Method for determining out-of-plane modality of cable structure with lateral spring support arranged in span

Abstract

The invention discloses a method for determining an out-of-plane mode of a cable structure with a lateral spring support arranged in a span, which comprises the following steps of: 1) establishing a single-gear cable-multi-point spring constraint system constrained by a laterally arranged spring to obtain a general solution of an out-of-plane natural vibration mode of the system; 2) establishing a local coordinate system for each cable section, and obtaining a local mode function of each cable section; (3) boundary conditions are established, wherein the boundary conditions include that the displacement of the two ends of the cable is zero and the displacement at the spring support is continuous; 4) obtaining a modal function of any cable section structure according to the boundary conditions in the step 3); the method comprises the steps of (1) obtaining a force balance at a spring constraint point, (2) obtaining a constraint condition according to the force balance at the spring constraint point, (6) establishing a transcendental equation by using the force balance at the spring constraint point, and (7) solving the transcendental equation to obtain the frequency and the vibration mode of the cable structure.According to the method, a theoretical basis is provided for vibration control design of the lateral constraint cable system, and the test and numerical simulation cost is remarkably reduced.

Description

Technical Field

[0001] This invention relates to structural engineering technology, and more particularly to a method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span. Background Technology

[0002] Cable structures, due to their efficient force transmission characteristics, are widely used in long-span buildings, curtain wall systems, and engineering fields such as cable-stayed bridges and suspension bridges. However, cable structures face significant vibration problems under dynamic loads such as wind loads and earthquakes, directly affecting structural safety and durability. Existing research mostly focuses on cable systems with hinged cables at both ends of a single span or with vertical spring constraints within the span, such as Irvine's classical theory for the analysis of the natural frequency of a single-span cable, and Zhou Haijun et al.'s analysis of the dynamic characteristics of cable-spring constrained systems. However, in engineering, to suppress out-of-plane vibrations, lateral spring supports are often used to constrain the cables at multiple points, and the dynamic characteristic analysis of such systems still lacks systematic theoretical support.

[0003] The existing technology has the following limitations: 1. Insufficient model simplification: Traditional methods ignore the coupling effect of lateral spring constraints and cannot accurately reflect the complex influence of multi-point weak constraints on cable-plane vibration. 2. Limited computational accuracy: Existing studies mostly rely on numerical simulations or simplified assumptions, making it difficult to analyze the quantitative relationship between spring stiffness, position, and cable vibration; 3. Poor engineering applicability: The methods for calculating the mode shapes and frequencies of lateral spring-constrained systems are not yet perfect, which restricts the optimized design and vibration control of such structures; Therefore, an analytical method for out-of-plane modes of lateral spring-constrained cable structures is needed to provide a high-precision calculation tool for engineering practice. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for determining the out-of-plane modes of a cable structure with lateral spring supports set inside the span, in order to address the deficiencies in the prior art.

[0005] The technical solution adopted by this invention to solve its technical problem is: a method for determining the out-of-plane modes of a cable structure with lateral spring supports set within the span, comprising the following steps: 1) Based on the cable structure with lateral spring supports in the span, establish a single-span cable-multi-point spring constraint system with lateral spring constraints, and obtain the general solution of the out-of-plane natural vibration modes of the system. Suppose a single cable lies in the XOY plane under its own weight, where x is the chordal coordinate of the cable and y is the vertical coordinate of the cable when it is at rest. n-1 spring supports are placed at arbitrary positions along the out-of-plane (z-axis) direction of the single cable, with spring stiffness . The cable is divided into n segments, and the spacing between each segment is... Where 1≤k≤n; The out-of-plane modal displacement equation of the cable is:

[0006] Its general solution for:

[0007] In the formula , It is the integration constant; Let ω be the angular frequency of the cable's out-of-plane vibration, H be the horizontal tension of the cable when it is at rest under its own weight, and m be the mass per unit length of the cable. Let be the wave number of the out-of-plane vibration of the cable; 2) Establish a local coordinate system for each cable segment and obtain the local modal functions of each cable segment; 3) Establish boundary conditions, including: the displacements at both ends of the cable are 0 and the displacement at the spring support is continuous; 4) Based on the boundary conditions in step 3), obtain the modal functions of any cable segment structure; 5) Based on the force balance at the spring constraint point, the constraint conditions are obtained; 6) Establish transcendental equations using the force balance at the spring constraint points; 7) Solve the transcendental equations to obtain the frequencies and mode shapes of the cable structure.

[0008] According to the above scheme, in step 2), the local modal functions of each cable segment are as follows; For the k-th segment, 1≤k≤n, establish a local coordinate system with the origin at the left end of the segment, and its modal function is:

[0009] in, Let k be the local mode function of the k-th cable segment. and The integral constant is determined by the boundary conditions.

[0010] According to the above scheme, the boundary conditions in step 3) are as follows: ; ; ; ; ... ; ; ... ; ; in, Let be the out-of-plane modal displacement value of the k-th cable segment at its left endpoint in the local coordinate system. Let be the out-of-plane modal displacement value of the k-th cable segment at its right endpoint in the local coordinate system. Let be the out-of-plane displacement amplitude at the k-th spring support.

[0011] According to the above scheme, in step 4), the modal functions of the arbitrary cable segment structure are obtained as follows;

[0012] According to the above scheme, in step 5), based on the force balance at the spring constraint point, the tension of the spring is balanced with the z-direction component force generated at the spring end by the left and right cable segments. This results in n-1 constraints: .

[0013] According to the above scheme, in step 6), substituting the modal function from step 4) into the constraint condition from step 5), we get:

[0014] eliminate , get about The transcendental equation.

[0015] According to the above scheme, the process of step 7) is as follows: Solving the transcendental equations yields the orders of... ; Let be the wave number of the i-th out-of-plane vibration; Calculate the frequency and mode shape of the structure.

[0016] According to the above scheme, in step 7), Calculate the frequency of the structure: ; in, Let be the i-th order out-of-plane natural frequency.

[0017] According to the above scheme, in step 7), the vibration mode of the structure is calculated; Will Substituting the boundary conditions from step 6), we can solve for each... ; Substituting back into step 4), we can obtain the modal functions for any cable segment structure.

[0018] The beneficial effects of this invention are: The method of this invention can accurately calculate out-of-plane frequencies and mode shapes of each order, and clarify the quantitative relationship between spring stiffness k, span L and tension H; The method of this invention provides a theoretical basis for the vibration control design of lateral constraint cable systems, and significantly reduces the cost of experiments and numerical simulations. Attached Figure Description

[0019] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings: Figure 1 This is a flowchart of a method according to an embodiment of the present invention; Figure 2 This is a schematic diagram of a single-cable-multi-point spring constraint system according to an embodiment of the present invention; Figure 3 This is a comparison chart of the numerical calculation results of the first-order out-of-plane mode theory of the dual-spring according to an embodiment of the present invention; Figure 4 This is a comparison chart of the numerical calculation results of the second-order out-of-plane mode theory of the dual-spring according to an embodiment of the present invention; Figure 5 This is a comparison chart of the numerical calculation results of the dual-spring third-order out-of-plane mode theory according to an embodiment of the present invention; Figure 6 This is a comparison chart of the numerical calculation results of the eight-order out-of-plane mode theory of the dual springs in an embodiment of the present invention. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0021] like Figure 1 As shown, a method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span includes the following steps: 1) For a single-cable-multi-point spring constraint system with lateral spring constraints, the general solution of the out-of-plane natural modes of the system is obtained; like Figure 2 A single cable is in a state of self-weight. XOY In the plane, where x Let the coordinates be the chord coordinates of the cable. y Let be the vertical coordinate when the cable is at rest. L For the span. At any position along the surface of a single-span cable ( z n-1 spring supports are set in the axial direction, and the spring stiffness is The cable is divided into n segments, and the spacing between each segment is... express.

[0022] The partial differential equation for the out-of-plane vibration of the cable is:

[0023] In the formula For the out-of-plane displacement of the cable, H The horizontal tension of the cable when it is at rest under its own weight. m The mass per unit length of the cable, Let be the damping coefficient in the out-of-plane direction. The method of separation of variables is used to solve this problem, assuming the solution is in the form of... Ignoring the effect of damping, according to equation (1), the out-of-plane modal displacement equation of the cable and its general solution can be obtained as follows:

[0024]

[0025] Mode middle , Let be the integration constant. Let be the wave number of the out-of-plane vibration of the cable.

[0026] Assume the elongation of the spring changes with time as follows: ,in Let be the amplitude. Establish a local coordinate system for each cable segment; the modal function of any cable segment satisfies the following expression:

[0027] in, and The integral constant is determined by the boundary conditions.

[0028] Pick Substitute the boundary conditions:

[0029] in, Let be the out-of-plane modal displacement value of the k-th cable segment at its left endpoint in the local coordinate system. Let be the out-of-plane modal displacement value of the k-th cable segment at its right endpoint in the local coordinate system. Let be the out-of-plane displacement amplitude at the k-th spring support.

[0030] like , by formula The frequencies and mode shapes of the structure are as follows:

[0031]

[0032] To determine the parameters To determine the vibration modes and frequencies of the structure, it is necessary to examine the boundary conditions at the spring constraints. At the spring constraint points, the tension of the spring... The z-axis component force generated at the spring ends by the left and right cable segments Phase equilibrium leads to n-1 boundary conditions:

[0033] The formula Differentiate and substitute into the expression ,have to:

[0034] eliminate You can get information about The transcendental equations are solved to obtain , Substitution Each can be obtained , Substitution Japanese style This allows us to obtain the out-of-plane frequencies and mode shapes of each order of the continuous cable.

[0035] like ,have to:

[0036] have to ,in for The greatest common divisor, The structure's frequencies and mode shapes are:

[0037]

[0038] Figures 3 to 6 The comparison diagrams demonstrate that the modal calculations of the method of the present invention are accurate.

[0039] This invention utilizes precise modal calculations to optimize spring parameter configurations, reduce vibration risks, and extend structural lifespan. This method can be developed into a software module for modal analysis of cable structures or integrated into existing finite element platforms; it can also be combined with sensor technology to form an integrated "detection-analysis-optimization" solution for engineering vibration monitoring and reinforcement.

[0040] It should be understood that those skilled in the art can make improvements or modifications based on the above description, and all such improvements and modifications should fall within the protection scope of the appended claims.

Claims

1. A method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span, characterized in that, Includes the following steps: 1) Based on the cable structure with lateral spring supports in the span, establish a single-span cable-multi-point spring constraint system with lateral spring constraints, and obtain the general solution of the out-of-plane natural vibration modes of the system. Suppose a single cable lies in the XOY plane under its own weight, where x is the chordal coordinate of the cable and y is the vertical coordinate of the cable when it is at rest; n-1 spring supports are set at any position of the single cable along the out-of-plane direction, and the spring stiffness is . The cable is divided into n segments, and the spacing between each segment is... Where 1≤k≤n; The out-of-plane modal displacement equation of the cable is: Its general solution for: In the formula , It is the integration constant; Let ω be the angular frequency of the cable's out-of-plane vibration, H be the horizontal tension of the cable when it is at rest under its own weight, and m be the mass per unit length of the cable. Let be the wave number of the out-of-plane vibration of the cable; 2) Establish a local coordinate system for each cable segment and obtain the local modal functions of each cable segment; 3) Establish boundary conditions, including: the displacements at both ends of the cable are 0 and the displacement at the spring support is continuous; 4) Based on the boundary conditions in step 3), obtain the modal functions of any cable segment structure; 5) Based on the force balance at the spring constraint point, the constraint conditions are obtained; 6) Establish transcendental equations using the force balance at the spring constraint points; 7) Solve the transcendental equations to obtain the frequencies and mode shapes of the cable structure.

2. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span, as described in claim 1, is characterized in that... In step 2), the local mode functions of each cable segment are as follows; For the k-th segment, 1≤k≤n, establish a local coordinate system with the origin at the left end of the segment, and its modal function is: in, Let k be the local mode function of the k-th cable segment. and The integral constant is determined by the boundary conditions.

3. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span, as described in claim 1, is characterized in that... In step 3), the boundary conditions are as follows: ； ； ； ； …… ； ； …… ； ； in, Let be the out-of-plane modal displacement value of the k-th cable segment at its left endpoint in the local coordinate system. Let be the out-of-plane modal displacement value of the k-th cable segment at its right endpoint in the local coordinate system. Let be the out-of-plane displacement amplitude at the k-th spring support. and is the integration constant.

4. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span, as described in claim 2, is characterized in that... In step 4), the modal functions of the arbitrary cable segment structure are obtained as follows; 。 5. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span according to claim 1, characterized in that, In step 5), based on the force balance at the spring constraint point, the tension of the spring is balanced with the z-direction component force generated at the ends of the spring by the left and right cable segments. This results in n-1 constraints: 。 6. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span according to claim 1, characterized in that, In step 6), the modal function from step 4) is substituted into the constraint conditions from step 5) to obtain the equation: Then, by simplifying and eliminating variables, we obtain information about... The transcendental equation.

7. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span according to claim 1, characterized in that, The process of step 7) is as follows: Solving the transcendental equations yields the orders of... ; Let be the wave number of the i-th out-of-plane vibration; Calculate the frequency and mode shape of the structure.

8. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span, as described in claim 7, is characterized in that... In step 7), Calculate the frequency of the structure: ； in, Let be the i-th order out-of-plane natural frequency.

9. The method for determining the out-of-plane modes of a cable structure with lateral spring supports within its span, as described in claim 7, is characterized in that... In step 7), the vibration modes of the structure are calculated; Will Substituting the boundary conditions from step 6), we can solve for each... ; Substituting back into step 4), we can obtain the modal functions for any cable segment structure.

10. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 9.

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