Method for estimating bending vibration mode, method for estimating bending rigidity of structure, program, and recording medium readable by computer

Abstract

To estimate the bending vibration mode of vibration of a structure from vibration data observed at a single point.SOLUTION: A method for estimating a bending vibration mode includes a displacement time history waveform derivation step, a pick-up step, a non-stationary amplitude spectrum derivation step, and an estimation step. In the displacement time history waveform derivation step, a displacement time history waveform of vibration viewed in two directions in-plane in a coordinate plane in which vibration mode movements appear is derived from the data observed. In the pick-up step, a time range in which the influence of the movements is determined to be predominant in the displacement time history waveform is picked up. In the non-stationary amplitude spectrum derivation step, a non-stationary amplitude spectrum of vibration viewed in the direction in which the movements appear is derived. In the estimation step, the vibration mode is estimated from data of spectrum characteristics in the non-stationary amplitude spectrum corresponding to the time range that is picked up in the pick-up step.SELECTED DRAWING: Figure 3

Description

[Technical field]

[0001] The present disclosure relates to a method for estimating a bending vibration mode, a method for estimating the bending stiffness of a structure, a program, and a computer-readable recording medium. [Background technology]

[0002] Conventionally, research has been conducted on structures such as bridges, attempting to grasp vibration characteristics such as bending vibration modes of structures from observed vibration data (see, for example, Non-Patent Document 1 below). [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] Toshiaki Okubo, Shinya Matsumoto, and Fumio Miyamoto: "Report: Application and Effectiveness of a Vibration Measurement System Using Wireless Acceleration Sensors to Aging Bridges," Proceedings of the Japan Concrete Institute, Vol. 36, No. 2, 2014 Summary of the Invention [Problem to be solved by the invention]

[0004] In the conventional technology used in the research described in the above-mentioned non-patent document 1, in order to grasp the bending vibration mode of a structure, it was necessary to set up observation devices at multiple points on the structure and analyze the observation data from all of these observation devices.

[0005] The present disclosure makes it possible to estimate the bending vibration mode of a structure from observed vibration data observed at a single point set on the structure. [Means for solving the problem]

[0006] The present disclosure includes a method for estimating a bending vibration mode in vibration of a structure, which estimates the bending vibration mode in the vibration of the structure from observation data of the vibration observed at one point set on the structure. The method for estimating the bending vibration mode includes a displacement time history waveform derivation step, a pick-up step, a non-stationary amplitude spectrum derivation step, and an estimation step, which will be described later. Here, in the displacement time history waveform derivation step, a displacement time history waveform of the vibration seen in two directions in a coordinate plane in which the movement of the vibration mode appears is derived from the observation data. In addition, in the pick-up step, a time range in which the influence of the above movement is determined to be prominent in light of a predetermined criterion is picked up in the displacement time history waveform. In addition, in the non-stationary amplitude spectrum derivation step, a non-stationary amplitude spectrum of the vibration seen in the direction in which the above movement appears is derived. In addition, in the estimation step, a vibration mode is estimated from data of spectrum characteristics corresponding to the time range picked up in the pick-up step in the above non-stationary amplitude spectrum.

[0007] According to the above method, it is possible to pick up a time range when the influence of the movement of the vibration mode to be estimated predominates (when the structure is moving roughly in line with the movement of the vibration mode) from the displacement time history waveform of the vibration observed at one point. Then, it is possible to estimate the bending vibration mode in the vibration of the structure from the spectral characteristics of the non-stationary amplitude spectrum of the vibration corresponding to this time range.

[0008] In one preferred embodiment, the displacement time history waveform deriving step is performed using observation data obtained by observing points at locations that can be regarded as nodes in even-order vibration modes.

[0009] In this case, by removing the influence of the motion due to the even-order vibration modes from the displacement time history waveform of the vibration, the displacement time history waveform can be made simpler. Then, in the pick-up step, the accuracy of picking up the time range in which the motion of the vibration mode is dominant can be improved, and the accuracy of estimating the vibration mode can be improved.

[0010] In another preferred embodiment, the picking step picks up a plurality of the time ranges, and the estimation step estimates a first mode of the vibration modes from a peak of a spectrum characteristic obtained by smoothing each data corresponding to the plurality of time ranges picked up in the picking step.

[0011] In this case, multiple timings at which the influence of the movement of the vibration mode to be estimated is prominent (timings at which the structure moves roughly in line with the movement of the vibration mode) are picked up, and a peak of the spectral characteristics common to these timings is found. This peak of the spectral characteristics is highly likely to be a peak of the spectral characteristics corresponding to the main movement of the vibration mode to be estimated. On the other hand, the bending vibration mode in the vibration of a structure generally appears in its first mode in a form that is easiest to understand. Therefore, by estimating the first mode of the vibration mode from the peak of the spectral characteristics, this first mode can be estimated with higher accuracy.

[0012] In the above preferred embodiment, it is preferable that the estimation step includes a searching step, a first mode estimation step, and an a-th mode estimation step, each of which will be described later. Here, in the searching step, the plurality of peaks are searched for from the spectrum characteristics. In the first mode estimation step, the maximum peak among the peaks found in the searching step is estimated to be the first mode. In the a-th mode estimation step, a is a natural number equal to or greater than 2, and a peak having a vibration frequency a times that of the peak estimated to be the first mode among the peaks found in the searching step is estimated to be the a-th mode among the vibration modes.

[0013] Generally, bending vibration modes have the property that the lower the order, the larger the peak of the spectral characteristics of the corresponding non-stationary amplitude spectrum. In addition, the a-th mode, where a is a natural number equal to or greater than 2, has the property that the vibration frequency of the corresponding peak is a times the vibration frequency of the peak corresponding to the first mode. In the above case, by estimating that the maximum peak in the spectral characteristics of the non-stationary amplitude spectrum is the lowest order (first) vibration mode, the first mode of the vibration modes can be estimated with a high degree of probability. Furthermore, based on the estimated first mode, the second and higher vibration modes can be estimated with a higher degree of accuracy.

[0014] There is also provided a method for estimating the bending stiffness of a structure, which uses the above method to obtain a peak vibration frequency estimated to be the a-th mode, and estimates the bending stiffness of the structure based on the obtained vibration frequency.

[0015] Generally, the bending vibration mode of a structure has the property that the higher the order (the higher the vibration frequency), the greater the fluctuation in vibration frequency due to changes in the bending rigidity of the structure. According to the above method, the bending rigidity of a structure can be estimated with higher accuracy by estimating the bending rigidity based on the vibration frequency of a vibration mode higher than the first order.

[0016] Also disclosed is a program for making a computer realize a function of estimating a bending vibration mode in vibration of a structure from observation data of vibration observed at one point set on the structure. This program makes a computer realize a displacement time history waveform derivation function, a pick-up function, a non-stationary amplitude spectrum derivation function, and an estimation function, which will be described later. Here, the displacement time history waveform derivation function is a function of deriving a displacement time history waveform of vibration seen in two directions in a coordinate plane in which the movement of the vibration mode appears from the observation data. Also, the pick-up function is a function of picking up a time range in the displacement time history waveform in which the influence of the above movement is determined to be prominent in light of a predetermined criterion. Also, the non-stationary amplitude spectrum derivation function is a function of deriving a non-stationary amplitude spectrum of vibration seen in the direction in which the above movement appears. Also, the estimation function is a function of estimating a vibration mode from data of spectrum characteristics corresponding to the time range picked up by the pick-up function in the above non-stationary amplitude spectrum.

[0017] The above program allows the computer to realize a function for picking out a time range when the influence of the movement of the vibration mode to be estimated is dominant (when the structure is moving roughly in line with the movement of the vibration mode) from the displacement time history waveform of the vibration observed at one point.Then, the computer can realize a function for estimating the bending vibration mode in the vibration of the structure from the spectral characteristics of the non-stationary amplitude spectrum of the vibration corresponding to this time range.

[0018] Also disclosed is a computer-readable recording medium having a program recorded thereon that causes a computer to realize a function of estimating a bending vibration mode in vibration of a structure from observation data of vibration observed at one point set on the structure. This computer-readable recording medium has a program recorded thereon that causes a computer to realize a displacement time history waveform derivation function, a pick-up function, a non-stationary amplitude spectrum derivation function, and an estimation function, each of which will be described later. Here, the displacement time history waveform derivation function is a function of deriving a displacement time history waveform of vibration seen in two directions in a coordinate plane in which the movement of the vibration mode appears from the observation data. Also, the pick-up function is a function of picking up a time range in the displacement time history waveform in which the influence of the above movement is determined to be prominent in light of a predetermined criterion. Also, the non-stationary amplitude spectrum derivation function is a function of deriving a non-stationary amplitude spectrum of vibration seen in the direction in which the above movement appears. Also, the estimation function is a function of estimating a vibration mode from data of spectrum characteristics corresponding to the time range picked up by the pick-up function in the above non-stationary amplitude spectrum.

[0019] According to the above computer-readable recording medium, a computer that has read the program from the recording medium can be made to realize a function of picking out a time range when the influence of the movement of the vibration mode to be estimated is dominant (when the structure is moving roughly in line with the movement of the vibration mode) from the displacement time history waveform of the vibration observed at one point.Then, the computer can be made to realize a function of estimating the bending vibration mode in the vibration of the structure from the spectral characteristics of the non-stationary amplitude spectrum of the vibration corresponding to this time range. Effect of the Invention

[0020] According to the present disclosure, the bending vibration mode in vibration of a structure can be estimated from observation data of vibration observed at one point set on the structure. [Brief description of the drawings]

[0021] [Figure 1] 1 is a photograph of a bridge 10 that is a target for estimating a bending vibration mode in a method for estimating a bending vibration mode according to a first embodiment of the present disclosure. [Diagram 2] FIG. 2 is a diagram showing a schematic diagram of a measurement position 10A in FIG. [Diagram 3] 4 is a flowchart showing a series of steps executed by a computer 20 in a method for estimating a bending vibration mode according to a first embodiment of the present disclosure. [Figure 4] 4 is a flowchart showing a subroutine 1 of FIG. 3. [Diagram 5] 3 is a graph showing an example of an acceleration waveform in the Y-axis direction (perpendicular to the bridge axis) observed by accelerometer 20A in FIG. 2. [Figure 6] 3 is a graph showing an example of an acceleration waveform in the Z-axis direction (up-down direction) observed by accelerometer 20A of FIG. 2. [Figure 7] This is a graph showing the displacement time history waveform of vibration in the Y-axis direction (perpendicular to the bridge axis), derived from the acceleration waveform in Figure 5. [Figure 8] 7 is a graph showing a displacement time history waveform of vibration as viewed in the Z-axis direction (up-down direction), derived from the acceleration waveform in FIG. 6. [Figure 9] 3 is a graph in which the trajectory of the movement of measurement position 10A shown in FIG. 2 is plotted on the YZ plane. [Figure 10] 3 is a diagram in which the movement direction of the measurement position 10A shown in FIG. 2 is plotted on the YZ plane. [Figure 11] 11 is a graph showing the change over time in the direction of movement shown in FIG. 10. [Figure 12] 2 is a spectrogram showing an example of a non-stationary amplitude spectrum of vibration in the bridge 10 of FIG. 1. [Figure 13] 5 is a graph showing an example of the spectral characteristics smoothed in step S130 of FIG. 4. [Figure 14] 3 is a graph showing an example of a change in bending stiffness over time at a measurement position 10A of the bridge 10 in FIG. 2. [Figure 15] FIG. 11 is a plan view showing a building 40 as a target for estimating a bending vibration mode in a method for estimating a bending vibration mode according to a third embodiment of the present disclosure. [Figure 16] 13 is a graph showing an example of an acceleration waveform in the X-axis direction acquired from an accelerometer in the third embodiment. [Figure 17] 13 is a graph showing an example of an acceleration waveform in the Y-axis direction acquired from the accelerometer in the third embodiment. [Figure 18] 17 is a graph showing a displacement time history waveform of vibration as viewed in the X-axis direction, derived from the acceleration waveform in FIG. 16. [Figure 19] 18 is a graph showing a displacement time history waveform of vibration as viewed in the Y-axis direction, derived from the acceleration waveform in FIG. 17. [Figure 20] 16 is a graph in which the trajectory of the movement of measurement position 40A shown in FIG. 15 is plotted on the XY plane. [Figure 21] 16 is a graph showing the direction of movement of measurement position 40A shown in FIG. 15 in the form of a change in rise angle from the X-axis direction over time. [Figure 22] 16 is a graph showing the direction of movement of measurement position 40A shown in FIG. 15 in the form of a rise angle from the Y-axis direction changed over time. [Figure 23] 16 is a spectrogram showing an example of a non-stationary amplitude spectrum of building 40 in FIG. 15 as viewed in the X-axis direction. [Figure 24] 16 is a spectrogram showing an example of a non-stationary amplitude spectrum of building 40 in FIG. 15 as viewed in the Y-axis direction. [Diagram 25] 24 is a graph illustrating a spectral feature 60 extracted from the non-stationary amplitude spectrum of FIG. 23. [Figure 26] 25 is a graph illustrating a spectral feature 70 extracted from the non-stationary amplitude spectrum of FIG. 24. [Figure 27] 17 is a graph showing a Fourier amplitude spectrum 80 derived from the acceleration waveform of FIG. 16. [Figure 28]18 is a graph showing a Fourier amplitude spectrum 90 derived from the acceleration waveform of FIG. 17. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0022] Hereinafter, embodiments of the present disclosure will be described with reference to the drawings.

[0023] First Embodiment First, a method for estimating a bending vibration mode according to a first embodiment of the present disclosure (hereinafter, also simply referred to as the "first embodiment") will be described with reference to Figs. 1 to 13. The first embodiment is a method for estimating a bending vibration mode in a bridge 10 (see Fig. 1), which is a structure spanning a river, from observation data of vibration observed at one measurement position 10A set on the bridge 10. The bridge 10 in this embodiment is, for example, a linear prestressed concrete three-span continuous rigid frame box girder bridge. The measurement position 10A is set at a point that is the center of the deck 10B of the bridge 10. This point is a location that can be regarded as a node in an even-order (secondary, fourth, etc.) bending vibration mode as viewed in the bridge axis direction of the bridge 10. The above point is set at a roadside strip 10D (see Fig. 2) provided at a position toward the end of the deck 10B as viewed in a horizontal direction perpendicular to the bridge axis of the bridge 10.

[0024] As shown in FIG. 2, the vibration acting on the bridge 10 is observed in the form of an acceleration waveform of three axes (X-axis, Y-axis, and Z-axis) by the accelerometer 20A installed at the measurement position 10A. In this embodiment, the accelerometer 20A is installed on the shoulder 10D of the bridge 10, so that it does not prevent vehicles (not shown) and the like from passing over the bridge 10. Therefore, the accelerometer 20A observes an acceleration waveform of vibration that combines constant microtremors, which are weak vibrations that constantly act on the bridge 10, with traffic vibrations caused by vehicles and the like. The accelerometer 20A is installed so that its X-axis direction is the horizontal direction along the bridge axis of the bridge 10, its Y-axis direction is the horizontal direction perpendicular to the bridge axis of the bridge 10, and its Z-axis direction is the up-down direction. Therefore, the bending vibration mode of the bridge 10 can appear as a movement of the measurement position 10A in the YZ plane.

[0025] The acceleration waveform observed by the accelerometer 20A is recorded in a computer-readable form as a discrete function of time on a recording medium 20B of the computer 20. A program for causing the computer 20 to realize the function of executing the first embodiment is also recorded in a computer-readable form on this recording medium 20B. All of the various data calculated or output when this program is executed by the computer 20 is recorded in a computer-readable form on the recording medium 20B.

[0026] Here, a series of steps executed by the computer 20 when the function of executing the first embodiment is realized by the computer 20 will be described mainly with reference to the flowcharts shown in Figures 3 and 4. In this series of steps, the computer 20 first executes step S10.

[0027] In step S10, the computer 20 performs initial settings required to execute each step described below, and then the computer 20 advances the process to step S20.

[0028] In step S20, computer 20 acquires observed data of the acceleration waveform of accelerometer 20A recorded in recording medium 20B. Then, computer 20 advances the process to step S30.

[0029] In step S30, the computer 20 derives the displacement time history waveforms of the vibration of the bridge 10 as viewed in the two in-plane directions of the Y-axis direction and the Z-axis direction in the YZ plane from the observation data acquired in the immediately preceding step S20. Then, the computer 20 advances the process to step S40.

[0030] Here, the YZ plane is a coordinate plane in which the movement of the bending vibration mode in the bridge 10 appears as a movement in the Z-axis direction (up and down direction). Therefore, step S30 corresponds to the "displacement time history waveform derivation step" in this disclosure, and the function realized by the computer 20 to execute step S30 corresponds to the "displacement time history waveform derivation function" in this disclosure.

[0031] In this embodiment, the computer 20 derives a displacement time history waveform of vibration viewed in the Y-axis direction (see FIG. 7) from the acceleration waveform in the Y-axis direction (direction perpendicular to the bridge axis, see FIG. 5) of the observation data acquired in the immediately preceding step S20. The computer 20 also derives a displacement time history waveform of vibration viewed in the Z-axis direction (see FIG. 8) from the acceleration waveform in the Z-axis direction (up and down direction, see FIG. 6) of the observation data acquired in the immediately preceding step S20. Note that since the observation data of the acceleration waveform recorded in the recording medium 20B of the computer 20 is in the form of a discrete function of time, the displacement time history waveform derived in the displacement time history waveform derivation step also takes the form of a discrete function of time.

[0032] In step S40, the computer 20 derives the trajectory of the movement of the measurement position 10A in the YZ plane from the displacement time history waveforms of the vibration in the Y-axis direction and the Z-axis direction derived in the immediately preceding step S30. Then, the computer 20 advances the process to step S50.

[0033] In this embodiment, the movement trajectory 21 of the measurement position 10A is not plotted on a graph. However, if the movement trajectory 21 is plotted on the YZ plane, a graph such as that shown in FIG. 9 is obtained.

[0034] In step S50, the computer 20 analyzes the trajectory of the movement of the measurement position 10A derived in the immediately preceding step S40. As a result, the computer 20 derives the direction of the movement of the measurement position 10A on the YZ plane (hereinafter, in the description of the first embodiment, this will also be simply referred to as the "movement direction") in the form of a function of time. The computer 20 then advances the process to step S60.

[0035] In this embodiment, the movement direction is not plotted on a graph, but when the movement direction is represented as a vector 21A starting from the measurement position 10A, a graph such as that shown in FIG. 10 is obtained. In this figure, the magnitude of each vector 21A corresponds to the speed of movement at the measurement position 10A. The cross section of the deck shown in FIG. 10 is displayed for the sake of convenience to make the direction of each vector 21A easier to understand, and does not represent the exact position of the measurement position 10A. When the movement direction is represented as the time change in the rise angle from the direction perpendicular to the bridge axis (Y-axis direction), a graph such as that shown in FIG. 11 is obtained.

[0036] In step S60, the computer 20 picks up a time range in the displacement time history waveform in which the influence of the movement due to the bending vibration mode is determined to be dominant in light of a predetermined criterion. Then, the computer 20 advances the process to step S70. Here, step S60 corresponds to the "pickup step" in this disclosure, and the function realized by the computer 20 to execute step S60 corresponds to the "pickup function" in this disclosure.

[0037] In this embodiment, the computer 20 picks up a time range in which the condition "the magnitude of the rise angle of the motion direction derived from the displacement time history waveform with respect to the direction perpendicular to the bridge axis (Y-axis direction) is 85° or more" is satisfied as a time range in which it is determined that the influence of the motion due to the bending vibration mode is dominant. This pick-up condition is equivalent to the condition "when the motion direction is expressed in the form of a vector 21A with the measurement position 10A as the starting point, this vector 21A falls within an angle range 21B in which the inclination angle with respect to the Z-axis direction (up and down direction) is ±5° (see FIG. 10)."

[0038] In step S60, the computer 20 picks up a time range from the displacement time history waveform that is in the form of a discrete function of time. Therefore, the time range picked up by the computer 20 is a set of a plurality of discrete time ranges.

[0039] In step S70, the computer 20 derives, from the observation data acquired in the most recent execution of step S20, a non-stationary amplitude spectrum of the vibration of the bridge 10 as viewed in the Z-axis direction, in which the bending vibration mode movement of the bridge 10 appears (including the bending mode component of the bridge 10). This non-stationary amplitude spectrum is defined in the form of a two-variable function with time and vibration frequency as variables. The computer 20 then advances the process to step S80. Here, step S70 corresponds to the "non-stationary amplitude spectrum derivation step" in this disclosure, and the function realized by the computer 20 to execute step S70 corresponds to the "non-stationary amplitude spectrum derivation function" in this disclosure.

[0040] In this embodiment, the derived unsteady amplitude spectrum is not plotted on a graph or the like. However, when this unsteady amplitude spectrum is plotted on a spectrogram, a spectrogram such as that shown in FIG. 12 is obtained.

[0041] In step S80, the computer 20 extracts spectral characteristics from the unsteady amplitude spectrum derived in the immediately preceding step S70, and estimates the bending vibration mode of the bridge 10 (see FIG. 1). Then, the computer 20 ends the process.

[0042] Here, the spectral characteristics extracted from the non-stationary amplitude spectrum in step S80 are the spectral characteristics corresponding to the time range picked up in the most recently executed step S60 (pickup step). Therefore, step S80 corresponds to the "estimation step" in this disclosure, and the function realized by computer 20 to execute step S80 corresponds to the "estimation function" in this disclosure.

[0043] In this embodiment, the estimation step is executed in the form of a subroutine shown in Fig. 4. In this subroutine, the computer 20 first executes step S110.

[0044] In step S110, computer 20 performs initial settings required to execute each step described below. This initial settings includes a process in which computer 20 reads order information of the vibration mode of bending to be estimated (e.g., 1st, 3rd, and 5th orders) from recording medium 20B. Then, computer 20 advances the process to step S120.

[0045] In step S120, the computer 20 acquires the multiple time ranges picked up in the most recently executed step S60. The computer 20 also acquires the unsteady amplitude spectrum derived in the most recently executed step S70. Next, the computer 20 extracts multiple spectral characteristics corresponding to the multiple time ranges from the acquired unsteady amplitude spectrum. The computer 20 then advances the process to step S130.

[0046] Here, the time ranges acquired by the computer 20 are, for example, a plurality of time ranges corresponding to the times of the vectors 21A circled in Fig. 11. These time ranges are, for example, a plurality of time ranges included in the time period 21C or the time period 21D in Fig. 12. Moreover, the spectral characteristic extracted in step S120 is a time cross section obtained by cutting the unsteady amplitude spectrum by one of a plurality of discrete time ranges, and is defined in the form of a one-variable function with the vibration frequency as a variable.

[0047] In step S130, the computer 20 smoothes the multiple spectral characteristics extracted in the immediately preceding step S120. In this embodiment, the computer 20 smoothes the spectral characteristics by stacking the multiple extracted spectral characteristics into one. The computer 20 then advances the process to step S140.

[0048] In this embodiment, the smoothed spectral characteristic is not plotted on a graph. However, if this spectral characteristic is plotted on a graph, a graph of a spectral characteristic 30 such as that shown in FIG. 13 is obtained.

[0049] In step S140, the computer 20 analyzes the spectral characteristics smoothed in the immediately preceding step S130, and searches for multiple peaks (maximum values) from the spectral characteristics. The computer 20 then advances the process to step S150. Here, step S140 corresponds to a "searching step" in this disclosure. Also, the function realized by the computer 20 to execute step S140 is also referred to as a "searching function" in this disclosure.

[0050] In step S150, the computer 20 estimates that the maximum peak among the peaks found in the immediately preceding step S140 (realization of the "search function") is the first mode among the bending vibration modes of the bridge 10. In other words, in step S80 (estimation step), the computer 20 estimates the first mode among the bending vibration modes of the bridge 10 from the peak of the spectral characteristics obtained by smoothing each data of the spectral characteristics corresponding to the multiple time ranges picked up in step S60 (pickup step). Then, the computer 20 advances the process to step S160. Here, step S150 corresponds to the "first mode estimation step" in this disclosure. Also, the function realized by the computer 20 to execute step S150 is also referred to as the "first mode estimation function" in this disclosure.

[0051] In step S160, the computer 20 acquires the vibration frequency corresponding to the peak estimated to be the first mode in the immediately preceding step S150 (i.e., the dominant vibration frequency of the first mode). Then, the computer 20 advances the process to step S170.

[0052] In step S170, the computer 20 repeatedly executes a series of processes from step S180 to step S200, which will be described later. This series of processes is repeatedly executed while changing the value of the counter variable a within the range of the degree information read by the computer in the initial setting in step S110, until there is no more to which the value of the counter variable a should be changed. When there is no more to which the value of the counter variable a should be changed, the computer 20 stops the repeated process of step S170 and proceeds to step S210.

[0053] In step S180, the computer 20 changes the value of the counter variable a within the range of the degree information read by the computer in the initial setting in step S110, excluding "1st degree." At this time, the computer 20 excludes the value of the degree information used as the counter variable a in the repeated processing of step S170 from the change destination of the counter variable a. As a result, in the repeated processing of step S170, processing is performed once for each degree included in the degree information, excluding "1st degree." Furthermore, the counter variable a is always a natural number equal to or greater than 2.

[0054] After executing step S180, the computer 20 advances the process to step S190. However, if there is no value of degree information to which the counter variable a should be changed when executing step S180 (for example, if all values ​​of degree information are excluded from the values ​​to which the counter variable a should be changed), the computer 20 stops the repeated process of step S170 and advances the process to step S210.

[0055] In step S190, the computer 20 extracts a peak having a vibration frequency a times the peak estimated to be the first mode by the most recently executed step S150 (realization of the "first mode estimation function") from among the multiple peaks found in the most recently executed step S140 (searching step). Then, the computer 20 estimates the extracted peak to be the a-th mode of the bending vibration modes of the bridge 10. Here, "a" is the order of the counter variable a changed in the most recently executed step S180. Therefore, step S190 corresponds to the "a-th mode estimation step" in this disclosure. Also, the function realized by the computer 20 to execute step S190 is also referred to as the "a-th mode estimation function" in this disclosure. After executing step S190, the computer 20 advances the process to step S200.

[0056] In step S200, the computer 20 acquires the vibration frequency corresponding to the peak estimated to be the a-th mode in the immediately preceding step S190 (i.e., the dominant vibration frequency of the a-th mode). Then, the computer 20 advances the process to step S180.

[0057] In step S210, the computer 20 outputs the estimation results of each bending vibration mode of the bridge 10 obtained by repeating the above-mentioned first mode estimation step (step S150) and a-order mode estimation step (step S190) to the recording medium 20B in a state in which the estimation results are linked to the vibration frequencies corresponding to the peaks of each vibration mode. Then, the computer 20 ends the processing of the subroutine of the estimation step (step S80).

[0058] Here, an example of estimation of the bending vibration mode of the bridge 10, which is realized by repeating the above-mentioned first mode estimation step and a-th mode estimation step, will be described with reference to Fig. 13. In this estimation example, the orders of the bending vibration mode to be estimated are three orders, namely, first order, third order, and fifth order.

[0059] In step S140 (realization of the "search function"), the computer 20 searches for a plurality of peaks (maximum values) of the smoothed spectral characteristic 30. Some of the peaks found may correspond to the bending vibration mode to be estimated, while others may correspond to noise unrelated to the bending vibration mode.

[0060] In step S150 (implementation of the “first mode estimation function”), the computer 20 estimates that the maximum peak 31 among the peaks found is the first mode of the bending vibration modes of the bridge 10.

[0061] In step S160, the computer 20 obtains the vibration frequency 31A corresponding to the peak 31 estimated to be the first mode.

[0062] In the series of processes executed for the first time in step S170, the computer 20 changes the value of the counter variable a to either "third order" or "quintic order" (step S180, hereinafter, the description will be given assuming that the value has been changed to "third order"). Next, the computer 20 estimates that, among the multiple peaks found in the most recently executed step S140 (searching step), the peak 32 having a vibration frequency 32A that is three times the vibration frequency 31A of the peak 31 is the third mode among the bending vibration modes of the bridge 10. Then, the computer 20 acquires the vibration frequency 32A corresponding to the peak 32 (step S200).

[0063] In the series of processes executed for the second time in step S170, the computer 20 changes the value of the counter variable a to a value ("5th order") that was not selected the first time (step S180). Next, the computer 20 estimates that, among the multiple peaks found in the most recently executed step S140 (searching step), the peak 33 having a vibration frequency 33A that is five times the vibration frequency 31A of the peak 31 is the 5th order mode of the bending vibration modes of the bridge 10. Then, the computer 20 acquires the vibration frequency 33A corresponding to the peak 33 (step S200).

[0064] When the third series of processes is executed in step S170, the value of the order information to which the counter variable a should be changed does not exist because it has been excluded. Therefore, the computer 20 stops the repeated process of step S170. Then, the computer 20 outputs the estimation results of each bending vibration mode (1st, 3rd, and 5th) of the bridge 10 to the recording medium 20B in a state in which the peaks 31, 32, and 33 of each vibration mode are linked to the corresponding vibration frequencies 31A, 32A, and 33A, respectively.

[0065] According to the above-described first embodiment, it is possible to pick up a time range when the influence of the motion of the vibration mode to be estimated is dominant (when the bridge 10 is moving roughly in line with the motion of the vibration mode) from the displacement time history waveform of the vibration observed at one measurement position 10A. Then, it is possible to estimate the bending vibration mode in the vibration of the bridge 10 from the spectral characteristics of the non-stationary amplitude spectrum of the vibration corresponding to this time range.

[0066] Moreover, according to the first embodiment, the measurement position 10A is a location that can be regarded as a node in the even-order (secondary, fourth, etc.) bending vibration mode of the bridge 10. Therefore, by removing the influence of the movement due to the even-order vibration mode from the displacement time history waveform of the vibration, this displacement time history waveform can be made simpler. Then, in the pick-up step (step S60), the accuracy of picking up the time range in which the movement of the vibration mode is dominant can be improved, and the accuracy of estimating the vibration mode can be improved.

[0067] Furthermore, according to the first embodiment, multiple timings at which the influence of the movement of the vibration mode to be estimated is prominent (timings at which the bridge 10 moves in accordance with the movement of the vibration mode) are picked up, and a peak of the spectral characteristics common to these timings is found. This peak of the spectral characteristics is highly likely to be a peak of the spectral characteristics corresponding to a major movement among the movements of the vibration mode to be estimated. On the other hand, the bending vibration mode in the vibration of a structure generally appears in a form that is most easily understood as its first mode. Therefore, by estimating the first mode of the vibration mode from the peak of the spectral characteristics, this first mode can be estimated with higher accuracy.

[0068] Generally, bending vibration modes have the property that the lower the order of the mode, the larger the peak of the spectrum characteristics of the corresponding unsteady amplitude spectrum. In addition, the a-th mode, where a is a natural number of 2 or more, has the property that the vibration frequency of the corresponding peak is a times the vibration frequency of the peak corresponding to the 1st mode.

[0069] According to the first embodiment, the maximum peak (e.g., peak 31 in FIG. 13) in the spectrum characteristic of the unsteady amplitude spectrum is estimated to be the lowest order (first order) vibration mode, so that the first order mode of the vibration mode can be estimated with high probability. Furthermore, based on the estimated first order mode, the second order or higher vibration modes can be estimated with higher accuracy.

[0070] Furthermore, according to the recording medium 20B, the computer 20 that has read the program from the recording medium 20B can realize a function of picking out a time range when the influence of the movement of the vibration mode to be estimated is dominant (when the bridge 10 is moving roughly in line with the movement of the vibration mode) from the displacement time history waveform of the vibration observed at one measurement position 10A. Then, the computer 20 can realize a function of estimating the bending vibration mode in the vibration of the bridge 10 from the spectral characteristics of the non-stationary amplitude spectrum of the vibration that corresponds to this time range.

[0071] Second embodiment Next, a method for estimating the bending stiffness of a structure according to a second embodiment of the present disclosure (hereinafter, also simply referred to as the "second embodiment") will be described. The second embodiment is a method in which an engineer (not shown) estimates the bending stiffness of a point (measurement position 10A in FIG. 1) that is the center of a deck 10B of a bridge 10 (see FIG. 1), which is a structure, by using the output obtained by the above-mentioned first embodiment.

[0072] In the second embodiment, an engineer first obtains the estimated results of each bending vibration mode of the bridge 10, which are output to the recording medium 20B by the first embodiment. The estimated results are linked to the vibration frequencies corresponding to the peaks of each bending vibration mode (i.e., the predominant vibration frequencies of each bending vibration mode. Hereinafter, these will also be referred to as "predominant vibration frequencies f"). The engineer obtains the predominant vibration frequencies f of each bending vibration mode, and solves an equation equivalent to the following (Equation 1) for each obtained predominant vibration frequency f, thereby obtaining an estimated result of the bending stiffness EI of each bending vibration mode.

[0073]

number

[0074] The above (Equation 1) is an equation for determining the bending stiffness EI of each bending vibration mode of a simple beam, in the form of a function of the predominant vibration frequency f, by regarding deck 10B (see Figure 1) of bridge 10 as a simple beam with uniform load distribution. In (Equation 1), π is the circular constant, g is the acceleration of gravity, W is the total load applied to deck 10B (see Figure 1), and L is the span length of bridge 10. This L is equal to twice the distance from the support (not shown) of pier 10C to measurement position 10A in bridge 10 shown in Figure 1.

[0075] Each dominant vibration frequency f acquired in the second embodiment includes a peak vibration frequency estimated to be the a-order mode in the a-order mode estimation step (step S190 in FIG. 4) in the first embodiment. Therefore, the second embodiment can be said to be a method for estimating the bending stiffness of a structure, which acquires the peak vibration frequency estimated to be the a-order mode using the first embodiment and estimates the bending stiffness EI of the structure (bridge 10) based on the acquired vibration frequency.

[0076] Generally, the bending vibration mode of a structure has a property that the higher the order (the higher the vibration frequency), the larger the fluctuation amount of the vibration frequency due to the change in the bending rigidity of the structure. In contrast, according to the second embodiment, the bending rigidity of the structure is estimated based on the vibration frequency of a vibration mode of a higher order than the first order, so that the bending rigidity can be estimated with higher accuracy.

[0077] Furthermore, according to the second embodiment, an engineer can repeatedly estimate the bending stiffness at the measurement position 10A (see FIG. 1) of the bridge 10 in a non-destructive manner. Therefore, the engineer can derive the change in bending stiffness over time at the measurement position 10A (see FIG. 14) by repeating the estimation of bending stiffness according to the second embodiment at intervals. Furthermore, when the bending stiffness at the measurement position 10A falls below a predetermined threshold value (see the circle in FIG. 14), the engineer can determine that deterioration of the bridge 10 has progressed and check the details of the deterioration.

[0078] Third embodiment Next, a method for estimating a bending vibration mode according to a third embodiment of the present disclosure (hereinafter, also simply referred to as the "third embodiment") will be described with reference mainly to Figs. 15 to 28. The third embodiment is a method for estimating a bending vibration mode in a seven-story building 40 (see Fig. 15) from observation data of constant microtremors (weak vibrations that constantly act on a structure) observed at one measurement position 40A set on the roof of the building 40. The building 40 of this embodiment is a structure that can be approximated by a sway rocking model. In addition, the measurement position 40A is set at a point that is a corner part on the roof of the building 40. This point can be regarded as a node in the even-order (secondary, fourth, etc.) bending vibration mode of the building 40, but is a location where the influence of the component in the torsional direction (see Fig. 15) in the vibration of the building 40 cannot be ignored.

[0079] In the third embodiment, an accelerometer and a computer that perform the same processing as the accelerometer 20A and the computer 20 (see FIG. 2) used in the first embodiment are used. However, in the third embodiment, the computer performs processing while displaying various data processed by the program in graphs as the program proceeds with processing.

[0080] In the third embodiment, the accelerometer is installed so that its Y-axis direction is the horizontal direction along the tangent direction of the building 40, its X-axis direction is the horizontal direction perpendicular to the tangent direction of the building 40, and its Z-axis direction is the up-down direction. Therefore, the bending vibration mode of the building 40 can appear as a movement of the measurement position 40A in the XY plane.

[0081] In the third embodiment, the computer derives a displacement time history waveform (see FIG. 18) of the vibration of the building 40 viewed in the X-axis direction from the observation data in the X-axis direction (acceleration waveform, see FIG. 16) acquired from the accelerometer. The computer also derives a displacement time history waveform (see FIG. 19) of the vibration of the building 40 viewed in the Y-axis direction from the observation data in the Y-axis direction (acceleration waveform, see FIG. 17) acquired from the accelerometer. These processes correspond to the "displacement time history waveform derivation step" or "realization of the displacement time history waveform derivation function" in this disclosure.

[0082] Next, the computer derives a trajectory 51 (see FIG. 20) of the movement of measurement position 40A on the XY plane from the displacement time history waveforms of the vibrations viewed in the X-axis and Y-axis directions. The computer also derives the direction of movement of measurement position 40A on the XY plane (hereinafter, in the description of the third embodiment, this will also be simply referred to as the "movement direction") in the form of a function of time. At this time, the computer expresses the direction of movement as a graph of the time change in the rise angle from the X-axis direction (see FIG. 21) and a graph of the time change in the rise angle from the Y-axis direction (see FIG. 22).

[0083] Next, the computer picks up a time range in the displacement time history waveform in which the influence of the motion due to the vibration mode of bending in the Y-axis direction is determined to be predominant in light of a predetermined criterion. The computer also picks up a time range in the displacement time history waveform in which the influence of the motion due to the vibration mode of bending in the X-axis direction is determined to be predominant in light of a predetermined criterion. These processes correspond to the "pickup step" or "realization of the pick-up function" in this disclosure.

[0084] In this embodiment, the computer picks up time range 51A (see FIG. 21) where the condition "the magnitude of the rise angle of the motion direction derived from the displacement time history waveform with respect to the X-axis direction is 88° or more" is satisfied as a time range determined to be dominated by the influence of motion due to the bending vibration mode in the Y-axis direction. Also, the computer picks up time range 51B (see FIG. 22) where the condition "the magnitude of the rise angle of the motion direction derived from the displacement time history waveform with respect to the Y-axis direction is 88° or more" is satisfied as a time range determined to be dominated by the influence of motion due to the bending vibration mode in the X-axis direction.

[0085] The computer also derives the unsteady amplitude spectrum (see spectrogram in FIG. 23) of the vibration of the building 40 seen in the X-axis direction from the observation data in the X-axis direction (acceleration waveform, see FIG. 16) acquired from the accelerometer. The computer also derives the unsteady amplitude spectrum (see spectrogram in FIG. 24) of the vibration of the building 40 seen in the Y-axis direction from the observation data in the Y-axis direction (acceleration waveform, see FIG. 17) acquired from the accelerometer. These unsteady amplitude spectra are determined in the form of a two-variable function with time and vibration frequency as variables. Here, the X-axis direction and the Y-axis direction are directions in which the movement of the bending vibration mode appears in the vibration of the building 40, respectively. Therefore, each of the above processes corresponds to the "unsteady amplitude spectrum derivation step" or the "realization of unsteady amplitude spectrum derivation function" in this disclosure.

[0086] The computer also analyzes the spectral characteristics 60 (see FIG. 25) extracted from the unsteady amplitude spectrum of the vibration of building 40 seen in the X-axis direction (see the spectrogram in FIG. 23) to estimate the bending vibration mode in the X-axis direction of building 40 (see FIG. 15). The computer also analyzes the spectral characteristics 70 (see FIG. 26) extracted from the unsteady amplitude spectrum of the vibration of building 40 seen in the Y-axis direction (see the spectrogram in FIG. 24) to estimate the bending vibration mode in the Y-axis direction of building 40 (see FIG. 15). These processes correspond to the "estimation step" or "realization of the estimation function" in this disclosure.

[0087] In this embodiment, the computer graphs the spectrum characteristic 60 and the spectrum characteristic 70 in a normalized state. The computer also estimates the first mode (see peak 61 and vibration frequency 61A in FIG. 25) and the third mode (see peak 62 and vibration frequency 62A in FIG. 25) of the vibration modes of bending in the X-axis direction of the building 40 from the spectrum characteristic 60. The computer also estimates the first mode (see peak 71 and vibration frequency 71A in FIG. 26) and the third mode (see peak 72 and vibration frequency 72A in FIG. 26) of the vibration modes of bending in the Y-axis direction of the building 40 from the spectrum characteristic 60.

[0088] The present inventor verified the usefulness of the above-mentioned third embodiment by deriving Fourier amplitude spectra 80, 90 for the acceleration waveform in the X-axis direction (see FIG. 16) and the acceleration waveform in the Y-axis direction (see FIG. 17) used in the third embodiment, and comparing these Fourier amplitude spectra 80, 90 with the spectral characteristics 60, 70.

[0089] The graph of the spectrum characteristic 60 (see FIG. 25) is compared with the graph of the Fourier amplitude spectrum 80 (see FIG. 27) derived from the acceleration waveform in the X-axis direction. Three peaks 81, 82, and 83 can be confirmed from the graph of the Fourier amplitude spectrum 80. The peak 81 exists at the same vibration frequency 81A as the vibration frequency 61A of the peak 61 estimated to be the first-order bending mode in the third embodiment. The peak 82 exists at the same vibration frequency 82A as the vibration frequency 62A of the peak 62 estimated to be the third-order bending mode in the third embodiment. The peak 83 exists at a vibration frequency 83A that is larger than the vibration frequency 81A and smaller than both twice the vibration frequency 81A and the vibration frequency 82A. In the graph of the spectrum characteristic 60, a peak 63 smaller than the peak 61 exists at the vibration frequency 63A equal to the vibration frequency 83A. The peaks 63 and 83 are estimated to be the peaks of vibration in the torsional direction (see FIG. 15) of the building 40.

[0090] Here, peak 82 is clearly smaller than peak 83. On the other hand, peak 62 is approximately the same as or slightly smaller than peak 63. This means that the accuracy with which the third embodiment distinguishes between peak 62 of the third bending mode and peak 63 of the torsional vibration can be higher than the accuracy with which peak 82 of the third bending mode and peak 83 of the torsional vibration can be distinguished from the Fourier amplitude spectrum 80 of the acceleration waveform.

[0091] Similarly, the graph of the spectrum characteristic 70 (see FIG. 26) is compared with the graph of the Fourier amplitude spectrum 90 (see FIG. 29) derived from the acceleration waveform in the Y-axis direction. Three peaks 91, 92, and 93 can be confirmed from the graph of the Fourier amplitude spectrum 90. The peak 91 exists at the same vibration frequency 91A as the vibration frequency 71A of the peak 71 estimated to be the first mode of bending in the third embodiment. The peak 92 exists at the same vibration frequency 92A as the vibration frequency 72A of the peak 72 estimated to be the third mode of bending in the third embodiment. The peak 93 exists at a vibration frequency 93A that is larger than the vibration frequency 91A and smaller than both twice the vibration frequency 91A and the vibration frequency 92A. In the graph of the spectrum characteristic 70, the peak 73, which is smaller than both the peak 71 and the peak 72, exists at the vibration frequency 73A that is equal to the vibration frequency 93A. These peaks 73 and 93 are estimated to be the peaks of vibration in the torsional direction (see FIG. 15) in building 40.

[0092] Here, peak 92 is clearly smaller than peak 91. On the other hand, peak 72 is approximately the same as or slightly smaller than peak 71. This means that the accuracy of finding peak 62 of the third bending mode by the third embodiment can be higher than the accuracy of finding peak 92 of the third bending mode from Fourier amplitude spectrum 90 of the acceleration waveform.

[0093] Other embodiments The above describes the embodiments for carrying out the present disclosure by using the first to third embodiments described above. However, it is clear to those skilled in the art that various substitutions, modifications, and changes are possible without departing from the purpose of the present disclosure. In other words, the embodiments for carrying out the present disclosure may include all substitutions, modifications, and changes that do not depart from the spirit and purpose of the claims attached to this specification. For example, the following various embodiments can be carried out as embodiments for carrying out the present disclosure.

[0094] (1) In the present disclosure, the observation device for observing vibrations at points on a structure is not limited to a triaxial accelerometer. In other words, any observation device capable of acquiring vibration observation data, such as a speedometer for observing the speed of vibration or a displacement meter for observing the displacement of vibration, can be used as the observation device. In addition, the observation device may be any device capable of observing vibrations on at least two axes, and may be, for example, an assembly combining multiple biaxial observation devices or multiple uniaxial observation devices.

[0095] (2) In the first and third embodiments, the vibration mode of bending in the axial direction where the observation device observes the vibration is estimated. However, in the present disclosure, for example, a method can be adopted in which waveform data of vibrations viewed in two appropriately selected directions within a plane is derived from three-axis waveform data observed by the observation device, and each step is executed by regarding the derived waveform data of vibration as observed data of vibration. This method can eliminate the need to install the observation device in accordance with the coordinate plane on which the movement of the vibration mode appears.

[0096] (3) In the present disclosure, the points on a structure at which vibrations are observed are not limited to locations that can be considered nodes in even vibration modes, but may be set at any appropriately selected location.

[0097] (4) In the present disclosure, the vibration observation data for estimating the bending vibration mode is not limited to observation data of microtremors, but may be any appropriately selected vibration observation data, such as observation data of natural earthquakes that have occurred in the past or observation data of artificially generated earthquakes.

[0098] (5) In the present disclosure, the method of smoothing the multiple data picked up in the pick-up step is not limited to the stacking described above, and can be modified in various ways, for example, by simple averaging or by applying an appropriately selected window function. [Explanation of symbols]

[0099] 10 Bridge (Structure) 10A Measurement position (point) 10B floor slab 10C Bridge Pier 10D roadside strip 20 Computers 20A accelerometer 20B Recording medium 21 Trajectory 21A Vector 21B Angle range 21C Time Zone 21D Time Zone 30 Spectral characteristics 31 Peak 31A Vibration Frequency 32 Peak 32A Vibration Frequency 33 Peak 33A Vibration Frequency 40 Building (Structure) 40A Measurement position (point) 51 Trajectory 51A Time Range 51B Time range 60 Spectral Characteristics 61 Peak 61A Vibration Frequency 62 Peak 62A Vibration Frequency 63 Peak 63A Vibration Frequency 70 Spectral Characteristics 71 Peak 71A Vibration Frequency 72 Peak 72A Vibration Frequency 73 Peak 73A Vibration Frequency 80 Fourier Magnitude Spectrum 81 Peak 81A Vibration Frequency 82 Peak 82A Vibration Frequency 83 Peak 83A Vibration Frequency 90 Fourier Magnitude Spectrum 91 Peak 91A Vibration Frequency 92 Peak 92A Vibration Frequency 93 Peak 93A Vibration Frequency

Claims

1. A method for estimating a bending vibration mode of a structure from observation data of the vibration observed at a point set on the structure, comprising: a displacement time history waveform deriving step of deriving, from the observation data, a displacement time history waveform of the vibration as viewed in two directions in a coordinate plane on which the motion of the vibration mode appears; A picking step of picking out a time range in the displacement time history waveform in which the influence of the movement is determined to be prominent in light of a predetermined criterion; a non-stationary amplitude spectrum deriving step of deriving a non-stationary amplitude spectrum of the vibration seen in a direction in which the movement occurs; an estimation step of estimating the vibration mode from data of spectral characteristics corresponding to the time range picked up in the picking step in the unsteady amplitude spectrum; It has A method for estimating bending vibration modes.

2. 2. A method for estimating bending vibration modes according to claim 1, comprising the steps of: performing the displacement time history waveform deriving step by using the observation data observed at locations that can be regarded as nodes in the vibration modes of even orders as the points; A method for estimating bending vibration modes.

3. A method for estimating a bending vibration mode according to claim 1 or 2, comprising: In the picking step, a plurality of the time ranges are picked up; In the estimation step, a first mode of the vibration modes is estimated from a peak of the spectrum characteristic obtained by smoothing each of the data corresponding to the plurality of time ranges picked up in the picking step. A method for estimating bending vibration modes.

4. A method for estimating bending vibration modes according to claim 3, comprising the steps of: The estimation step: a locating step of locating a plurality of said peaks from said spectral characteristics; a first mode estimation step of estimating the maximum peak among the peaks found in the finding step as the first mode; an a-th mode estimating step of estimating, among the peaks found in the finding step, a peak having a vibration frequency a times the peak estimated to be the first mode, as an a-th mode among the vibration modes, where a is a natural number equal to or greater than 2; It has A method for estimating bending vibration modes.

5. Obtaining the vibration frequency of the peak estimated to be the a-th order mode using the method for estimating the bending vibration mode according to claim 4; Estimating the bending stiffness of the structure based on the acquired vibration frequency; A method for estimating the bending stiffness of a structure.

6. A program for causing a computer to realize a function of estimating a bending vibration mode in vibration of a structure from observation data of the vibration observed at one point set on the structure, the program comprising: a displacement time history waveform deriving function for deriving, from the observation data, a displacement time history waveform of the vibration as viewed in two directions in a coordinate plane on which the motion of the vibration mode appears; A pick-up function for picking up a time range in which the influence of the movement is determined to be prominent in the displacement time history waveform in light of a predetermined criterion; A non-stationary amplitude spectrum deriving function for deriving a non-stationary amplitude spectrum of the vibration seen in a direction in which the movement occurs; an estimation function for estimating the vibration mode from data of a spectrum characteristic corresponding to a time range picked up by the pickup function in the unsteady amplitude spectrum; The computer is caused to realize the above. program.

7. A computer-readable recording medium having a program recorded thereon for causing a computer to realize a function of estimating a bending vibration mode in vibration of a structure from observation data of the vibration observed at one point set on the structure, a displacement time history waveform deriving function for deriving, from the observation data, a displacement time history waveform of the vibration as viewed in two directions in a coordinate plane on which the motion of the vibration mode appears; A pick-up function for picking up a time range in which the influence of the movement is determined to be prominent in the displacement time history waveform in light of a predetermined criterion; A non-stationary amplitude spectrum deriving function for deriving a non-stationary amplitude spectrum of the vibration seen in a direction in which the movement occurs; an estimation function for estimating the vibration mode from data of a spectrum characteristic corresponding to a time range picked up by the pickup function in the unsteady amplitude spectrum; A program for causing the computer to realize the above is recorded. A computer-readable recording medium.

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