Building roof drainage optimization method and structure based on viscous resistance type brachistochrone

Abstract

The invention discloses a building roof drainage optimization method and structure based on a viscous resistance type brachistochrone, and relates to the technical field of constructional engineering and fluid mechanics. The method aims at solving the problem that the efficiency is poor due to the fact that viscous resistance is ignored in actual drainage application of a traditional resistance-free brachistochrone theory. The method comprises the following steps: establishing a mass point viscous resistance model; deducing and solving a brachistochrone parameter equation under the model; and designing a roof drainage path or a tile curved surface shape according to the obtained equation. By introducing the resistance factor, the drainage path better conforms to the real physical environment. Experiments show that compared with a traditional resistance-free brachistochrone design, the design has the advantages that the roof drainage time can be effectively shortened, the optimization rate exceeds 40%, the drainage efficiency is remarkably improved, and the design has important application value in optimization of roof drainage systems for repairing large buildings and ancient buildings.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of building engineering and fluid mechanics, specifically to a method for optimizing roof drainage based on the theory of viscous drag type brachistochrone, and roof tiles and roof structures using this method. Background Technology

[0002] The brachistochrone problem is a classic variational problem used to find the path of a particle that falls along a curve connecting two points under the sole influence of gravity, with the shortest possible time. This path has been proven to be a cycloid (or cycloid). The Bernoulli brothers, Newton, Leibniz, and other scholars have provided analytical solutions for the case of no resistance using variational methods. This curve possesses important properties such as isochronism.

[0003] In the field of architecture, scholars have noted the similarity between the concave shape of the roofs of traditional Chinese buildings and the brachistochrone (maximum descent curve) without resistance, and have interpreted this from aesthetic and cultural perspectives. Recent studies, through numerical calculations, have found that the brachistochrone model considering viscous resistance shows a higher degree of agreement with examples of ancient Chinese architecture, indicating that resistance is a non-negligible factor in actual drainage processes.

[0004] However, existing technologies still have significant shortcomings: firstly, the traditional brachistochrone theory does not consider the viscous resistance between air, water flow, and the roof, resulting in suboptimal drainage efficiency in practical applications; secondly, there is a lack of a systematic design method based on a viscous resistance model to quantitatively optimize roof drainage paths and tile shapes. Therefore, there is an urgent need in this field for a new technical solution to address the problem of maximizing roof drainage efficiency under viscous resistance conditions. Summary of the Invention

[0005] (a) Technical problems to be solved

[0006] The present invention aims to address the aforementioned deficiencies in the prior art by providing a method and structure for optimizing building roof drainage based on a viscous resistance-type brachistochrone line, thereby significantly improving the drainage efficiency of building roofs in real-world environments (where viscous resistance exists).

[0007] (II) Technical Solution

[0008] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0009] A method for optimizing building roof drainage based on viscous resistance-type brachistochrone lines, the core of which is:

[0010] By establishing and solving the brachistochrone dynamic model considering viscous resistance, its parametric equations are obtained, and the roof drainage path or tile surface shape is designed accordingly. This method includes the following steps:

[0011] S1. Establish a dynamic model of the brachistochrone considering viscous drag. Consider a particle of mass m moving along the track from rest, and the viscous drag f acting on the particle on the track. k Satisfy f k = k·v, where k is the viscous drag coefficient and v is the linear velocity of the particle.

[0012] S2. Based on the aforementioned dynamic model, derive and solve the parametric equations of the brachistochrone of the viscous drag type. Write the tangential and normal dynamic equations of the mass in the natural coordinate system, introducing the constraint that the angular velocity ω is constant as a necessary and sufficient condition for the brachistochrone of the viscous drag type to hold. Solve the simultaneous equations, where g is the acceleration due to gravity, m is the mass, and π is pi. For parameters, First, the functional expression of the linear velocity v with respect to the horizontal angle θ is obtained as follows:

[0013]

[0014] Subsequently, through integration, the parametric equation is finally obtained, which is expressed as:

[0015]

[0016]

[0017] S3. Based on the parametric equation, determine the roof drainage path and / or tile surface shape of the building roof so that the water flow time is minimized when flowing along the path.

[0018] A building roof structure includes a roof truss and a plurality of tiles laid thereon, the tiles being interlocked to form a continuous drainage channel based on the viscous resistance type brachistochrone.

[0019] (III) Beneficial Effects

[0020] Compared with the prior art, the technical solution provided by the present invention has the following significant advantages:

[0021] Significantly improved drainage efficiency: By introducing a viscous resistance model, the drainage path provided by this invention better reflects the real physical environment. Experimental results show that, compared to the traditional resistance-free brachistochrone design, the viscous resistance-based brachistochrone design provided by this invention can effectively reduce roof drainage time, with an optimization rate exceeding 40%.

[0022] Highly innovative in theory: This invention derives the parametric equations of the brachistochrone of viscous drag, providing a new theoretical basis for the application of the brachistochrone theory in drag environments.

[0023] High application guidance value: This invention transforms abstract mathematical theory into specific engineering design methods, providing a quantitative and implementable solution for optimizing drainage systems of building roofs (especially large public buildings and ancient building restoration), which helps to extend the service life of buildings and improve safety. Attached Figure Description

[0024] This specification includes several accompanying drawings to further explain the invention. These drawings are an important part of the specification, but they are submitted as separate pages and will not be repeated in this text. Brief descriptions of each drawing are as follows:

[0025] Figure 1 This is a schematic diagram of the force analysis model for the brachistochrone of the viscous resistance type.

[0026] Figure 2 This is a comparison chart of the brachistochrone curve of viscous resistance type and the brachistochrone curve of resistance-free type. The orange line represents the brachistochrone curve of resistance-free type, and the blue line represents the brachistochrone curve of viscous resistance type.

[0027] Figure 3 This is a velocity-height diagram, where the vertical axis represents the velocity v (m / s) and the horizontal axis represents the displacement y (m). The blue line represents the brachistochrone (maximum descent) under resistance-free conditions, and the orange line represents the brachistochrone (maximum descent) under viscous resistance conditions.

[0028] Figure 4 This is a velocity-time graph, with the vertical axis representing velocity v (m / s) and the horizontal axis representing time (s). The cyan line represents the brachistochrone (maximum descent) under drag-free conditions, and the red line represents the brachistochrone (maximum descent) under viscous drag conditions. Detailed Implementation

[0029] The specific implementation methods described below are elaborated in detail with reference to the technical solutions of this invention. Those skilled in the art should understand that the embodiments described herein are for illustration and explanation only and are not intended to limit the scope of protection of this invention.

[0030] A preferred embodiment of the present invention relates to the application of the drainage optimization method.

[0031] First, determine or estimate a reasonable range of values ​​for the viscous drag coefficient k based on the environmental parameters of the actual construction project (such as local rainfall intensity, water flow characteristics, etc.). Then, determine the basic gravitational acceleration g and the particle mass m.

[0032] At the same time, determine the required horizontal width x or vertical height y of the building's roof.

[0033] Subsequently, the value of k, the mass m of the particle, the gravitational acceleration g, and the lateral width x or longitudinal height y are substituted into the derived parametric equations.

[0034]

[0035] ,

[0036] in For parameters,

[0037] Given that x and y are fixed:

[0038] One of them is and Given a system of two equations, the constant angular velocity ω can be calculated using numerical iteration. If x, or y, is determined unilaterally, we can let... = Substituting these equations into one of the equations, we can obtain the constant angular velocity ω and simultaneously deduce the expression for y and the height. The reason for not directly providing the expression for the angular velocity is that this system of equations is too complex to obtain an analytical solution; numerical iteration is a better approach.

[0039] At this point, calculations can be performed to obtain a steepest descent curve optimized for this specific environment.

[0040] This curve can be used as a design benchmark in the design of roof drainage systems. For example, when planning the drainage path of the entire roof, the direction of its centerline or main drainage channels can be made to conform to the shape of this curve.

[0041] Based on theoretical calculations, the advantages of this invention can be visualized by directly plotting a scatter plot, as shown in the appendix figure.

[0042] Below is a calculation example: (Note: Units are omitted, and the default dimensions are used)

[0043] First, we select a theoretical proportion for Chinese architecture and perform calculations: The calculations are as follows:

[0044] Calculation of architecture: Taking Song Dynasty architecture as an example: Select a palace-style building with seven sides, three courtyards, and ten rafters, using second-class timber with double grooves.

[0045] Let x = 41.6914. = Taking k=1.05, m=1.3807, and g=9.832, we get ω=-0.2059.

[0046] Substituting into the equation for y, we get: y = -53.8672

[0047] Substituting the values ​​will yield the parametric equations. See the attached diagram for a comparison with the traditional descent curve.

[0048] In a specific experimental embodiment of the present invention, this parameter was used to calculate the specific curve shape.

[0049] Subsequently, a physical model of the curve was created using 3D printing technology to conduct water flow experiments.

[0050] The experiment used the controlled variable method. Under room temperature conditions, simulated rainfall (15mm) was allowed to flow through the resistanceless brachydescent model, the viscous resistance brachydescent model, and the straight line model, and the time required for the water to accumulate to 400mL in the beaker was measured.

[0051] Record experimental data in multiple experiments

[0052] The analysis and calculation of the average value, accurate to two decimal places, yielded the following results:

[0053] Curve type: Landing directly due to air resistance

[0054] Time delay: 0 seconds

[0055] Water volume: 400 mL

[0056] Curve type: Viscous resistance type steepest descent curve (k=1.05)

[0057] Time delay: 2.84 seconds

[0058] Water volume: 400 mL

[0059] Curve type: Resistance-free brachistochrone

[0060] Time delay: 5.71 seconds

[0061] Water volume: 400 mL

[0062] Curve type: Straight line

[0063] Time delay: 6.39 seconds

[0064] Water volume: 400 mL

[0065] After averaging multiple experiments, the data shows that the viscous resistance type brachistochrone model provided by this invention requires the shortest drainage time. Compared with the resistance-free model, the optimization time is shortened by more than 40%, which fully verifies the effectiveness and superiority of this invention.

[0066] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and not restrictive.

Claims

1. A method for optimizing building roof drainage based on viscous resistance-type brachistochrone lines, characterized in that, Includes the following steps: Establish a dynamic model of the steepest descent line considering viscous drag, where the viscous drag f on the particle satisfies f = k·v, where k is the viscous drag coefficient and v is the linear velocity of the particle. Based on the aforementioned dynamic model, the parametric equations for the brachistochrone of the viscous drag type are derived and solved, where g is the acceleration due to gravity, m is the mass, π is pi, and θ is a parameter. The parametric equation is expressed as Based on the parametric equations, determine the roof drainage path and / or tile surface shape of the building roof to minimize the time required for water to flow along that path.

2. The method according to claim 1, characterized in that, The process of deriving the parametric equations includes: Write the tangential and normal dynamic equations of a particle in the natural coordinate system; The constraint that the angular velocity ω is constant is introduced as a necessary and sufficient condition for the viscous drag type brachistochrone to hold. Solving the simultaneous equations yields the functional expression of the linear velocity v with respect to the horizontal angle θ; The parametric equations are finally obtained through integration.

3. A building roof structure, characterized in that, It includes a roof truss and multiple tiles laid on it, the water-facing surfaces of the multiple tiles being spliced ​​together to form a continuous drainage path defined by the parametric equation of the viscous resistance type brachistochrone as described in claim 1 or 2.

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