Vacuum tube magnetic levitation transportation line shape design optimization method

Abstract

The present application relates to a kind of vacuum tube maglev traffic line alignment design optimization method.The existing design system cannot meet the design requirements of superhigh-speed vacuum tube maglev traffic line.This method establishes vehicle motion force analysis model, obtains the calculation formula of guiding force and levitation force of any alignment;According to the alignment characteristic parameters of vehicle parameters and different alignment, the guiding force and levitation force of different alignment are calculated;Select the alignment combination that changes relatively gently, optimize the alignment characteristic parameters of different alignment;Establish the dynamic interaction model of vehicle and guide rail beam, calculate the dynamic response index of each alignment combination;Compare the dynamic response index and statistical distribution characteristics of each alignment combination, obtain the optimal alignment combination.This method can accurately consider the comfort and safety variation law under superhigh-speed running condition, realize the parameter determination and design scheme optimization of vacuum tube maglev traffic line, and provide protection for the engineering application of superhigh-speed vacuum tube maglev traffic.

Description

Technical Field

[0001] This invention relates to the field of railway engineering technology, specifically to a method for optimizing the route design of a vacuum tube maglev transportation line. Background Technology

[0002] Vacuum tube ultra-high-speed maglev transportation systems utilize the vacuum environment and supersonic shape to reduce air resistance, and maglev to reduce frictional resistance, theoretically enabling supersonic operation. Ultra-high-speed vacuum maglev transportation systems possess characteristics such as ultra-high speed, high safety, low energy consumption, low noise, and low pollution, making them a current research hotspot and trend both domestically and internationally, with broad future development prospects. Currently, domestic and international vacuum tube transportation technology solutions mainly include the Swiss SWISSMETRO scheme using electromagnetic levitation, the American Hyperloop system using permanent magnet levitation, and the Southwest Jiaotong University scheme using high-temperature superconducting maglev technology. It is worth noting that current research on pipeline engineering technology mainly focuses on the load-bearing structure, lacking systematic analysis of alignment design and optimization.

[0003] Currently, a design system for high-speed conventional maglev transportation lines with speeds of 600 km / h and below has been established, and corresponding maglev transportation design specifications and technical specifications have been formulated. The line design methods and parameter values ​​are determined based on comfort and the guiding characteristics of conventional levitation, and the design scheme is optimized by comprehensively considering comfort and safety. For vacuum tube maglev transportation with operating speeds up to 1000 km / h, the requirements for driving safety and operational comfort will be significantly different. Existing line design methods and parameter values ​​cannot meet the requirements of ultra-high-speed vacuum tube maglev transportation systems, and there is still a gap in this area. Summary of the Invention

[0004] The purpose of this invention is to provide a method for optimizing the route design of vacuum tube maglev transportation lines, so as to solve the problem that existing methods cannot meet the design requirements of ultra-high-speed vacuum tube maglev transportation lines.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for optimizing the route shape of a vacuum tube maglev transportation line, wherein the route shape includes straight lines, circular curves, and transition curves, and several route shape combinations are proposed as design schemes, wherein the route shape combinations include continuous different route shapes;

[0007] The method includes:

[0008] Establish a vehicle motion force analysis model to obtain the calculation formulas for guiding force and suspension force for arbitrary alignment;

[0009] Based on vehicle parameters and the characteristic parameters of different alignments, calculate the guiding force and suspension force for different alignments;

[0010] Based on the calculation results of guiding force and suspension force, a relatively gradual change in line shape combination is selected, and the line shape characteristic parameters of different line shapes are optimized.

[0011] Establish a dynamic interaction model between the vehicle and the guide beam, and calculate the dynamic response indexes for each linear combination;

[0012] By comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the optimal linear combination can be obtained.

[0013] Furthermore, a vehicle motion force analysis model is established to obtain the calculation formulas for guiding force and suspension force for arbitrary alignment, including:

[0014] The vehicle is simplified as a rigid body rotating about the longitudinal axis of the forward direction. The force analysis model of the vehicle's motion can be obtained by the theorem of motion of the center of mass and the theorem of angular momentum about the center of mass. Based on the force analysis model of the vehicle's motion, the calculation formulas of guiding force and suspension force are derived.

[0015] Furthermore, the formula for calculating the guiding force is:

[0016]

[0017] The formula for calculating levitation force is:

[0018]

[0019]

[0020] In the formula:

[0021] F D This refers to the lateral guiding force exerted on the wheelset along the top surface of the rail.

[0022] m represents the vehicle mass;

[0023] D is the distance from the center of mass to the top surface of the orbit;

[0024] H is the actual set ultra-high value;

[0025] R S The radius of the vertical curve;

[0026] k S The curvature of the vertical curve;

[0027] g is the acceleration due to gravity;

[0028] S represents the track gauge;

[0029] k p Curvature of a plane curve;

[0030] v represents the vehicle's speed;

[0031] F C1 FC2 The levitation force perpendicular to the top surface of the rail is exerted on the inner and outer wheels;

[0032] ρ0 is the radius of inertia of the vehicle about its center of mass.

[0033] Furthermore, based on the calculation results of guiding force and levitation force, a relatively gradual change in alignment combination is selected, and the alignment characteristic parameters of different alignments are optimized, including:

[0034] Select line combinations with relatively gentle changes;

[0035] By using fixed threshold values ​​for guiding force and levitation force, the values ​​of the radius of the circular curve and the length of the transition curve are calculated in reverse, thereby optimizing the linear characteristic parameters.

[0036] Furthermore, the dynamic interaction model between the vehicle and the guide beam includes a vehicle dynamics model and a guide beam dynamics model;

[0037] The vehicle body, bogie, and magnet device are all considered as rigid bodies, and the suspension system is approximated as a spring-damping element. Each rigid body is considered to have heave, pitching, lateral movement, head-shaking, and roll motions. The vehicle motion equation is established based on the multi-rigid-body dynamics theory, and the additional forces caused by the alignment and track irregularities are considered to obtain the vehicle dynamics model.

[0038] Based on the influence of structural vibration, the beam vibration equation is established using the modal method or finite element method to obtain the dynamic model of the guide rail beam.

[0039] Based on experiments and magnetic field analysis, mathematical relationships between suspension gap and suspension force, as well as between guide gap and guide force, are established. The vehicle dynamics model is linked to the guide rail beam dynamics model. The motion equations of the vehicle-guide rail dynamic interaction system are solved by cross-iteration numerical integration, thus obtaining the vehicle-guide rail beam dynamic interaction model.

[0040] Furthermore, the dynamic response indices for each linear combination are calculated, including:

[0041] Using a dynamic interaction model between the vehicle and the guide beam, the dynamic response indices for each linear combination are calculated. These indices include vehicle body vibration acceleration, suspension force, guiding force, suspension clearance, guiding clearance, guide beam deformation, and guide beam vibration acceleration.

[0042] Furthermore, by comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the optimal linear combination is obtained, including:

[0043] Statistical analysis of the time history results of the dynamic response indices of each linear combination yields the statistical distribution characteristics of each index in different segments of the combination, including maximum and minimum values, average value, standard deviation, 99% probability value, and 95% probability value.

[0044] By comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the linear combination with the smallest statistical distribution characteristic value and the best dynamic performance is selected as the optimal linear combination.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0046] The design scheme for ultra-high-speed vacuum tube maglev transportation lines determined using this method can accurately consider the changes in comfort and safety under ultra-high-speed operation conditions, realize the determination of vacuum tube maglev transportation line parameters and optimization of design schemes, and provide a guarantee for the engineering application of ultra-high-speed vacuum tube maglev transportation. Attached Figure Description

[0047] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other embodiments can be obtained from these drawings without creative effort.

[0048] Figure 1 This is a force analysis diagram of vehicles on a curved section.

[0049] Figure 2 This is a side view of the dynamic interaction model between the vehicle and the guide beam.

[0050] Figure 3 This is a cross-sectional view of the dynamic interaction model between the vehicle and the guide beam.

[0051] Figure 4 This is a flowchart of the method of the present invention.

[0052] Figure 5 This is the line cross-section design drawing for Example 2.

[0053] Figure 6 This is the longitudinal section design drawing of Example 2.

[0054] Figure 7 This is a time history curve diagram of Scheme 1 in Example 2.

[0055] Figure 8 This is a time history curve diagram of Scheme 2 in Example 2.

[0056] Figure 9 This is a time history curve diagram of Scheme 3 in Example 2.

[0057] Figure 10 This is a chart comparing the statistical data of the three options. Detailed Implementation

[0058] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0059] It should be noted that similar reference numerals and letters indicate similar items; therefore, once an item is defined in one embodiment, it does not need to be further defined and explained in subsequent embodiments. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or apparatus that includes a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0060] Vacuum tube line engineering must possess high safety, sealing, and reliability. The line shape is closely related to the load-bearing structure of the vacuum tube maglev transportation system. Overall, the line shape has the following characteristics: (1) The structure of the vacuum tube ultra-high-speed maglev line must meet the requirements of train operation at speeds of 1000 km / h. The load-bearing structure of the line must achieve high precision during manufacturing, installation, and maintenance, and the smoothness of the line and structural deformation must be strictly controlled. (2) The train acts directly on the load-bearing beam of the line through levitation force and guiding force. Changes in the line shape can be more clearly reflected in the load-bearing structure, and the acceleration and impact caused by the line shape will be more significant. (3) High-precision manufacturing and positioning technology makes it possible to realize and maintain complex lines on the load-bearing structure.

[0061] Key parameters for the route shape design of vacuum tube maglev transportation lines include the type of transition curve, the minimum curve radius, and the minimum length of the transition curve. For vacuum tube maglev transportation with operating speeds up to 1000 km / h, the requirements for driving safety and operational comfort will be significantly increased. Existing wheel-rail railway line design methods and parameter values ​​cannot meet the requirements of ultra-high-speed vacuum tube maglev transportation systems, and there is still a gap in this area.

[0062] Example 1:

[0063] This embodiment provides a method for optimizing the route design of a vacuum tube maglev transportation line. The route shape includes straight lines, circular curves, and transition curves. Several route shape combinations are proposed as design schemes, and these combinations include continuous different route shapes. When designing the route, straightness should be the primary consideration, striving to include longer straight sections to shorten the route length and improve operating conditions. However, to adapt to the terrain and avoid features to reduce engineering work and investment, circular curves and transition curves of a certain length must be included. Straight lines offer advantages such as good visibility, simple driving force, clear direction, and ease of surveying. Their characteristic parameters include the minimum length of the straight section. Circular curves are characterized by easy adaptation to terrain, aesthetic appeal, and ease of surveying. Their characteristic parameters include the minimum curve radius and curve length. Transition curves feature continuous curvature changes, gradual changes in centrifugal acceleration, and gradual changes in superelevation and widening. Their characteristic parameters include the minimum length of the transition curve and the type of transition curve.

[0064] The method specifically includes the following steps:

[0065] S1: Establish a vehicle motion force analysis model to obtain the calculation formulas for guiding force and suspension force for arbitrary alignment.

[0066] After entering the transition curve, the vehicle exhibits various motion postures, including roll, pitching, and yaw. Research has found that the total angular displacement of pitching on the transition curve is significantly smaller than that of yaw and roll. The wheel-rail forces associated with yaw are three orders of magnitude smaller than those causing roll. Therefore, the pitching and yaw motions can be ignored, and the vehicle can be simplified as a rigid body rotating about its longitudinal axis in the direction of travel. The force analysis of any ultra-high-speed vehicle is as follows: Figure 1 As shown, the side closer to the center of curvature is defined as the inner side.

[0067] The vehicle is simplified as a rigid body rotating about the longitudinal axis of the forward direction. The force analysis model of the vehicle's motion can be obtained by the theorem of motion of the center of mass and the theorem of angular momentum about the center of mass. Based on the force analysis model of the vehicle's motion, the calculation formulas of guiding force and suspension force are derived.

[0068] The force analysis model for vehicle motion is as follows:

[0069] mgsinα+F D =m(k p v 2 cosα+Lα”cosθ+L(α') 2 sinθ)

[0070] mgcosα-F C1 -F C2 =m(-k) p v 2 sinα-Lα”sinθ+L(α') 2 cosθ)

[0071]

[0072] In the formula:

[0073] sinα≈α=HR s k s / S

[0074] cosα≈1

[0075] sinθ=S / (2L)

[0076] cosθ=D / L

[0077] α' = HR s k′ s / S

[0078] α” = HR s k″ s / S

[0079] The vehicle parameters and linear characteristic parameters in the formula include:

[0080] m represents the vehicle mass;

[0081] g is the acceleration due to gravity;

[0082] α is the transverse slope angle;

[0083] F D This refers to the lateral guiding force exerted on the wheelset along the top surface of the rail.

[0084] k p Curvature of a plane curve;

[0085] v represents the vehicle's speed;

[0086] L is the distance from the vehicle's center of gravity O to the line connecting the inner wheel-rail contact point;

[0087] θ is the angle between the line connecting the vehicle's center of mass to the inner wheel-rail contact point and the y-axis;

[0088] F C1 F C2 The levitation force perpendicular to the top surface of the rail is exerted on the inner and outer wheels;

[0089] S is the track gauge;

[0090] D is the distance from the center of mass to the top surface of the orbit;

[0091] ρ0 is the radius of inertia of the entire vehicle about its center of mass;

[0092] H is the actual set ultra-high value;

[0093] R S The radius of the vertical curve;

[0094] k S The curvature is the vertical curve.

[0095] By transforming and deriving the above model, we obtain:

[0096] Formula for calculating guiding force:

[0097]

[0098] Formula for calculating levitation force:

[0099]

[0100]

[0101] S2: Calculate the guiding force and suspension force for different alignments based on vehicle parameters and alignment characteristic parameters of different alignments.

[0102] Given the vehicle parameters and the characteristic parameters of different alignments, substitute them into the relevant formulas in step S1 to calculate the guiding force and suspension force.

[0103] S3: Based on the calculation results of guiding force and levitation force, select a relatively gradual change in the linear combination and optimize the linear characteristic parameters of different linear shapes.

[0104] Based on the law of force variation with distance, a linear combination with relatively gentle changes is selected, where the curvature of the curve changes continuously without obvious abrupt changes.

[0105] By using fixed threshold values ​​for guiding force and levitation force, the values ​​of the radius of the circular curve and the length of the transition curve are calculated in reverse, thereby optimizing the linear characteristic parameters.

[0106] S4: Establish a dynamic interaction model between the vehicle and the guide beam, and calculate the dynamic response indexes for each linear combination.

[0107] S401: Establish a dynamic interaction model between the vehicle and the guide beam. The dynamic interaction model between the vehicle and the guide beam includes a vehicle dynamics model and a guide beam dynamics model.

[0108] like Figure 2 and Figure 3As shown, the car body, bogie, and magnet assembly are all considered as rigid bodies, and the suspension system is approximated as a spring-damped element. Each rigid body is considered for heave, pitching, lateral movement, head-swaying, and roll motions. Based on multi-rigid-body dynamics theory, the vehicle's motion equations are established, taking into account the additional forces caused by alignment and track irregularities, resulting in a vehicle dynamics model. Considering the influence of structural vibration, modal analysis or finite element method is used to establish the beam vibration equations, resulting in a guide rail beam dynamics model. Based on experiments and magnetic field analysis, mathematical relationships between suspension clearance and suspension force, and between guide clearance and guide force, are established, linking the vehicle model and the guide rail beam model. The motion equations of the vehicle-guide rail dynamic interaction system are solved using cross-iterative numerical integration, thus obtaining the vehicle-guide rail beam dynamic interaction model.

[0109] S402: Calculate the dynamic response indices for each linear combination, including:

[0110] Using a dynamic interaction model between the vehicle and the guide beam, the dynamic response indices for each linear combination are calculated. These indices include vehicle body vibration acceleration, suspension force, guiding force, suspension clearance, guiding clearance, guide beam deformation, and guide beam vibration acceleration.

[0111] S5: Compare the dynamic response indices and statistical distribution characteristics of each linear combination to obtain the optimal linear combination.

[0112] Statistical analysis of the time history results of the dynamic response indices of each linear combination yields the statistical distribution characteristics of each index in different segments of the combination, including maximum and minimum values, average value, standard deviation, 99% probability value, and 95% probability value.

[0113] By comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the linear combination with the smallest statistical distribution characteristic value and the best dynamic performance is selected as the optimal linear combination.

[0114] After obtaining the dynamic response indices and statistical distribution characteristics of each linear combination, with the aim of improving comfort and reducing the frequency distribution of dynamic responses with large amplitudes, the section with poor dynamic performance in the linear combination can be selected to determine the sensitive linear parameters that affect the dynamic response. The sensitive linear parameters are then adjusted to form a new linear combination. The dynamic response indices and statistical distribution characteristics of each linear combination are then compared again. This process of adjustment and comparison is repeated to obtain a satisfactory optimization scheme.

[0115] The steps of this embodiment can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, they can be executed in a different order.

[0116] Example 2:

[0117] There are three different route design schemes, and the route horizontal and vertical profile design drawings are as follows: Figure 5 and Figure 6 As shown in Table 1, the parameter designs for different schemes are as follows: l1 is the length of the first transition curve, R1 is the radius of the first circular curve, l0 is the length of the straight line, l2 is the length of the second transition curve, R2 is the radius of the second circular curve, l3 is the length of the straight line, i is the slope, and L is the length of the slope segment. Based on different design schemes, specific parameter values ​​such as the length of the straight line, the length of the transition curve, the length and radius of the circular curve, the slope, and the length of the slope segment are input into the established dynamic interaction model between the vehicle and the guide beam. This yields six dynamic response indicators: vertical acceleration of the vehicle body, lateral acceleration of the vehicle body, suspension clearance, guide clearance, suspension force, and guide force. The time history results of the dynamic response for the three schemes are statistically analyzed, and the time history curves are shown in the table below. Figure 7-9 As shown, the extreme values, average values, standard deviations, 99% probability values, and 95% probability values ​​of various indicators are compared across different schemes. Figure 10 As shown in the figure, statistical data comparison reveals that Scheme 3 exhibits the smallest statistical distribution characteristic value of the dynamic response index compared to Schemes 1 and 2, indicating better dynamic performance. Therefore, Scheme 3 is superior.

[0118] <![CDATA[l1 / m]]> <![CDATA[R1 / m]]> <![CDATA[l0 / m]]> <![CDATA[l2 / m]]> <![CDATA[R2 / m]]> <![CDATA[l3 / m]]> <![CDATA[i1 / ‰]]> <![CDATA[i2 / ‰]]> <![CDATA[i3 / ‰]]> Option 1 500 10000 1000 500 10000 1000 1 2 3 Option 2 500 10000 1000 1120 23000 1000 1 2 3 Option 3 1120 23000 1000 1120 23000 1000 1 2 3

[0119] The length of the circular curve is equal to the length of the transition curve, and the points where the longitudinal profile changes slope are all located at the midpoint of the circular curve.

[0120] Based on a relatively objective consideration of passenger comfort and safety under different route conditions, this invention proposes a design method for vacuum tube maglev transportation lines based on dynamic analysis, which has significant theoretical and engineering practical value.

[0121] The above examples illustrate the present invention only to aid in understanding it and are not intended to limit the scope of the invention. Those skilled in the art can make various simple deductions, modifications, or substitutions based on the principles of this invention.

Claims

1. A method for optimizing the route shape of a vacuum tube maglev transportation line, characterized by: The linear shapes include straight lines, circular curves, and transition curves. Several combinations of linear shapes are proposed as design schemes, and the combinations of linear shapes include continuous different linear shapes. The method includes: For non-contact maglev systems, a vehicle motion force analysis model is established to obtain calculation formulas for guiding force and levitation force for arbitrary linear shapes, including vertical curve radius and vertical curve curvature. Based on vehicle parameters and the characteristic parameters of different alignments, calculate the guiding force and suspension force for different alignments; Based on the calculation results of guiding force and suspension force, a relatively gradual change in line shape combination is selected, and the line shape characteristic parameters of different line shapes are optimized. Establish a dynamic interaction model between the vehicle and the guide beam, and calculate the dynamic response indexes for each linear combination; By comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the optimal linear combination can be obtained. This includes establishing a vehicle motion force analysis model to obtain calculation formulas for guiding force and suspension force for arbitrary alignments, including: Simplifying the vehicle as a rigid body rotating about its longitudinal axis in the direction of travel, the force analysis model of the vehicle's motion can be obtained using the theorem of motion of the center of mass and the theorem of angular momentum about the center of mass. Based on the force analysis model of the vehicle's motion, the calculation formulas for guiding force and levitation force are derived. The force analysis model of the vehicle's motion is as follows: In the formula: The vehicle parameters and linear characteristic parameters in the formula include: m represents the total vehicle mass; g is the acceleration due to gravity; α is the transverse slope angle; F D This refers to the lateral guiding force exerted on the wheelset along the top surface of the rail. k p Curvature of a plane curve; v represents the vehicle's speed; L is the distance from the vehicle's center of gravity O to the line connecting the inner wheel-rail contact point; θ is the angle between the line connecting the vehicle's center of mass to the inner wheel-rail contact point and the y-axis; F C1 F C2 The levitation force perpendicular to the top surface of the rail is exerted on the inner and outer wheels; S represents the track gauge; D is the distance from the center of mass to the top surface of the orbit; ρ0 is the radius of inertia of the entire vehicle about its center of mass; H is the actual set ultra-high value; R S The radius of the vertical curve; k S The curvature of the vertical curve; By deforming and deriving the force analysis model of vehicle motion, we obtain: The formula for calculating guiding force is: The formula for calculating levitation force is: Based on the calculation results of guiding force and levitation force, a relatively gradual change in alignment combination is selected, and the alignment characteristic parameters of different alignments are optimized, including: Select line combinations with relatively gentle changes; By using fixed threshold values ​​for guiding force and levitation force, the values ​​of the radius of the circular curve and the length of the transition curve are calculated in reverse, thereby optimizing the linear characteristic parameters. The dynamic interaction model between the vehicle and the guide beam includes the vehicle dynamics model and the guide beam dynamics model. The vehicle body, bogie, and magnet device are all considered as rigid bodies, and the suspension system is approximated as a spring-damping element. Each rigid body is considered to have heave, pitching, lateral movement, head-shaking, and roll motions. The vehicle motion equation is established based on the multi-rigid-body dynamics theory, and the additional forces caused by the alignment and track irregularities are considered to obtain the vehicle dynamics model. Based on the influence of structural vibration, the beam vibration equation is established using the modal method or finite element method to obtain the dynamic model of the guide rail beam. Based on experiments and magnetic field analysis, mathematical relationships between suspension gap and suspension force, as well as between guide gap and guide force, are established. The vehicle dynamics model is linked to the guide rail beam dynamics model. The motion equations of the vehicle-guide rail dynamic interaction system are solved by cross-iteration numerical integration, thus obtaining the vehicle-guide rail beam dynamic interaction model.

2. The method according to claim 1, characterized in that: Calculate the dynamic response indices for each linear combination, including: Using a dynamic interaction model between the vehicle and the guide beam, the dynamic response indices for each linear combination are calculated. These indices include vehicle body vibration acceleration, suspension force, guiding force, suspension clearance, guiding clearance, guide beam deformation, and guide beam vibration acceleration.

3. The method according to claim 2, characterized in that: By comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the optimal linear combination is obtained, including: Statistical analysis of the time history results of the dynamic response indices of each linear combination yields the statistical distribution characteristics of each index in different segments of the combination, including maximum and minimum values, average value, standard deviation, 99% probability value, and 95% probability value. By comparing the dynamic response indices and statistical distribution characteristics of various linear combinations, the linear combination with the smallest statistical distribution characteristic value and the best dynamic performance is selected as the optimal linear combination.

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