A method for calculating the natural frequency of partial vibration modes of point-supported building curtain wall panels

Abstract

The present invention discloses a method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel, comprising the following steps: S1, selecting a support point, defining the distance from the support point to the adjacent edge of the curtain wall panel along a first direction as a support distance x1, and the total length of the curtain wall panel along the first direction x0, which satisfies: x1 = (1 / a) x0; S2, simplifying the curtain wall panel into a model of two-end overhanging beams along the first direction, wherein the total length of the two-end overhanging beams is x0, and the length of the free overhanging end of the two-end overhanging beams is x 1; S3. Simplify the mid-span part of the overhanging beams at both ends into a simply supported beam model, and simplify the free overhanging ends into a cantilever beam model, and derive the homogeneous equations of the simply supported beam and the homogeneous equations of the cantilever beam; S4. Combine the homogeneous equations of the simply supported beam and the homogeneous equations of the cantilever beam to derive the homogeneous equations of the overhanging beams at both ends; S5. Substitute x1=(1 / a)x0 into the homogeneous equations of the overhanging beams at both ends to obtain the natural frequency solution of the partial vibration mode of the curtain wall panel along the first direction.

Description

Technical Field

[0001] The present invention relates to the field of point-supported building curtain walls, and in particular to a method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel. Background Art

[0002] Point-supported building curtain walls have the advantages of being beautiful, energy-saving, flexible, easy to maintain, and having strong wind and earthquake resistance. They have been widely used in modern architectural design and have become an important component of many important buildings.

[0003] In most cases of point-supported curtain wall failure, connection failure and improper construction are the primary causes, with most damage occurring at the connections between the panels and the hangers. When a point-supported curtain wall is damaged, corresponding physical parameters change, allowing inference of structural damage through changes in its vibration characteristics.

[0004] Among them, the natural frequency is relatively easy to obtain among the structural modal parameters, and the identification accuracy is high. In the related art, the natural frequency of the point-supported building curtain wall is mainly obtained by dynamic test or finite element simulation analysis. Although a relatively accurate natural frequency can be obtained through dynamic test, the test steps are relatively complex, there are many interference factors, and the test efficiency is subject to the arrangement and removal of sensors. When using finite element simulation analysis to support the building curtain wall, the modeling, boundary conditions and corresponding constraint forms of the curtain wall model have certain deviations compared with the actual curtain wall structure, which in turn affects the accuracy of the dynamic characteristics of the curtain wall panel obtained. Summary of the Invention

[0005] One object of the present invention is to provide a method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel, so as to obtain a theoretical solution for the natural frequency of a partial vibration mode of a point-supported building curtain wall panel.

[0006] To achieve the above objectives, the present invention adopts a technical solution: a method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel, comprising the following steps:

[0007] S1. Select a support point, define the distance from the support point to the adjacent edge of the curtain wall panel along the first direction as the support distance x1, and the total length x0 of the curtain wall panel along the first direction satisfies: x1 = (1 / a)x0, where a>1;

[0008] S2. Simplify the curtain wall panel into a model of two overhanging beams along a first direction, wherein the total length of the two overhanging beams along the first direction is x0, and the length of the free overhanging ends of the two overhanging beams along the first direction is x1;

[0009] S3. Simplify the mid-span portion of the cantilever beams at both ends into a simply supported beam model, and simplify the free overhanging ends of the cantilever beams into a cantilever beam model. Based on the free vibration equilibrium equations of the uniform cross-section beam, derive the homogeneous equations for the simply supported beam according to the boundary conditions of the simply supported beam, and derive the homogeneous equations for the cantilever beam according to the boundary conditions of the cantilever beam.

[0010] S4. Based on the free vibration equilibrium equations of the uniform cross-section beam, according to the boundary conditions and displacement continuity conditions of the cantilever beams at both ends, and combined with the homogeneous equations of the simply supported beam and the cantilever beam, the homogeneous equations of the cantilever beam at both ends are derived;

[0011] S5. Substitute x1=(1 / a)x0 into the homogeneous equations of the cantilever beams at both ends to obtain the natural frequency solution of the cantilever beams at both ends along the first direction as the natural frequency solution of the partial vibration mode of the curtain wall panel along the first direction.

[0012] As a preference, step S2 includes the steps of:

[0013] S21a, simplifying the curtain wall panel into a simply supported plate model with four outwardly extending sides and four locally constrained supports, wherein the distance from the locally constrained supports to the adjacent side along the first direction is x1, and the distance from the two sides of the simply supported plate model along the first direction is x0;

[0014] S22a, applying equivalent linear constraints to the four-point local constraints of the simply supported plate model along a second direction perpendicular to the first direction, simplifying the model into a transition model, wherein the distance from the linear constraint to the adjacent edge along the first direction is x1, and the distance from the two opposite edges of the transition model along the first direction is x0;

[0015] S23a, simplifying the transition model into a model of an overhanging beam at both ends, wherein the total length of the overhanging beam at both ends is x0, and the length of the free overhanging end of the overhanging beam at both ends is x1.

[0016] As a preferred embodiment, step S1 also includes the distance from the support point to the adjacent side of the curtain wall panel along the second direction perpendicular to the first direction as the support distance y1, and the total length y0 of the curtain wall panel along the second direction satisfies: y1 = (1 / b)y0, where b>1; step S2 also includes simplifying the curtain wall panel into a model of an overhanging beam at both ends along the second direction, the total length of the overhanging beam at both ends along the second direction is y0, and the length of the free overhanging end of the overhanging beam at both ends along the second direction is y1; step S5 also includes substituting y1 = (1 / b)y0 into the homogeneous equation group of the overhanging beam at both ends to obtain the natural frequency solution of the overhanging beam at both ends along the second direction as the natural frequency solution of the partial vibration mode of the curtain wall panel along the second direction.

[0017] As a preference, step S2 includes the steps of:

[0018] S21b, simplifying the curtain wall panel into a simply supported plate model with four outwardly extending sides and four local constrained supports, wherein the distance from the local constrained supports to the adjacent side along the second direction is y1, and the distance from the two sides of the simply supported plate model along the second direction is y0;

[0019] S22b, the four-point local constraints of the simply supported plate model are made into equivalent linear constraints along the first direction to simplify the model into a transition model, wherein the distance from the linear constraint to the adjacent edge along the second direction is y1, and the distance from the two sides of the transition model along the second direction is y0;

[0020] S23b, simplifying the transition model into a model of an overhanging beam at both ends along the second direction, wherein the total length of the overhanging beam at both ends along the second direction is y0, and the length of the free overhanging end of the overhanging beam at both ends along the second direction is y1.

[0021] As a preference, in step S3, the free vibration equilibrium equation of the uniform cross-section beam is:

[0022] Where E is the elastic modulus of the material, I is the moment of inertia of the section, is the mass per unit length of the beam, z is the deflection, x is the position of any point, and t is the time.

[0023] As a preference, in step S4, the boundary conditions of the endpoints of the free overhanging ends of the two-end overhanging beams are: the section bending moment and shear force of the endpoints of the free overhanging ends are both 0, and the homogeneous equations of the endpoints of the free overhanging ends of the two-end overhanging beams are:

[0024]

[0025] The boundary conditions of the first node on the cantilever beam at both ends corresponding to one of the supports are: the cross-sectional bending moments and rotation angles on the left and right sides of the first node are equal, and the vertical deflection of the first node is 0. To meet the continuity conditions, the homogeneous equations of the first node of the cantilever beam at both ends are:

[0026] The boundary conditions of the second node corresponding to the other support on the cantilever beam at both ends are: the cross-sectional bending moments and rotation angles on the left and right sides of the second node are equal, and the deflection of the second node is 0. To meet the continuity conditions, the homogeneous equations of the second node of the cantilever beam at both ends are:

[0027] Where Z(x) is the amplitude curve of the beam, M is the bending moment, θ is the rotation angle, C1 to C 12 is a constant, and λ is an introduced parameter.

[0028] As a preferred embodiment, the solution of the first n-order frequencies of the curtain wall panel along the first direction is: Among them, λ i is the parameter of the i-th order frequency solution.

[0029] As a preferred embodiment, in step S3, the separation of variables method is used to further derive the free vibration equilibrium equation of the equal-section beam to obtain the intermediate equation group:

[0030] T″(t)+ω 2 T(t)=0;Z Ⅳ (x)-λ 4 Z(x)=0; Where Z(x) is the amplitude curve of the beam, T(t) is the displacement amplitude of the beam as a function of time, ω 2 is a constant, and λ is an introduced parameter;

[0031] The general solution of the intermediate equations is obtained as:

[0032] T(t)=C1sinωt+C2cosωt; Z(x)=C1coshλx+C2sinhλx+C3cosλx+C4sinλx, where C1 to C4 are unknown constants and λ is an introduced parameter.

[0033] As a preferred embodiment, in step S3, after substituting the boundary conditions, the homogeneous equations of the simply supported beam are:

[0034]

[0035] As a preferred embodiment, in step S3, after substituting the boundary conditions, the homogeneous equations of the cantilever beam are:

[0036]

[0037] Compared with the prior art, the beneficial effects of the present invention are as follows: by performing reasonable constraint equivalent processing on the curtain wall panel and simplifying the cantilever beam model at both ends corresponding to the movable curtain wall panel in step S2, a homogeneous equation group of the cantilever beams at both ends is derived through steps S3 and S4, and then substituting the position relationship of the support points in step S1 into the equation group, the theoretical solution of the natural frequency of the partial vibration mode of the curtain wall panel is obtained, which reduces the workload compared with the conventional reliance on dynamic test and finite element simulation analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a three-dimensional structural diagram of the curtain wall panels of some embodiments of the present application.

[0039] Figure 2 It is a three-dimensional structural diagram of curtain wall panels according to other embodiments of the present application.

[0040] Figure 3 Schematic diagram of a simply supported plate model according to some embodiments of the present application.

[0041] Figure 4is a schematic diagram of a transition model of some embodiments of the present application.

[0042] Figure 5 Schematic diagram of a beam model with two ends extended in some embodiments of the present application.

[0043] Figure 6 It is a schematic diagram of simplifying the two-end extended beam model of some embodiments of the present application into a simply supported beam and a cantilever beam.

[0044] Figure 7A 1 is a vibration mode diagram of a beam model with two ends extended in the first direction at a frequency of 1 according to an embodiment of the present application.

[0045] Figure 7B is a mode shape diagram of the transition model of an embodiment of the present application along the first direction at frequency 1.

[0046] Figure 7C 1 is a vibration mode diagram of a simply supported plate model in an embodiment of the present application along a first direction at a frequency of 1.

[0047] Figure 8A 1 is a vibration mode diagram of a beam model with two ends extended in the first direction at a frequency of 2 according to an embodiment of the present application.

[0048] Figure 8B is a mode shape diagram of the transition model of an embodiment of the present application along the first direction at frequency 2.

[0049] Figure 8C is a vibration mode diagram of the simply supported plate model of an embodiment of the present application along the first direction at frequency 2.

[0050] Figure 9A 3 is a mode shape diagram of the two-end cantilevered beam model of an embodiment of the present application along the first direction at frequency 3.

[0051] Figure 9B 3 is a mode shape diagram of the transition model of an embodiment of the present application along the first direction at frequency 3.

[0052] Figure 9C 3 is a mode shape diagram of the simply supported plate model of an embodiment of the present application along the first direction at frequency 3.

[0053] Figure 10A 4 is a mode shape diagram of the two-end extended beam model in the embodiment of the present application along the second direction at frequency 4.

[0054] Figure 10B 4 is a mode shape diagram of the transition model of an embodiment of the present application along the second direction at frequency 4.

[0055] Figure 10C 4 is a mode shape diagram of the simply supported plate model of an embodiment of the present application along the second direction at frequency 4.

[0056] Figure 11A 3 is a mode shape diagram of the two-end cantilevered beam model of an embodiment of the present application along the second direction at a frequency of 5.

[0057] Figure 11B 5 is a mode shape diagram of the transition model of an embodiment of the present application along the second direction at a frequency of 5.

[0058] Figure 11C 5 is a mode shape diagram of the simply supported plate model of an embodiment of the present application along the second direction at a frequency of 5.

[0059] Figure 12 This is a diagram of a test device for performing a power test in an embodiment of the present application.

[0060] Figure 13 This is a physical diagram of the distribution of measuring points for the dynamic test of the embodiment of the present application.

[0061] Figure 14 This is a schematic diagram of the distribution of measurement points for the dynamic test of the embodiment of the present application.

[0062] Figure 15 This is a spectrum diagram obtained by performing a power test on an embodiment of the present application.

[0063] In the figure: 1. Curtain wall panel; 2. Back bolt; 3. Connector; 4. Beam; 5. Column. DETAILED DESCRIPTION

[0064] The present invention will be further described below in conjunction with specific implementation methods. It should be noted that, under the premise of no conflict, the various embodiments or technical features described below can be arbitrarily combined to form new embodiments.

[0065] In the description of the present invention, it should be noted that, for directional words, such as the terms "center", "horizontal", "longitudinal", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise" and the like, indicating directions and positional relationships, are based on the directions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific direction, be constructed and operated in a specific direction, and cannot be understood as limiting the specific scope of protection of the present invention.

[0066] It should be noted that the terms "first", "second", etc. in the description and claims of this application are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence.

[0067] The terms "comprises" and "having" and any variations thereof in the specification and claims of this application are intended to cover non-exclusive inclusions. For example, a process, method, system, product or apparatus that includes a series of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, product or apparatus.

[0068] A method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel comprises the following steps:

[0069] S1. Select a support point and define the distance from the support point to the adjacent edge of the curtain wall panel along the first direction (i.e., the X direction) as the support distance x1. The total length x0 of the curtain wall panel along the first direction satisfies: x1 = (1 / a)x0, where a > 1.

[0070] S2. Simplify the curtain wall panel into a model of two overhanging beams along the first direction. The total length of the two overhanging beams along the first direction is x0, and the length of the free overhanging ends of the two overhanging beams along the first direction is x1. Figure 5 As shown;

[0071] S3. Simplify the mid-span part of the cantilever beam at both ends into a simply supported beam model, and simplify the free cantilever ends of the cantilever beam at both ends into a cantilever beam model, as shown in the following example: Figure 6 As shown in the figure, based on the free vibration equilibrium equation of the uniform cross-section beam, the homogeneous equations of the simply supported beam are derived according to the boundary conditions of the simply supported beam, and the homogeneous equations of the cantilever beam are derived according to the boundary conditions of the cantilever beam;

[0072] S4. Based on the free vibration equilibrium equations of the uniform cross-section beam, according to the boundary conditions and displacement continuity conditions of the cantilever beams at both ends, and combined with the homogeneous equations of the simply supported beam and the cantilever beam, the homogeneous equations of the cantilever beam at both ends are derived;

[0073] S5. Substitute x1=(1 / a)x0 into the homogeneous equations of the cantilever beams at both ends to obtain the natural frequency solution of the cantilever beams at both ends along the first direction as the natural frequency solution of the partial vibration mode of the curtain wall panel along the first direction.

[0074] Among them, the mid-span part refers to the part between the two supports on the cantilever beams at both ends, and the free cantilever end refers to the part of the cantilever beams at both ends that extends outward from the supports.

[0075] Furthermore, step S2 includes the steps of:

[0076] S21a, simplify the curtain wall panel into a simply supported plate model with four outward extensions and four local constraint supports, where the distance from the local constraint support to the adjacent edge along the first direction is x1, and the distance from the two sides of the simply supported plate model along the first direction is x0, as shown in Figure 3It is worth mentioning that in the related art, a simply supported plate model is used to perform finite element simulation analysis to obtain the natural frequency of the curtain wall panel.

[0077] S22a, the four-point local constraints of the simply supported plate model are made into equivalent linear constraints along the second direction (i.e., Y direction) perpendicular to the first direction (i.e., X direction), and simplified into a transition model, where the distance from the linear constraint to the adjacent edge along the first direction is x1, and the distance from the transition model to the two sides along the first direction is x0, as shown in the following example: Figure 4 As shown in the figure, it is understandable that after the curtain wall panel is installed, the structural sealant on the four sides will produce an embedded constraint effect on the curtain wall panel. Therefore, the four-point local constraints in the simply supported slab model are used as equivalent linear constraints to simplify the simulation of the force exerted by the structural sealant on the curtain wall panel.

[0078] S23a. Simplify the transition model to a model with two-end cantilevered beams. The total length of the two-end cantilevered beams is x0, and the length of the free cantilevered ends of the two-end cantilevered beams is x1. Figure 5 shown.

[0079] It is understood that building curtain walls usually include back-bolted stone curtain walls, such as Figure 1 As shown, the back-bolt stone curtain wall includes a curtain wall panel 1, a connector 3, and a keel consisting of a beam 4 and a column 5. The connector 3 is connected to the keel by bolts, and the connector 3 is connected to the curtain wall panel 1 with a hole through a back bolt 2. Building curtain walls usually also include glass curtain walls, such as Figure 2 It is understood that the curtain wall panel 1 of a building curtain wall is usually fixedly connected to other components via four supporting points, and the supporting points are symmetrically arranged relative to the center line of the curtain wall panel 1 so that the curtain wall panel 1 is evenly stressed.

[0080] Through steps S21a to S23a, the curtain wall panel is subjected to reasonable constraint equivalent processing and simplification to obtain the overhanging beam model of the curtain wall panel at both ends along the first direction. Through steps S3 and S4, the homogeneous equation group of the overhanging beam at both ends is derived, and then the position relationship of the support points in step S1 is substituted into it to obtain the natural frequency solution of some vibration modes of the curtain wall panel along the first direction.

[0081] It is worth mentioning that by using the same constraint equivalent processing and simplification as above, the natural frequency solution of some vibration modes of the curtain wall panel along the second direction can also be obtained.

[0082] Specifically, step S1 also includes: the distance from the support point to the adjacent side of the curtain wall panel along the second direction perpendicular to the first direction is the support distance y1, and the total length of the curtain wall panel along the second direction y0 satisfies: y1 = (1 / b) y0, where b > 1; step S2 also includes simplifying the curtain wall panel into a model of an overhanging beam at both ends along the second direction, the total length of the overhanging beam at both ends along the second direction is y0, and the length of the free overhanging end of the overhanging beam at both ends along the second direction is y1; step S5 also includes substituting y1 = (1 / b) y0 into the homogeneous equation group of the overhanging beam at both ends to obtain the natural frequency solution of the overhanging beam at both ends along the second direction as the natural frequency solution of the partial vibration mode of the curtain wall panel along the second direction.

[0083] Furthermore, step S2 includes the steps of:

[0084] S21b. Simplify the curtain wall panel into a simply supported plate model with four outwardly extending sides and four locally constrained supports, wherein the distance from the locally constrained support to the adjacent side along the second direction is y1, and the distance from the two sides of the simply supported plate model along the second direction is y0.

[0085] S22b. The four-point local constraints of the simply supported plate model are made into equivalent linear constraints along the first direction and simplified into a transition model, wherein the distance from the linear constraint to the adjacent edge along the second direction is y1, and the distance from the transition model to the two sides along the second direction is y0.

[0086] S23b, simplifying the transition model into a model of an overhanging beam at both ends along the second direction, wherein the total length of the overhanging beam at both ends along the second direction is y0, and the length of the free overhanging end of the overhanging beam at both ends along the second direction is y1.

[0087] That is to say, through steps S21b to S23b, the curtain wall panel is subjected to reasonable constraint equivalent processing and simplification to obtain the overhanging beam model of the curtain wall panel at both ends along the first direction, and the homogeneous equation group of the overhanging beams at both ends is derived through steps S3 and S4, and then substituted into the position relationship of the support points in step S1 to obtain the natural frequency solution of some vibration modes of the curtain wall panel along the second direction.

[0088] It can be understood that, compared with the conventional reliance on dynamic test and finite element simulation analysis, the present application reduces the workload by simplifying the curtain wall panel into two end cantilever beams along the first direction and two end cantilever beams along the second direction, and through the homogeneous equations of the two end cantilever beams, the natural frequency solutions of some vibration modes of the curtain wall panel along the first direction and some vibration modes along the second direction are derived.

[0089] Specifically, in step S3, the free vibration equilibrium equation of the uniform cross-section beam is:

[0090]

[0091] Where E is the elastic modulus of the material, I is the moment of inertia of the section, is the mass per unit length of the beam, z is the deflection, x is the position of any point, and t is the time.

[0092] Using the separation of variables method, it is assumed that the solution of the deflection z is the product of two independent functions of the position x and time t at any point, that is:

[0093] Z(x,t)=Z(x)·T(t) (2)

[0094] Where Z(x) is the amplitude curve of the beam, and T(t) is the displacement amplitude of the beam as a function of time. Substituting equation (2) into equation (1) yields the intermediate equations:

[0095] T″(t)+ω 2 T(t)=0 (3-1)

[0096] Z Ⅳ (x)-λ 4 Z(x)=0 (3-2)

[0097]

[0098] Among them, ω 2 is a constant, and λ is an introduced parameter;

[0099] The general solution of the intermediate equations is:

[0100] T(t)=C1sinωt+C2cosωt (4-1)

[0101] Z(x)=C1coshλx+C2sinhλx+C3cosλx+C4sinλx (4-2)

[0102] Among them, C1 to C4 are unknown constants, and λ is an introduced parameter.

[0103] It can be understood that for a system with infinite degrees of freedom, the characteristic equation has infinite roots and thus infinite frequencies ω n , n is a natural number greater than 0. For each frequency, a set of ratios of C1, C2, C3, and C4 can be obtained, and the corresponding main vibration mode function Z can be obtained. n (x).

[0104] Furthermore, in step S3, the mid-span portion of the overhanging beams at both ends is simplified to a simply supported beam model, and the homogeneous equations of the simply supported beam are derived based on the boundary conditions of the simply supported beam.

[0105] Specifically, the boundary conditions of the simply supported beam are that the deflection and section bending moment at the endpoints of the simply supported beam are both 0. Combining equations (4-1) and (4-2), the homogeneous equations of the simply supported beam are derived as follows:

[0106]

[0107] It is worth mentioning that according to the homogeneous equations (5) of the simply supported beam, combined with the characteristic equation sinλx0=0, the natural frequency solution of the simply supported beam is The corresponding main vibration mode of the simply supported beam is

[0108] Furthermore, in step S3, the free overhanging end of the overhanging beam at both ends is simplified to a cantilever beam model, and the homogeneous equations of the cantilever beam are derived based on the boundary conditions of the cantilever beam. Specifically, the boundary conditions of the cantilever beam are that the deflection and cross-sectional rotation of the fixed end of the cantilever beam are both 0, and the cross-sectional bending moment and shear force of the free overhanging end are both 0. Combining equations (4-1) and (4-2), the homogeneous equations of the cantilever beam are derived as follows:

[0109]

[0110] It is worth mentioning that according to the homogeneous equations of the cantilever beam (6), let the determinant corresponding to the homogeneous equations be 0, and we can deduce:

[0111] cosλx0·coshλx0+1=0 (7)

[0112] The trial algorithm is used to solve equation (7) and we get: λ1x0=1.88,λ2x0=4.69,λ3x0=7.86,λ n x0=(n-1 / 2)π, and the corresponding main vibration mode of the cantilever beam is

[0113] In step S4, based on the boundary conditions and displacement continuity conditions of the cantilever beams, and in combination with the homogeneous equations of the simply supported beam and the homogeneous equations of the cantilever beam, the homogeneous equations of the cantilever beams are derived.

[0114] Specifically, the boundary conditions of the endpoints of the free overhanging ends of the two-end overhanging beam are: the section bending moment and shear force of the endpoints of the free overhanging ends are both 0. Combining the homogeneous equations of the simply supported beam (5) and the homogeneous equations of the cantilever beam (6), the homogeneous equations of the endpoints of the free overhanging ends of the two-end overhanging beam are derived as follows:

[0115]

[0116] The boundary conditions of the first node on the cantilever beam at both ends corresponding to one of the supports are: the cross-sectional bending moments and rotation angles on the left and right sides of the first node are equal, and the vertical deflection of the first node is 0. In order to meet the continuity condition, the homogeneous equations of the first node of the cantilever beam at both ends are derived by combining the homogeneous equations of the simply supported beam (5) and the homogeneous equations of the cantilever beam (6):

[0117]

[0118] The boundary conditions of the second node corresponding to the other support on the cantilever beam at both ends are: the cross-sectional bending moments and rotation angles on the left and right sides of the second node are equal, and the deflection of the second node is 0. In order to meet the continuity condition, the homogeneous equations of the second node of the cantilever beam at both ends are derived by combining the homogeneous equations of the simply supported beam (5) and the homogeneous equations of the cantilever beam (6):

[0119]

[0120] Where Z(x) is the amplitude curve of the beam, M is the bending moment, θ is the rotation angle, C1 to C12 are unknown constants, and λ is an introduced parameter.

[0121] Furthermore, the solution of the first n-order frequencies of the curtain wall panel along the first direction is: Among them, λi is the parameter of the i-th order frequency solution. [Specific embodiment]

[0123] The technical solutions of this application will be described below in conjunction with specific embodiments. Obviously, the embodiments described are only part of the embodiments of this application, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this invention.

[0124] In a specific embodiment, the support distance x1 of the support point along the first direction is equal to (1 / 5) x0, and the support distance y1 of the support point along the second direction is equal to (1 / 4) y0.

[0125] Substituting x1=(1 / 5)x0 into equations (8), (9), and (10), the first three natural frequencies of the curtain wall panel along the first direction are obtained as follows:

[0126]

[0127] Substituting y1=(1 / 4)y0 into equations (8), (9), and (10), it can be understood that when y0 is substituted into x0 and y1 is substituted into x1, the first two-order natural frequencies of the curtain wall panel along the second direction are solved as follows:

[0128]

[0129] In order to verify the feasibility of using the two-end cantilever beam model to obtain the natural frequency solution of the curtain wall panel, finite element simulation analysis and dynamic test tests were used for auxiliary verification.

[0130] Specifically, finite element models were established for the cantilever beam model, transition model, and simply supported plate model. The cantilever beams at both ends were constructed using beam3 elements. The curtain wall panels in the transition model and simply supported plate model were 900 mm × 600 mm × 25 mm in size and used shell281 elements. These elements were set to linear elasticity and isotropy, with a Young's modulus of 29.5 GPa, a Poisson's ratio of 0.125, and a density of 2.67 g / cm. 3 The Lanczos algorithm is used to perform modal analysis to analyze the natural vibration characteristics of each finite element model. The analysis results are as follows: Figures 7A-11C shown.

[0131] Use DH5922N dynamic signal test and analysis system and hardware facilities to conduct dynamic test, such as Figure 12 and Figure 13 Specifically, the excitation device uses a stainless steel hammer to excite 70 measuring points, and the vibration pickup device uses a 1A116E piezoelectric accelerometer. The reference point is selected at Figure 14 The modal test uses a multiple-input, multiple-output (MIMO) approach at measurement points 18, 33, 35, and 47. The excitation force signals at the different measurement points and the vibration response signal at the reference point are input into an acquisition card. The natural frequencies and mode shapes of the curtain wall panels are then determined using the DH5922N dynamic test and analysis software.

[0132] It is worth mentioning that the spectrum obtained by using the polyLSCF (Polynomial Least Squares with Constraints) modal parameter identification method is as follows: Figure 15 As shown, Figure 15 The mid-wave peak appears when the frequency, damping and modal participation factor are all stable, and all three parameters are within the given error range.

[0133] Table 1 shows the natural frequencies of the curtain wall panels obtained from theoretical derivation of the two-end cantilever beams, finite element simulation analysis, and dynamic testing. The values ​​in parentheses are the errors between the natural frequencies obtained from finite element simulation analysis and dynamic testing and those derived from the theoretical derivation of the two-end cantilever beams.

[0134] Table 1

[0135]

[0136]

[0137] The first five natural frequencies obtained from the cantilever beam model are generally consistent with the natural frequencies derived from the two-end cantilever beam theory, with a maximum error of only -3.18%. The first five natural frequencies obtained from the transition model, compared with the natural frequencies derived from the two-end cantilever beam theory, also have a maximum error of only -6.20%. The first five natural frequencies obtained from the simply supported plate model, compared with the natural frequencies derived from the two-end cantilever beam theory, have a maximum error of -7.82%. It is understandable that the main reason for the gradual increase in error is: during the simplification process in step S2, the difference caused by the simplified equivalence between the four-point local constraints in the simply supported plate model and the linear constraints in the transition model. The first five natural frequencies obtained from the dynamic test test, compared with the natural frequencies derived from the two-end cantilever beam theory, have a maximum error of -8.72%.

[0138] In summary, the use of the two-end cantilever beam theory to calculate the natural frequencies of some vibration modes of curtain wall panels has good feasibility and reference value.

[0139] The above describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the foregoing embodiments. The foregoing embodiments and description merely illustrate the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed by the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel, characterized in that: Including steps: S1. Select a support point, define the distance from the support point to the adjacent edge of the curtain wall panel along the first direction as the support distance x1, and the total length of the curtain wall panel along the first direction as x0, satisfying: x1=(1 / a)x0, where a>1; wherein the first direction is the X direction; S2. Simplify the curtain wall panel into a model of two overhanging beams along a first direction, wherein the total length of the two overhanging beams along the first direction is x0, and the length of the free overhanging ends of the two overhanging beams along the first direction is x1; S3. Simplify the mid-span portion of the cantilever beams at both ends into a simply supported beam model, and simplify the free overhanging ends of the cantilever beams into a cantilever beam model; based on the free vibration equilibrium equations of the uniform cross-section beam, derive the homogeneous equations for the simply supported beam according to the boundary conditions of the simply supported beam, and derive the homogeneous equations for the cantilever beam according to the boundary conditions of the cantilever beam; S4. Based on the free vibration equilibrium equations of the uniform cross-section beam, according to the boundary conditions and displacement continuity conditions of the cantilever beams at both ends, and combined with the homogeneous equations of the simply supported beam and the cantilever beam, the homogeneous equations of the cantilever beam at both ends are derived; S5. Substitute x1=(1 / a)x0 into the homogeneous equations of the cantilever beams at both ends to obtain the natural frequency solution of the cantilever beams at both ends along the first direction as the natural frequency solution of the partial vibration mode of the curtain wall panel along the first direction.

2. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to claim 1, characterized in that: Step S2 includes the steps of: S21a, simplifying the curtain wall panel into a simply supported plate model with four outwardly extending sides and four locally constrained supports, wherein the distance from the locally constrained supports to the adjacent side along the first direction is x1, and the distance from the two sides of the simply supported plate model along the first direction is x0; S22a, applying equivalent linear constraints to the four-point local constraints of the simply supported plate model along a second direction perpendicular to the first direction, simplifying the model into a transition model, wherein the distance from the linear constraint to the adjacent edge along the first direction is x1, and the distance from the two opposite edges of the transition model along the first direction is x0; S23a, simplifying the transition model into a model of an overhanging beam at both ends, wherein the total length of the overhanging beam at both ends is x0, and the length of the free overhanging end of the overhanging beam at both ends is x1.

3. The method for calculating the natural frequency of the partial vibration mode of the point-supported building curtain wall panel according to claim 1, characterized in that: In step S1, the distance from the support point to the adjacent side of the curtain wall panel along a second direction perpendicular to the first direction is also included as the support distance y1, and the total length of the curtain wall panel along the second direction is y0, satisfying: y1=(1 / b)y0, where b>1; in step S2, the curtain wall panel is also simplified into a model of an overhanging beam at both ends along the second direction, the total length of the overhanging beam at both ends along the second direction is y0, and the length of the free overhanging end of the overhanging beam at both ends along the second direction is y1; in step S5, y1=(1 / b)y0 is also included as the substitution of y1=(1 / b)y0 into the homogeneous equation group of the overhanging beam at both ends to obtain the natural frequency solution of the overhanging beam at both ends along the second direction as the natural frequency solution of the partial vibration mode of the curtain wall panel along the second direction.

4. The method for calculating the natural frequency of the partial vibration mode of the point-supported building curtain wall panel according to claim 3 is characterized in that: Step S2 includes the steps of: S21b, simplifying the curtain wall panel into a simply supported plate model with four outwardly extending sides and four local constrained supports, wherein the distance from the local constrained supports to the adjacent side along the second direction is y1, and the distance from the two sides of the simply supported plate model along the second direction is y0; S22b, the four-point local constraints of the simply supported plate model are made into equivalent linear constraints along the first direction to simplify the model into a transition model, wherein the distance from the linear constraint to the adjacent edge along the second direction is y1, and the distance from the two sides of the transition model along the second direction is y0; S23b, simplifying the transition model into a model of an overhanging beam at both ends along the second direction, wherein the total length of the overhanging beam at both ends along the second direction is y0, and the length of the free overhanging end of the overhanging beam at both ends along the second direction is y1.

5. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to any one of claims 1 to 4, characterized in that: In step S3, the free vibration equilibrium equation of the uniform cross-section beam is: , where E is the elastic modulus of the material, I is the moment of inertia of the section, is the mass per unit length of the beam, z is the deflection, x is the position of any point, and t is the time.

6. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to claim 5, characterized in that: In step S4, the boundary conditions of the endpoints of the free overhanging ends of the two-end overhanging beams are: the section bending moment and shear force of the endpoints of the free overhanging ends are both 0, and the homogeneous equations of the free overhanging ends of the two-end overhanging beams are: ； The boundary conditions of the first node on the cantilever beam at both ends corresponding to one of the supports are: the cross-sectional bending moments and rotation angles on the left and right sides of the first node are equal, and the vertical deflection of the first node is 0. To meet the continuity conditions, the homogeneous equations of the first node of the cantilever beam at both ends are: ； The boundary conditions of the second node corresponding to the other support on the cantilever beam at both ends are: the cross-sectional bending moments and rotation angles on the left and right sides of the second node are equal, and the deflection of the second node is 0. To meet the continuity conditions, the homogeneous equations of the second node of the cantilever beam at both ends are: , where Z(x) is the amplitude curve of the beam, M is the bending moment, θ is the rotation angle, C1 to C 12 is an undetermined constant, To introduce parameters.

7. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to claim 6, characterized in that: The solution of the first n-order frequencies of the curtain wall panel along the first direction or along the second direction is: ;in, is the parameter of the i-th order frequency solution.

8. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to claim 5, characterized in that: In step S3, the separation of variables method is used to further derive the free vibration equilibrium equation of the equal-section beam to obtain the intermediate equation group: ; ; ; Where Z(x) is the amplitude curve of the beam, T(t) is the displacement amplitude of the beam as a function of time, ω 2 is a constant, To introduce parameters; The general solution of the intermediate equations is obtained as: ; , where C1 to C4 are unknown constants, To introduce parameters.

9. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to claim 8, characterized in that: In step S3, the homogeneous equations of the simply supported beam are: 。 10. The method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel according to claim 8, characterized in that: In step S3, the homogeneous equations of the cantilever beam are: 。

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