Vibration characteristic analysis method

Abstract

To propose a vibration characteristic analysis method to improve the prediction accuracy of vibration characteristics at arbitrary points in an architectural structure.SOLUTION: The present invention measures the frequency response function at a measurement point of an architectural structure (step S1), identifies from the measured frequency response function the damped natural angular frequency, the mode attenuation rate, and the characteristic mode of the vibration in the excitation direction at the excitation point and the response point (step S2), approximates the characteristic mode at an arbitrary point of the architectural structure with a series of a predetermined sequence of functions (step S3), calculates the coefficients of the series by the weighted least squares method (step S6), and predicts the frequency response function at an arbitrary point of the architectural structure (step S8).SELECTED DRAWING: Figure 4

Description

Technical Field

[0001] The present invention relates to a vibration characteristic analysis method.

Background Art

[0002] In view of the fact that vibrations occurring in structures such as buildings cause various problems, technical development for analyzing the vibration characteristics of structures has been actively carried out. When analyzing vibration characteristics, it is necessary to represent the target structure as a mathematical model. As a method of representing it as a mathematical model, experimental mode analysis is known. Experimental mode analysis is a method of applying vibrations to the target structure for measurement and identifying the natural frequency (damped natural angular frequency), mode damping ratio (mode damping rate), natural mode, or parameters indirectly corresponding to these from the frequency response function obtained as the measurement result.

[0003] The natural frequency and mode damping ratio are parameters that determine time or frequency characteristics, and can be accurately identified by mode characteristic identification methods such as the multi-point variational method and the subspace method. On the other hand, the natural mode is a parameter (function) that determines the spatial characteristics of vibration. Regarding the excitation points and response points set when measuring the vibration characteristics of the target structure, the natural mode can be identified even by the above-mentioned mode characteristic identification methods. However, there is a problem that the natural mode cannot be identified for points other than the excitation points and response points. Due to time and space constraints during measurement, the number of excitation points and response points that can be set is often limited and cannot be increased. However, if the vibration characteristics of any point other than these can be analyzed from the information of the limited excitation points and response points, it is very useful for grasping the vibration characteristics of the target structure.

[0004] In addition, through the analysis of the vibration characteristics of a structure, vibration control measures using a vibration control device or the like may be required. When actually implementing vibration control measures, the following procedures are often taken: (1) measuring the vibration characteristics (frequency response function) of the structure before the measures; (2) mathematically modeling the vibration characteristics from the measurement results; (3) setting the target values of the vibration characteristics; (4) considering vibration control measures; (5) predicting the effects after the vibration control measures using the model; (6) determining whether the prediction results meet the target values. However, conventionally, there has been a problem that the natural modes cannot be identified for points other than the excitation points and response points as described above, so there is room for improvement in the consideration of vibration control measures.

[0005] Patent Document 1 discloses a method for identifying the vibration of a slab of a building, a vibration control device, a method for arranging the vibration control device, a building floor structure, and a vibration measuring device. In Patent Document 1, since the measurement points of the vibration spectrum are limited to four points, it is impossible to accurately identify higher-order natural modes, natural modes with an asymmetric shape, or natural modes with a complex shape. Also, since the natural modes are not identified in a positive form, when predicting the effects of vibration control measures, it is impossible to predict the effects at any point other than the measurement points.

Prior Art Documents

Patent Documents

[0006]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0007] From such a perspective, an object of the present invention is to propose a vibration characteristic analysis method for improving the prediction accuracy of the vibration characteristics at any point of a building.

Means for Solving the Problems

[0008] In order to solve the above problems, the present invention includes steps of measuring a frequency response function at a measurement point of a building, identifying a damping natural angular frequency, a mode damping ratio, and a natural mode of vibration in the excitation direction at the excitation point and the response point from the measured frequency response function, approximating the natural mode at an arbitrary point of the building by a series of a predetermined sequence of functions, calculating coefficients of the series by a weighted least squares method using the identified first natural mode and the series, and predicting a frequency response function at an arbitrary point of the building using the calculated coefficients and the approximated natural mode. This is a vibration characteristic analysis method.

[0009] According to the present invention, by approximating and expressing the natural mode at an arbitrary point other than the measurement point, the vibration characteristics at an arbitrary point can be predicted without measurement. Also, the order of the series can be appropriately set according to the required prediction accuracy. Further, the number of measurement points can be appropriately set according to the required prediction accuracy. Therefore, the prediction accuracy of the vibration characteristics at an arbitrary point of the building can be improved. Even when vibration control measures are taken, the same approach of obtaining a frequency response function for the vibration system when a vibration control device is installed at an arbitrary position of the building can be adopted. That is, not only obtaining the frequency response function at the measurement point of the building, but also if the frequency response function of the vibration control device is obtained by measurement or theoretically, the vibration characteristics at an arbitrary point can be predicted without measurement using these frequency response functions. For this reason, the effect of the vibration control measures can be predicted.

[0010] In the step of calculating the coefficients of the series, in addition to the response points where the frequency response function is actually measured, it is preferable to use a point where the vibration in the excitation direction becomes substantially zero as a virtual response point. In the step of calculating the coefficients of the series, the number of response points needs to be larger than the number of coefficients. However, according to such a configuration, since it is a very rigid part in the structure of the building, it hardly vibrates in the excitation direction and the natural mode can be set to 0 without measurement. By using such a point as a response point, the number of response points can be substantially increased. Therefore, it is possible to prepare a relational expression necessary for calculating all the coefficients of the series of function sequences approximating the natural mode. As a result, even under conditions where the number of excitation points and response points that can be set is often limited due to time and space constraints during measurement and cannot be increased, the natural mode at any point other than the measurement points can be approximated and expressed.

Advantages of the Invention

[0011] According to the present invention, the prediction accuracy of the vibration characteristics at any point of a building can be improved.

Brief Description of the Drawings

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Mode for Carrying Out the Invention

[0013] Hereinafter, the mode for carrying out the present invention will be described in detail with appropriate reference to the drawings. Each figure is only schematically shown to the extent that the present invention can be sufficiently understood. Therefore, the present invention is not limited only to the illustrated examples. In each figure, common components and similar components are denoted by the same reference numerals, and their overlapping descriptions are omitted.

[0014] [Calculation Method] First, a calculation method for obtaining the frequency response function of an arbitrary point when vibration is applied to the structure will be described.

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[0026] The mean squared error ε of Equation (4) can be expressed in matrix form as follows.

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[0035] Therefore, data of points that become nodes of the natural mode is added from the structural characteristics of the structure. Specifically, data of points where the columns of the building are located is added. Fig. 3 is an explanatory diagram of the positions that become nodes of the natural mode in the floor structure of the building. The floor structure of Fig. 3 includes a floor slab 1, a plurality of small beams 2 and a large beam 3 erected with respect to the floor slab 1. Further, a plurality of columns 4 are erected at arbitrary positions of the floor slab 1.

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[0046] [Processing] The analysis device of this embodiment is a computer that executes the above-described calculation method. The analysis device includes hardware such as an input unit, an output unit, a control unit, and a storage unit. For example, when the control unit is composed of a CPU (Central Processing Unit), the information processing by the computer including the control unit is realized by the program execution processing by the CPU. Further, the storage unit included in the computer stores various programs for realizing the functions of the computer according to the instructions of the CPU. Thereby, the cooperation of software and hardware is realized. The program can be provided by being recorded on a recording medium or via a network.

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[0050] [Example] An example of predicting the vibration characteristics of the floor structure will be described. FIG. 5 is a schematic diagram of the floor structure of this example. The floor structure of FIG. 5 includes a floor slab 1, secondary beams 2, and main beams 3, and columns 4 are erected, similar to the floor structure of FIG. 3. The horizontal dimension of the floor structure: L x is 22200 mm, and the vertical dimension: L y is 11500 mm.

[0051] First, when the point X11Y21 is excited with an impulse hammer, the frequency response function of the (transfer) compliance at each response point (■ in FIG. 5) is measured. In this example, the calculation result by numerical calculation is adopted as the measurement result so that the identification result according to the present invention can be evaluated. The same procedure can be applied to the actual measurement results. FIG. 6 is an example of the frequency response function measured in this example. FIG. 6 shows the actual measurement results when the point X11Y21 is the excitation point and the point X21Y21 is the response point. The graph format of FIG. 6 is the same as that of FIG. 2. From the perspective of consistency with the mode characteristic identification method to be performed later, at least one of the plurality of measurement points needs to be the same as the excitation point.

[0052] Next, the analysis device identifies the damped natural angular frequency, the mode damping ratio, the excitation point, and the natural modes at each response point by a mode characteristic identification method such as the multi-point partial differentiation method or the subspace method. The vibration of the identified natural mode is the vibration in the excitation direction when the point X11Y21 is excited by an impulse hammer, and is the vertical vibration of the floor slab 1. In this embodiment, the subspace method is used with the degree of freedom n = 30. FIG. 7 is a table showing the results identified by the subspace method in this embodiment. For convenience of illustration, in FIG. 7, the natural frequencies f r [Hz] (corresponding to the damped natural angular frequency ω dr ), and the mode damping ratio ζ r (corresponding to the mode damping rate σ r ) are shown. FIG. 8 is a graph of the frequency response function from the point X11Y21 to the point X21Y21 identified by the subspace method in this embodiment. The format of the graph in FIG. 8 is the same as that in FIG. 2. As shown in FIG. 8, the identification result (dashed line) of the subspace method faithfully reproduces the measured result (solid line). Note that the r-th natural frequency f r [Hz] is related to the r-th natural angular frequency Ω r [Hz] by f r = 2πΩ r , and the r-th damped natural angular frequency ω dr is ω dr = Ω r (1 - ζ r 2 ) 0.5 .

[0053] Next, the analysis device sets the sequence of functions in Equation (2). In this embodiment, it is expressed as follows by the product of the Chebyshev polynomials in the x direction and the y direction.

[0054]

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[0055] The truncation order N in the x and y directions x , N y are each N x = 5, N y = 3. As an example, the Chebyshev polynomial T up to the third order i (−1 ≤ ζ ≤ 1) is expressed as follows.

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[0061] Therefore, an equation for the points that become the nodes of the natural mode, that is, the points where the vibration displacement in the excitation direction becomes almost 0, is added from the structural characteristics of the target structure. The analysis device acquires the data of the added equation. The addition of the equation may be executed, for example, by input (specification) from the user, or may be automatically input and executed for the corresponding points. FIG. 9 is an explanatory diagram of virtual measurement points that become the nodes of the natural mode. The floor structure shown in FIG. 9 is the same as that in FIG. 5. In FIG. 9, since the column 4 is stiffer than other parts and becomes a node of the natural mode, the analysis device sets 12 virtual measurement points (■ in FIG. 9) at the position of the column 4, assumes that the natural mode values of the 12 points become 0, and adds 12 relational expressions.

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[0064] By using Equation (1), the analysis device can also predict the frequency response function at points where measurements are not performed in this embodiment. FIG. 12 is an explanatory diagram of points where measurements are not performed in the floor structure of this embodiment. FIG. 13 is a graph of the transfer compliance from the point X31Y22 where no measurement is taken to the point X11Y21. FIG. 14 is a graph of the driving-point compliance at the point X31Y22 where no measurement is taken. The graph formats of FIGS. 13 and 14 are the same as those of FIG. 2. In FIG. 13, the identification result (dashed line) of the subspace method predicted using Equation (1) faithfully reproduces the calculation result (solid line) obtained from the existing calculation model, indicating that it can be predicted with high accuracy. In FIG. 14, the point X31Y22 is treated as an excitation point and a measurement point, that is, as a driving point. In FIG. 14, the identification result (dashed line) of the subspace method predicted using Equation (1) faithfully reproduces the calculation result (solid line) obtained from the existing calculation model, indicating that it can be predicted with high accuracy.

[0065] [Prediction of the Effect of Vibration Control Measures] By using the present invention, it is possible to predict the vibration reduction effect at any position when vibration control measures are taken at any position of the actually measured floor structure. Here, an example of predicting the vibration control effect when a tuned mass damper (hereinafter, TMD (Tuned Mass Damper)), which is a vibration control device, is installed in the floor structure will be shown. FIG. 15 is an explanatory diagram of the point where the TMD is installed in the floor structure for explaining the prediction of the effect of the vibration control measures. In order to suppress the first natural mode (7.37 Hz) of the floor structure shown in FIG. 15, the TMD is installed at point X31Y22. The analysis device predicts the frequency response functions at points X21Y22, X31Y22, and X41Y22 when point X21Y22 is excited. Note that since points X21Y22, X31Y22, and X41Y22 do not correspond to any of the measurement points shown in FIG. 5, the frequency response functions before and after the vibration control measures at points X21Y22, X31Y22, and X41Y22 cannot be directly obtained.

[0066] FIG. 16 is a block diagram of the entire vibration system composed of the floor structure and the TMD. In FIG. 16, G is the frequency response function of the floor structure, and H is the frequency response function of the TMD. Also, U is the excitation force applied to the floor, and Y is the response (vibration displacement) of the vibration system. When the excitation force U applied to the floor acts, the response Y of this vibration system can be written as follows in the frequency domain.

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Equation

[0068] Here, G is as shown in Equation (1). Also, when the mass, damping, and stiffness of the TMD are m, c, and k, respectively, they are represented by the following equation. Note that j is the imaginary unit, and ω is the angular frequency.

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Equation

[0070] The frequency response functions of the points X21Y22, X31Y22, and X41Y22 predicted by the analysis device using Equation (21) are shown in FIGS. 17, 18, and 19, respectively. FIG. 17 is a graph of the driving point acceleration at the point X21Y22. FIG. 18 is a graph of the transfer acceleration from the point X21Y22 to the point X31Y22. FIG. 19 is a graph of the transfer acceleration from the point X21Y22 to the point X41Y22. The horizontal axis of the graphs in FIGS. 17 to 19 is the frequency [Hz]. In FIGS. 17 to 19, the frequency and the transfer acceleration (acceleration) are shown on a logarithmic scale. In each of FIGS. 17 to 19, a graph (solid line) without a vibration control device and a graph (dashed line) with a vibration control device are shown.

[0071] In any of the graphs in FIGS. 17 to 19, it can be seen that the response near the first natural mode of 7.37 Hz can be reduced by the effect of the TMD. By using the present invention in this way, it becomes possible to quantitatively predict not only the frequency response function before the vibration control measures at points where measurements are not being taken, but also the effect (the change in the frequency response function) when the vibration control measures are implemented.

[0072] [Summary] According to the present embodiment, by approximately expressing the natural mode at an arbitrary point other than the measurement point, the vibration characteristics at an arbitrary point can be predicted without measurement. Also, the degree of the series can be appropriately set according to the required prediction accuracy. Further, the number of measurement points can be appropriately set according to the required prediction accuracy. Therefore, the prediction accuracy of the vibration characteristics at an arbitrary point of a building can be improved. Even when taking vibration control measures, the same approach of obtaining the frequency response function for the vibration system when a vibration control device is installed at an arbitrary position of the building can be taken. That is, not only the frequency response function at the measurement points of the building is obtained, but also the frequency response function of the vibration control device is obtained. Using these frequency response functions, the vibration characteristics at an arbitrary point can be predicted without measurement. For this reason, the effect of the vibration control measures can be predicted. In addition, due to the structure of the building, it is a very rigid part and hardly vibrates in the excitation direction, so a response point can be set at a point where the natural mode can be set to 0 without measurement, and the number of response points can be substantially increased. Therefore, relational expressions necessary for calculating all the coefficients of the series of the function sequence approximating the natural mode can be prepared. As a result, even when the number of excitation points and response points that can be set is often limited due to time and space constraints during measurement and cannot be increased, the natural mode at any point other than the measurement point can be approximated and expressed.

[0073] [Modification Example] (a): In this embodiment, the case of constructing a model of the two-dimensional frequency response function for the vertical vibration generated in the same plane of the floor structure, that is, a two-dimensional structure, has been described. However, the present invention can also be applied to the case of constructing a model of the three-dimensional frequency response function for the vibration generated in a three-dimensional structure. For example, the present invention can also be applied to the case of predicting the acoustic mode at an arbitrary three-dimensional point in a room.

[0074] (b): It is also possible to realize a technique in which various techniques described in this embodiment are appropriately combined. (c): In addition, the components of the present invention can be appropriately changed without departing from the gist of the present invention.

Description of Reference Numerals

[0075] 1 Floor slab 2 Secondary beam 3 Main beam 4 Column

Claims

1. Measuring the frequency response function at the measurement points of the building; Identifying the damped natural angular frequency, the mode decay rate, and the natural mode of vibration in the excitation direction at the excitation point and the response point from the measured frequency response function; Approximating the natural mode at any point of the building by a series of a predetermined sequence of functions; Calculating the coefficients of the series by the weighted least squares method using the identified natural mode and the series; Predicting the frequency response function at any point of the building using the calculated coefficients and the approximated natural mode. A vibration characteristic analysis method comprising the steps.

2. The vibration characteristic analysis method according to claim 1, wherein, in the step of calculating the coefficients of the series, in addition to the response points where the frequency response function is actually measured, points where the vibration in the excitation direction becomes substantially zero are used as virtual response points.

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