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 present invention relates to the field of point-supported building curtain walls, and particularly to a calculation method for the natural frequencies of partial vibration modes of the curtain wall panels of point-supported building curtain walls. Background Art

[0002] Point-supported building curtain walls have the advantages of being beautiful, energy-saving, flexible and diverse, and easy to maintain, and also have strong wind and earthquake resistance capabilities. They have been widely used in modern architectural designs 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 reasons for the damage of point-supported building curtain walls, and most of the damaged parts are 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 of the curtain wall structure can be inferred through the change of the structural vibration characteristics.

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

[0005] An object of the present invention is to provide a calculation method for the natural frequencies of partial vibration modes of the curtain wall panels of point-supported building curtain walls, so as to obtain the theoretical solutions of the natural frequencies of partial vibration modes of the curtain wall panels of point-supported building curtain walls.

[0006] To achieve the above object, the technical solution adopted by the present invention is: a calculation method for the natural frequencies of partial vibration modes of the curtain wall panels of point-supported building curtain walls, including the following steps:

[0007] S1. Select the support points, and define the distance from the support points to the adjacent side of the curtain wall panel in the first direction as the support distance x 1 , the total length x 0 of the curtain wall panel in the first direction, satisfying: x 1 = (1 / a)x 0 , where a > 1;

[0008] S2. Simplify the curtain wall panel into a model of a beam with overhanging ends at both ends in the first direction, and the total length of the beam with overhanging ends at both ends in the first direction is x 0, the free overhanging end length of the beam overhanging at both ends along the first direction is x 1 ;

[0009] S3. Simplify the middle part of the beam overhanging at both ends into a simply supported beam model, and simplify the free overhanging ends of the beam overhanging at both ends into a cantilever beam model. Based on the free vibration equilibrium equation of the prismatic beam, derive the homogeneous equation set of the simply supported beam according to the boundary conditions of the simply supported beam, and derive the homogeneous equation set of the cantilever beam according to the boundary conditions of the cantilever beam;

[0010] S4. Based on the free vibration equilibrium equation of the prismatic beam, according to the boundary conditions and displacement continuity conditions of the beam overhanging at both ends, and combining the homogeneous equation set of the simply supported beam and the homogeneous equation set of the cantilever beam, derive the homogeneous equation set of the beam overhanging at both ends;

[0011] S5. Substitute x 1 =(1 / a)x 0 into the homogeneous equation set of the beam overhanging at both ends to obtain the natural vibration frequency solution of the beam overhanging at both ends along the first direction as the natural vibration 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. Simplify the curtain wall panel into a simply supported plate model supported by four-sided overhangs and four-point local constraints, where the distance from the local constraint to the adjacent side along the first direction is x 1 , and the distance between the opposite sides of the simply supported plate model along the first direction is x 0 ;

[0014] S22a. Make the four-point local constraints of the simply supported plate model equivalent linear constraints along the second direction perpendicular to the first direction and simplify it into a transition model, where the distance from the linear constraint to the adjacent side along the first direction is x 1 , and the distance between the opposite sides of the transition model along the first direction is x 0 ;

[0015] S23a. Simplify the transition model into a beam overhanging at both ends model, the total length of the beam overhanging at both ends is x 0 , and the free overhanging end length of the beam overhanging at both ends is x 1 .

[0016] As a preference, in step S1, it further 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 y 1 , and the total length y 0 of the curtain wall panel along the second direction satisfies: y 1 =(1 / b)y 0, where b > 1; in step S2, it further includes simplifying the curtain wall panel into a model of two-end extended beams along the second direction, and the total length of the two-end extended beams along the second direction is y 0 , and the free extended end length of the two-end extended beams along the second direction is y 1 ; in step S5, it further includes substituting y 1 = (1 / b)y 0 into the homogeneous equations of the two-end extended beams to obtain the natural vibration frequency solution of the two-end extended beams along the second direction as the natural vibration frequency solution of the partial vibration modes of the curtain wall panel along the second direction.

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

[0018] S21b. Simplify the curtain wall panel into a simply supported plate model with four-side extensions and four-point local constraint supports, where the distance from the local constraint support to the adjacent side along the second direction is y 1 , and the distance between the opposite sides of the simply supported plate model along the second direction is y 0 ;

[0019] S22b. Make the four-point local constraints of the simply supported plate model equivalent linear constraints along the first direction and simplify it into a transition model, where the distance from the linear constraint to the adjacent side along the second direction is y 1 , and the distance between the opposite sides of the transition model along the second direction is y 0 ;

[0020] S23b. Simplify the transition model into a model of two-end extended beams along the second direction, and the total length of the two-end extended beams along the second direction is y 0 , and the free extended end length of the two-end extended beams along the second direction is y 1 .

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

[0022] where E is the elastic modulus of the material, I is the moment of inertia of the cross-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 extended ends of the two-end extended beams are: the sectional bending moment and shear force at the endpoints of the free extended ends are both 0, and the homogeneous equations of the endpoints of the free extended ends of the two-end extended beams:

[0024]

[0025] The boundary conditions for the first node corresponding to one of the supports on the beam with both ends extended are as follows: the sectional moments on both sides of the first node are equal, the rotations are equal, and the vertical deflection of the first node is 0 to meet the continuity condition. The homogeneous equations for the first node of the beam with both ends extended are:

[0026] The boundary conditions for the second node corresponding to the other support on the beam with both ends extended are as follows: the sectional moments on both sides of the second node are equal, the rotations are equal, and the deflection of the second node is 0 to meet the continuity condition. The homogeneous equations for the second node of the beam with both ends extended are:

[0027] Among them, Z(x) is the amplitude curve of the beam, M is the bending moment, θ is the rotation angle, C 1 to C 12 are constants, and λ is an introduced parameter.

[0028] As a preference, the solutions for the first n natural frequencies of the curtain wall panel in the first direction are: Among them, λ i is the parameter of the i-th natural frequency solution.

[0029] As a preference, in step S3, using the method of separation of variables, the intermediate equations are further derived from the free vibration equilibrium equation of the beam with a constant cross-section:

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

[0031] Finding the general solution of the said intermediate equations is:

[0032] T(t) = C 1 sinωt + C 2 cosωt; Z(x) = C 1 coshλx + C 2 sinhλx + C 3 cosλx + C 4 sinλx, where C 1 to C 4 are undetermined constants, and λ is an introduced parameter.

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

[0034]

[0035] As a preference, in step S3, the homogeneous equation system of the cantilever beam after substituting the boundary conditions:

[0036]

[0037] Compared with the prior art, the beneficial effects of the present invention are as follows: By reasonably performing constraint equivalent processing and simplification on the curtain wall panel in step S2 to simplify it into an outer extended beam model corresponding to the movable curtain wall panel, and by deriving the homogeneous equation system of the outer extended beam at both ends through steps S3 and S4, and then substituting the positional relationship of the support points in step S1, the theoretical solution of the natural vibration frequency of the partial vibration mode of the curtain wall panel is obtained. Compared with the conventional method relying on dynamic test measurement and finite element simulation analysis, the workload is reduced. Description of the Drawings

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

[0039] Figure 2 is a three-dimensional structure diagram of the curtain wall panel of other embodiments of the present application.

[0040] Figure 3 is a schematic diagram of the simply supported plate model of some embodiments of the present application.

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

[0042] Figure 5 is a schematic diagram of the outer extended beam model at both ends of some embodiments of the present application.

[0043] Figure 6 is a schematic diagram of the outer extended beam model at both ends of some embodiments of the present application being simplified into a simply supported beam and a cantilever beam.

[0044] Figure 7A is a vibration mode diagram of the outer extended beam model at both ends of the embodiment of the present application in the frequency 1 state along the first direction.

[0045] Figure 7B is a vibration mode diagram of the transition model of the embodiment of the present application in the frequency 1 state along the first direction.

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

[0047] Figure 8A is a vibration mode diagram of the outer extended beam model at both ends of the embodiment of the present application in the frequency 2 state along the first direction.

[0048] Figure 8BIt is the vibration mode diagram of the transition model of the embodiment of the present application along the first direction in the frequency 2 state.

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

[0050] Figure 9A It is the vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the first direction in the frequency 3 state.

[0051] Figure 9B It is the vibration mode diagram of the transition model of the embodiment of the present application along the first direction in the frequency 3 state.

[0052] Figure 9C It is the vibration mode diagram of the simply supported plate model of the embodiment of the present application along the first direction in the frequency 3 state.

[0053] Figure 10A It is the vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the second direction in the frequency 4 state.

[0054] Figure 10B It is the vibration mode diagram of the transition model of the embodiment of the present application along the second direction in the frequency 4 state.

[0055] Figure 10C It is the vibration mode diagram of the simply supported plate model of the embodiment of the present application along the second direction in the frequency 4 state.

[0056] Figure 11A It is the vibration mode diagram of the two-end cantilever beam model of the embodiment of the present application along the second direction in the frequency 5 state.

[0057] Figure 11B It is the vibration mode diagram of the transition model of the embodiment of the present application along the second direction in the frequency 5 state.

[0058] Figure 11C It is the vibration mode diagram of the simply supported plate model of the embodiment of the present application along the second direction in the frequency 5 state.

[0059] Figure 12 It is the test device diagram for the dynamic test of the embodiment of the present application.

[0060] Figure 13 It is the physical diagram of the measuring point distribution for the dynamic test of the embodiment of the present application.

[0061] Figure 14 It is the schematic diagram of the measuring point distribution for the dynamic test of the embodiment of the present application.

[0062] Figure 15 It is the frequency spectrum diagram obtained from the dynamic test of the embodiment of the present application.

[0063] In the figure: 1, curtain wall panel; 2, back bolt; 3, connecting piece; 4, cross beam; 5, column. Specific embodiments

[0064] Next, in combination with specific embodiments, the present invention will be further described. It should be noted that, on the premise of non - conflict, the following described embodiments or technical features can be arbitrarily combined to form new embodiments.

[0065] In the description of the present invention, it should be noted that for orientation terms, such as the terms "center", "horizontal", "vertical", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "clockwise", "counterclockwise", etc., the orientation and position relationships indicated are based on the orientation or position relationships shown in the drawings. It is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and should not be construed as limiting the specific protection scope of the present invention.

[0066] 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 do not necessarily need to describe a specific order or sequence.

[0067] The terms "comprising" and "having" in the description and claims of the present application, and any variations thereof, are intended to cover non - exclusive inclusion. For example, a process, method, system, product, or device that includes a series of steps or units does not necessarily have to be limited to those steps or units clearly listed, but may include other steps or units not clearly listed or inherent to these processes, methods, products, or devices.

[0068] A calculation method for the natural vibration frequency of partial vibration modes of a point - supported building curtain wall panel includes the following steps:

[0069] S1. Select the support points, and define the distance from the support points along the first direction (i.e., the X - direction) to the adjacent edge of the curtain wall panel as the support distance x 1 , the total length of the curtain wall panel along the first direction is x 0 , satisfying: x 1 =(1 / a)x 0 , where a > 1;

[0070] S2. Simplify the curtain wall panel into a model of a beam with overhanging ends at both ends along the first direction. The total length of the beam with overhanging ends at both ends along the first direction is x 0 , the length of the free overhanging ends of the beam with overhanging ends at both ends along the first direction is x 1 , as Figure 5 shown;

[0071] S3. Simplify the middle part of the beam with overhanging ends at both ends into a simply supported beam model, and simplify the free overhanging ends of the beam with overhanging ends at both ends into a cantilever beam model. As Figure 6 shown, based on the free vibration equilibrium equation of the beam with constant cross-section, the homogeneous equation set of the simply supported beam is derived according to the boundary conditions of the simply supported beam, and the homogeneous equation set of the cantilever beam is derived according to the boundary conditions of the cantilever beam;

[0072] S4. Based on the free vibration equilibrium equation of the beam with constant cross-section, according to the boundary conditions and displacement continuity conditions of the beam with overhanging ends at both ends, and combining the homogeneous equation set of the simply supported beam and the homogeneous equation set of the cantilever beam, the homogeneous equation set of the beam with overhanging ends at both ends is derived;

[0073] S5. Substitute x 1 =(1 / a)x 0 into the homogeneous equation set of the beam with overhanging ends at both ends to obtain the natural vibration frequency solution of the beam with overhanging ends at both ends in the first direction as the natural vibration frequency solution of part of the vibration modes of the curtain wall panel in the first direction.

[0074] Among them, the middle part refers to the part of the beam with overhanging ends at both ends located between the two supports, and the free overhanging end refers to the part of the beam with overhanging ends at both ends that extends out of the support.

[0075] Furthermore, step S2 includes the steps:

[0076] S21a. Simplify the curtain wall panel into a simply supported plate model supported by four-point local constraints with overhanging edges on all four sides, where the distance from the local constraint along the first direction to the adjacent edge is x 1 , and the distance between the opposite sides of the simply supported plate model along the first direction is x 0 , as Figure 3 shown. It is worth mentioning that in the related technology, a simply supported plate model is used for finite element simulation analysis to obtain the natural vibration frequency of the curtain wall panel.

[0077] S22a. Make the four-point local constraints of the simply supported plate model equivalent linear constraints in the second direction (i.e., the Y direction) perpendicular to the first direction (i.e., the X direction), and simplify it into a transition model, where the distance from the linear constraint along the first direction to the adjacent edge is x 1 , and the distance between the opposite sides of the transition model along the first direction is x 0 , as Figure 4 shown. It can be understood that after the curtain wall panel is installed, the structural sealant located on the four sides will produce an embedding and constraint effect on the curtain wall panel. Therefore, the four-point local constraints in the simply supported plate model are made equivalent linear constraints to simplify the force exerted by the simulated structural sealant on the curtain wall panel.

[0078] S23a. Simplify the transition model into a beam model with overhanging ends at both ends, and the total length of the beam with overhanging ends at both ends is x0 , the free overhanging end length of the two-end overhanging beam is x 1 , as Figure 5 shown.

[0079] It can be understood that building curtain walls usually include back-bolted stone curtain walls, as Figure 1 shown. The back-bolted stone curtain wall includes a curtain wall panel 1, a connecting member 3, and a keel composed of a cross beam 4 and a column 5. The connecting member 3 is connected to the keel by bolts, and the connecting member 3 is connected to the perforated curtain wall panel 1 by back bolts 2. Building curtain walls usually also include glass curtain walls, as Figure 2 shown. It can be understood that the curtain wall panel 1 of the building curtain wall is usually fixedly connected to other components via 4 support points, and the support points are symmetrically arranged relative to the midline of the curtain wall panel 1 so that the curtain wall panel 1 is evenly stressed.

[0080] Through steps S21a to S23a, reasonable constraint equivalent processing and simplification are performed on the curtain wall panel to obtain a two-end overhanging beam model of the curtain wall panel in the first direction. Through steps S3 and S4, a homogeneous equation system of the two-end overhanging beam is derived, and then the position relationship of the support points in step S1 is substituted to obtain the natural vibration frequency solution of the partial vibration mode of the curtain wall panel in the first direction.

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

[0082] Specifically, step S1 also includes: the distance from the support point along the second direction perpendicular to the first direction to the adjacent edge of the curtain wall panel is the support distance y 1 , the total length of the curtain wall panel in the second direction is y 0 , satisfying: y 1 =(1 / b)y 0 , where b>1; in step S2, it also includes simplifying the curtain wall panel into a two-end overhanging beam model in the second direction, and the total length of the two-end overhanging beam in the second direction is y 0 , the free overhanging end length of the two-end overhanging beam in the second direction is y 1 ; in step S5, it also includes substituting y 1 =(1 / b)y 0 into the homogeneous equation system of the two-end overhanging beam to obtain the natural vibration frequency solution of the two-end overhanging beam in the second direction as the natural vibration frequency solution of the partial vibration mode of the curtain wall panel in 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-side overhangs and four-point local constraint supports, where the distance from the local constraint support to the adjacent edge along the second direction is y1 For the simply supported plate model, the distance between the two opposite sides in the second direction is y 0 .

[0085] S22b. Apply equivalent linear constraints to the four-point local constraints of the simply supported plate model in the first direction and simplify it to a transition model, where the distance from the linear constraint to the adjacent side in the second direction is y 1 For the transition model, the distance between the two opposite sides in the second direction is y 0 .

[0086] S23b. Simplify the transition model to a two-end cantilever beam model in the second direction, and the total length of the two-end cantilever beam in the second direction is y 0 For the two-end cantilever beam in the second direction, the length of the free cantilever end in the second direction is y 1 .

[0087] That is to say, through steps S21b to S23b, reasonable constraint equivalent processing and simplification are performed on the curtain wall panel to obtain a two-end cantilever beam model of the curtain wall panel in the first direction. By deriving the homogeneous equations of the two-end cantilever beam through steps S3 and S4, and then substituting the position relationship of the support points in step S1, the natural vibration frequency solutions of some vibration modes of the curtain wall panel in the second direction are obtained

[0088] It can be understood that compared with the conventional method relying on dynamic test and finite element simulation analysis, in this application, by simplifying the curtain wall panel into a two-end cantilever beam in the first direction and a two-end cantilever beam in the second direction, and through the homogeneous equations of the two-end cantilever beam, the natural vibration frequency solutions of some vibration modes of the curtain wall panel in the first direction and the natural vibration frequency solutions of some vibration modes of the curtain wall panel in the second direction are derived, reducing the workload

[0089] Specifically, in step S3, the free vibration equilibrium equation of the beam with constant cross-section is as follows

[0090]

[0091] where E is the elastic modulus of the material, I is the moment of inertia of the cross-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 method of separation of variables, assume that the solution of the deflection z is the product of two independent functions of the two parameters of the position x of any point and the time t, 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 function of the displacement amplitude of the beam with respect to time. Substitute equation (2) into equation (1) to obtain 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] where ω 2 is a constant and λ is an introduced parameter;

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

[0100] T(t) = C 1 sinωt + C 2 cosωt (4 - 1)

[0101] Z(x) = C 1 coshλx + C 2 sinhλx + C 3 cosλx + C 4 sinλx (4 - 2)

[0102] where C 1 to C 4 are undetermined 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 infinitely many roots, and thus there are infinitely many frequencies ω n , where n is a natural number greater than 0. For each frequency, a set of ratios of C 1 , C 2 , C 3 , C 4 can be obtained, and the corresponding principal mode function Z n (x) can be obtained.

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

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

[0106]

[0107] It is worth mentioning that according to the homogeneous system of equations (5) of the simply - supported beam, combined with the characteristic equation sinλx 0 = 0, the natural vibration frequency solution of the simply - supported beam can be solved as The main vibration modes of the corresponding simply supported beam are

[0108] Furthermore, in step S3, the free overhanging ends of the beam with overhanging ends at both ends are simplified into 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 the cross-section rotation angle at the end point of the fixed end of the cantilever beam are both 0, and the bending moment and shear force at the end point 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 (6) of the cantilever beam, let the determinant corresponding to the homogeneous equations be 0, and it is derived that:

[0111] cosλx 0 ·coshλx 0 +1 = 0 (7)

[0112] Using the trial method to solve equation (7), it is obtained that: λ 1 x 0 = 1.88, λ 2 x 0 = 4.69, λ 3 x 0 = 7.86, λ n x 0 = (n - 1 / 2)π, and the main vibration modes of the corresponding cantilever beam are

[0113] In step S4, according to the boundary conditions and displacement continuity conditions of the beam with overhanging ends at both ends, and combining the homogeneous equations of the simply supported beam and the homogeneous equations of the cantilever beam, the homogeneous equations of the beam with overhanging ends at both ends are derived.

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

[0115]

[0116] The boundary conditions at the first node corresponding to one of the supports on the beam with overhanging ends at both ends are: the bending moments on both sides of the first node are equal, the rotation angles are equal, and the vertical deflection of the first node is 0 to meet the continuity condition. Combining the homogeneous equations (5) of the simply supported beam and the homogeneous equations (6) of the cantilever beam, the homogeneous equations at the first node of the beam with overhanging ends at both ends are derived as follows:

[0117]

[0118] The boundary conditions of the second node corresponding to another support on the beam with both ends extended are as follows: the sectional moments on both sides of the second node are equal, the rotations are equal, and the deflection of the second node is 0 to meet the continuity condition. Combining the homogeneous equation system (5) of the simply supported beam and the homogeneous equation system (6) of the cantilever beam, the homogeneous equation system of the second node of the beam with both ends extended is derived as follows:

[0119]

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

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

[0123] Next, the technical solutions in the present application will be described in conjunction with specific embodiments. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present invention without creative efforts shall fall within the protection scope of the present invention.

[0124] In a specific embodiment, the support distance x of the support point along the first direction 1 =(1 / 5)x 0 , and the support distance y of the support point along the second direction 1 =(1 / 4)y 0 .

[0125] Substitute x 1 =(1 / 5)x 0 into equations (8), (9) and (10), and solve to obtain the first three natural frequency solutions of the curtain wall panel along the first direction as follows:

[0126]

[0127] Substitute y 1 =(1 / 4)y 0 into equations (8), (9) and (10). It can be understood that y 0 substituting into x in the equation 0 , y 1 substituting into x in the equation 1 , and solve to obtain the first two natural frequency solutions of the curtain wall panel along the second direction as follows:

[0128]

[0129] To verify the feasibility of obtaining the natural vibration frequency solution of the curtain wall panel using the two-span cantilever beam model, finite element simulation analysis and dynamic test are used for auxiliary verification.

[0130] Specifically, finite element models of the two-span cantilever beam model, the transition model, and the simply supported plate model are established respectively. Among them, beam3 elements are used for the two-span cantilever beam. The dimensions of the curtain wall panel in the transition model and the simply supported plate model are 900mm×600mm×25mm, and shell281 elements are used, which are set as linear elastic and isotropic, 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 natural vibration characteristics of each finite element model, and the analysis results are as Figures 7A - 11C shown.

[0131] The DH5922N dynamic signal test and analysis system and hardware facilities are used for dynamic test, as shown in Figure 12 and Figure 13 . Specifically, a stainless steel force hammer is used as the excitation device to excite 70 measuring points, a 1A116E piezoelectric acceleration sensor is used as the vibration pickup device, and the reference points are selected at the measuring points 18, 33, 35, and 47 shown in Figure 14 . The multi-input multi-output (MIMO) method is used for modal testing. The excitation force signals at different measuring points and the vibration response signals at the reference points are input into the acquisition card, and the natural vibration frequency and vibration mode of the curtain wall panel can be obtained through the DH5922N dynamic test analysis software.

[0132] It is worth mentioning that the spectrogram obtained by using the polyLSCF (Polynomial Least Squares with Constraints) modal parameter identification method is as shown in Figure 15 shown, Figure 15 where the peak in the middle wave appears at the stable state of the three parameters of frequency, damping, and modal participation factor, and all three parameters are within the given error range.

[0133] Table 1 shows the natural vibration frequencies of the curtain wall panel obtained by theoretical derivation of the two-span cantilever beam, finite element simulation analysis, and dynamic test respectively. Among them, the values in parentheses are the errors between the natural vibration frequencies obtained by finite element simulation analysis and dynamic test and the natural vibration frequency of the two-span cantilever beam obtained by theoretical derivation. Table 1

[0134] The first five natural frequencies obtained from the cantilever beam model are generally in good agreement with those obtained from the theoretical derivation of the cantilever beam with both ends, and the maximum error is only -3.18%. Compared with the first five natural frequencies obtained from the theoretical derivation of the cantilever beam with both ends, the maximum error of the transition model is only -6.20%; compared with the first five natural frequencies obtained from the theoretical derivation of the cantilever beam with both ends, the maximum error of the simply supported plate model is -7.82%. It can be understood that the main reason for the gradual increase in the error is the difference caused by the simplified equivalence between the four-point local constraint in the simply supported plate model and the linear constraint in the transition model during the simplification process in step S2. Compared with the first five natural frequencies obtained from the theoretical derivation of the cantilever beam with both ends, the maximum error of the first five natural frequencies obtained from the dynamic test is -8.72%.

[0135] In summary, it is feasible and referenceable to calculate the natural frequencies of some vibration modes of the curtain wall panel by using the theory of the cantilever beam with both ends.

[0136] The basic principle, main features and advantages of the present invention have been described above. Those skilled in the art should understand that the present invention is not limited by the above embodiments, and what is described in the above embodiments and the specification is only the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. 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: Includes 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 x0 of the curtain wall panel along the first direction satisfies: x1 = (1 / a) x0, where a>1; 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; 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. 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 the partial vibration mode of the point-supported building curtain wall panel according to claim 1 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 x1, and the distance from the two sides of the simply supported plate model along the first direction is x0; 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; 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.

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 is characterized in that: In step S1, 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 also included as 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; 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 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 comprises 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 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; 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; 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.

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 the partial vibration mode of the point-supported building curtain wall panel according to claim 5 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: Curve, M is the bending moment, θ is the rotation angle, C1 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, 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, 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 amplitude 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)=C1sinωt+C2cosωt;Z(x)=C1coshλx+C2sinhλx+C3cosλx+C4sinλx, where C1 to C4 are unknown constants 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, 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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