Method for calculating natural vibration frequency of partial vibration mode of point-supported building curtain wall panel

Abstract

The invention discloses a method for calculating the natural vibration frequency of a partial vibration mode of a point-supported building curtain wall panel, which comprises the following steps: S1, selecting a supporting point, defining the distance from the supporting point to the adjacent side of the curtain wall panel along a first direction as a supporting distance x1, and defining the total length x0 of the curtain wall panel along the first direction, which meets the condition that x1 is equal to (1 / a) x0; s2, the curtain wall panel is simplified into a two-end overhanging beam model in the first direction, the total length of two-end overhanging beams is x0, and the length of the free overhanging ends of the two-end overhanging beams is x1; s3, simplifying the midspan part of the two-end overhanging beam into a simply supported beam model, simplifying the free overhanging end into a cantilever beam model, and deducing to obtain a homogeneous equation set of the simply supported beam and a homogeneous equation set of the cantilever beam; s4, in combination with the homogeneous equation set of the simply supported beam and the homogeneous equation set of the cantilever beam, deriving a homogeneous equation set of overhanging beams at the two ends; s5, x1 = (1 / a) x0 is substituted into the homogeneous equation set of the overhanging beams at the two ends, so that the natural vibration frequency solution of the partial vibration mode of the curtain wall panel in the first direction is obtained.

Description

Technical Field

[0001] The 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 part of many important buildings.

[0003] In most cases of damage to point-supported building curtain walls, connection failure and unreasonable structure are the main causes of damage to point-supported building curtain walls, and most of the damage is concentrated at the connection between the curtain wall panels and the hangers. When a point-supported building curtain wall is damaged, the corresponding physical parameters will change accordingly, so the damage to the curtain wall structure can be inferred through the changes in the structural vibration characteristics.

[0004] Among them, the natural frequency is relatively easy to obtain in the structural modal parameters, and the identification accuracy is high. In the related technology, the natural frequency of the point-supported building curtain wall is mainly obtained by dynamic test or finite element simulation analysis. Although a more accurate natural frequency can be obtained through dynamic test, the test steps are relatively complicated, there are many interference factors, and the test efficiency is subject to the layout and removal of sensors. When the building curtain wall is supported by finite element simulation analysis, 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. 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 purpose, the technical solution adopted by the present invention is: 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, simplifying the curtain wall panel into a model of cantilevered beams at both ends along a first direction, wherein the total length of the cantilevered beams at both ends along the first direction is x0, and the length of the free cantilevered ends of the cantilevered beams at both ends along the first direction is x1;

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

[0010] S4. Based on the free vibration equilibrium equation 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 beams 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 preferred embodiment, step S2 comprises the steps of:

[0013] S21a, 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 support 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, making equivalent linear constraints on the four-point local constraints of the simply supported plate model along a second direction perpendicular to the first direction, simplifying it 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 sides of the transition model along the first direction is x0;

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

[0016] As a preferred embodiment, step S1 also includes that 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 y0 of the curtain wall panel along the second direction satisfies: y1=(1 / b)y0, wherein 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 preferred embodiment, step S2 comprises 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 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;

[0019] S22b, making the four-point local constraints of the simply supported plate model equivalent linear constraints along the first direction, simplifying it 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 two-end cantilevered beams along the second direction, wherein the total length of the two-end cantilevered beams along the second direction is y0, and the length of the free cantilevered ends of the two-end cantilevered beams along the second direction is y1.

[0021] As a preferred embodiment, 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 overhanging beams at both ends 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 overhanging beams at both ends 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 section bending moments and rotation angles on both 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]

[0027] The boundary conditions of the second node on the cantilever beam at both ends corresponding to the other support are: the section 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:

[0028] Among them, C1 to C 12 is a constant and λ is an introduced parameter.

[0029] 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 frequency solution.

[0030] As a preferred embodiment, in step S3, the variable separation method is used to further derive the free vibration equilibrium equation of the equal-section beam to obtain the intermediate equation group:

[0031] T″(t)+ω 2 T(t)=0;Z Ⅳ (x)-λ 4 Z(x)=0; Where Z(x) is the deflection curve of the beam, T(t) is the displacement amplitude of the beam as a function of time, ω 2 is a constant, λ is an introduced parameter;

[0032] The general solution of the intermediate system of equations is obtained as:

[0033] T(t)=C1sinωt+C2cosωt;Z(x)=C1chλx+C2shλx+C3cosλx+C4sinλx, where C1 to C4 are unknown constants and λ is an introduced parameter.

[0034] As a preferred embodiment, in step S3, after substituting the boundary conditions, the homogeneous equations of the simply supported beam are:

[0035]

[0036] As a preferred embodiment, in step S3, after substituting the boundary conditions, the homogeneous equations of the cantilever beam are:

[0037]

[0038] Compared with the prior art, the beneficial effect of the present invention lies in: 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, the homogeneous equation group of the cantilever beams at both ends is derived through steps S3 and S4, and then the position relationship of the support points in step S1 is substituted to obtain the theoretical solution of the natural frequency of the partial vibration mode of the curtain wall panel, which reduces the workload compared with the conventional reliance on dynamic test and finite element simulation analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 It is a three-dimensional structural diagram of curtain wall panels of some embodiments of the present application.

[0040] Figure 2 It is a three-dimensional structural diagram of curtain wall panels according to other embodiments of the present application.

[0041] Figure 3 is a schematic diagram of a simply supported plate model of some embodiments of the present application.

[0042] Figure 4is a schematic diagram of a transition model of some embodiments of the present application.

[0043] Figure 5 It is a schematic diagram of a two-end cantilever beam model of some embodiments of the present application.

[0044] Figure 6 It is a schematic diagram of simplifying the two-end cantilever beam model of some embodiments of the present application into a simply supported beam and a cantilever beam.

[0045] Fig. 7A It is a vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the first direction at the frequency 1 state.

[0046] Figure 7B It is a vibration mode diagram of the transition model of the embodiment of the present application along the first direction at the frequency 1 state.

[0047] Figure 7C is a vibration mode diagram of the simply supported plate model of an embodiment of the present application along the first direction at a frequency of 1.

[0048] Fig. 8A It is a vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the first direction at the frequency 2 state.

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

[0050] 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 a frequency of 2.

[0051] Fig. 9A It is a vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the first direction at the frequency 3 state.

[0052] Fig. 9B 3 is a vibration mode diagram of the transition model of an embodiment of the present application along the first direction at a frequency of 3.

[0053] Fig. 9C 3 is a vibration mode diagram of the simply supported plate model of the embodiment of the present application along the first direction at the frequency 3 state.

[0054] Fig. 10A It is a vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the second direction at the frequency 4 state.

[0055] Fig. 10B 4 is a vibration mode diagram of the transition model of the embodiment of the present application along the second direction at the frequency 4 state.

[0056] Fig. 10C 4 is a vibration mode diagram of the simply supported plate model of the embodiment of the present application along the second direction at a frequency of 4.

[0057] Fig.11A It is a vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the second direction at the frequency 5 state.

[0058] Fig. 11B 5 is a vibration mode diagram of the transition model of an embodiment of the present application along the second direction at a frequency of 5.

[0059] Fig. 11C 3 is a vibration mode diagram of the simply supported plate model of the embodiment of the present application at a frequency of 5 along the second direction.

[0060] Fig.12 It is a diagram of a test device for performing a dynamic test in an embodiment of the present application.

[0061] Fig.13 This is a physical diagram of the distribution of measuring points for the dynamic test of the embodiment of the present application.

[0062] Fig.14 It is a schematic diagram of the distribution of measuring points for the dynamic test of the embodiment of the present application.

[0063] Fig.15 This is a spectrum diagram obtained by performing a dynamic test on an embodiment of the present application.

[0064] In the figure: 1. Curtain wall panel; 2. Back bolt; 3. Connector; 4. Crossbeam; 5. Column. DETAILED DESCRIPTION

[0065] The present invention is 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 a new embodiment.

[0066] In the description of the present invention, it should be noted that directional words, such as the terms "center", "lateral", "longitudinal", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise", etc., indicating directions and positional relationships are based on the directions or positional relationships shown in the accompanying drawings, which are only for the convenience of narrating the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and cannot be understood as limiting the specific scope of protection of the present invention.

[0067] It should be noted that the terms "first", "second", etc. in the description and claims of the present application are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence.

[0068] The terms "including" 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 comprising 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 these processes, methods, products or apparatuses.

[0069] A method for calculating the natural frequency of a partial vibration mode of a point-supported building curtain wall panel comprises the following steps:

[0070] 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 (i.e., the X 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;

[0071] S2. Simplify the curtain wall panel into a model of two-end cantilever beams along the first direction. The total length of the two-end cantilever beams along the first direction is x0, and the length of the free cantilever end of the two-end cantilever beams along the first direction is x1. Figure 5 As shown;

[0072] S3. Simplify the mid-span part of the cantilever beam at both ends into a simply supported beam model, and simplify the free cantilever end of the cantilever beam at both ends into a cantilever beam model, such as Figure 6 As shown, 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;

[0073] S4. Based on the free vibration equilibrium equation 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 beams at both ends are derived;

[0074] 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.

[0075] Among them, the mid-span part refers to the part of the cantilever beams at both ends located between the two supports, and the free cantilever end refers to the part of the cantilever beams at both ends that extends out of the supports.

[0076] Further, step S2 includes the steps of:

[0077] S21a, simplify the curtain wall panel into a simply supported plate model with four outwardly extending sides and four local constrained supports, where the distance from the local constrained support 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, such as 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.

[0078] S22a, the four-point local constraints of the simply supported plate model are made into equivalent linear constraints along the second direction (i.e., the Y direction) perpendicular to the first direction (i.e., the 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, such as Figure 4 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 plate model are used as equivalent linear constraints to simplify the simulation of the force exerted by the structural sealant on the curtain wall panel.

[0079] S23a, simplify the transition model to a two-end cantilever beam model, the total length of the two-end cantilever beam is x0, and the length of the free cantilever end of the two-end cantilever beam is x1, such as Figure 5 shown.

[0080] 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 composed 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 by a back bolt 2. The building curtain wall usually also includes a glass curtain wall, such as Figure 2 It can be understood that the curtain wall panel 1 of the 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.

[0081] The curtain wall panel is subjected to reasonable constraint equivalent processing and simplification through steps S21a to S23a to obtain the cantilever beam model of the curtain wall panel at both ends along the first direction. The homogeneous equation group of the cantilever 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 first direction.

[0082] It is worth mentioning that by adopting 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.

[0083] 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 y0 of the curtain wall panel along the second direction satisfies: y1=(1 / b)y0, wherein 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.

[0084] Further, step S2 includes the steps of:

[0085] S21b. Simplify the curtain wall panel into a simply supported plate model with four extended sides and four local constrained supports, wherein the distance from the local constrained support to the adjacent edge 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.

[0086] S22b. The four-point local constraints of the simply supported plate model are made 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 two sides of the transition model along the second direction is y0.

[0087] S23b, simplifying the transition model into a model of the cantilevered beams at both ends along the second direction, wherein the total length of the cantilevered beams at both ends along the second direction is y0, and the length of the free cantilevered ends of the cantilevered beams at both ends along the second direction is y1.

[0088] That is to say, through steps S21b to S23b, the curtain wall panel is reasonably constrained equivalently processed and simplified to obtain the cantilever beam model of the curtain wall panel at both ends along the first direction, and the homogeneous equation group of the cantilever beams at both ends is derived through steps S3 and S4, 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 second direction.

[0089] 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.

[0090] Specifically, in step S3, the free vibration equilibrium equation of the uniform cross-section beam is:

[0091]

[0092] 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.

[0093] 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:

[0094] Z(x,t)=Z(x)·T(t) (2)

[0095] Where Z(x) is the deflection curve of the beam, and T(t) is the function of the displacement amplitude of the beam with respect to time. Substituting equation (2) into equation (1) yields the intermediate equation system:

[0096] T″(t)+ω 2 T(t)=0 (3-1)

[0097] Z Ⅳ (x)-λ 4 Z(x)=0 (3-2)

[0098]

[0099] Among them, ω 2 is a constant, λ is an introduced parameter;

[0100] The general solution of the intermediate system of equations is:

[0101] T(t)=C1sinωt+C2cosωt (4-1)

[0102] Z(x)=C1chλx+C2shλx+C3cosλx+C4sinλx (4-2)

[0103] Among them, C1 to C4 are unknown constants, and λ is an introduced parameter.

[0104] 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).

[0105] Furthermore, in step S3, the mid-span portion of the outrigger beams at both ends is simplified to a simply supported beam model, and the homogeneous equation group of the simply supported beam is derived according to the boundary conditions of the simply supported beam.

[0106] 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:

[0107]

[0108] 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

[0109] Furthermore, in step S3, the free extension end of the extension beam at both ends is simplified to a cantilever beam model, and the homogeneous equations of the cantilever beam are derived according to 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 end point of the fixed end of the cantilever beam are both 0, and the cross-sectional bending moment and shear force of the end point of the free extension end are both 0. Combining equations (4-1) and (4-2), the homogeneous equations of the cantilever beam are derived as follows:

[0110]

[0111] 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:

[0112] cosλx0·coshλx0+1=0 (7)

[0113] 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, the corresponding main vibration mode of the cantilever beam is

[0114] In step S4, based on the boundary conditions and displacement continuity conditions of the cantilever beams at both ends, 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 at both ends are derived.

[0115] Specifically, the boundary conditions of the endpoints of the free extension ends of the two-end extension beam are: the section bending moment and shear force of the endpoints of the free extension 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 extension ends of the two-end extension beam are derived as follows:

[0116]

[0117] The boundary conditions of the first node on the cantilever beam at both ends corresponding to one of the supports are: the section 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 equation group of the first node of the cantilever beam at both ends is derived by combining the homogeneous equation group (5) of the simply supported beam and the homogeneous equation group (6) of the cantilever beam:

[0118] (9)

[0119] The boundary conditions of the second node on the cantilever beam at both ends corresponding to the other support are: the section 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 equation group of the second node of the cantilever beam at both ends is derived by combining the homogeneous equation group (5) of the simply supported beam and the homogeneous equation group (6) of the cantilever beam:

[0120]

[0121] Among them, C1 to C12 are unknown constants, and λ is an introduced parameter.

[0122] 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]

[0124] The technical solution in this application will be described below in conjunction with specific embodiments. Obviously, the described embodiments are only part of the embodiments of this application, not all of them. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0125] In a specific embodiment, the support distance x1 of the support point along the first direction is =(1 / 5)x0, and the support distance y1 of the support point along the second direction is =(1 / 4)y0.

[0126] 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:

[0127]

[0128] 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 frequency solutions of the curtain wall panel along the second direction are obtained as follows:

[0129]

[0130] 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 were used for auxiliary verification.

[0131] Specifically, finite element models of the cantilever beam model, transition model and simply supported plate model are established. The cantilever beams at both ends use beam3 units. The curtain wall panels in the transition model and simply supported plate model have a size of 900mm×600mm×25mm, and use shell281 units, which are set to linear elasticity and isotropy, with a Young's modulus of 29.5GPa, a Poisson's ratio of 0.125, and a density of 2.67g / cm 3 The Lanczos algorithm is used for modal analysis to analyze the self-vibration characteristics of each finite element model. The analysis results are as follows: Figures 7A-11C shown.

[0132] Use DH5922N dynamic signal test and analysis system and hardware facilities to carry out dynamic test, such as Fig.12 and Fig.13 Specifically, the excitation device uses a stainless steel hammer to excite 70 measuring points, and the vibration pickup device uses a 1A116E piezoelectric acceleration sensor. The reference point is selected at Fig.14 The modal test adopts the multiple input multiple output (MIMO) method for the measuring points 18, 33, 35 and 47 shown in the figure. The excitation force signals at different measuring points and the vibration response signals at the reference point are input into the acquisition card, and the natural frequency and vibration mode of the curtain wall panel can be obtained through the DH5922N dynamic test analysis software.

[0133] It is worth mentioning that the spectrum obtained by using the polyLSCF (Polynomial Least Squares with Constraints) modal parameter identification method is as follows: Fig.15 As shown, Fig.15 The mid-wave peak appears when the frequency, damping and modal participation factor are stable at the same time, and all three parameters are within the given error range.

[0134] Table 1 shows the natural frequencies of the curtain wall panels obtained by theoretical derivation of the cantilever beams at both ends, finite element simulation analysis and dynamic test tests. The values ​​in brackets are the errors between the natural frequencies obtained by finite element simulation analysis and dynamic test and the natural frequencies derived by the cantilever beams at both ends. Table 1

[0135] The first five natural frequencies obtained from the cantilever beam model are generally consistent with the natural frequencies derived from the cantilever beam theory at both ends, with a maximum error of only -3.18%. The first five natural frequencies obtained from the transition model have a maximum error of only -6.20% compared with the natural frequencies derived from the cantilever beam theory at both ends; the first five natural frequencies obtained from the simply supported plate model have a maximum error of -7.82% compared with the natural frequencies derived from the cantilever beam theory at both ends. It can be understood that the main reason for the gradual increase in the error is: during the simplification process of 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 have a maximum error of -8.72% compared with the natural frequencies derived from the cantilever beam theory at both ends.

[0136] 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.

[0137] 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 above embodiments. The above embodiments and descriptions only describe the principles of the present invention. The present invention may be subject to various changes and improvements without departing from the spirit and scope of the present invention. These changes and improvements fall within the scope of the present invention to be protected. The scope of protection claimed by the present invention is defined by the attached 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. It is characterized in that Includes steps: 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 as the support distance x. 1 , the total length x of the curtain wall panel along the first direction 0 , satisfying: x 1 =(1 / a)x 0 , where a>1; S2, simplifying the curtain wall panel into a model of two outrigger beams along the first direction, where the total length of the two outrigger beams along the first direction is x 0 , the length of the free extension ends of the extension beams at both ends along the first direction is x 1 ; S3. Simplify the mid-span part of the cantilever beams at both ends into a simply supported beam model, and simplify the free cantilever ends of the cantilever beams at both ends into a cantilever beam model; on the basis of the free vibration equilibrium equation of the uniform cross-section beam, derive the homogeneous equations of the simply supported beam according to the boundary conditions of the simply supported beam, and derive the homogeneous equations of the cantilever beam according to the boundary conditions of the cantilever beam; S4. Based on the free vibration equilibrium equation 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 beams at both ends are derived; S5. x 1 =(1 / a)x 0 Substitute the homogeneous equations of the cantilever beams at both ends into the system to obtain the natural frequency solutions of the cantilever beams at both ends along the first direction as the natural frequency solutions of the partial vibration modes of the curtain wall panel along the first direction.

2. The method for calculating the natural frequency of the partial vibration mode of the point-supported building curtain wall panel according to claim 1, It is characterized in that Step S2 comprises the steps of: S21a, 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 support to the adjacent side along the first direction is x 1 , the distance between the two sides of the simply supported plate model along the first direction is x 0 ; S22a, the four-point local constraints of the simply supported plate model are made into equivalent linear constraints along a second direction perpendicular to the first direction, and simplified into a transition model, wherein the distance from the linear constraint to the adjacent edge along the first direction is x 1 , the distance between the transition model and both sides along the first direction is x 0 ; S23a, simplifying the transition model into a model of two-end cantilever beams, where the total length of the two-end cantilever beams is x 0 The length of the free extension of the extension beam at both ends is x 1 .

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, It is characterized in that In step S1, the distance from the supporting point to the adjacent side of the curtain wall panel along a second direction perpendicular to the first direction is a support distance y. 1 , the total length y of the curtain wall panel along the second direction 0 , satisfying: y 1 =(1 / b)y 0 , wherein b>1; in step S2, the curtain wall panel is simplified into a model of two outrigger beams along the second direction, and the total length of the two outrigger beams along the second direction is y 0 The length of the free extension ends of the two extension beams along the second direction is y 1 ; Step S5 also includes y 1 =(1 / b)y 0 Substitute into the homogeneous equation group of the cantilever beams at both ends to obtain the natural frequency solution of the cantilever beams 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, It is characterized in that Step S2 comprises the steps of: S21b, simplify the curtain wall panel into a simply supported plate model with four sides extended and four points of local constraint support, where the distance from the local constraint support to the adjacent edge along the second direction is y 1 , the distance between the two sides of the simply supported plate model along the second direction is y 0 ; 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 y 1 , the distance between the transition model and the two sides along the second direction is y 0 ; S23b, simplifying the transition model into a model of two-end cantilevered beams along the second direction, where the total length of the two-end cantilevered beams along the second direction is y 0 The length of the free extension ends of the two extension beams along the second direction is y 1 .

5. A 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, It is 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 the partial vibration mode of the point-supported building curtain wall panel according to claim 5, It is characterized in that In step S4, the boundary conditions of the endpoints of the free extension ends of the two-end extension beams are: the section bending moment and shear force of the endpoints of the free extension ends are both 0, and the homogeneous equations of the free extension ends of the two-end extension beams are: The boundary conditions of the first node on the cantilever beam at both ends corresponding to one of the supports are: the section bending moments and rotation angles on both 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 on the cantilever beam at both ends corresponding to the other support are: the section 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: Among them, C 1 To C 12 is a constant to be determined, and λ is an introduced parameter.

7. The method for calculating the natural frequency of the partial vibration mode of the point-supported building curtain wall panel according to claim 6, It is 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: Among them, λ i is the parameter of the i-th frequency solution.

8. The method for calculating the natural frequency of the partial vibration mode of the point-supported building curtain wall panel according to claim 5, It is characterized in that In step S3, the variable separation method is used to further derive the free vibration equilibrium equation of the equal-section beam to obtain the intermediate equation group: T″(t)+ω 2 T(t)=0;Z Ⅳ (x)-λ 4 Z(x)=0; Where Z(x) is the deflection curve of the beam, T(t) is the displacement amplitude of the beam as a function of time, ω 2 is a constant, λ is an introduced parameter; The general solution of the intermediate system of equations is obtained as: T(t)=C 1 sinωt+C 2 cosωt; Z(x)=C 1 chλx+C 2 shλx+C 3 cosλx+C 4 sinλx, where C 1 To C 4 is a constant to be determined, and λ is an introduced parameter.

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

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

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