JP3878626B2 - Impulse response calculation method, apparatus and program - Google Patents

Description

  The present invention relates to an impulse response calculation method, apparatus, and program, and in particular, converts dynamic stiffness representing the relationship between an external force that vibrates an object and the behavior of the object in the frequency domain into an impulse response representing the relationship in the time domain. The present invention relates to an impulse response calculation method, an impulse response calculation device to which the impulse response calculation method can be applied, and an impulse response calculation program for causing a computer to function as the impulse response calculation device.

  The ground motion (earthquake wave) that propagates from the ground to the ground during the earthquake is incident on the foundation of the building via the surface ground, part of which is reflected by the foundation and the rest is transmitted to the top of the building, causing the building to vibrate. The building that is vibrated by becomes a new epicenter and releases waves to the ground. In this way, the building and the ground are affected by each other and vibrate during an earthquake, so when analyzing and evaluating the behavior of the building and the seismic safety, etc. during an earthquake by seismic response analysis, Force characteristics) must also be considered. The relationship between seismic motion and ground behavior can be expressed by ground dynamic stiffness (also referred to as ground impedance). The dynamic stiffness of the ground is represented by a complex function in the frequency domain in which the values of the real part and the imaginary part change according to the change in the frequency of vibration, and the Poisson ratio ν, density ρ, damping rate h, layer thickness H of the ground Or the like based on the calculation.

  Earthquake response analysis is broadly divided into frequency response analysis in which response analysis is performed in the frequency domain and time history response analysis in which response analysis is performed in the time domain. As described above, since the dynamic stiffness of the ground is a complex function in the frequency domain, a frequency response analysis that can use the dynamic stiffness of the ground as it is is often used in the seismic response analysis (see, for example, Patent Document 1). . However, in the event of a large earthquake where a large amount of energy is input to the building, a part of the energy is consumed by causing cracks in the members that make up the building or by partially plasticizing each member. In addition, the behavior of the building has non-linearity because it undergoes a process in which the fracture strength of each member is repeatedly reduced due to the occurrence of cracks, partial plasticization, and the like. For this reason, in order to predict and analyze the behavior of a building at the time of an earthquake with high accuracy, the state of the building at each point in time of the earthquake (what force is applied in the past, It is necessary to analyze the behavior of the building at each point in time. In performing time history response analysis, it is necessary to convert the ground dynamic stiffness, which is a complex function in the frequency domain, into an impulse response represented in the time domain.

  The inventor of the present application uses a displacement-dependent and speed-dependent time delay component as a conversion method that can accurately convert the dynamic stiffness of the ground into an impulse response even when the frequency dependency of the dynamic stiffness of the ground is strong. Is set as a general solution for the impulse response F (t) (see the following equation (5)),

(Where u (t) is the displacement of the ground, u ′ (t) is the velocity, c (t ₀ ) and k (t ₀ ) are the simultaneous components of the impulse response, c (t ₁ ), c (t ₂ ), ... c (t _n ), k (t ₁ ), k (t ₂ ), ... k (t _n ) are time delay components of impulse response, t _j = Δt · j, and the first term of equation (5) Is the simultaneous component of the damping term (speed-dependent term) (component at time t = 0), the second term is the simultaneous component of the stiffness term (displacement-dependent term), and the third term is the time-lag component of the damping term (time t > 0 component), and the fourth term represents the time delay component of the stiffness term)
The general solution of the set impulse response, and the equation of the dynamic stiffness S (ω) of the ground expressed from the general solution using the simultaneous component and the time delay component of the impulse response (see the following equation (6))

Based on the above, a simultaneous equation having a coefficient matrix of 2N × 2N is established using the dynamic stiffness data of N (= n + 1) grounds at each frequency of ω _{0 to} ω _n (the following (7), (8 ))),

A conversion method for obtaining impulse response components by solving these simultaneous equations has been proposed (see Non-Patent Document 1). Further, the inventor of the present application converts the dynamic stiffness of the ground into the time domain and evaluates the characteristics of the impulse response, and performs the seismic response analysis (time history response analysis) of the structure. The effectiveness of the above conversion method has been confirmed.
JP-A-4-27829 Naohiro Nakamura, “Earthquake response analysis of structures embedded in stratified ground by time domain transformation of ground impedance”, Architectural Institute of Japan, May 2003, No. 567, p. 63-70

  The dynamic stiffness of the actual ground is different for each ground, but depending on the characteristics of the dynamic stiffness of the ground to be converted, the conversion method described in Non-Patent Document 1 etc. is applied to convert it into an impulse response. Even then, an accurate impulse response may not be obtained. In addition, the problem that an accurate impulse response may not be obtained is not limited to the case where the dynamic stiffness of the ground is converted into an impulse response. For example, a viscous damper used as a vibration control device for a building Even when the dynamic stiffness of other objects such as the above is converted into an impulse response, there is a problem that may occur depending on the characteristics of the dynamic stiffness to be converted.

  The present invention has been made in consideration of the above facts, and the dynamic stiffness representing the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain corresponds to the upper limit and the lower limit of the frequency range to be calculated. Impulse response calculation method capable of accurately obtaining an impulse response representing the relationship between the external force and the behavior of the object in the time domain from the dynamic stiffness even when the real part value has a difference in frequency. It is an object to obtain an impulse response calculation device and an impulse response calculation program.

As shown in Fig. 1 (A), the conditions for calculating the dynamic stiffness of the ground are the seismic wave propagation velocity Vs ₂ = 800 (m / s), Poisson's ratio ν = 0.45, and density ρ = 2.0 (t / m ³ ). , Stratum with attenuation h = 0% or 5%, seismic wave propagation velocity Vs ₂ = 400 (m / s), Poisson's ratio ν = 0.45, density ρ = 2.0 (t / m ³ ), attenuation h = 0 % Or 5%, assuming the condition that a 50 × 50 (m) square foundation is provided on the ground consisting of a layer of H = 20 (m), and according to this calculation condition, An example of the dynamic stiffness of the ground obtained for each of the cases of the damping rate h = 0% and the damping rate h = 5% is shown in FIG. In FIG. 1B, “Real” means the real part of the dynamic rigidity, and “Imag” means the imaginary part of the dynamic rigidity.

  The inventor of the present application uses the dynamic stiffness of the ground at the damping rate h = 0% and 5% shown in FIG. 1B, respectively, and 0 (Hz), 0.25 (in the frequency range of 0 to 20 (Hz). Complex data of ground dynamic stiffness is extracted for each of 20 types of frequencies (Hz), 1.0 (Hz), 2.0 (Hz), etc. (excluding the frequency 20 (Hz) corresponding to the upper limit). 20), and the extracted 20 complex data is used to convert the impulse response to the time domain under the condition of Δt = 0.05 seconds by the conversion method described in Non-Patent Document 1 (hereinafter referred to as “old conversion method”). In order to verify the accuracy of the calculated impulse response, the dynamic stiffness is reproduced by reconverting the calculated impulse response to the frequency domain, and the reproduced dynamic stiffness is restored to the original dynamic stiffness ( Comparison with dynamic rigidity before conversion). The reproduced dynamic stiffness is divided into a real part and an imaginary part and is expressed as “pre-transformation method” and is shown in FIGS. In the re-conversion to the frequency domain, the displacement-dependent time delay component (fourth term in the above equation (5)) and the speed-dependent time delay component (the above ( Of the third term in equation (5), when the number of data to be calculated is not limited (“all terms” in FIG. 2), limited to five (“5 terms” in FIG. 2), two In the case of limiting to (2 items in FIG. 4), each calculation was performed.

  2 to 4, the dynamic stiffness before conversion (data used for impulse response calculation) is shown as “data point”, but as is apparent from FIG. 2 to FIG. The dynamic stiffness reproduced by re-transforming the impulse response obtained by the transformation method into the frequency domain shows a change such that a component that vibrates with a large amplitude is added to the dynamic stiffness before the transformation ( In particular, the above change appears remarkably under the condition of damping rate h = 5%), and the deviation from the dynamic rigidity before conversion is very large. With the old conversion method, the characteristics as shown in FIG. It can be understood that the accuracy in converting the dynamic stiffness of the motor into the impulse response is low.

  In order to confirm the reason why the conversion accuracy may be lowered by the old conversion method, the inventor of the present application uses the rigidity term (displacement dependence term: the second term and the second term in the equation (5)) in the above equation (5). (4)) and the dynamic stiffness is reproduced only from the damping term (speed dependent terms: the first and third terms in equation (5)), and the reproduced dynamic stiffness and before conversion. Comparison with the dynamic rigidity of was performed over a range of 0 to 30 (Hz). The results are shown in FIG. As is clear from FIG. 5 (B), especially the dynamic stiffness reproduced only from the damping term has a significant disturbance around the frequency 20 (Hz), and this damping term (the dynamic stiffness reproduced from) is remarkable. It is inferred that the disturbance is a cause of a decrease in conversion accuracy in the old conversion method.

  In the dynamic stiffness before conversion shown in FIG. 1B, the real part has a certain value even when the frequency is 0 (Hz) (≠ 0), but the dynamic stiffness component reproduced from the attenuation term is the frequency. Since it becomes 0 at 0 (Hz), the value of the real part of the original dynamic stiffness at the frequency 0 (Hz) is reproduced only by the stiffness term in the equation (5). On the other hand, the dynamic stiffness component reproduced from the stiffness term has a period of 1 / j of 1 / Δt (Hz) on the frequency axis (in the above example, 20 (Hz) because Δt = 0.05 seconds). Since j is a periodic function repeated at a natural number), the value at 20 (Hz) is the same as the value at 0 (Hz). For this reason, the dynamic rigidity before conversion is a characteristic in which the difference in the value of the real part is large at frequencies corresponding to the upper and lower limits of the frequency range to be calculated (in this case, 0 (Hz) and 20 (Hz)). (The dynamic rigidity before conversion shown in FIG. 1B applies to this condition, and the dynamic rigidity of the ground having inhomogeneity also applies to this condition). The difference in the real part value of the dynamic stiffness before conversion at the frequency corresponding to the upper limit and the lower limit is reproduced by the real part of the attenuation term in the equation (5), which is reproduced from the attenuation term ( This is considered to be the reason why a significant disturbance occurs in the dynamic rigidity). In fact, in the old conversion method, if the calculation is performed using the frequency data corresponding to the upper limit of the frequency range to be calculated, the dynamic stiffness reproduced from the obtained impulse response is violently disturbed near the upper limit frequency, and divergence occurs. Therefore, without using data corresponding to the upper limit frequency (for example, if the frequency range to be calculated is 0 to 20 (Hz), 0 to 19 (Hz without using the upper limit frequency 20 (Hz) data) The actual situation is that the calculation is performed using the data of).

  The inventor of the present application obtains the above knowledge from the results shown in FIG. 5, and the dynamic stiffness before conversion at the frequency corresponding to the upper limit and the lower limit in the frequency range to be calculated is expressed in the general solution formula of the impulse response. By adding a term for reproducing the difference between the real part values, it is possible to convert the dynamic stiffness into an impulse response with high accuracy, and the present invention has been achieved.

  Based on the above, the impulse response calculation method according to the first aspect of the present invention converts the dynamic stiffness representing the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain to the impulse response representing the relationship in the time domain. In converting, the mathematical expression that defines the impulse response includes a stiffness term consisting of a simultaneous component and a time delay component depending on the displacement of the object, and an attenuation term consisting of a simultaneous component and a time delay component depending on the velocity of the object. The impulse response is obtained based on the value of the dynamic stiffness when the vibration is at each frequency, using a mathematical formula including a mass term including at least a simultaneous component depending on the acceleration of the object. .

  In the first aspect of the invention, the mathematical expression that defines the impulse response includes a mass term that includes at least a simultaneous component depending on the acceleration of the object (this mass term may also be composed of a simultaneous component and a time delay component). ) Is included, the dynamic stiffness that should be converted to the impulse response is a characteristic that has a difference in the value of the real part at the frequency corresponding to the upper and lower limits in the frequency range subject to impulse response calculation. However, when the dynamic stiffness is converted to the impulse response, the above difference is reproduced by the mass term, and the obtained impulse response (the dynamic stiffness reproduced using the) is remarkably caused by the above difference. Disturbance can be prevented from occurring. Therefore, according to the first aspect of the present invention, the dynamic stiffness representing the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain is the actual stiffness at the frequency corresponding to the upper limit and the lower limit in the frequency range to be calculated. Even in the case of the characteristic having a difference in the value of the part, it is possible to accurately obtain an impulse response representing the relationship between the external force that vibrates the object and the behavior of the object in the time domain from the dynamic rigidity.

According to a second aspect of the present invention, there is provided an impulse response calculation method for converting dynamic stiffness representing the relationship between an external force that vibrates an object and the behavior of the object in the frequency domain into an impulse response representing the relationship in the time domain. The simultaneous component of the impulse response depending on the displacement of the object is k (t ₀ ), the simultaneous component of the impulse response depending on the velocity of the object is c (t ₀ ), and the simultaneous component of the impulse response depending on the acceleration of the object is m (t ₀ ), k (t _j ) represents the time delay component of the impulse response that depends on the displacement of the object, and k (t _j ), and c (t _j ) represents the time delay component of the impulse response that depends on the velocity of the object. Where j is a natural number, t _j = Δt · j), the displacement of the object in the time domain is u (t), the velocity is u '(t), and the acceleration is u "(t). As a formula defining F (t),

The impulse response is obtained on the basis of the value of the dynamic stiffness when the vibration has N kinds of frequencies (N = n + 1) using the above equation (1).

In the second aspect of the invention, the first term “m (t ₀ ) · u” (t) ”of the mathematical expression (equation (1)) defining the impulse response F (t) is the above-mentioned first aspect. The second term is the simultaneous component of the damping term, the third term is the simultaneous component of the stiffness term, the fourth term is the time delay component of the damping term, and 5 corresponds to the time delay component of the rigidity term), so that the dynamic stiffness to be converted into the impulse response is the upper limit and the upper limit of the frequency range of the impulse response calculation target, as in the first aspect of the invention. Even when there is a difference in the value of the real part at the frequency corresponding to the lower limit, the above difference is reproduced by the mass term when converting from dynamic stiffness to impulse response. Reproducible dynamic stiffness) prevents significant disturbances due to the above differences. It can be. Therefore, also in the invention described in claim 2, the dynamic stiffness representing the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain is a real part at a frequency corresponding to the upper limit and the lower limit in the frequency range to be calculated. Can be obtained with high accuracy from the dynamic stiffness, the impulse response representing the relationship between the external force that vibrates the object and the behavior of the object in the time domain.

  In order to confirm the effect of the present invention, the inventor of the present application uses the dynamic stiffness of the ground at the damping rate h = 0% and 5% shown in FIG. Dynamic stiffness of the ground at a total of 21 frequencies (including frequency 20 (Hz) corresponding to the upper limit) of 0 (Hz), 0.25 (Hz), 1.0 (Hz), 2.0 (Hz), etc. within the range Each of the complex data is extracted (a total of 21), and the extracted 21 complex data is used under the condition of Δt = 0.05 seconds by the conversion method based on the above formula (1) (hereinafter referred to as “new conversion method”). After each impulse response was obtained by conversion to the time domain, the dynamic stiffness was reproduced by reconverting the obtained impulse response to the frequency domain, and the reproduced dynamic stiffness was compared with the original dynamic stiffness. The reproduced dynamic stiffness is divided into a real part and an imaginary part and is expressed as “new conversion method” and is shown in FIGS. In the re-conversion to the frequency domain, a displacement-dependent time delay component (the fifth term in equation (1)) and a speed-dependent time delay component (the fourth term in equation (1)) are subject to calculation. When the number of data is not limited (“all items” in FIG. 2), limited to five (“5 items” in FIG. 2), and limited to two (“2 items” in FIG. 4), respectively. Arithmetic was performed.

  As is clear from FIG. 2 to FIG. 4, the dynamic stiffness reproduced by re-transforming the impulse response obtained by the new transformation method into the frequency domain is the pre-conversion shown as “data point”. Compared with the old conversion method, the accuracy at the time of converting the dynamic rigidity with the characteristics shown in Fig. 1 (B) into an impulse response is greatly improved. I can understand that.

  In addition, as in the case of the old conversion method described above, the present inventor reproduces the dynamic stiffness only from the stiffness term (displacement dependence term: the third term and the fifth term in the formula (1)) in the formula (1). In addition, the dynamic stiffness is reproduced only from the damping term (speed-dependent term: the second term and the fourth term in the equation (1)), and the comparison between the reproduced dynamic stiffness and the dynamic stiffness before conversion is 0 to The measurement was performed over a range of 30 (Hz). The results are shown in FIG. As is clear from FIG. 5 (B), in the old conversion method, the dynamic rigidity reproduced from the damping term alone was significantly disturbed near the frequency of 20 (Hz). However, the difference in the real part of the dynamic stiffness at the frequency corresponding to the upper limit and the lower limit in the frequency range to be calculated is replaced with the damping term in the old conversion method. It can be seen that this is reproduced by the added mass terms.

  In addition, the calculation of the impulse response in the invention of claim 2 is specifically based on the equation (1) when the angular frequency of the vibration is ω as described in claim 3, for example. , As a formula defining the dynamic stiffness S (ω) of the object,

Using the above equation (2), N complex data S (ω ₁ ),..., S (ω representing the value of the dynamic stiffness when the vibration has N frequencies from the data of the dynamic stiffness of the object. _N ) and extract the extracted N complex data

The calculation can be performed by substituting into the above expressions (3) and (4) derived from the expressions (1) and (2).

Further, in the invention according to claim 2 or claim 3, when the obtained impulse response is used for the time history response analysis of the object, for example, as described in claim 4, of the obtained impulse responses, Reproduce the dynamic stiffness of an object by selecting an arbitrary number of data from the data of the time delay component k (t _j ) depending on the displacement and the time delay component c (t _j ) depending on the speed of the object. Based on the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness, the number of data of the time delay components k (t _j ) and c (t _j ) used for the time history response analysis is calculated. It is preferable to set.

  As is clear from comparison of FIGS. 2 to 4, the impulse response obtained by applying the impulse response calculation method according to the present invention is also processed in the subsequent stage (in the example of FIGS. The accuracy of the impulse response represented by the impulse response data used for the subsequent processing depends on the number of time delay component data used for reconversion (reproduction of dynamic stiffness) (in the examples of FIGS. 2 to 4, Response accuracy is replaced with dynamic stiffness). On the other hand, time history response analysis has the property that the computation load increases dramatically as the number of time delay component data increases, and the accuracy of the impulse response represented by the impulse response data used for time history response analysis, that is, The analysis accuracy of time history response analysis and the computation load have a trade-off relationship.

On the other hand, in the invention according to claim 4, an arbitrary number of data is selected from the data of the time delay components k (t _j ) and c (t _j ) among the obtained impulse responses, and the object is detected. Data of time delay components k (t _j ) and c (t _j ) used for time history response analysis based on the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness. Therefore, while maintaining the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness, that is, the accuracy of the impulse response represented by the impulse response data used for time history response analysis, the data of the time delay component It is possible to set the number of time delay component data so that the number of time delays is as small as possible, and the analysis accuracy of time history response analysis is extremely reduced, and the computation load of time history response analysis is extremely increased. Can be prevented.

  Further, in the invention according to any one of claims 1 to 4, for example, as described in claim 5, the ground can be applied as the object, and seismic motion can be applied as the external force. The obtained ground impulse response can be used, for example, for a time history earthquake response analysis of a building.

The impulse response calculation device according to the invention of claim 6 is an impulse that converts a dynamic stiffness that represents a relationship between an external force that vibrates an object and a behavior of the object in a frequency domain into an impulse response that represents the relationship in a time domain. A response arithmetic unit comprising N complex data S (ω ₁ ),... Representing dynamic stiffness values when the vibration has N kinds of frequencies (N = n + 1) from the dynamic stiffness data. The extraction means for extracting S (ω _N ), the simultaneous component of the impulse response depending on the displacement of the object k (t ₀ ), the simultaneous component of the impulse response depending on the velocity of the object c (t ₀ ), The simultaneous component of the impulse response that depends on the acceleration is m (t ₀ ), the time delay component of the impulse response that depends on the displacement of the object is k (t _j ), and the impulse response that depends on the velocity of the object The time delay component is c (t _j ) (where j is a natural number and t _j = Δ t · j), the N complex data extracted by the extracting means

As in the first and second aspects of the present invention, the apparatus includes an arithmetic means for obtaining the impulse response by substituting into the formulas (3) and (4). Furthermore, the dynamic stiffness representing the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain is a characteristic in which the value of the real part at the frequency corresponding to the upper limit and the lower limit of the frequency range to be calculated is different. Even in this case, an impulse response that represents the relationship between the external force that vibrates the object and the behavior of the object in the time domain can be accurately obtained from the dynamic rigidity.

According to a seventh aspect of the present invention, there is provided an impulse response calculation program that converts a computer to an impulse response that represents a dynamic stiffness representing a relationship between an external force that vibrates an object and a behavior of the object in a frequency domain, and representing the relationship in a time domain. An impulse response calculation program for functioning as an impulse response calculation device for conversion, wherein the computer calculates a value of dynamic stiffness when the vibration has N frequencies (N = n + 1) from the dynamic stiffness data. Extracting means for extracting _N complex data S (ω ₁ ),..., S (ω _N ) to be represented, and simultaneous component of impulse response depending on the displacement of the object is k (t ₀ ), depending on the speed of the object C (t ₀ ), the simultaneous component of the impulse response depending on the acceleration of the object m (t ₀ ), and the time delay component in Δt increments of the impulse response depending on the displacement of the object k (t _j ), When the time delay component in increments of Δt of the impulse response that depends on the speed of the object is c (t _j ) (where j is a natural number, t _j = Δt · j), N extracted by the extraction means Complex data

By substituting into the above equations (3) and (4) and calculating, it is made to function as a calculation means for obtaining the impulse response.

  Since the impulse response calculation program according to the seventh aspect of the invention is a program for causing a computer to function as the extraction means and the calculation means, the computer executes the impulse response calculation program according to the seventh aspect of the invention. By doing so, the computer functions as the impulse response calculation device according to claim 6, and similarly to the invention according to claim 6, the relationship between the external force that vibrates the object and the behavior of the object is expressed in the frequency domain. Even when the dynamic stiffness is a characteristic that has a difference in the value of the real part at the frequency corresponding to the upper limit and the lower limit of the frequency range to be calculated, the external force that vibrates the object and the behavior of the object from the dynamic stiffness. An impulse response that expresses the relationship in the time domain can be obtained with high accuracy.

  As described above, the present invention is a mathematical expression that defines the impulse response that expresses the relationship between the external force that vibrates an object and the behavior of the object in the time domain. And an external force that oscillates the object and the behavior of the object using a mathematical expression that includes a damping term that depends on the speed of the object and consists of a simultaneous component and a time delay component, and a mass term that depends on the acceleration of the object and that includes at least the simultaneous component. Since the impulse response is obtained based on the value of the dynamic stiffness when the vibration has N kinds of frequencies among the dynamic stiffnesses in the frequency domain, the external force that vibrates the object and the behavior of the object Even if the dynamic stiffness representing the relationship in the frequency domain is a characteristic in which there is a difference in the value of the real part at the frequency corresponding to the upper and lower limits in the frequency range to be calculated, the dynamic stiffness An impulse response that represents the relationship between the behavior of the external force and the object to vibrate in the time domain can be obtained with high accuracy, has an excellent effect that.

  Hereinafter, an example of an embodiment of the present invention will be described in detail with reference to the drawings. FIG. 1 shows a personal computer (PC) 10 to which the present invention can be applied. The PC 10 is configured by connecting a CPU 10A, a ROM 10B, a RAM 10C, and an input / output port 10D to each other via a bus 10E including a data bus, a control bus, an address bus, and the like. The input / output port 10D is a CD 12 for reading out information from a display 12, a keyboard 14, a mouse 16, a printer 18, a hard disk drive (HDD) 20, and a CD-ROM 22 as various input / output devices. -ROM drives 24 are connected to each other.

  An earthquake response analysis program for performing an earthquake response analysis process, which will be described later, is installed in the HDD 20 of the PC 10. The earthquake response analysis program includes an impulse response calculation program according to the seventh aspect of the invention. There are several methods for installing (transferring) the earthquake response analysis program to the PC 10. For example, the earthquake response analysis program is recorded on the CD-ROM 22 together with the setup program, and the CD-ROM 22 is stored in the CD-ROM drive 24. If the CPU 10A is instructed to execute the setup program, the earthquake response analysis program is sequentially read from the CD-ROM 22, and the read earthquake response analysis program is sequentially written to the HDD 20, thereby providing an impulse. Installation of earthquake response analysis program including response calculation program. The PC 10 functions as an impulse response calculation device according to the invention of claim 6 by the CPU 10A executing an earthquake response analysis program (impulse response calculation program).

  The computer according to claim 7 is not limited to the PC 10, and may be, for example, a workstation or a general-purpose large computer.

  Next, as an operation of the present embodiment, the execution of the earthquake response analysis program is instructed via the keyboard 14 or the mouse 16 by the operator who desires to execute the earthquake response analysis of the building to be analyzed, whereby the CPU 10A of the PC 10 The earthquake response analysis process executed in step 1 will be described with reference to the flowchart of FIG.

  In step 100, calculation condition data for calculating the dynamic stiffness of the ground (the ground to be calculated) in the planned construction site of the building to be analyzed is acquired. As the calculation condition data, for example, as shown in FIG. 1 (A), the structure of the ground to be calculated, the layer thickness H of each layer, the propagation velocity Vs of the seismic wave, the Poisson's ratio ν, the density ρ, the attenuation rate h, Data such as the shape and size of the foundation of the building to be analyzed are listed. Among these data, various data related to the ground to be calculated can be obtained by, for example, performing drilling at the planned construction site of the building to be analyzed, and performing a predetermined test on the sample obtained by this boring. . The calculation condition data is not limited to using the data obtained by performing a predetermined test on the sample as it is. For example, the calculation condition data of the building to be analyzed for aftershocks after a relatively strong earthquake has occurred. If you want to analyze the behavior, assume that the characteristics of the ground subject to calculation will change due to a relatively strong earthquake, and change the value corresponding to the characteristic change to the data obtained by the predetermined test. The obtained data may be used as calculation condition data. In step 100, the calculation condition data is acquired by allowing the operator to input the calculation condition data through the keyboard 14 or by reading the calculation condition data from a recording medium (for example, a CD-ROM) in which the calculation condition data is recorded in advance. Is temporarily stored in the memory (RAM 10C) or HDD 20.

  In the next step 102, the calculation condition data acquired in step 100 is read from the memory or the HDD 20, and based on the read calculation condition data, the dynamic rigidity of the ground to be calculated is calculated using a calculation method such as a thin layer element method. Applying and calculating, the dynamic rigidity data obtained by the calculation is temporarily stored in the memory or the HDD 20. Thereby, as shown in FIG. 1B as an example, it is possible to obtain dynamic stiffness data representing the relationship between the external force (earthquake motion) that vibrates the ground and the ground behavior in the frequency domain. The dynamic stiffness data is not limited to being obtained by calculation, but can be obtained by performing an experiment.

In Step 104, the dynamic stiffness data of the calculation target ground obtained by the calculation in Step 102 is read from the memory or the HDD 20, and the read dynamic stiffness data is read from within the preset frequency range of the calculation target. N species frequency (N kinds of angular frequency omega _1, ..., omega _N) N pieces of complex data S (omega ₁₎ representing the value of the dynamic stiffness at, ..., extracting respectively S (omega _N), The extracted complex data is stored in the memory or the HDD 20. This step 104 corresponds to the extracting means according to the present invention. In addition, as a frequency range of calculation object, the range of 0-20 (Hz) is applicable, for example. The N types of frequencies for extracting the complex data are, for example, frequencies corresponding to the upper and lower limits of the frequency range to be calculated (for example, if the frequency range to be calculated is 0 to 20 (Hz), the upper limit frequency. 20 (Hz) and the lower limit frequency 0 (Hz)) can be set. Further, the complex data extracted from the dynamic stiffness data of the ground is used for impulse response calculation, and at this time t = 0 and time t = Δt · j (j = 1, 2,...) Impulse response data representing the impulse response of the ground is obtained. The number of impulse response data obtained is determined according to the number of complex data used in the calculation (that is, j maximum value jmax = number of complex data−1 (= n )), The time range of the impulse response of the ground represented by the obtained impulse response data is also determined according to the number of complex data used in the calculation (for example, if the number of complex data is 21 and Δt = 0.05 seconds, tmax = Δt · jmax = 0.05 × 20 = 1 second, and 21 impulse response data representing the impulse response of the ground in the time range of time t = 0 to 1 second is obtained), so the dynamic stiffness of the ground The number of complex data to be output (the number of types of frequencies to extract the complex data) can be predetermined by taking into consideration the length of the time range to be calculated the impulse response of the soil.

In the next step 106, the simultaneous equations according to the present invention for converting the dynamic stiffness representing the relationship between the ground motion and the ground behavior in the frequency domain into the impulse response representing the relationship between the ground motion and the ground behavior in the time domain. (Formulas (3) and (4) having a 2N × 2N coefficient matrix) are read out from the HDD 20, and the N complex data S (ω ₁ ),. , S (ω _N ) and substituting the simultaneous equations, the impulse response data representing the ground impulse response is calculated in increments of Δt set in advance. This step 106 corresponds to the computing means according to the present invention. As a result of this calculation, as impulse response data representing the impulse response of the ground, a simultaneous component k (t ₀ ) of the impulse response that depends on the displacement of the ground, a simultaneous component c (t ₀ ) of the impulse response that depends on the speed of the ground, The data of the impulse response simultaneous component m (t ₀ ) depending on the acceleration is obtained, and the time delay component k (t _j ) data of the impulse response depending on the ground displacement is n in increments of Δt (n = N-1) is obtained, and n-1 pieces of data of the time delay component c (t _j ) of the impulse response depending on the ground speed are obtained in increments of Δt. The obtained impulse response data is temporarily stored in the memory or the HDD 20.

The simultaneous equations of Equations (3) and (4) are equations that define the impulse response of the ground, and include a mass term consisting of only simultaneous components depending on the acceleration of the ground (Equation (1) above) , The mass term corresponds to the first term of the equation (1)), and the simultaneous components k (t ₀ ), c (t ₀ ), m () of the equation (1) and the impulse response in the equation (1) t _0), the component k (t ₁ the delay time of the impulse _{response), k (t 2),} ..., k (t n) and _{c (t 1), c (} t 2), ..., c (t n-1 ) Is used to define the dynamic stiffness S (ω) of the ground, so the dynamic stiffness of the ground subject to computation is the upper limit within the frequency range subject to computation. Even in the case of a characteristic having a difference in the value of the real part at the frequency and the lower limit frequency, the above difference is reproduced by the mass term upon conversion from the dynamic stiffness to the impulse response by the calculation in Step 106, Up It is possible to obtain a highly accurate impulse response (data) free from disturbance caused by the difference.

In step 108, among the impulse response data obtained by the calculation in step 106, the time delay component data k (t _j ) of the impulse response depending on the displacement of the ground and the time delay component of the impulse response depending on the velocity of the ground. From the data c (t _j ), a predetermined number of data are selected as calculation targets, starting from the data at time t ₁ . In step 108, the selected time delay component data and the impulse response simultaneous component data k (t ₀ ), c (t ₀ ), m (t ₀ ) are read from the memory or HDD 20 and the ground represented by the read data is displayed. The data representing the dynamic stiffness of the ground corresponding to the impulse response of the ground represented by these data is obtained by reconverting the impulse response of the above to the frequency domain, and the obtained data is temporarily stored in the memory or the HDD 20.

  In the next step 110, data of the dynamic stiffness of the ground (hereinafter referred to as “reproduced dynamic stiffness”) obtained by the calculation in step 108 is read from the memory or the HDD 20, and Data of dynamic stiffness obtained by calculation (hereinafter referred to as “original dynamic stiffness”) is read from the memory or the HDD 20, and the reproduced dynamic stiffness and the original dynamic stiffness represented by the both read data are read. And the calculated degree of coincidence is stored in a memory or the like. As the degree of coincidence, for example, an arbitrary physical quantity such as the reciprocal of the result of accumulating the absolute value of the deviation between the reproduced dynamic stiffness and the original dynamic stiffness at each frequency within the frequency range to be calculated is applied. be able to. In step 110, the number of selected time delay component data in the previous step 108 is evaluated based on the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness. In the next step 112, the time delay is evaluated. It is determined whether or not it is necessary to change the selected number of component data. If the determination at step 112 is affirmed, the routine proceeds to step 114, where the selected number of time delay component data is changed and set, and then the routine returns to step 108. In this case, processing such as calculation of dynamic stiffness and calculation of coincidence is repeated.

  For example, if the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness is greater than or equal to a predetermined value, it can be determined that the impulse response of the ground can be accurately expressed by setting the current data selection number, There is a possibility that the degree of coincidence between the reproduced dynamic rigidity and the original dynamic rigidity becomes a predetermined value or more even if the number of selected data is smaller. Therefore, in the above case, the determination in step 112 is affirmed and the routine proceeds to step 114, where the number of selected time delay component data is reduced and the routine returns to step 108. Also, for example, if the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness is less than a predetermined value, the current impulse response of the ground cannot be accurately expressed by the setting of the number of selected data. Since there is a possibility that the number of time delay component data selected at 108 is too small, the determination at step 112 is affirmed and the routine proceeds to step 114 where the number of selected time delay component data is increased and then the processing returns to step 108. . As a result, the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness, that is, the time delay while ensuring the accuracy of the impulse response represented by the impulse response data excluding the data of the time delay component exceeding the set number of selections. The selected number of time delay component data can be set so that the selected number of component data is as small as possible. The steps 108 to 114 correspond to the invention according to claim 4.

For example, as apparent from comparing FIG. 2 with FIGS. 3 and 4, the accuracy of the impulse response represented by the impulse response data does not necessarily improve as the number of selected time delay component data increases. In the example of FIG. 2, the impulse response data represents the selected number of time delay component data even though the selected number is the maximum (number of selected data s (data k (t _j ) s = number of data k (t _j ) n)). Impulse response is much smaller than the “old conversion method” but oscillates in a relatively short period, and the impulse response is smaller than the example of FIGS. The accuracy of the impulse response represented by the data is low, and the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness is also lowered. For this reason, the dynamic stiffness calculation and the coincidence calculation are repeated while sequentially increasing the selected number of time delay component data from the minimum value. You may make it set as the final selection number of the data of a delay component. In addition, by selecting the time delay component data from the maximum value to the minimum value or changing from the minimum value to the maximum value, the dynamic stiffness calculation and the coincidence calculation are repeated to select the time delay component data. The relationship between the number and the degree of coincidence may be obtained, and the minimum selection number that makes the degree of coincidence equal to or greater than a predetermined value in the obtained relationship may be set as the final selection number of the time delay component data.

  When the number of selected time delay component data is determined by performing the above-described processing, the determination in step 112 is denied and the process proceeds to step 116, where an impulse is obtained by excluding data of the time delay component exceeding the determined selection number. The response data is read from the memory or the HDD 20, and the time history earthquake response analysis of the building to be analyzed is performed using the read impulse response data. Since the impulse response data used for the time history earthquake response analysis is data calculated by applying the present invention, the dynamic stiffness of the ground to be calculated is the actual frequency at the upper limit frequency and the lower limit frequency within the frequency range to be calculated. Even if there is a difference in the value of the part, there is no disturbance due to the above difference, and the data accurately represents the relationship between the seismic motion and the ground behavior in the time domain. The time history earthquake response analysis can be performed accurately. In addition, since the number of data of the time delay component is reduced as much as possible in the range in which the above-described impulse response data does not adversely affect the accuracy of the data by the processing of steps 108 to 114 described above, the time of the building to be analyzed It is also possible to prevent a large load from being applied to the PC 10 (CPU 10A) during the historical earthquake response analysis.

  In the above description, an example in which a predetermined number N of complex data is extracted from dynamic rigidity data to be calculated has been described. However, the present invention is not limited to this. For example, if the dynamic stiffness of the calculation target has a characteristic that changes in a complex manner with changes in frequency, the reproduced dynamic stiffness and original Since the degree of coincidence with the dynamic stiffness of the motor does not exceed a predetermined value (that is, the accuracy of the impulse response data obtained by applying the present invention is insufficient), such a condition is satisfied. In this case, the impulse response (data) may be calculated again after increasing the number of complex data extracted from the dynamic rigidity data to be calculated.

  In the earthquake response analysis process described above, the dynamic stiffness is obtained by re-converting the impulse response represented by the impulse response data in steps 108 to 114 into the frequency domain, and the obtained dynamic stiffness (reproduced dynamic stiffness) is obtained. Stiffness) and the original dynamic stiffness are calculated to set the number of selected time delay component data. However, in the present invention, the above processing is not essential, and the number of selected time delay component data is selected. May be fixedly set in advance.

  In the above, N = n + 1 in the equation (1) that defines the impulse response F (t) of the object. However, the present invention is not limited to this, and N> n + 1, and the equation can be calculated from the number of unknowns. It is also possible to obtain an impulse response by setting up simultaneous equations with many numbers and solving them by applying the least square method or the like.

  Further, in the above description, the time delay component (fourth term) of the attenuation term is integrated over a period of j = 1 to (n−1) and the time delay component of the stiffness term as a mathematical expression that defines the impulse response of the object. (5) is integrated using the equation (1) over a period of j = 1 to n. However, the present invention is not limited to this, and the following ( As shown in Eq. 9), using an equation in which the integration period of the time lag component of the attenuation term and the time lag component of the stiffness term is reversed, this equation (9) and the motion of the object derived from equation (9) are used. The impulse response of the object may be obtained based on the following equation (10) that defines the mechanical rigidity.

Further, in the above, the equation (1) including the mass term consisting only of the simultaneous component (first term) is used as the equation defining the impulse response of the object. However, the present invention is not limited to this. As a formula for defining the response, a formula including a mass term composed of a simultaneous component (first term) and a time delay component (fourth term) is used as shown in the following formula (11) (note that formula (11) 2N = n ₁ + n ₂ + n ₃ +3), the impulse response of the object on the basis of this equation (11) and the following equation (12) that defines the dynamic stiffness of the object derived from the equation (11) May be requested.

The invention according to claim 1 includes the scope of the right to obtain the impulse response of an object using the above-described equations (9), (10) or (11), (12).

  In the above description, the present invention is applied from the dynamic rigidity of the ground to determine the impulse response of the ground, and the example of using the determined impulse response for the time history earthquake response analysis of the building to be analyzed has been described. Needless to say, the present invention can also be applied when converting the dynamic stiffness of another object such as a viscous damper used as a vibration damping device of a building into an impulse response.

(A) is an image figure which shows an example of the calculation conditions of the dynamic rigidity of a ground, (B) is a diagram which shows an example of the dynamic rigidity of the ground calculated | required by calculation according to the calculation conditions of (A). (A) is the damping rate h = 0%, (B) is the impulse response obtained from the dynamic stiffness of the ground calculated under the condition of the damping rate h = 5% by the old conversion method and the new conversion method. Is a diagram showing dynamic rigidity characteristics reproduced by re-converting to the frequency domain without limiting the number of data of time delay components. (A) is the damping rate h = 0%, (B) is the impulse response obtained from the dynamic stiffness of the ground calculated under the condition of the damping rate h = 5% by the old conversion method and the new conversion method. Is a diagram showing dynamic rigidity characteristics reproduced by limiting the number of time-delayed component data to 5 and reconverting them to the frequency domain. (A) is the damping rate h = 0%, (B) is the impulse response obtained from the dynamic stiffness of the ground calculated under the condition of the damping rate h = 5% by the old conversion method and the new conversion method. Is a diagram showing dynamic rigidity characteristics reproduced by limiting the number of time-delayed component data to two and re-converting them into the frequency domain. Of the impulse responses obtained by the old conversion method and the new conversion method, (A) is a diagram showing the stiffness characteristic, and (B) is a diagram showing the dynamic stiffness characteristics reproduced only from the damping term. It is a block diagram which shows schematic structure of PC concerning this embodiment. It is a flowchart which shows the content of an earthquake response analysis process.

Explanation of symbols

10 PC
12 Display 14 Keyboard 16 Mouse 20 HDD

Claims (7)

In converting the dynamic stiffness that expresses the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain into an impulse response that expresses the relationship in the time domain,
Formulas defining the impulse response include a stiffness term consisting of a simultaneous component and a time delay component depending on the displacement of the object, an attenuation term consisting of a simultaneous component and a time delay component depending on the velocity of the object, An impulse response calculation method for obtaining the impulse response based on a value of the dynamic stiffness when the vibration is at each frequency, using a mathematical formula including a mass term that depends on acceleration and includes at least a simultaneous component. . In converting the dynamic stiffness that expresses the relationship between the external force that vibrates the object and the behavior of the object in the frequency domain, to the impulse response that expresses the relationship in the time domain,
The simultaneous component of the impulse response that depends on the displacement of the object is k (t ₀ ), the simultaneous component of the impulse response that depends on the velocity of the object is c (t ₀ ), and the simultaneous component of the impulse response that depends on the acceleration of the object is m ( t ₀ ), the time delay component of the impulse response depending on the displacement of the object by k (t _j ), and the time delay component of the impulse response depending on the velocity of the object by the step of Δt c (t _j ) (where j is a natural number, t _j = Δt · j), the displacement of the object in the time domain is u (t), the velocity is u ′ (t), and the acceleration is u ″ (t). t)

An impulse response calculation method characterized in that the impulse response is obtained based on the value of the dynamic stiffness when the vibration has N types of frequencies (N = n + 1) using the above equation (1). Assuming that the angular frequency of the vibration is ω, on the basis of the equation (1), the mathematical formula defining the dynamic stiffness S (ω) of the object is as follows:

Using the above equation (2), N complex data S (ω ₁ ),..., S (ω representing the value of the dynamic stiffness when the vibration has N frequencies from the data of the dynamic stiffness of the object. _N ) and extract the extracted N complex data

3. The impulse response according to claim 2, wherein the impulse response is obtained by substituting the calculation into the equations (3) and (4) derived from the equations (1) and (2). Calculation method. The determined impulse response is used for time history response analysis of the object,
Of the obtained impulse responses, an arbitrary number of data are selected as calculation targets from the data of the time delay component k (t _j ) depending on the displacement of the object and the time delay component c (t _j ) depending on the velocity of the object. Then, an operation for reproducing the dynamic stiffness of the object is performed, and based on the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness, time delay components k (t _j ), c used for the time history response analysis are performed. 4. The impulse response calculation method according to claim 2, wherein the number of data of (t _j ) is set.   The impulse response according to any one of claims 1 to 4, wherein the object is ground, and the external force is seismic motion, and the obtained impulse response of the ground is used for a time history earthquake response analysis of a building. Calculation method. An impulse response computing device that converts dynamic stiffness representing a relationship between an external force that vibrates an object and a behavior of the object in a frequency domain into an impulse response representing the relationship in a time domain,
From the dynamic stiffness data, N complex data S (ω ₁ ),..., S (ω _N ) representing dynamic stiffness values when the vibration has N frequencies (N = n + 1) are extracted. Extraction means to
The simultaneous component of the impulse response that depends on the displacement of the object is k (t ₀ ), the simultaneous component of the impulse response that depends on the velocity of the object is c (t ₀ ), and the simultaneous component of the impulse response that depends on the acceleration of the object is m ( t ₀ ), the time delay component of the impulse response depending on the displacement of the object by k (t _j ), and the time delay component of the impulse response depending on the velocity of the object by the step of Δt c (t _j ) (where j is a natural number and t _j = Δt · j), N complex data extracted by the extracting means are

An arithmetic means for obtaining the impulse response by substituting into the above formulas (3) and (4) and calculating,
An impulse response calculation device comprising: An impulse response calculation program that causes a computer to function as an impulse response calculation device that converts a dynamic stiffness that represents the relationship between an external force that vibrates an object and the behavior of the object in the frequency domain into an impulse response that represents the relationship in the time domain. There,
The computer,
From the dynamic stiffness data, N complex data S (ω ₁ ),..., S (ω _N ) representing dynamic stiffness values when the vibration has N frequencies (N = n + 1) are extracted. Extraction means to
The simultaneous component of the impulse response depending on the displacement of the object is k (t ₀ ), the simultaneous component of the impulse response depending on the velocity of the object is c (t ₀ ), and the simultaneous component of the impulse response depending on the acceleration of the object is m (t ₀ ), k (t _j ) represents the time delay component of the impulse response that depends on the displacement of the object, and k (t _j ), and c (t _j ) represents the time delay component of the impulse response that depends on the velocity of the object. However, when j is a natural number and t _j = Δt · j), the N complex data extracted by the extracting means

An impulse response calculation program which functions as a calculation means for obtaining the impulse response by calculating by substituting into the above equations (3) and (4).

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