, is the vertical displacement of the main beam, represents the fourth-order derivative of the vertical displacement of the beam with respect to its local coordinates, represents the second-order derivative of the vertical displacement of the beam with respect to its local coordinates, represents the second-order derivative of the vertical displacement of the beam with respect to time, is the axial force.

[0009] Technical effect: It is easy to use to solve the vertical displacement function of the beam.

[0010] Preferably, as an improvement, the unit dynamic stiffness matrix of the main beam is:

[0011] in, is the unit node force of the main beam, is the bending stiffness of the main beam, is the unit dynamic stiffness matrix of the main beam, is the main beam unit node displacement, D is the main beam unit node force matrix, and C is the main beam unit node displacement matrix.

[0012] Technical effect: It is easy to establish the relationship between the main beam unit node force and node displacement.

[0013] Preferably, as an improvement, the amplitude function of the vertical vibration of the main beam unit is:

[0014] in, (i is 1, 2, 3, 4) as coefficients, and is the coefficient of x.

[0015] Technical effect: It is easy to obtain the vertical displacement function of the main beam unit.

[0016] Preferably, as an improvement, the vertical vibration differential equation of the main cable unit is:

[0017] in, is the bending stiffness of the main cable, is the horizontal force of the main cable under the action of the dead load, The additional horizontal force generated by the vibration of the main cable, is the mass per unit length of the main cable, is the vertical displacement of the main cable, is the fourth-order derivative of the vertical displacement of the main cable with respect to its local coordinates, is the second-order derivative of the vertical displacement of the main cable with respect to its local coordinates, It is the second-order derivative of the vertical displacement of the main cable with respect to time.

[0018] Technical effect: It is convenient to solve the vertical displacement function of the main cable.

[0019] Preferably, as an improvement, the amplitude function of the vertical vibration of the main cable unit is:

[0020]

[0021] in, is the coefficient, is the mass per unit length of the main cable, is the horizontal force of the main cable under the action of the dead load, is the main cable span, is the vertical span ratio of the main cable, , It is the additional horizontal force generated by the vibration of the main cable.

[0022] Technical effect: It is easy to obtain the vertical displacement function of the main cable unit.

[0023] Preferably, as an improvement, the additional horizontal force model of the main cable unit is:

[0024] in, , .

[0025] Technical effect: It is easy to obtain additional horizontal force of the single-span main cable.

[0026] Preferably, as an improvement, the vertical vibration equation of the main cable taking into account the additional horizontal force is:

[0027] in, .

[0028] Technical effect: It is easy to take the additional horizontal force into account in the main cable vertical vibration equation.

[0029] Preferably, as an improvement, the overall unit dynamic stiffness matrix of the main cable is:

[0030] in:

[0031]

[0032]

[0033] The dynamic stiffness matrix of all the units of the main cable is assembled according to the displacement coding to obtain the overall dynamic stiffness matrix of the main cable.

[0034] Technical effect: It is easy to obtain the overall unit stiffness matrix of the main cable.

[0035] Preferably, as an improvement, the overall dynamic stiffness matrix takes into account the suspension rod connection, and the overall dynamic stiffness matrix is:

[0036] in:

[0037]

[0038]

[0039] in, is the overall dynamic stiffness matrix of the main cable, is the overall dynamic stiffness matrix of the main beam.

[0040] Technical effect: It is easy to obtain the overall dynamic stiffness matrix of the suspension bridge considering the hanger connection. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a schematic diagram of the beam unit of a suspension bridge under the action of axial force; Figure 2 The span of the suspension bridge is L and the vertical span ratio is Schematic diagram of a single-span main cable; Figure 3 This is a schematic diagram of the suspender connection between the main cable and the main beam of the suspension bridge; Figure 4 Schematic diagram of the finite element model of the suspension bridge example. DETAILED DESCRIPTION

[0042] The following is further described in detail through specific implementation methods: The embodiment is basically as shown in the attached Figure 1 , Figure 2 , Figure 3 , Figure 4 As shown: A simplified calculation method for the dynamic characteristics of a suspension bridge includes: S1, approximate the dead load line of the main cable to a parabola, and discretize the main beam and the main cable as a whole to obtain the main beam unit and the main cable unit, respectively obtain the vertical vibration differential equation of the main beam unit according to the bending stiffness, mass per unit length, vertical displacement and axial force, and obtain the vertical vibration differential equation of the main cable unit according to the bending stiffness, horizontal force, mass per unit length, and vertical displacement. The vertical displacement is associated with the amplitude function, and the amplitude functions of the main beam unit and the main cable unit are derived based on the vertical vibration differential equation. Specifically: Based on axial force Under the action, the length is The vertical vibration differential equation of the Euler Bernoulli beam is:

[0043] in, is the bending stiffness of the main beam, is the mass per unit length of the main beam, is the vertical displacement of the main beam, represents the fourth-order derivative of the vertical displacement of the beam with respect to its local coordinates, represents the second-order derivative of the vertical displacement of the beam with respect to its local coordinates, represents the second-order derivative of the vertical displacement of the beam with respect to time, is the axial force.

[0044] The vertical displacement of the main beam is:

[0045] in, is the amplitude function of the main beam at coordinate x, i represents virtuality and reality, is the circular frequency, t is the time. In order to solve the vertical vibration differential equation of the main beam, it is necessary to set the general solution first. Therefore, let

[0046] Among them, A and r are unknown variables to be determined.

[0047] That is, the amplitude function of the vertical free vibration of the main beam unit can be obtained, and the amplitude function is:

[0048] In the above formula, (i is 1, 2, 3, 4) as coefficients, and is the coefficient of x. Rewriting it into vector form is:

[0049] Among them, A is the unknown number to be found, and A are the vector forms of the two matrices in the formula.

[0050]

[0051]

[0052]

[0053] The vertical vibration differential equation of the main cable unit is:

[0054] in, is the bending stiffness of the main cable, is the horizontal force of the main cable under the action of the dead load, The additional horizontal force generated by the vibration of the main cable, is the mass per unit length of the main cable, is the vertical displacement of the main cable, is the fourth-order derivative of the vertical displacement of the main cable with respect to its local coordinates, is the second-order derivative of the vertical displacement of the main cable with respect to its local coordinates, is the second-order derivative of the vertical displacement of the main cable with respect to time, The y-coordinate of the main cable in the parabolic shape.

[0055] Based on the parabola assumption, the horizontal force of the main cable is:

[0056] Among them, g is the acceleration of gravity and e is the vertical span ratio of the main cable.

[0057] Separating variables and considering the vertical displacement of the main cable is as follows:

[0058] in, It represents the x-coordinate of the j-th main cable after the main cable is divided into several units.

[0059] Substituting it into the vertical vibration differential equation of the main cable unit, we get:

[0060] in, express The second order inverse of express The fourth order inverse of It is the additional horizontal force generated by the vibration of the main cable.

[0061] For the homogeneous equation above, let , then:

[0062] In the above formula , , the solution of the homogeneous equation is:

[0063]

[0064] After adding the general solution to the special solution, we get the formula The solution is:

[0065] In the above formula (i is 1, 2, 3, 4) as coefficients, , , is the vector form of the three matrices in the formula.

[0066] When the main cable has a small cross section and the bending stiffness of the main cable does not need to be considered, the vertical free vibration differential equation of the main cable is:

[0067] Similarly, the solution to this equation is:

[0068] Its vector form is:

[0069] in, (i is 1, 2) is the coefficient, and Represents the vector form of the matrix in the formula, is the mass per unit length of the main cable, is the horizontal force of the main cable under the action of the dead load, is the main cable span, is the vertical span ratio of the main cable, , It is the additional horizontal force generated by the vibration of the main cable.

[0070] S2, based on the amplitude function, obtain the node force matrix and node displacement matrix of the main beam unit and the main cable unit respectively, and obtain the unit dynamic stiffness matrix based on the correlation between the node force matrix and the node displacement matrix. Specifically: The amplitude function of the main beam is derived to obtain the relationship between the node displacement vector and the node displacement matrix of the main beam unit as follows:

[0071] In addition, the expressions of shear force and bending moment of the main beam are:

[0072] At both ends of the main beam unit ij The shear force and bending moment at the node are:

[0073] Separating the time term from the above equation, we get the node force matrix of the main beam element: .

[0074]

[0075] in , ,

[0076] The nodal force of the main beam element is: .

[0077] By combining the main beam unit node displacement vector and the nodal displacement matrix The relationship between the main beam unit and the node force matrix , and the node displacement of the main beam unit is obtained and nodal forces The relationship function of , that is, the unit dynamic stiffness matrix of the main beam is: ; Based on the additional horizontal force calculation formula given by Irvine, the additional horizontal force model of the single-span main cable after structural discretization is derived as follows:

[0078] Substituting the main cable unit amplitude function derived in the previous article, the additional horizontal force model of the main cable is obtained as follows:

[0079] in, is the main cable span, represents the span of the i-th section after the main cable is discretized, , The additional horizontal force generated by the vibration of the main cable, is the coefficient.

[0080]

[0081]

[0082] in, , E is the elastic modulus.

[0083] The calculation formula for the additional horizontal force of the main cable is shifted and multiplied by , and then bring it into the corresponding amplitude function to get the dynamic stiffness matrix model of the main cable:

[0084] in: , is the coefficient, , Represents the span of the jth section after the main cable is discretized.

[0085] S3, assembling the unit dynamic stiffness matrix to obtain the overall dynamic stiffness matrix, solving the nonlinear eigenvalue to obtain the dynamic characteristic parameters of the suspension bridge, the dynamic characteristic parameters of the suspension bridge including the natural frequency and vibration mode. Specifically: After obtaining the unit dynamic stiffness matrix of all main beam units, the overall stiffness matrix assembly method in the finite element method is assembled and the boundary conditions are introduced to obtain the overall dynamic stiffness matrix of the main beam. for:

[0086]

[0087] right The negative elements on the main diagonal of the diagonal matrix obtained by LDLT decomposition are ; By solving the frequency value when both ends of the unit are fixed, calculate the number of fixed end frequencies of all main beam units Then sum up to get the total fixed end frequency number J0 of the structure, combined with the LDLT decomposition of the overall dynamic stiffness matrix , the natural frequency of the main beam is calculated using the Wittrick-Williams algorithm.

[0088] Establish the relationship between the node displacement matrix and the node force matrix of the single-span main cable unit:

[0089] in:

[0090]

[0091]

[0092] It is the horizontal force of the main cable under the action of dead load.

[0093] The above formula can be abbreviated as: .

[0094] Represents the force vector of all unit nodes and the displacement vectors of all unit nodes The relationship between , so the overall dynamic stiffness matrix of the main cable is: ; The above formula is an implicit expression of the overall dynamic stiffness matrix of the main cable. When there are many discrete units, the operation speed of using the implicit expression to form the overall dynamic stiffness matrix is ​​slow. Perform elementary transformation of the block matrix to obtain the explicit expression of the overall dynamic stiffness matrix of the main cable and then insert it into , the dynamic stiffness matrix of all units of the main cable is obtained as:

[0095] in: ;

[0096]

[0097] To form the overall dynamic stiffness matrix of the main cable, it is necessary to assemble all unit dynamic stiffness matrices according to displacement coding. No. Line and Add the columns to OK Column, then remove the Line and Columns, from 2 to Perform the above operations on nodes and After the above processing and addition, the overall dynamic stiffness matrix of the main cable can be obtained: :

[0098]

[0099]

[0100]

[0101] After obtaining the overall dynamic stiffness matrix of the main cable, the number of negative elements on the main diagonal of the diagonal matrix obtained by LDLT decomposition is ; By solving the frequency value when both ends of the unit are fixed, the number of fixed end frequencies of all beam units is calculated and then summed up, the total number of fixed end frequencies of the structure J0 is obtained, combined with the LDLT decomposition of the overall dynamic stiffness matrix , the natural frequency of the main beam is calculated using the Wittrick-Williams algorithm.

[0102] Obtain the overall dynamic stiffness matrix of the main cables and main beams and After that, the block diagonal matrix is ​​formed to obtain the overall dynamic stiffness matrix without considering the hanger connection. :

[0103] The hanger is considered as a massless spring connection with The axial stiffness of the hanger at , is the node displacement, then the main cable and main beam are at the node The vertical forces generated by the stretching of the hanger at are:

[0104]

[0105] After synthesizing all the boom forces into a matrix form and arranging them, the matrix of the boom influence is:

[0106] Combination Matrix and The overall dynamic stiffness matrix expression considering the hanger connection stiffness is obtained as follows: .

[0107] Taking a three-span suspension bridge as an example, the accuracy of a simplified calculation method for the dynamic characteristics of a suspension bridge is verified. The spans of the side cables and the middle cables of the suspension bridge are , The vertical span ratio of the side span is 0.038, and that of the middle span is 0.112. The spacing of the hangers is 24 m, and they are arranged equidistantly along the span direction, with a total of 105 hangers. The height from the anchor to the top of the tower is 175 m. The cross-sections and material properties of the main components are shown in Table 1.

[0108] Table 1 Cross-section and material properties of main components of suspension bridge

[0109] The dynamic characteristics of the bridge are calculated by using the finite element method, analytical method and dynamic stiffness method. The comparison of the natural frequency results of the three methods is shown in Table 2. Among them, the dynamic stiffness method is solved by the inverse power iteration method, and the amplitude with the largest absolute value is used to normalize the vibration mode.

[0110] Table 2 Comparison of natural frequency of finite element method, analytical method and dynamic stiffness method

[0111] As shown in Table 2, the results of the dynamic stiffness method and the finite element method are in good agreement, with the average absolute value of the error being 1.4395%; the average absolute value of the error between the analytical model and the finite element model is 1.555%. In addition, the frequency errors of most dynamic stiffness methods are close to or slightly lower than those of the analytical method. The dynamic stiffness method and the analytical method have similar errors compared to the finite element method because both use the parabola assumption and thus have similar initial state deviations. However, the dynamic stiffness method has a smaller error than the analytical method because the latter solves the amplitude function from the span dimension by assuming that the main beam and main cable meet their boundary conditions, while the former solves the amplitude function of the unit directly from the unit dimension by solving the dynamic differential equation, thereby obtaining a finer-scale unit dynamic stiffness matrix. It can be seen that in the preliminary design, although the proposed dynamic stiffness method and traditional analytical method are both superior to the finite element method, can avoid form-finding analysis and have higher efficiency, the former can maintain higher accuracy by using unit division, while the latter can only achieve higher accuracy in structures with uniform stiffness distribution along the span. This shows that the dynamic stiffness method of suspension bridge absorbs the respective advantages of the finite element method and the analytical method: it not only has higher accuracy and is suitable for programming, but also has a wider application prospect, and is therefore more suitable for the preliminary design of any suspension bridge than the analytical method.

[0112] The above is only an embodiment of the present invention, and the common knowledge such as the known specific technical solutions and / or characteristics in the solution is not described in detail here. It should be pointed out that for those skilled in the art, without departing from the technical solution of the present invention, several modifications and improvements can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicality of the patent. The scope of protection required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the specification can be used to interpret the content of the claims.

Claims

1. A simplified calculation method for the dynamic characteristics of a suspension bridge, characterized in that: include: S1, respectively obtaining the vertical vibration differential equation of the main beam unit according to the bending stiffness, the mass per unit length, the vertical displacement and the axial force, and obtaining the vertical vibration differential equation of the main cable unit according to the bending stiffness, the horizontal force, the mass per unit length and the vertical displacement, wherein the vertical displacement is associated with the amplitude function, and the amplitude functions of the main beam unit and the main cable unit are respectively derived based on the vertical vibration differential equation; S2, based on the amplitude function, the node force matrix and the node displacement matrix of the main beam unit and the main cable unit are obtained respectively, and the unit dynamic stiffness matrix of the main beam and the main cable is obtained based on the correlation between the node force matrix and the node displacement matrix; S3, assembling unit dynamic stiffness matrices to obtain an overall dynamic stiffness matrix, solving nonlinear eigenvalues ​​to obtain dynamic characteristic parameters of the suspension bridge, wherein the dynamic characteristic parameters of the suspension bridge include natural frequency and vibration mode.

2. A simplified calculation method for the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The vertical vibration differential equation of the main beam unit is: in, is the bending stiffness of the main beam, is the mass per unit length of the main beam, is the vertical displacement of the main beam, represents the fourth-order derivative of the vertical displacement of the beam with respect to its local coordinates, represents the second-order derivative of the vertical displacement of the beam with respect to its local coordinates, represents the second-order derivative of the vertical displacement of the beam with respect to time, is the axial force.

3. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1 is characterized in that: The unit dynamic stiffness matrix of the main beam is: in, is the unit node force of the main beam, is the bending stiffness of the main beam, is the unit dynamic stiffness matrix of the main beam, is the main beam unit node displacement, D is the main beam unit node force matrix, and C is the main beam unit node displacement matrix.

4. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1 is characterized in that: The amplitude function of the vertical vibration of the main beam unit is: in, (i is 1, 2, 3, 4) as coefficients, and is the coefficient of x.

5. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The vertical vibration differential equation of the main cable unit is: in, is the bending stiffness of the main cable, is the horizontal force of the main cable under the action of the dead load, The additional horizontal force generated by the vibration of the main cable, is the mass per unit length of the main cable, is the vertical displacement of the main cable, is the fourth-order derivative of the vertical displacement of the main cable with respect to its local coordinates, is the second-order derivative of the vertical displacement of the main cable with respect to its local coordinates, It is the second-order derivative of the vertical displacement of the main cable with respect to time.

6. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The amplitude function of the vertical vibration of the main cable unit is: in, is the coefficient, is the mass per unit length of the main cable, is the horizontal force of the main cable under the action of the dead load, is the main cable span, is the vertical span ratio of the main cable, , It is the additional horizontal force generated by the vibration of the main cable.

7. The simplified calculation method for the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The additional horizontal force model of the main cable unit is: in, , .

8. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The vertical vibration equation of the main cable taking into account the additional horizontal force is: in, .

9. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The overall unit dynamic stiffness matrix of the main cable is: in: The dynamic stiffness matrix of all the units of the main cable is assembled according to the displacement coding to obtain the overall dynamic stiffness matrix of the main cable.

10. The simplified calculation method of the dynamic characteristics of a suspension bridge according to claim 1, characterized in that: The overall dynamic stiffness matrix is: in: in, is the overall dynamic stiffness matrix of the main cable, is the overall dynamic stiffness matrix of the main beam.

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