JP7628529B2 - Layer stiffness estimation method and layer stiffness estimation device for a given layer - Google Patents

Description

The present invention relates to a method and device for estimating story stiffness of a specific story in a building.

In recent years, importance has been placed on structural health monitoring technology to ensure the safety of buildings. For example, a method has been proposed to identify the story stiffness of each story of a building in order to evaluate damage to buildings caused by earthquakes, etc. (Patent Document 1, Non-Patent Documents 1 and 2). When a member of a certain story of a building is damaged by an earthquake, etc., the stiffness of that story decreases, so the level of damage can be evaluated by comparing the story stiffness before and after the earthquake.

JP 2016-194514 A

Mitsuru Nakamura, Yuzuru Yasui, "Evaluation of Story Damage in Earthquake-damaged Steel-framed Buildings Based on Microtremor Measurements," Journal of the Architectural Institute of Japan, Structural Engineering and Systems, Vol. 517, pp. 61-68, 1999 Yoshimura, Tomotake, Mita, Akira, "Online Identification of Architectural Structural Parameters Based on Multi-Input Multi-Output Model," Journal of the Architectural Institute of Japan, Vol. 574, pp. 39-44, 2003

In general, in the case of high-rise buildings, damage to structural members due to earthquakes is more likely to occur in the lower stories. However, in both the methods of Patent Document 1 and Non-Patent Documents 1 and 2, in order to calculate the story stiffness of the story being evaluated, it is necessary to calculate the story stiffness based on the acceleration in at least all stories above the story being evaluated.

The present invention therefore aims to provide a layer stiffness estimation method and device for estimating layer stiffness for a given layer that can significantly reduce the number of acceleration measurement points compared to conventional methods.

The present invention has been made to solve at least some of the problems described above, and can be realized in the following aspects or application examples.

[1] One aspect of the method for estimating layer stiffness of a predetermined layer according to the present invention comprises:
A method for estimating a story stiffness of a predetermined story in a multi-story building using a multi-mass system model, comprising the steps of:
A step of performing a Fourier transform on the acceleration of the i-th mass point counting from below the mass point corresponding to the roof of the building and the accelerations of mass points above and below the i-th mass point to calculate a Fourier spectrum of each acceleration;
a step of calculating a layer stiffness (k i ) of the predetermined layer between the i-th mass point and the (i-1 ₎ -th mass point by the following formula (1) and formula (2) using the Fourier spectrum of each acceleration obtained in the step and the mass of the i-th mass point;
extracting only layer stiffnesses having an estimated S/(S+N) ratio obtained by the following formula (3) that is equal to or greater than a predetermined reference value close to 1 from the layer stiffnesses (k _i ) of the predetermined layer;
The present invention is characterized by comprising:

[2] One aspect of the layer stiffness estimation device according to the present invention is
A layer stiffness estimation device for executing one aspect of the layer stiffness estimation method for a predetermined layer,
A calculation unit and a storage unit are provided,
The calculation unit is
A process of acquiring an acceleration at the i-th mass point and an acceleration of mass points above and below the i-th mass point;
A process of performing a Fourier transform on each of the acquired accelerations to calculate a Fourier spectrum of each acceleration;
A process of calculating a story stiffness (k _i ) of the predetermined story using the calculated Fourier spectrum of each acceleration and the mass of the i-th mass point stored in the storage unit;
extracting only layer stiffnesses having an estimated S/(S+N) ratio close to 1 and equal to or greater than a predetermined reference value from the calculated layer stiffnesses (k _i ) of the predetermined layer;
The present invention is characterized by carrying out the following steps.

According to one aspect of the layer stiffness estimation method for a specified layer and one aspect of the layer stiffness estimation device of the present invention, the number of acceleration measurement points can be significantly reduced compared to the conventional method.

FIG. 2 is a schematic diagram of a building used in the story stiffness estimation method according to the present embodiment, and a conceptual diagram of a multi-mass system model thereof. 1 is a block diagram showing an example of the configuration of a layer stiffness estimation device for a predetermined layer according to the present embodiment. FIG. 10 is a flowchart illustrating an example of a method for estimating layer stiffness of a predetermined layer according to the present embodiment. 4 is a flowchart showing an example of processing for each section in FIG. 3 . 4 is a flowchart showing an example of an S/(S+N) estimation process in FIG. 3 . Input (simulated ground motion u _g ...) in the embodiment. FIG. 13 is a diagram showing the results of a verification calculation of the layer stiffness estimation method of the first embodiment. FIG. 13 is a diagram showing the results of a verification calculation of the layer stiffness estimation method of the second embodiment.

Below, preferred embodiments of the present invention will be described in detail with reference to the drawings. Note that the embodiments described below do not unduly limit the content of the present invention described in the claims. Furthermore, not all of the configurations described below are necessarily essential components of the present invention.

1. Story Stiffness Estimation Device A story stiffness estimation device 2 according to one embodiment of the present invention will be described with reference to Figures 1 and 2. Figure 1 is a schematic diagram (a) of a building 1 used in a story stiffness estimation method according to this embodiment, and a conceptual diagram (b) of a multi-mass system model 1a thereof, and Figure 2 is a block diagram showing an example of the configuration of the story stiffness estimation device 2 for a predetermined story according to this embodiment.

The multi-story building 1 shown in (a) of FIG. 1 can be replaced with a multi-mass system model 1a shown in (b) of the same figure. The masses of the mass points corresponding to the floors of each floor can be represented as m ₀ , m ₁ , m ₂ , ..., m _i-1 , m _i , m _i+1 , ..., m _n-1 from the mass point corresponding to the rooftop toward the first floor, and n mass points are arranged at intervals in the vertical direction from the ground (GL) at the initial system position. Mass m ₁ indicates the mass of the first mass point counting from the mass point below the mass point on the rooftop (corresponding to the floor of the 10th floor if the building is 10 stories), and mass m _i indicates the mass of the i-th mass point counting from the mass point below the mass point on the rooftop (corresponding to the floor of the 10-(i-1)th floor if the building is 10 stories). A spring model having stiffness k _1...n and damping (or viscosity) c _1...n of each story is placed between adjacent mass points above and below, and both ends are fixed to the mass points above and below. Story stiffness k ₁ is the stiffness of the first story counting from the top floor, and story stiffness k _i is the stiffness of the i-th story counting from the top floor. In this embodiment, the story stiffness (k _i ) of the i-th story is estimated, in other words, the story stiffness (k i ) of a given story between the i-th mass point (hereinafter "i-th mass point") and the i-1-th mass point (hereinafter "i-1-th mass point") counting from the mass point (1st) below the mass point ( _0th ) corresponding to the rooftop is estimated.

In addition, in Fig. 1(b), u _i is the time distribution u _i (t) of the horizontal displacement of the i-th mass point, and u _g is the time distribution u _g (t) of the displacement of the ground (GL) of the building 1. Furthermore, the time derivative (velocity) of the displacement time distribution u _i (t) is represented by one dot as u _i ·(t) or u _g ·(t), and similarly the second derivative (acceleration) with respect to time is represented by two dots as u _i ·.(t) or u _g ·.(t).

In the building 1, acceleration sensors 14 are attached to at least the floors of three floors corresponding to the i-th mass point and the mass points above and below it. The acceleration sensors 14 can measure at least the horizontal acceleration of the floors of each floor, that is, can measure the acceleration at the i-1th, i-th, and i+1th mass points. The building 1 may be provided with acceleration sensors 14 on the floors of all floors. The acceleration sensors 14 may be three-axis accelerometers.

In the multi-mass system model 1a, the masses m ₀ ...m _n-1 of each mass point can be obtained in advance from, for example, the design specifications of the building 1, and the absolute accelerations (hereinafter simply referred to as accelerations) u _i ...(t) + u _g ...(t), u i-1 ...(t) + u g ...(t), and u i+1 ...(t) + u _g ...(t) at the i _-1th , i _{-th, and i+1th mass} points can be obtained from the output of the acceleration sensor 14.

At this time, the equation of motion (force balance equation) for the multi-mass system model 1a is as shown in the following equation (4).

Based on this equation, the following formulas (1) and (2) can be derived as described below.

The layer stiffness estimation device 2 according to the present embodiment shown in Fig. 2 is a device that executes an embodiment of the layer stiffness estimation method for a predetermined layer, which will be described later. The layer stiffness estimation device 2 can estimate the layer stiffness (k _i ) of the predetermined layer by using the above formulas (1) and (2).

The layer stiffness estimation device 2 includes a calculation unit 21 and a storage unit 22. The layer stiffness estimation device 2 is, for example, a personal computer or a server, and includes a CPU (Central Processing Unit) (not shown), memories such as ROM and RAM, a storage device such as a hard disk drive, and a communication interface for communicating with external devices. The calculation unit 21 can be composed of a CPU, RAM, etc., and executes a program stored in the storage unit 22. The storage unit 22 can be composed of a storage device such as a hard disk drive, and stores the mass of each mass point of the multi-mass point model 1a, the above formula (1) and the above formula (2), etc. The storage unit 22 may store measurement data acquired from the acceleration sensor 14.

The story stiffness estimation device 2 can connect to a plurality of acceleration sensors 14 installed in the building 1 via a communication network and receive measurement data. As long as it is possible to acquire the measurement data of the acceleration sensors 14, a method other than a communication network, for example, importing the measurement data using a USB memory, may be used. The story stiffness estimation device 2 may also include known input devices such as a keyboard and a mouse for operating the story stiffness estimation device 2, and known output devices such as a display and a printer for outputting the estimation results.

The calculation unit 21 executes a process of acquiring at least the acceleration u _i (t) +u _g (t) of the i-th mass point and the accelerations u _i-1 (t) +u _g (t) and u _i+1 (t) +u _g (t) of the upper and lower mass points, a process of Fourier transforming the acquired accelerations to calculate the Fourier spectra u _i (ω) +u _g (ω), u _i-1 (ω) +u _g (ω), and u _i+1 (ω) +u _g (ω), and a process of calculating the story stiffness (k _i ) of a predetermined story using the calculated Fourier spectra of the accelerations and the mass m _i of the i-th mass point stored in the storage unit 22. Each process will be described in the story stiffness estimation method described later.

The story stiffness (k _i ) of a specified story calculated by the calculation unit 21 is the story stiffness (k _i ) when the building 1 is replaced with the multi-mass system model 1a, and is therefore an estimated value of the story stiffness of a specified story in the actual building 1.

If the story stiffness estimation device 2 can estimate the story stiffness (k _i ) of a given story from the results of earthquake observation or vibration measurement of the building 1, the degree of damage or deterioration of the building 1 can be grasped. Moreover, the story stiffness estimation device 2 does not require acceleration data of many stories as in the past, and only requires acceleration data at at least three mass points (floors of three floors), so that in the case of a super-high rise building, the number of acceleration measurement points can be significantly reduced. If the story stiffness can be estimated, for example, if there is a story whose story stiffness has decreased before and after an earthquake, it is possible to know that the story has been damaged. Furthermore, even if there is no record before the earthquake, if there is a story whose story stiffness is unnaturally small by comparing the story stiffness of each story, it is possible to know that the story has been damaged. Furthermore, if the story stiffness is estimated by periodically performing vibration measurement not only during earthquakes but also during other events, the degree of progress of aging deterioration of the building 1 can be grasped.

Next, we will explain how the above formula (1) and formula (2) can be derived from the above formula (4). Hereinafter, where necessary, the mass matrix (matrix in which mi is arranged) in the above formula (4) will be written as M, the damping matrix (matrix in which ci is arranged) as C, and the stiffness matrix (matrix in which ki is arranged) as K.

First, the time distribution of the displacement, velocity, and acceleration of a mass point is Fourier transformed to obtain the frequency distribution of the displacement, velocity, and acceleration. This can be written in symbols such as u _i (ω), u _i (ω), u _i (ω), u _g (ω) where time t is replaced by angular frequency ω. The above formula (4) also holds true for u _i (ω), u _i (ω), _{u i (} ω), u _g (ω), but due to the nature of the Fourier transform, the following formulas (5) and (6) hold true between u _i (ω), u _i (ω), u i (ω), u _g (ω).

Then, by substituting the above formulas (5) and (6) into the above formula (4), the following formulas (7) and (8) hold.

In addition, the j-th order natural angular frequency in the multi-mass system model 1a in Fig. 1 is denoted as _λj , and the j-th order eigenvector is denoted as { _φ0 , _φ1j , ..., φn _-1 , _j }. In this case, due to the properties of the natural angular frequency and the eigenvalue vector, the eigenvector { _φ0 , _φ1j , ..., φn _-1 , _j } corresponds to a displacement vector under the condition that the angular frequency is _λj and the ground displacement is zero (u _g (ω=λ _j )=0), and therefore, from the above formula (8), the following formula (9) holds as an equation of motion in the eigenspace.

Here, the masses of all mass points are known from, for example, design specifications, and the accelerations at mass points of three floors including the predetermined floor (actually, since this is building 1, the floors of three consecutive floors) can be obtained from the acceleration sensor 14. In actual measurements, the accelerations u _i ...(t) are recorded at regular time intervals. If this time interval is dt, and t _p = dt(p-1) (p = 1, 2, 3, ...), the time distribution of the actual acceleration records is not continuous u _i ...(t) but discrete u _i ...(tp), and the Fourier transform of this also becomes a discrete value (written as u _i ...( _ωq ) (q = 1, 2, 3, ...)).

If we collect the known terms in the equations of motion (4), (7), and (9) above on the right-hand side, we get the following equations (10) to (12).

In addition, to simplify the notation, the symbols can be defined as in the following formula (2) and formula (13).

At this time, the relationships in the following formulas (14) and (15) hold.

The right-hand sides of the above formula (14) and formula (15) can be written as the following formula (16) and formula (17).

The conventional technology, Non-Patent Document 1, uses the above formula (10), Non-Patent Document 2 uses the above formula (12), and Patent Document 1 uses the above formula (11).

In the present invention, the above formula (11) is modified as follows based on the Fourier transform integration rule (above formula (5)):

If we extract only the i+1th line from the above formula (18), we get the following formula (19).

Rearranging the above formula (19), the following formula (20) holds true.

Since _ωq is included in the definition equation of s _i (above equation (2)), there is frequency dependency, and to begin with, the fact that k _i and c _i do not change with time or frequency is a property of the multi-mass system model 1a in FIG. 1 and is not necessarily the case in an actual building 1. However, if we assume here that the changes in s _i and s _i+1 are sufficiently small in the angular frequency range from ωq _-b to ωq _+b , then the following equation (21) holds from the above equation (20).

Furthermore, if the matrix of the displacement differences on the left side of the above equation (21) is defined as D _i (ω _q ), then s _i and s _i+1 are determined as shown in the following equation (1) from the above equation (21).

As explained above, it can be seen that the above formula (1) and the above formula (2) hold true from the above formula (4). And, s _i , s _i+1 are determined from the above formula (1) or the above formula (21), so that, similarly to Non-Patent Documents 1, 2 and Patent Document 1, the definition of s _i (above formula (2)) determines the layer rigidity (k _i ) of a given layer as well as k _i+1 , c _i , c _i+1 . At this time, if the frequency width b used in the calculation is at least 1 or more, a solution can be obtained. However, if the frequency width b is too small, the solution is not stable, and if it is too large, an average result that ignores the frequency change of s _i , s _i+1 will be obtained, or for the reasons described later, the calculation accuracy will be adversely affected, so it is preferable to appropriately set b, for example, between about 10 and 100.

Then, the calculation unit 21 executes a process of extracting only layer stiffnesses (ki) of the predetermined layer that have an estimated S/(S+N) ratio equal to or greater than a predetermined reference value close to 1 from the calculated layer stiffnesses (ki). Details of this process will be described in "2. Layer stiffness estimation method."

2. Story Stiffness Estimation Method A story stiffness estimation method for a predetermined story according to one embodiment of the present invention will be described with reference to Figures 1 to 5. Figure 3 is a flowchart showing an example of a story stiffness estimation method for a predetermined story according to this embodiment, Figure 4 is a flowchart showing an example of processing for each section in Figure 3, and Figure 5 is a flowchart showing an example of S/(S+N) estimation processing in Figure 3. In the following description, the story stiffness of a predetermined story of the building 1 in Figure 1(a) is estimated by the story stiffness estimation device 2 in Figure 2 using the multi-mass system model 1a shown in Figure 1(b). Portions that overlap with the description of the story stiffness estimation device 2 will be omitted.

The method for estimating story stiffness of a given story (i-th story) shown in Fig. 3 is a method for estimating the story stiffness (k _i ) of a given story in a multi-story building 1 using a multi-mass system model 1a, and includes at least a step (S21) of calculating the Fourier spectrum of acceleration, a step (S31) of calculating the story stiffness (k _i ) of the given story, and a step (S41) of extracting only story stiffnesses equal to or greater than a given reference value. In the example of Fig. 3, the processes of S10, S11, S12, S21, S23, S27, S29, S31, S32, and S41 are executed.

S10: In the step of acquiring acceleration data (S10), the story stiffness estimation device 2 acquires acceleration measurement data (hereinafter, "acceleration data") from the acceleration sensors 14 of a predetermined story and two stories adjacent thereto above and below. The acceleration data may be acquired directly from data output from the acceleration sensors 14 due to a currently occurring earthquake or vibration, or may be acquired by saving past acceleration data in the storage unit 22. The acceleration data is the acceleration u i...(t)+u _{g...(t) of the i-th mass point counting from the bottom of the mass point corresponding to the roof of the building 1, and the accelerations u i-1...(t)+u g ... ( t} ) and u _i+1 ...(t)+u _g ...(t) of the mass points above and below the i-th mass point.

Next, before describing each step, in order to extract a highly reliable value from the layer stiffness value calculated from the acceleration data, the above formula (1) is used to define H _i (ω _q ), which is an evaluation index of reliability. First, the above formula (1) can be transformed to obtain the following formula (22).

Multiplying each row on both sides of the above equation (22) by the complex conjugate of the acceleration Fourier spectrum gives the following equation (23).

If the matrix on the left side of the above formula (23) is defined as H _i (ω _q ), the following formula (24) is obtained.

In this embodiment, the S/(S+N) ratio of H _i (ω _q ) is used to extract a layer stiffness value that is less affected by measurement errors. The SN (signal/noise) ratio (S/(S+N) ratio is used in this embodiment) is a value generally defined for a scalar quantity, and represents the ratio of the signal component S (true value) and the noise component N (measurement error, deviation between the measurement and the true value) contained in the squared value of the absolute value of the scalar quantity to be evaluated. In this embodiment, it is considered that it is more desirable to consider the SN ratio of each element constituting H _i (ω _q ) rather than the SN ratio of H _i (ω _q ), but in that case, a band with an excessively high SN ratio (relative to the low estimation accuracy of the layer stiffness) will appear, making it difficult to extract only the estimated value close to the true value. In this embodiment, not only the SN ratio of each element of the matrix is considered individually, but also the relationship between the elements will be considered. Therefore, in this embodiment, det(H _i ^* (ω _q )H _i (ω _q )) (where detA represents the determinant of matrix A) is defined as the “power of matrix H _i (ω _q ) (hereinafter referred to as powH _i (ω _q ))” as the amount equivalent to the square of the absolute value of matrix H _i (ω _q ), and a highly reliable estimate of the layer stiffness can be extracted by estimating the ratio of signal components S and noise components N contained in powH _i (ω _q ).

S11: Section length setting value (S11) provides data on a previously set section length setting value to step S12. The section length is set, for example, between 10 and 100 times the primary natural period of the building 1 under consideration.

S12: Data division into sections (S12) is performed by dividing the acceleration data acquired by the calculation unit 21 in S10 by the section length setting value retrieved in S11. For example, if there is 100 seconds of acceleration data, it is divided into five sections with a section length of 20 seconds.

S21: The process for each section (S21) includes a step of calculating the Fourier spectrum of the acceleration. In the process for each section (S21), the calculation unit 21 calculates the Fourier spectrum of each acceleration obtained in the step of S12, and creates H _i (ω _q ) on the left side of the above formula (23) in order to obtain the SN (signal/noise) ratio of the data. The detailed process will be described later with reference to FIG. 5.

S23: The process of checking whether all sections have been processed (S23) proceeds to S25 if all sections have been processed (Yes) in S21, and returns to S21 if all sections have not been processed (No) in S21.

S27: In the process of estimating the S/(S+N) ratio of H _i (ω _q ) (S27), the calculator 21 calculates an estimated value R _m hat, which is the S/(S+N) ratio of powH _i (ω _q ) using H _i (ω _q ) of the m sections filtered in S25. Note that the detailed process will be described later with reference to FIG. 7.

S29: In the ensemble average process of H _i (ω _q ) (S29), the calculation unit 21 calculates the ensemble average of H _i (ω _q ) filtered in S25. By calculating the ensemble average of H _i (ω _q ), the influence of measurement errors (noise components) can be reduced.

S31: The estimation process (S31) of the layer stiffness (k _i ) is a process of calculating the layer stiffness of a predetermined layer. In the estimation process (S31) of the layer stiffness (k _i ), the calculation unit 21 calculates the layer stiffness (k _{i ) of a predetermined layer between the i-th mass point and the (i-1} )-th mass _point estimated by the following formula (25) and the following formula (2) using the ensemble average of H _i (ω _q ) obtained in the process of S29 and the mass m i of the i-th mass point. The following formula (25) is a basic equation obtained by modifying the above formula (1), and is obtained by executing the above-mentioned process (S29). Then, s _i is obtained by the following formula (25), and the layer stiffness (k _i ) of the predetermined layer is calculated from the real part of s _i by the following formula (2), and the damping c _i can be calculated from the imaginary part.

S32: The reference value of S/(S+N) (S32) provides data of a predetermined reference value set in advance to the S41 step. By setting the predetermined reference value to a value close to 1, for example 0.97, a value close to the true value of the estimated layer stiffness can be extracted in the S41 step.

S41: Extraction process (S41) of the layer stiffness at the frequency where S/(S+N) is equal to or greater than the reference value
From the layer stiffnesses (k _i ) of the predetermined layer calculated in step S31, the calculation unit 21 extracts only layer stiffnesses whose estimated S/(S+N) ratios obtained by the following formula (3) are equal to or greater than a predetermined reference value close to 1 set in step S32. Step S41 makes it possible to extract layer stiffness values that are less affected by measurement errors.

According to one aspect of the layer stiffness estimation method for a specified layer and one aspect of the layer stiffness estimation device of the present invention, the number of acceleration measurement points can be significantly reduced compared to the conventional method.

2.1 Processing for Each Section The processing for each section (S21) will be described with reference to Fig. 4. The processing for each section (S21) shown in Fig. 4 executes the processes of S210, S212, S214, S216, S218, and S220.

S210: In the baseline correction process (S210), the calculation unit 21 performs baseline correction on the acceleration data for each section divided in S12 of FIG. 3.

S212: In the process of applying a window function (S212), the calculation unit 21 applies a window function to the acceleration data for each section that has been baseline corrected in S210.

S214: In the Fourier transform process (S214), the calculation unit 21 performs a Fourier transform (u _i ..(ω)+u _g ..(ω)) on the acceleration data (u _i ..(t)+u _g ..(t)) for each interval to which the window function was applied in S212.

S216: In the inter-story displacement calculation process (S216), the calculation unit 21 creates H i ( _{ω q} ) for all ω _q in the verification target band using the acceleration (u _i ··(ω)+u _g ··(ω)) Fourier transformed in S214.

2.2. Estimation process of S/(S+N) ratio of H _i (ω _q ) The estimation process (S27) of the S/(S+N) ratio of H _i (ω _q ) described above will be described with reference to Fig. 5. In the estimation process (S27) of the S _/ (S+ _N ) ratio of H _{i (ω q ) shown in Fig. 5, the calculation unit 21 executes the processes of S290, S292, S294, S296, S298, and S300 for H i} (ω _q ) reconstructed using the frequencies filtered in S266.

In the step S290, the calculation unit 21 reads H _i (ω _q ) of m intervals from the storage unit 22. In the step S292, the calculation unit 21 calculates the ensemble average of m H _i ^* (ω _q ) H _i (ω _q ). In the step S294, the calculation unit 21 multiplies the ensemble average of m H _i ^* (ω _q ) by the ensemble average of m H _i (ω _q ). In the step S296, the calculation unit 21 calculates N by the following formula (26), in the step S298, the calculation unit 21 calculates S by the following formula (27), and in the step S300, the calculation unit 21 calculates R _m hat by the following formula (28). Here, R _m hat is an estimate of the S/(S+N) ratio of powH _i (ω _q ) using H _i (ω _q ) of m intervals.

2.3 Derivation of Estimation Equation for S/(S+N) Ratio of powH _i (ω _q ) The process of deriving the above equation (28) will be described below.

(1) Definition of symbols In the following, we distinguish between measured values (values containing noise) and true values by adding "~" to the measured values. First, we assume that the absolute acceleration has a sufficiently high S/(S+N) ratio, and that the measured value and the true value are approximately equal.

On the other hand, for relative displacement, since the power of the signal (true value) decreases as the ground motion component (u _g (ω _q )) is eliminated by taking the difference, the influence of noise cannot be ignored and the relationship between signal and noise is set as follows:

For simplicity of notation, each element of H _i (ω _q ) is written as follows:

Furthermore, h _k1 ~ (k=1, 2, . . . , n), l=1, 2) can be divided into a signal component h _kl and a noise component ε _kl and written as follows:

From the definition, the following properties can be assumed for noise:

Furthermore, if each element of <H _i (ω _q )> _m is written as <h _kl ∼> _m +<ε _kl ∼> _m , the following relationship holds due to the nature of a random variable with an expected value of zero.

Furthermore, n is assumed to be sufficiently large, and the following approximation holds for any variable x _l related to H _i (ω _q ):

Here, from the above formula (31), H _i ^* (ω _q ) and H _i (ω _q ) are as follows.

Combining the properties of the above formulas (32) to (37), <H _i ^* (ω _q )H _i (ω _q )> _m and <H _i ^* (ω _q )> _m <H _i (ω _q )> _m are expressed as follows:

(2) Estimation Equation for S/(S+N) Ratio of <H _i (ω _q )> First, the following equation can be derived by taking the difference between the above equation (39) and equation (40).

Using the above formula (41), the following approximation holds for the matrix of the noise component N in the above formula (26):

Therefore, the following approximation holds for the matrix of signal component S in the above formula (27):

From the above equations (40) and (43), since S is a form in which noise components have been removed from <H _i ^* ( _ωq )>m<H _i ( _ωq )>m, it can be derived that the S/(S+N) ratio of <H _i ( _ωq )>m can be estimated by the following equation (28).

(Verification target and input data)
In this example, the estimated value of story stiffness calculated using the story stiffness estimation method according to the present invention was verified. The multi-mass system model to be verified had five masses ( _m0 = _m1 = ... = _m4 = 1t), the story stiffness (kN/m) of each story was as shown in Table 1, and the damping was h1 = h2 = ... = h4 = 0.05. The lower subscript number indicates the upper part of the system. The primary natural frequency was set to 3.3 Hz.

The input was white noise as shown in Figure 6. The sampling rate was 100 Hz and the duration was 163.84 seconds.

(Example 1) The frequency response of the multi-mass system model for the input in FIG. 6 was calculated by frequency response calculation, and the result was converted back to the time domain to obtain the true value of the mass system response waveform. The true value was used as the simulated earthquake observation record as it is to obtain the acceleration data of Example 1. The simulated earthquake observation record was then Fourier transformed to obtain s _i with a frequency width of 0.5 Hz (±0.25 Hz relative to the center frequency) using the following formula (1), and the estimated value of story stiffness _ki ( _ki hat) was calculated from the real part of s _i using the following formula (2), and the estimated value of damping _ci ( _ci hat) was calculated from the imaginary part. Furthermore, for damping _ci , the imaginary part of s _i hat was divided by twice _ki hat to obtain the estimated value of damping constant ( _hi hat) from the relationship between the viscosity coefficient and the damping constant. The calculated estimated value of story stiffness ( _ki hat) and the estimated value of damping constant ( _hi hat) are shown in FIG. 7.

In Fig. 7, the true value is indicated by a dashed line, the estimated value of the story stiffness estimated with the lower point as the reference is indicated by a light-colored wide line, and the estimated value of the story stiffness estimated with the upper point as the reference is indicated by a dark-colored narrow line. In Fig. 7, the lower point reference and the upper point reference indicate whether the estimation was performed with the mass point located at the upper part of the estimation target story as the reference point (mass point i in the above formula (1) and formula (2)) or with the lower mass point as the reference. The story stiffness of the nth story is determined from both s _i+1 when i=n-1 and s _{_i} when i=n, so here, the story stiffness determined at i=n-1 is the upper point reference, and the story stiffness determined at i=n is the lower point reference. As shown in Fig. 7, the true value and the estimated result are exactly the same, and the validity of the theory was confirmed.

(Example 2) Next, Gaussian noise with a standard deviation of 1% of the RMS value of the ground motion (approximately 0.79 m/ ^s2 ) was added to the true value of the mass system response waveform in Example 1, and the ground motion and each point response were used as a simulated earthquake observation record to obtain acceleration data for Example 2 (S10). Then, processing for each section (S21) was performed for all sections (S22) according to the flowcharts described using Figures 3 to 5.
After the ensemble average of _{H i} (ω _q ) was calculated (S29), s _i was calculated using the following formula (25) with a frequency width of 0.5 Hz (±0.25 Hz relative to the center frequency), and an estimate (k _i hat) of the layer stiffness k _i (ω _q ) was calculated from the real part of s _i using the above formula (2) (S31). In addition, an estimate (R _m hat) of the S/(S+N) ratio of H _i (ω _q ) was calculated using the following formula (3) (S27). The reference value of S/(S+N) was set to 0.97 (S32), and only layer stiffnesses with an _estimate (R _m hat) of the S/(S+N) ratio of 0.97 or more were extracted from the layer stiffness k i (S41). The estimated values of the S/(S+N) ratio of the calculated H _i (ω _q ) are shown on the right side of Figure 8, and among them, points with estimated values of 0.97 or more are shown on the left side of Figure 8 as estimated values of the left layer stiffness (k _i hat).

In Fig. 8, the true value is shown by a dashed line, the estimated value of the story stiffness estimated based on the lower point is shown by a light-colored wide line (or dot), and the estimated value of the story stiffness estimated based on the upper point is shown by a dark-colored narrow line (or dot). The estimated value of the story stiffness ( _ki ) is distributed in the vicinity of the true value (dashed line), and only the estimated value of the story stiffness ( _ki ) with high reliability could be extracted using the estimated value ( _Rm ) of the S/( _S +N) ratio of H _i (ω q ).

The present invention is not limited to the above-described embodiment, and various modifications are possible. For example, the present invention includes a configuration that is substantially the same as the configuration described in the embodiment (for example, a configuration with the same function, method, and result, or a configuration with the same purpose and effect). The present invention also includes a configuration in which non-essential parts of the configuration described in the embodiment are replaced. The present invention also includes a configuration that has the same action and effect as the configuration described in the embodiment, or can achieve the same purpose. The present invention also includes a configuration in which publicly known technology is added to the configuration described in the embodiment.

1...Building, 1a...Multi-mass system model, 11...Story, 14...Acceleration sensor, 2...Story stiffness estimation device, 21...Calculation unit, 22...Memory unit

Claims (2)

A method for estimating a story stiffness of a predetermined story in a multi-story building using a multi-mass system model, comprising the steps of:
A step of performing a Fourier transform on the acceleration of the i-th mass point counting from below the mass point corresponding to the roof of the building and the accelerations of mass points above and below the i-th mass point to calculate a Fourier spectrum of each acceleration;
a step of calculating a layer stiffness (k i ) of the predetermined layer between the i-th mass point and the (i-1 ₎ -th mass point by the following formula (1) and formula (2) using the Fourier spectrum of each acceleration obtained in the step and the mass of the i-th mass point;
extracting only layer stiffnesses having an estimated S/(S+N) ratio obtained by the following formula (3) that is equal to or greater than a predetermined reference value close to 1 from the layer stiffnesses (k _i ) of the predetermined layer;
A method for estimating story stiffness of a predetermined story, comprising:

A layer stiffness estimation device for executing the layer stiffness estimation method for a predetermined layer according to claim 1,
A calculation unit and a storage unit are provided,
The calculation unit is
A process of acquiring an acceleration at the i-th mass point and an acceleration of mass points above and below the i-th mass point;
A process of performing a Fourier transform on each of the acquired accelerations to calculate a Fourier spectrum of each acceleration;
A process of calculating a story stiffness (k _i ) of the predetermined story using the calculated Fourier spectrum of each acceleration and the mass of the i-th mass point stored in the storage unit;
extracting only layer stiffnesses having an estimated S/(S+N) ratio close to 1 and equal to or greater than a predetermined reference value from the calculated layer stiffnesses (k _i ) of the predetermined layer;
A layer stiffness estimation device, comprising:

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