Double-hoist damping device and design method
By installing dampers and concentrated mass blocks between the cables and optimizing their parameters, the problem of multimodal vibration control of suspension bridges or under-deck arch bridges was solved, especially the control of in-phase vibration, which achieved an effective vibration reduction effect on double cables.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NANJING TECH UNIV
- Filing Date
- 2023-11-29
- Publication Date
- 2026-05-19
AI Technical Summary
There are challenges in controlling the multimodal vibration of suspension bridges or through-arch bridges, especially in controlling in-phase vibration. Existing measures cannot effectively increase the vibration difference between cables, resulting in poor performance of dampers in controlling multimodal vibration.
Two slings are connected by a damper, and a concentrated mass block is installed at the connection between one of the slings and the damper. By optimizing the parameters of the damper-concentrated mass block device, the difference in vibration characteristics between the slings is increased, thereby achieving control of in-phase and out-of-phase modal vibrations.
It effectively controls the multimodal vibration of the sling, enhances the energy dissipation and vibration reduction effect of the damper, and optimizes the design to meet vibration reduction requirements and reduce costs.
Smart Images
Figure CN117552318B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of cable structures and relates to vibration control technology for cable structures, especially multimodal vibration control technology for suspension bridges or under-deck arch bridges. Background Technology
[0002] In recent years, the vibration problem of suspension bridge or through-arch bridge cables has become prominent, with vibration phenomena observed in bridge cables at bridges such as the Dadaidong Bridge and the Xihoumen Bridge. In long-span suspension bridges or through-arch bridges, the cable system typically consists of several cables. Therefore, connecting vibration-damping separators / damperes between the cables is a practical method and has been applied in many engineering projects. Rigid separators are the most commonly used measure for cable vibration control; they couple multiple cables together at the connection point, enhancing the overall stiffness of the system. However, since the parameters of adjacent cables are essentially the same, the modal frequencies of in-phase vibration modes do not change compared to a single cable. Furthermore, to enhance system damping, the method of installing dampers between cables has been proposed, for example, using high-damping rubber dampers on the Akashi Kaikyo Bridge in Japan to reduce vortex-induced vibration of the cables. The strategy of installing tuned mass dampers through separators has also been used in cable vibration control research. Similarly, vibration dampers, as a special type of tuned mass damper, have also been used for cable vibration reduction in recent years. Due to the frequency sensitivity of tuned mass dampers, multiple dampers are typically required to achieve multimodal vibration control of slings. Furthermore, controlling low-frequency vibrations presents challenges, especially for devices like vibration dampers that require significant vibration acceleration to trigger. In conclusion, the aforementioned measures have both advantages and disadvantages, and multimodal vibration control of slings remains a challenge.
[0003] Cable-to-cable dampers offer an advantage in vibration reduction by eliminating the need for support structures. However, a major drawback is their inability to control in-phase vibration modes. Therefore, increasing the vibration difference between the cables connected to the damper is crucial for its energy dissipation and vibration reduction function. Theoretically, increasing the differences in length, mass per unit length, or tension between cables in a cable-damper system can increase the difference in mode shapes, but this is difficult to achieve in practice. In contrast, incorporating concentrated mass blocks on the cables is more feasible. Previous studies have analyzed the vibration of cable systems with concentrated mass, demonstrating its significant impact on modal frequencies and mode shapes. However, previous research has focused on single cables, lacking studies on multi-cable systems. Therefore, this invention proposes a damper-concentrated mass block device to suppress the vibration of two cables, outlining specific damping devices and their arrangement. Furthermore, a theoretical analysis model is established, and an optimized design method for the damper-concentrated mass block device for controlling multimodal vibrations of two cables is proposed. Summary of the Invention
[0004] To overcome the problem of cable vibration, this invention proposes a dual-cable damper-lumen mass block vibration reduction device and design method, which can effectively control multimodal vibration of cables, especially solving the problem of controlling in-phase vibration between cables.
[0005] To achieve the above objectives, the solution of the present invention is:
[0006] A dual-sling damper-concentrated mass block vibration reduction device includes a damper installed between adjacent slings and a concentrated mass block installed on one sling.
[0007] Furthermore, the two ends of the damper are connected to adjacent slings via cable clamps.
[0008] Preferably, the concentrated mass block is installed on the upper side of the cable clamp and close to the cable clamp to prevent the mass block from sliding downwards.
[0009] Furthermore, the damper is a viscous damper or a high-damping rubber damper.
[0010] Preferably, the damper is a damper with low stiffness, such as a viscous damper, whose stiffness is almost zero.
[0011] Preferably, the concentrated mass block is a circular ring, longitudinally cut into two halves along its diameter, with the inner diameter of the ring being the diameter of the cable, and is installed on the sling by connecting bolts; the mass of the concentrated mass block is changed by altering the outer diameter or height of the circular ring.
[0012] A design method for a dual-cable damper-lumen mass block vibration reduction device includes the following steps:
[0013] Step 1: Establish a system model of double cable-damper-lumen mass block, derive the characteristic frequency equation, and solve for multimodal damping;
[0014] Step 2: Considering the multimodal vibration characteristics of the suspension cable, determine the vibration reduction index and construct the objective function using multimodal damping;
[0015] Step 3: Use a multi-parameter optimization algorithm to optimize the damping coefficient, mass of the mass block, and installation position parameters of the damper-lumen mass block vibration reduction device.
[0016] Preferably, step one involves establishing a system model of a double-sling-damper-lumen mass block, and the specific steps are as follows:
[0017] The sling is numbered j (=1, 2), H j The term "sling tension" (m) represents the tension in the sling. j L represents the mass per unit length of the sling. jThe length of the sling is indicated by ; the two slings are connected by a damper, whose stiffness and damping coefficient are denoted by k and c, respectively. A concentrated mass block with mass M is located at the junction of sling 1 and the damper. Each sling is divided into left and right segments by the damper, and its length is denoted by l. p (p=1,2), the dynamic displacement of each sling segment is expressed as v j，p (x j，p x, t), where t represents time, x j，p The coordinate axes are along the slings, with the origin located at the endpoints of each sling. The equations of motion for the sling segments are as follows:
[0018]
[0019] Considering the free in-plane vibration of the sling, the solution to equation (1) is expressed as:
[0020]
[0021] Where ω is the complex frequency of the system. The variable with a tilde represents the amplitude of the corresponding time variable. Substituting equation (2) into equation (1) yields...
[0022]
[0023] in It is the complex wave number;
[0024] The solution to equation (3) has the following form:
[0025]
[0026] in This indicates the amplitude of the sling at the damper;
[0027] The internal force balance of the sling at the damper connection point is expressed as:
[0028]
[0029]
[0030] Substituting equation (2) into equations (5) and (6) yields
[0031]
[0032]
[0033] in
[0034]
[0035] The parameters of the lumped mass and damper are dimensionless.
[0036]
[0037] Finally, combining equations (7) and (8), we obtain the matrix form of the equation:
[0038] Sφ=0 (11)
[0039] in
[0040]
[0041] The coefficient matrix S is
[0042]
[0043] make
[0044]
[0045] The characteristic equation of the system can be obtained by setting the determinant of S equal to zero, i.e.
[0046]
[0047] The complex frequency ω can be numerically calculated from equation (14), and the formula for calculating the damping ratio of the nth mode of the system is as follows.
[0048]
[0049] Preferably, step two, which determines the vibration reduction index, includes the following specific steps:
[0050] The Reynolds number is a measure of the ratio of wind-induced inertial forces to viscous forces, as shown in the following formula:
[0051]
[0052] In the formula, ρ is the air density (kg / m³). 3 V is the wind speed (m / s), D is the cable diameter (m), and μ is the air viscosity (g / ms).
[0053] The Strauhal number is a dimensionless parameter, and its calculation formula is as follows:
[0054]
[0055] In the formula N s The vortex-induced frequency;
[0056] The Strouhal number remains constant over the extended range of wind speeds, while the Reynolds number remains constant at 1 × 10⁻⁶. 4 ~3×10 5 Within the specified range, the Strouhal value for the sling is set to S = 0.2;
[0057] When the vortex shedding frequency coincides with the natural frequency of the sling, vortex-induced vibration occurs in the sling. Substituting the natural frequency of the sling into equation (17), the vortex-induced wind speed V is calculated. The corresponding vortex-induced wind speed is calculated by the following formula.
[0058]
[0059] In the formula N s It is equal to the natural frequency f of the sling;
[0060] The formula for calculating the sling frequency is as follows:
[0061]
[0062] In the formula, H is the cable force (N), m is the mass per unit length (kg / m), and L is the length of the cable (m).
[0063] Once the wind speed range at the bridge site is known, the range of modes in which the suspension cable is prone to vibration is determined according to equations (18) and (19). In order to control the vortex-induced vibration of the suspension cable, the logarithmic decay rate of the corresponding mode is generally required to reach 0.01.
[0064] Preferably, step three employs a genetic optimization algorithm to optimize multiple parameters, and the specific steps are as follows:
[0065] (1) Construct the optimization objective function
[0066] The objective function is written as
[0067] minimize g(x1, x2, x3) = |ζ min -ζ obj | (20)
[0068]
[0069] In the formula, ζ min ζ is the minimum target-order modal damping ratio. obj For the target damping ratio (ζ) obj ×2π=0.01), x1, x2, x3 are the damping coefficient, mass of the mass block, and installation position parameters of the damper-lumen mass block vibration reduction device, respectively; the objective function is defined to make the minimum target modal damping ratio as close as possible to the target damping ratio, so as to meet the additional damping requirements of vibration reduction and reduce the vibration reduction cost. In other words, a small lump mass block and a lower installation height are used.
[0070] (2) Multi-parameter optimization
[0071] Based on the installation height limitations of the vibration damping device and the actual range of the mass block parameters, the upper and lower bounds of the variables in the objective optimization problem are determined. The objective optimization problem is solved using the genetic optimization algorithm (NSGA-II) to obtain the optimal damper-lumped mass block parameters.
[0072] Compared with the prior art, the beneficial effects of the present invention are as follows:
[0073] This invention proposes a vibration reduction strategy that uses a damper to connect two suspension cables, with a counterweight mass block installed at the connection point between one of the cables and the damper. The installation of the mass block increases the difference in vibration characteristics between the two adjacent cables, resulting in a displacement difference between the cable displacements at the connection points of the damper when the two cables vibrate in the same phase, thus inducing energy dissipation and vibration reduction by the damper. Through parameter optimization of the damper-mass block device, simultaneous control of in-phase and out-of-phase modal vibrations of two cables with similar dynamic characteristics can be achieved. Furthermore, considering the multimodal vibration of the cables, the damping coefficient, mass block mass, and installation position of the damper-mass block vibration reduction device are optimized. Attached Figure Description
[0074] Figure 1 This is a model of the double cable-damper-lumen mass block system of the present invention;
[0075] Figure 2 This is a schematic diagram of the actual bridge installation layout of the present invention. Detailed Implementation
[0076] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.
[0077] The present invention provides a damper-lumen mass block combined vibration reduction device, comprising a damper installed between adjacent slings and a lump mass block (e.g., a lumen mass block installed on one sling) Figure 2 (As shown). The damper-centralized mass block vibration reduction device is not limited to dual-songline vibration reduction, but can be used for multiple-songline vibration reduction; and it is not limited to songline vibration reduction, but can be installed between adjacent stay cables.
[0078] The damper and lumped mass block are optimized for multiple parameters in response to the sling's volatile modes. The specific steps are as follows:
[0079] Step 1: Establish a system model
[0080] The model of the double cable-damper-lumen mass block system is as follows: Figure 1 As shown in the model. In this model, the two cables are hinged at both ends, and their bending stiffness and inherent damping are negligible. The cables are numbered j (=1, 2), H… j The term "sling tension" (m) represents the tension in the sling.j L represents the mass per unit length of the sling. j The length of the sling is indicated. The two slings are connected by a damper, whose stiffness and damping coefficient are denoted by k and c, respectively. A concentrated mass block with mass M is located at the junction of sling 1 and the damper. Each sling is divided into left and right segments by the damper, and its length is denoted by l. p (p = 1, 2). The dynamic displacement of each sling segment is expressed as v. j，p (x j，p x, t), where t represents time, x j，p It is along the coordinate axis of the slings, with the origin located at the end of each sling, such as... Figure 1 As shown.
[0081] The equations of motion for the cable segment are as follows:
[0082]
[0083] Considering the free in-plane vibration of the sling, the solution to equation (1) can be expressed as:
[0084]
[0085] Where ω is the complex frequency of the system. The variable marked with a tilde represents the amplitude of the corresponding time variable. Substituting equation (2) into equation (1) yields...
[0086]
[0087] in It is the complex wave number.
[0088] The solution to equation (3) has the following form:
[0089]
[0090] in This indicates the amplitude of the suspender cable at the damper.
[0091] The internal force balance of the sling at the damper connection point is expressed as:
[0092]
[0093]
[0094] Substituting equation (2) into equations (5) and (6) yields
[0095]
[0096]
[0097] in
[0098]
[0099] The parameters of the lumped mass and damper are dimensionless.
[0100]
[0101] Finally, combining equations (7) and (8), we obtain the matrix form of the equation:
[0102] Sφ=0 (11)
[0103] in
[0104]
[0105] The coefficient matrix S is
[0106]
[0107] make
[0108]
[0109] The characteristic equation of the system can be obtained by setting the determinant of S equal to zero, i.e.
[0110]
[0111] The complex frequency ω can be numerically calculated from equation (14), and the formula for calculating the damping ratio of the nth mode of the system is as follows.
[0112]
[0113] Step 2: Determine the vibration reduction target
[0114] Based on the cable parameters and the wind environment at the bridge site, the cable's easily vibrating modes and modal damping parameters were determined. The most common vibration of the cable is vortex-induced vibration, which can be controlled by adding additional damping to the cable.
[0115] A key parameter describing the flow of compressible fluid around an object is the Reynolds number, which is a measure of the ratio of inertial forces to viscous forces, as shown in the following equation:
[0116]
[0117] In the formula, ρ is the air density (kg / m³). 3 V is the wind speed (m / s), D is the cable diameter (m), and μ is the air viscosity (g / ms).
[0118] The Strauhal number is a dimensionless parameter, and its calculation formula is as follows:
[0119]
[0120] In the formula N s The vortex-induced frequency is denoted as .
[0121] According to the US Federal Highway Administration's report, the Strouhal number remains constant across the extended range of wind speeds. When the Reynolds number is at 1×10⁻⁶... 4 ~3×10 5 When within the specified range, the Strouhal value of the sling can be set to S = 0.2.
[0122] When the vortex shedding frequency coincides with the natural frequency of the sling, vortex-induced vibration occurs in the sling. Substituting the natural frequency of the sling into equation (17), the vortex-induced wind speed V can be calculated. The corresponding vortex-induced wind speed is calculated by the following formula.
[0123]
[0124] In the formula N s It is equal to the natural frequency f of the sling.
[0125] The formula for calculating the sling frequency is as follows:
[0126]
[0127] In the formula, H is the cable force (N), m is the mass per unit length (kg / m), and L is the length of the cable (m).
[0128] Therefore, once the wind speed range at the bridge site is known, the range of modal orders in which the suspension cables are prone to vibration can be determined according to equations (18) and (19). To control vortex-induced vibration of the suspension cables, the logarithmic decay rate of the corresponding mode is generally required to reach 0.01.
[0129] Step 3: Multi-parameter optimization design
[0130] The damping coefficient, mass of the mass block, and installation position of the damper-lumen mass block vibration reduction device were optimized to ensure that the damping ratio of all easily vibrating modes of the suspension cable met the specification requirements (logarithmic attenuation rate not less than 0.01). A genetic optimization algorithm was used for multi-parameter optimization, and the steps are as follows:
[0131] (1) Construct the optimization objective function
[0132] The objective function is written as
[0133]
[0134] In the formula, ζ min ζ is the minimum target-order modal damping ratio. obj For the target damping ratio (ζ) obj×2π=0.01). x1, x2, and x3 are the damping coefficient, mass of the mass block, and installation position parameters of the damper-lumen mass block vibration reduction device, respectively. The objective function is defined to make the minimum target modal damping ratio as close as possible to the target damping ratio. This satisfies the additional damping requirements of vibration reduction while reducing vibration reduction costs. In other words, it uses a small lump mass block and a low installation height.
[0135] (2) Multi-parameter optimization
[0136] Determine the range of damper parameters. Based on the installation height constraints of the damping device and the actual range of parameters such as the mass of the lumped mass block, determine the upper and lower bounds of the variables in the objective optimization problem. Use the Genetic Optimization Algorithm (NSGA-II) to solve the objective optimization problem and obtain the optimal damper-lumped mass block parameters.
[0137] Example:
[0138] Based on the example description of the actual bridge suspension cables, the cable parameters used are shown in Table 1, and the bridge-ground environmental parameters are shown in Table 2.
[0139] Table 1 Sling Parameters
[0140] serial number Cable length (m) Mass per unit length (kg / m) Cable weight (kN) Cable diameter (mm) Fundamental frequency (Hz) 1 36 9.6 142 55 1.69 2 36 9.6 142 55 1.69
[0141] Table 2 Environmental Parameters
[0142] project parameter <![CDATA[ρ is the air density (kg / m 3 )]]> 1.25 V represents wind speed (m / s) 0.2～2.7 μ is the air viscosity (g / ms). <![CDATA[1.875×10 -5 ]]>
[0143] Based on the parameters in Tables 1 and 2, the vortex-induced vibration of the sling within the wind speed range can be calculated using equation (18) to determine the 1st to 5th modes. The damper-lumen mass block vibration reduction device is optimized for the easily vibrating modes.
[0144] In equation (20), ζ min =min(ζ1, ζ2, ..., ζ5), where x1 is the damping coefficient of the damper-lumen mass block vibration reduction device. x2 represents the mass of the mass block. x3 represents the installation height l1 / L1. During optimization, x1 is set to the range [0, 5], x2 to [0, 0.05], and x3 to [0, 0.5]. The initial population is set to 200, and the maximum number of generations is 100.
[0145] After optimization, The dimensional parameters of the damper-lumen mass block vibration reduction device are shown in Table 3. The damping characteristics of the first five modal modes of the sling system are shown in Table 4. As can be seen from Table 4, the damping characteristics of the first five modal modes of the sling system all meet the vibration suppression design requirements.
[0146] Table 3 Parameters of Damper-Lumped Mass Block
[0147] project Quantity Damping coefficient (Ns / m) 154.1 Mass of the mass block (kg) 16.6 Installation height (m) 10.6
[0148] Table 4 shows the damping of the first five modalities of the system.
[0149] Modal order Modal damping ratio Logarithmic decay rate 1 0.0048 0.03 2 0.0502 0.32 3 0.0176 0.11 4 0.0198 0.12
[0150]
[0151] The above description of the embodiments is provided to enable those skilled in the art to understand and apply the present invention. It will be apparent to those skilled in the art that various modifications can be made to these embodiments, and the general principles described herein can be applied to other embodiments without inventive effort. Therefore, the present invention is not limited to the above embodiments, and any improvements and modifications made by those skilled in the art based on the disclosure of the present invention without departing from the scope of the present invention should be within the protection scope of the present invention.
Claims
1. A design method for a double-cable damper-concentrated mass block vibration reduction device, characterized in that: Includes the following steps: Step 1: Establish a system model of double cable-damper-lumen mass block, derive the characteristic frequency equation, and solve for multimodal damping; The sling number is , Indicates the cable tension. Indicates the mass per unit length of the sling. Indicates the length of the sling; the two slings are connected by a damper, whose stiffness and damping coefficient are expressed as follows: and The concentrated mass block is located at the junction of sling 1 and damper, and has a mass of [missing information]. Each cable is divided into two sections by a damper, and its length is expressed as follows: The dynamic displacement of each sling segment is expressed as: , Indicates time, The coordinate axes are along the slings, with the origin located at the endpoints of each sling. The equations of motion for the sling segments are as follows: Considering the free in-plane vibration of the sling, the solution to equation (1) is expressed as: in For the complex frequency of the system, The variable with a tilde represents the amplitude of the corresponding time variable. Substituting equation (2) into equation (1) yields... in It is the complex wave number; The solution to equation (3) has the following form. in This indicates the amplitude of the sling at the damper; The internal force balance of the sling at the damper connection point is expressed as: Substituting equation (2) into equations (5) and (6) yields in The parameters of the lumped mass and damper are dimensionless. Finally, combining equations (7) and (8), we obtain the matrix form of the equation: in coefficient matrix for make The characteristic equation of the system can be obtained by letting The determinant of the matrix equals zero, that is, The complex frequency can be numerically calculated from equation (14). Then the system's first The formula for calculating the first-order modal damping ratio is as follows: Step 2: Considering the multimodal vibration characteristics of the suspension cable, determine the vibration reduction index and construct the objective function using multimodal damping; The Reynolds number is a measure of the ratio of wind-induced inertial forces to viscous forces, as shown in the following formula: In the formula air density (kg / m³) 3 ), Wind speed (m / s) The diameter of the cable is (m). Air viscosity (kg / (m*s)); The Strauhal number is a dimensionless parameter, and its calculation formula is as follows: In the formula The vortex-induced frequency; The Strouhal number remains constant over the extended range of wind speeds, while the Reynolds number remains constant at 1 × 10⁻⁶. 4 ~3×10 5 When within range, the Strauhal value of the sling is set to ; When the vortex shedding frequency coincides with the natural frequency of the sling, the sling experiences vortex-induced vibration. Substituting the natural frequency of the sling into equation (17), the vortex-induced wind speed V is calculated. The corresponding vortex-induced wind speed is calculated by the following formula. In the formula Equal to the natural frequency of the sling ; The formula for calculating the sling frequency is as follows: In the formula It is the cable tension (N). It is the mass per unit length (kg / m). It is the length of the sling (m). Once the wind speed range at the bridge site is known, the range of modal orders in which the suspension cable is prone to vibration is determined according to equations (18) and (19). In order to control the vortex-induced vibration of the suspension cable, the logarithmic decay rate of the corresponding mode is generally required to reach 0.
01. Step 3: Use a multi-parameter optimization algorithm to optimize the damping coefficient, mass of the mass block, and installation position parameters of the damper-lumen mass block vibration reduction device; Construct the optimization objective function The objective function is written as In the formula, The target modal damping ratio is the minimum. For the target damping ratio ( ), These are the damping coefficient, mass of the mass block, and installation position parameters of the damper-lumen mass block vibration reduction device, respectively. The objective function is defined to make the minimum target modal damping ratio as close as possible to the target damping ratio. This satisfies the additional damping requirements of vibration reduction while reducing vibration reduction costs. In other words, it uses a small lump mass block and a lower installation height. Multi-parameter optimization Based on the installation height limitations of the vibration damping device and the actual range of the mass block parameters, the upper and lower bounds of the variables in the objective optimization problem are determined. The objective optimization problem is solved using the genetic optimization algorithm (NSGA-II) to obtain the optimal damper-lumped mass block parameters.
2. The design method of a double-cable damper-concentrated mass block vibration reduction device as described in claim 1, characterized in that: The damper is installed between adjacent slings, and the concentrated mass block is installed on one sling.
3. The design method of a double-cable damper-concentrated mass block vibration reduction device as described in claim 2, characterized in that: The two ends of the damper are connected to the adjacent slings via cable clamps.
4. The design method of a double-cable damper-concentrated mass block vibration reduction device as described in claim 2, characterized in that: The concentrated mass block is installed on the upper side of the cable clamp and close to the cable clamp to prevent the mass block from sliding downwards.
5. A design method for a double-cable damper-concentrated mass block vibration reduction device as described in any one of claims 1 to 4, characterized in that: The damper is a viscous damper or a high-damping rubber damper.
6. A design method for a double-cable damper-concentrated mass block vibration reduction device as described in any one of claims 1 to 4, characterized in that: The concentrated mass block is a circular ring, which is longitudinally cut into two halves along its diameter. The inner diameter of the ring is the diameter of the cable, and it is installed on the sling by connecting bolts. The mass of the concentrated mass block can be changed by changing the outer diameter or height of the ring.