JP2000081485A - Earthquake response spectrum computing device - Google Patents

Abstract

PROBLEM TO BE SOLVED: To perform processing at high speed so as to assume damage in real time by computing an acceleration of an earth motion from sampled measurement signal data and computing an earthquake response spectrum according to a method using z-conversion. SOLUTION: This earthquake response spectrum computing device is used for a data recording device 2 in a measurement system 1 in which a plurality of seismographs 3 for example are arranged so as to be connected to a central computer 4 via a communication line 5 for assuming damage from seismic intensity data. The data from the seismograph 3 are sampled for computing an acceleration of an earth motion, and then, an earthquake response spectrum is computed by using a mathematical expression represented by a formula (in the formula, U (z), V (z), W (z) represent respective response spectra of displacement, speed, and acceleration, while a0, a1, a2 represent a natural angular frequency of a building, an attenuation constant, and a constant found from a sampling interval). In this way, required accuracy can be provided by less calculation, and processing can be carried out in a short time even by an inexpensive numeric coprocessor for preinstallation in equipment.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an apparatus for calculating a response spectrum due to earthquake motion.

[0002]

2. Description of the Related Art Hitherto, ground surface acceleration, seismic intensity, and the like have been used to estimate damage when an earthquake occurs.

However, ground surface acceleration and seismic intensity are vectors and scalar quantities representing the magnitude and magnitude of earthquake motion. Therefore, in recent years, the evaluation of “earthquake response spectrum” correlated with building damage has been increasing.

[0004] The seismic response spectrum indicates the response of a structure to a mass point system with respect to a seismic wave input. ), When α (t) is given, the displacement y (t)

[0005]

(Equation 2)

[0006]

(Equation 3)

Is expressed as a function of ω ₀ .

[0008]

(Equation 4)

Here, ω ₀ is the natural frequency of the structure, and h is the damping constant.

[0010] Therefore, by looking at the response spectrum, it is possible to know how much the structure is shaken by the seismic wave of what period, and it is possible to estimate the damage to the building due to the generated earthquake.

Further, if response spectra are taken for structures having different natural periods, damage for buildings having different natural periods can be estimated.

Therefore, if the response spectrum can be calculated together with the occurrence of the earthquake, accurate damage estimation can be performed, and quick rescue can be performed.

Conventionally, the analysis of the response spectrum has been performed by two methods. The first method uses a Fourier transform.

The Fourier transforms of y (t) and α (t) are

[0015]

(Equation 5)

Then, from equation (2),

[0017]

(Equation 6)

Is obtained,

[0019]

(Equation 7)

## EQU1 ## Inverting Fourier transform of displacement, velocity and acceleration response is

[0021]

(Equation 8)

Is given by

Equations (4) and (5) are exact solutions, but the actual calculation is performed as follows. First, a spectrum A (ω) is calculated from the observed waveform α (t). When the number of data is N, the time axis of the equation (3) is calculated by the trapezoidal rule by dividing the frequency axis into N equal parts.

[0024]

(Equation 9)

## EQU1 ## U (kΔω) is calculated from the definition equation (4). For example, an inverse transform for obtaining a displacement spectrum is

[0026]

(Equation 10)

[0027]

However, in order to calculate equation (6) for all j, the sum of products of N ² must be calculated except for the calculation of trigonometric functions. However, if FFT is used, N
The number of calculations of ² can be completed by the sum of products of Nlog ₂ N.

The second method is a method using convolution.

The inverse Fourier transform of U (ω) is

[0031]

[Equation 11]

Is as follows. Using the convolution theorem, the solution (5) on the frequency axis can be rewritten as the following equation.

[0033]

(Equation 12)

Here, it is assumed that α (t) = 0 when t <0.

Equation (8) is an exact solution. By approximating the convolution integral equation (8) with the simplest integration formula, the trapezoidal rule, for example, for the displacement response,

[0036]

(Equation 13)

Is obtained. Here, Δt indicates a sample interval, and 'indicates that ｋ is multiplied when k = 0 and j.

In the above equation (9), α (jΔt) is the observation data itself. u (kΔt) must be calculated according to equation (7). When the data length is N, (9)
It requires approximately N ^2/2 calculations of the product-sum in Equation calculations.

[0039]

The calculation formulas (6) and (9) conventionally used are formulas based on exact solutions, but have a problem that the amount of calculation increases.

For this reason, the response spectrum must be calculated over a long period of time at a workstation or the like after obtaining the earthquake waveform, and can be used only for diagnosing the building resistance after the earthquake.

Therefore, in order to perform damage estimation in real time as described above, it is necessary to perform high-speed response spectrum calculation for a huge amount of seismic waveform data, and it is difficult to prepare even an expensive dedicated computer for numerical calculation. Met.

Accordingly, an object of the present invention is to provide a high-speed earthquake response spectrum calculation device capable of calculating an earthquake response spectrum without using a dedicated computer.

[0043]

According to a first aspect of the present invention, a seismometer observation signal is sampled, ground motion acceleration α is calculated from the sampled data, and the calculated acceleration α is calculated as follows. Z transformation U of equation (1)
An arrangement is employed in which an earthquake response spectrum is calculated using any one, two or all of (z), V (z) and W (z).

[0044]

[Equation 14]

By adopting such a configuration, the response spectrum is approximated by the equation (1), and the processing can be performed at high speed. Since the concept of response spectrum is a rough approximation in the first place, in such an earthquake analysis as to estimate damage in real time, it is sufficient if the response can be calculated at a high speed with a certain level of accuracy.

Then, the frequency response such as U (ω), V (ω), W (ω) was converted into z-transform by the following equation and approximated.

[0047]

(Equation 15)

Here, z is a delay operator. To distinguish from the angular frequency ω of the analog data, if the angular frequency of the digital data is σ,

[0049]

(Equation 16)

Therefore, from the above equation,

[0051]

[Equation 17]

The following relationship is obtained.

Therefore, when | σΔt | ≪1,
As ω ≒ σ, the frequency responses of the analog filter and the digital filter become equal. In other words, at frequencies well below the Nyquist frequency, this conversion is allowed. Further, since this transformation projects the inside of the unit circle of the complex z plane onto the upper half plane of the complex ω plane, the minimum phase property is preserved by the transformation. By performing this conversion, U (ω), V
(Ω), W (ω) and the like are the z-transforms of the equation (1).

The actual calculation is represented by Y (z) =-U (z) A (z) in the z-transform. Since the z-transform is of a recursive type, it can be transformed into a differential format.

For example, in the case of a displacement response spectrum,

[0056]

(Equation 18)

Thus, a solution can be obtained for a certain j by four sums of products. When the number of data is N, the displacement response spectrum can be calculated by the product sum of 4N. The same applies to the velocity response spectrum V (z) and the acceleration response spectrum W (z).

[0058]

Embodiments of the present invention will be described below.

In this embodiment, first, an evaluation experiment of the calculation speed was performed.

For the evaluation, a simulation using a workstation was performed. The prepared workstation is programmed with the method according to the present invention and the conventional method to measure the execution time.

At this time, there are two conventional methods, a method using “convolution” in equation (9) and a method using “FFT” in equation (6). The amount of calculation and the memory required to calculate was calculated. The result is shown in FIG.

Here, N represents the number of data. In the case of FFT, it represents a number obtained by adding a sufficient 0 to a power of two. Also, u (t) at the time of convolution
And the amount of calculation such as U (ω) at the time of FFT are not included here.

The calculation amount ratio is such that the data length is N = 16384
= 2 ¹⁴ (82 seconds in real time), the other calculation method is shown with the calculation amount of the present invention (hereinafter referred to as recursive filter) as 1.

From this result, it can be seen that the convolution is 2048 times that of the recursive filter, so that the performance is overwhelmingly low. Although the FFT is several times that of the recursive filter, this estimation does not include the calculation of the trigonometric function and the like, so the ratio may be several times more. Since this cannot be understood unless it is actually calculated, the calculation of the FFT and the recursive filter is performed.

Based on the results, the workstation was programmed with the recursive filter equation (9) obtained by z-transforming the displacement response spectrum and the FFT equation (6).

Then, each program was read out and operated under the same input conditions using the same computer. FIG. 2 shows the calculation results.

As is apparent from the results, the calculation time was about 10 times faster for the recursive filter on average.

This is because the dynamic characteristic of the recursive filter in the above equation is expressed by a state equation of U (z) and an output equation.
It is clear from the fact that the analog simulation circuit shown in FIG. That is, as shown in FIG.
This is because the recursive operation by the delay operator z is performed, so that the product-sum operation can be reduced.

As described above, it has been proved that the operation speed of the recursive filter is superior, but it cannot be used if the calculation accuracy is low.

Therefore, it was next verified whether the required accuracy could be obtained. This verification was confirmed by comparing the time axis and the frequency axis.

Then, this time, the response V (σ) to the seismic wave sampled by setting Δt = 5 ms to the z-transformed velocity response spectrum V (z) and the Fourier-transformed velocity response spectrum V (ω) are Δt = 5ms
V (ω) for the seismic wave sampled as
The results were plotted and compared.

The results are shown in FIGS.

(A) of each figure is a result calculated by a recursive filter, (b) is a result obtained by inverse Fourier transform of V (ω) A (ω), and (c) is an error between the two.
(D) shows the amplitude and (e) shows the phase.

As can be seen from FIGS. 4 and 5, the amplitudes and phases of the two responses almost coincide. The two responses are completely indistinguishable. The difference between the two amplitudes is shown in FIG.
The average was 1.6%, and in FIG. 5C, the average was 1.3%.

It has been found that this approximation error increases as the resonance frequency f ₀ increases and the attenuation constant h decreases, and the error increases at high frequencies.
In particular, it can be seen that the response to the impulse differs greatly between the two.

However, in FIG. 5, as can be seen from the time series of the error, the error where the response is large is rather small. This is because the characteristics of the impulse response of the filter shown in FIG. 5C appear as they are because the input data suddenly starts at a certain time. This is more likely to occur when the bandwidth of the filter is reduced.

Therefore, the first and last time 1 of the input data
The output was calculated by performing a linear taper process between / f ₀ . FIG. 6 shows the result.

The error shown in FIG. 6C is の of that when no taper is applied. However, even if a taper is applied, the initial error remains large. In actual operation, it is possible to improve this by taking a sufficiently long approach period than the wave group of interest and applying a taper.

As described above, it has been found that the response spectrum can be calculated in a short time with the required accuracy.

As described above, since the response spectrum can be calculated in a short time, it is possible to provide an arithmetic unit capable of calculating the response spectrum at the same time as the occurrence of an earthquake and accurately estimating damage.

At this time, since the amount of calculation is small, even an inexpensive built-in numerical processor (hereinafter referred to as DSP) can be processed in a short time.

FIG. 7 shows an earthquake observation system 1 in which damage estimation is performed in real time.

In this observation system 1, a plurality of seismometers 3 are installed at high density over a wide area and connected to a center computer 4 via a communication line 5 to estimate damage from seismic intensity data. A data recording device 2 is installed at each observation point.

The recording device 2 for each observation point incorporates a DSP 6 for data processing, as shown in FIG. 8, and processes the observation signal from the seismograph 3 converted by the A / D converter 7. It is to send. Therefore, processing by the recursive filter is added to the recording device 2,
The response spectrum is calculated on the observation point side, and the calculation result is transmitted to the center computer 4.

In this way, the response spectrum at each observation point is obtained by the center computer 4 in a short time (within one minute) after the occurrence of the earthquake, and the damage is estimated in real time.

In the embodiment, the method in which the arithmetic unit is constituted by the program means has been described, but the present invention is not limited to this. It is apparent that the hardware may be constituted by a circuit means having a sampler and a pulse transfer function instead of the program means.

[0087]

According to the present invention, a high-speed seismic response spectrum computing device capable of performing damage estimation in real time can be provided because the response spectrum is approximated and processed at a high speed as configured above. it can.

[Brief description of the drawings]

FIG. 1 is a table showing calculation results according to an embodiment;

FIG. 2 is a table of calculation results according to the embodiment;

FIG. 3 is a simulation circuit diagram of the embodiment;

FIG. 4 is a table showing calculation results according to the embodiment;

FIG. 5 is a table showing calculation results according to the embodiment;

FIG. 6 is a table showing calculation results according to the embodiment;

FIG. 7 is a block diagram showing a usage form of the embodiment;

FIG. 8 is a block diagram showing a use mode of the embodiment.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Earthquake observation system 2 Data recording device 3 Seismograph 4 Center computer 5 Communication line 6 DSP 7 A / D converter

Claims (1)

[Claims] 1. An observation signal of a seismometer is sampled, an acceleration α of ground motion is calculated from the sampled data, and the calculated acceleration α is converted into a Z-transform U (z),
An arithmetic unit that calculates an earthquake response spectrum using one, two, or all of V (z) and W (z). (Equation 1)