Calculation method of tension of linear body

Abstract

To provide a calculation method of tension of a linear body without affected by modeling errors of a damper.SOLUTION: A calculation method of tension of a linear body includes: detecting vibrations at three or more points on a cable 120 to acquire mode shape measurement values at the three or more points in a plurality of vibration modes; deriving mode shape theoretical values at the three or more points on the basis of a vibration equation for the cable 120 under tension and a boundary condition of the damper 110 being placed; selecting at least two sets of any two points from the three or more points; calculating a ratio of the mode shape measurement values corresponding to the two selected points in each of the two sets and calculating the ratio of the mode shape theoretical values corresponding to any two selected points in each of the two sets; and calculating the cable tension using a constraint condition that the ratio of the mode shape measurement values is equal to the ratio of the mode shape theoretical values in each set.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to a method for calculating the tension of a linear body.

Background Art

[0002] Conventionally, as disclosed in Patent Documents 1 and 2 below, a method for calculating the tension generated in a linear body provided with a damper is known. In the tension calculation methods disclosed in these documents, the following three conditions are given to a model of a beam in a state where a damper is installed and tension is applied. (1) Boundary conditions at both ends of the beam (deflection = 0, bending moment = 0) (2) Continuity conditions of the beam at the damper installation position (deflection, deflection angle, bending moment) (3) Equilibrium condition of forces at the damper installation position Then, using the total eight boundary conditions obtained from these three conditions, a constraint equation for estimating the tension and the like from the natural frequency of the beam is derived.

Prior Art Documents

Patent Documents

[0003]

Patent Document 1

Patent Document 2

Summary of the Invention

Problems to be Solved by the Invention

[0004] In Patent Documents 1 and 2, by taking into account the boundary conditions at the damper installation position, it has become possible to calculate the tension generated in the linear body even when a damper is installed. However, since a complex stiffness model of the damper is used as the equilibrium condition (3), there is a problem that if this model does not accurately reflect the characteristics of the actual damper, the influence will appear in the tension calculation result.

[0005] Therefore, the present invention has been made in view of the above prior art, and an object thereof is to provide a method for calculating the tension of a linear body that is not affected by the modeling error of a damper.

Means for Solving the Problems

[0006] To achieve the above object, a method for calculating the tension of a linear body according to the present invention is a method for calculating the tension of a linear body to which a damper is attached, the method including detecting vibrations at any three or more points on the linear body, obtaining mode shape measurement values at the three or more points in a plurality of vibration modes based on the detected vibrations, deriving mode shape theoretical values at the three or more points on the linear body in a plurality of vibration modes based on a vibration equation representing the relationship between the displacement, the tension, and the bending rigidity of the linear body under tension and boundary conditions representing that the damper is disposed on the linear body, selecting at least two sets of any two points from the three or more points, calculating the ratio of the mode shape measurement values in the plurality of vibration modes corresponding to the selected any two points for each of the two sets, calculating the ratio of the mode shape theoretical values in the plurality of vibration modes corresponding to the selected any two points for each of the two sets, and calculating the tension of the linear body using a constraint condition that the ratio of the mode shape measurement values and the ratio of the mode shape theoretical values are equal in each set.

[0007] In the method for calculating tension according to the present invention, vibrations at any three or more points on the linear body are detected, and from the vibration waveforms at each point, mode shape measurement values at any three or more points are obtained. Also, theoretical mode shape values at any three or more points are derived. Then, at least two sets of any two points are selected from the three or more points, and in each set, a constraint condition that the ratio of the mode shape measurement values and the ratio of the theoretical mode shape values are equal at each of the plurality of mode orders is applied. Therefore, since at least two constraint conditions are obtained, even if the attenuation coefficient of the linear body derived from the complex natural vibration frequency based on the vibration equation of the linear body is included as an unknown, it is possible to estimate the unknown. Moreover, in order to calculate the ratio of the theoretical mode shape values corresponding to any two selected points, even if the general solution of the mode function obtained from the vibration equation includes integration constants, it is possible to avoid being affected by them. Moreover, the influence of the complex natural vibration frequency based on the vibration equation can also be eliminated. For this reason, since there is no need to model the damper and use its characteristic values as constraint conditions, the tension of the linear body can be calculated without being affected by the modeling error of the damper.

[0008] In the above calculation method, when deriving the theoretical mode shape value, the theoretical mode shape value is derived by adjusting the amplitudes at the three or more points in the plurality of vibration modes so that the maximum value becomes 1, and when obtaining the mode shape measurement value, the mode shape measurement value may be obtained by adjusting the amplitudes at the three or more points in the plurality of vibration modes so that the maximum value becomes 1.

Advantages of the Invention

[0009] As described above, according to the present invention, the tension of the linear body can be calculated without being affected by the modeling error of the damper.

Brief Description of the Drawings

[0010]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Mode for Carrying Out the Invention

[0011] FIG. 1 is a schematic diagram of a measuring device 100 used for measuring the tension of a cable 120 (linear body) to which a damper 110 is attached. Using the measuring device 100, the tension of the cable 120 is calculated.

[0012] The measuring device 100 includes a pair of support portions 131 and 132 arranged at positions spaced apart left and right, and the cable 120 extends between these support portions 131 and 132. Both ends of the cable 120 are attached to the support portions 131 and 132.

[0013] A damper 110 is attached to the cable 120. A portion of the cable 120 on one side of the damper 110 is referred to as the first span 121, and a portion of the cable 120 on the other side of the damper 110 is referred to as the second span 122. The first span 121 is the length portion from the left end of the cable 120 to the damper 110. In the following description, the length of the first span 121 is represented by the symbol "l1". Also, the second span 122 is the length portion from the right end of the cable 120 to the damper 110, and is longer than the first span 121. In the following description, the length of the second span 122 is represented by the symbol "l2" (l1≦l2). The sum of the lengths of the two spans 121 and 122 is the total length of the cable 120. In the following description, the total length of the cable 120 is represented by the symbol "L".

[0014] To detect the vibration of the cable 120, three accelerometers 135, 136, and 137 are attached to the second span 122. The accelerometers 135, 136, and 137 are each configured to output a signal representing the measured acceleration. The accelerometers 135, 136, and 137 can be installed at any position on the cable 120 as long as they can detect the acceleration of the vibration of the cable 120. However, the accelerometers 135, 136, and 137 are arranged to detect the vibration of the cable 120 at different positions in the longitudinal direction on the cable 120.

[0015] A data processing unit 140 is connected to the accelerometers 135, 136, and 137, and the signals of the accelerometers 135, 136, and 137 are input to the data processing unit 140. The data processing unit 140 is configured to perform a predetermined function by executing a stored computer program. This predetermined function includes a measured value processing unit 141 and a tension calculation unit 142.

[0016] Based on the signals output from the accelerometers 135, 136, and 137, the measurement value processing unit 141 is configured to record data on the time change of the acceleration of vibration (time history response value) and perform predetermined processing on the recorded data. This predetermined processing includes obtaining the recorded data as an acceleration waveform and performing a conversion process of Fourier-transforming this waveform, and from the Fourier-transformed frequency waveform, deriving the intensity at the frequency where a peak appears as the peak intensity (Fourier amplitude) at each mode order, and a measurement value derivation process of deriving a measurement value of the mode shape from the derived Fourier amplitude.

[0017] Data regarding the theoretical value of the mode shape based on the theoretical amplitude at the positions where the accelerometers 135, 136, and 137 are installed is stored in the tension calculation unit 142. The tension calculation unit 142 calculates the tension of the cable 120 using this stored data and the measurement value of the mode shape derived in the measurement value processing unit 141.

[0018] (Tension calculation theory) Hereinafter, the theory used in the method for calculating the tension of the cable 120 will be specifically described.

[0019] As a premise for tension calculation, the cable 120 in the measuring device 100 is modeled as a one-dimensional beam with both ends supported and a tension T applied, as shown in FIG. 2. Assuming that the cross-sectional area of this beam is A, the second moment of area is I, the elastic modulus is E, and the unit volume mass is ρ, the vibration equation of the beam with the tension T acting is as shown in the following equation (1). Note that EI is the flexural rigidity of the beam. A rightward positive x-axis is taken, and the downward displacement is denoted as y.

[0020]

Equation

[0021] This equation is regarded as the vibration equation of the cable 120 considering the flexural rigidity EI.

[0022] Next, for the differential equation of formula (1), when the variable y is set as the following formula (2) and solved by the variable separation method, formula (1) can be rewritten as formula (3).

[0023]

Number

[0024]

Number

[0025] Substituting the following formula (4) into this formula (3) results in formula (5).

[0026]

Number

[0027]

Number

[0028] Solving this formula (5) for λ gives

[0029]

Number

[0030] where α and β are as follows.

[0031]

Number

[0032] In this case, the general solution of the deflection mode represented by formula (3) is as follows in formula (9).

[0033]

Number

[0034] Equation (1) is the vibration equation for the cable 120 supported at both ends, and this vibration equation can also be similarly applied to each of the first span 121 and the second span 122 that make up the cable 120. That is, by regarding the first span 121 and the second span 122 as simply supported beams respectively, the vibration equations can be obtained for each of the first span 121 and the second span 122. At that time, for example, as shown in FIG. 3, for the first span 121, a rightward coordinate axis x1 is taken, and the downward displacement is y1, and for the second span 122, a rightward coordinate axis x2 is taken, and the downward displacement is y2.

[0035] The coordinate axis x1 is set with reference to the support portion 131 on the left side of the cable 120, and the position where "x1 = 0" is the position of the left end of the cable 120 (the left end of the first span 121). Also, the position where "x1 = l1" is the position where the damper 110 is installed. The coordinate axis x2 is set with reference to the damper 110, and the position where "x2 = 0" is the position where the damper 110 is installed. Also, the position where "x2 = l2" is the position of the right end of the cable 120 (the second span 122).

[0036] In this case, since Equation (9), which is the general solution of the deflection mode, also holds for each of the first span 121 and the second span 122, it can be written as the following Equations (10) and (11) respectively. These Equations (10) and (11) are the general solutions of the mode shapes representing the vibration shapes of each span when vibrating at the circular frequency ω. Note that α and β in the equations are the same as those shown in Equations (7) and (8).

[0037]

Number

[0038] Substitute the boundary conditions into this equation. In the model of FIG. 3, in the first span 121, since it is assumed that the left end is simply supported, the following boundary conditions are set.

[0039]

Mathematics

[0040] Also, in the second span 122, since it is assumed that the right end is simply supported, the following boundary conditions are set.

[0041]

Mathematics

[0042] Also, for the position where the damper 110 is installed, the following boundary conditions indicating that the deflection, deflection angle, and curvature of the cable 120 are continuous are set.

[0043]

Mathematics

[0044] Substituting these boundary conditions into equations (10) and (11) and arranging the integration constants in the general solution around B1, the following equations (15) and (16) are obtained.

[0045]

Mathematics

[0046]

Mathematics

[0047] Thus, since seven boundary conditions are applied to eight integration constants, one integration constant remains. Therefore, as a new constraint condition, a constraint condition using the mode shape will be introduced. That is, while obtaining the theoretical value of the mode shape, the measured value of the mode shape is obtained, and a constraint condition for comparing these is set.

[0048] First, focusing on Y2 among the general solutions Y1 and Y2 of the mode shapes represented by Equations (10) and (11), and substituting Equation (16) into the integration constants A2 to D2, an equation for Y2 (also referred to as a mode function), Equation (17), is obtained. That is, Equation (17) is an equation using the tension T, bending rigidity EI, unit volume mass ρ of the cable 120, cross-sectional area A, and circular frequency ω, and represents the shape (mode shape) formed when the amplitudes at any point x2 on the second span 122 (cable 120) are arranged in the cable length direction. However, in deriving this Equation (17), since the sin and cos functions containing α and the hyperbolic functions containing β may diverge to infinity and cause a decrease in accuracy, equation transformation is performed.

[0049] [Number]

[0050] Note that sn(αL) in Equation (17) is expressed as follows. The same applies to sn(αl1), sn(αl2), etc.

[0051] [Number]

[0052] Using this Equation (17), the absolute values of Y2(p1) and Y2(p2) at the two points where x2 = p1 and p2, respectively, at each mode order are adjusted by the following Equations (20) and (21) so that the maximum value becomes 1 (also referred to as normalization). The values defined by these Equations (20) and (21) are respectively taken as the theoretical values (mode shape theoretical values) for the mode shape. That is, in the present embodiment, Y2 representing the amplitudes at x2 = p1 and p2 is adjusted so that the maximum value becomes 1 and is taken as the mode shape theoretical value. Note that the reason for performing such normalization is that, instead of Equation (29) described later, Equation (31) based on Equation (30) is used as the objective function to find the minimum value. Also, the reason for considering the ratio of Y2(p1) and Y2(p2) at the two points is to eliminate the integration constants, as will be described later.

[0053] [Number]

[0054] Here, i is the mode order, and t is a subscript indicating a theoretical value.

[0055] The theoretical value Φ of the mode shape i1 t , Φ i2 t includes the integration constant B1, but if we take the ratio of Φ i1 t to Φ i2 t , the integration constant B1 can be eliminated. Therefore, we will use the ratio of Φ i1 t to Φ i2 t This ratio can be said to be the ratio of the theoretical values of the mode shapes between two points p1 and p2.

[0056] On the other hand, consider the case of obtaining the acceleration waveforms of the cable 120 at two points where x2 = p1 and p2. In this case, the obtained waveforms are Fourier-transformed, and the frequencies at which the Fourier amplitudes are dominant at each of the two points are read in ascending order, and are taken as the first natural frequency, the second natural frequency, ···. Then, by reading the values of the Fourier amplitudes at each natural frequency, we obtain the Fourier amplitudes at each mode order. The Fourier amplitude Y i1 m , Y i2 m corresponds to the Fourier amplitude of the i-th mode at x2 = p1 and p2 obtained from the acceleration waveform. Here, m is a subscript indicating a measured value. And by normalizing the Fourier amplitudes Y i1 m , Y i2 m we obtain the measured value of the mode shape. That is, similar to the theoretical value of the amplitude obtained from the mode function, for the mode shape Φ i1 m , Φ i2 mIt is defined by the following formulas (22) and (23) so that the maximum value becomes 1. In this embodiment, these are respectively used as mode shape measurement values.

[0057]

Equation

[0058] And for the mode shape measurement values Φ i1 m , Φ i2 m , the ratio is obtained in the same way as for the mode shape theoretical values Φ i1 t , Φ i2 t . Since the ratio of the mode shape theoretical values Φ i1 t , Φ i2 t should be equal to the ratio of the mode shape measurement values Φ i1 m , Φ i2 m , the following constraint conditions are satisfied.

[0059]

Equation

[0060] From this formula (24), the following formula (25) showing the constraint conditions is obtained. Formula (25) is a formula considering that the denominator does not become 0.

[0061]

Equation

[0062] By using this constraint condition, it becomes possible to calculate the tension of the cable 120 without modeling the damper 110. However, since the formula (25) contains T, EI, and further contains complex natural frequencies, the attenuation constant of the cable 120 is not exactly known. Therefore, it is necessary to calculate the tension with the imaginary-real ratio Hi of each mode order i as an unknown. Note that this imaginary-real ratio Hi is the ratio of the imaginary part to the real part in the complex natural frequency, which is defined by the following formula using the attenuation ratio hi.

[0063]

Number

[0064] When calculating the tension with the imaginary-real ratio Hi as an unknown, if we use the mode order i up to the nth order, the number of unknowns becomes 2 + n, so a larger number of constraint equations are required. As described above, when the number of measurement points of the Fourier amplitude (mode shape) is 2, the number of constraint equations is the number of mode orders i, that is, n, which is less than the number of unknowns (2 + n). Therefore, it is necessary to increase the number of measurement points of the Fourier amplitude. If the number of measurement points of the Fourier amplitude is 3, the number of constraint equations corresponding to the formula (25) can be made 2n. Therefore, by setting the mode order i to 2 or more, it becomes possible to estimate the unknowns. Therefore, in this embodiment, it is assumed to obtain the measured values of the Fourier amplitude at 3 or more points.

[0065] Accordingly, regarding the theoretical amplitude Y2, it is also necessary to obtain it at least at three points x2 = p1, p2, p3, so we denote them as Y2(p1), Y2(p2), and Y2(p3) respectively. Select two sets of any two points from these three points. Here, for example, at least two sets are selected, such as the first set that selects Y2(p1) and Y2(p2), and the second set that selects Y2(p1) and Y2(p3).

[0066] In each of the first group and the second group, following the formulas (20) and (21), for the absolute values of Y2(p1), Y2(p2), and Y2(p3), they are normalized so that the maximum value becomes 1, and these are respectively taken as the theoretical mode shape values Φ i1 t , Φ i2 t , Φ i3 t . Then, for the first group, the ratio between Φ i1 t and Φ i2 t is calculated, and for the second group, the ratio between Φ i1 t and Φ i3 t is calculated. The data regarding these ratios are stored in the tension calculation unit 142.

[0067] On the other hand, the measured value processing unit 141 acquires the acceleration waveforms of the cable 120 by the accelerometers 135, 136, and 137 at three points where x2 = p1, p2, p3 in order to obtain the mode shape measured values. In the measured value processing unit 141, a process of respectively performing Fourier transform on the acquired waveforms (transformation process) is carried out. Also, in the measured value processing unit 141, the intensity at the frequency where a peak appears is derived as the peak intensity (Fourier amplitude) at each mode order (amplitude derivation process).

[0068] The Fourier amplitudes Y i1 m , Y i2 m , Y i3 m will be read as absolute values, but in the amplitude derivation process, two out of these three Fourier amplitudes Y i1 m , Y i2 m , Y i3 m are selected. At this time, in the measured value processing unit 141, two Fourier amplitudes Y i1 m , Y i2 mFor the first set of selections, normalization is performed according to equations (22) and (23) such that the maximum value becomes 1, and each is stored as a measured value of the mode shape (measured value derivation process). Also, for the two Fourier amplitudes Y i1 m 、Y i3 m For the second set of selections, normalization is performed according to the following equations (27) and (28) such that the maximum value becomes 1, and each is stored as a mode shape measured value.

[0069]

Number

[0070] Then, the measured value processing unit 141 calculates the ratio of the mode shape measured value Φ i1 m and the mode shape measured value Φ i2 m for the first set, and calculates the ratio of the mode shape measured value Φ i1 m and the mode shape measured value Φ i3 m for the second set.

[0071] In both the case of the first set and the case of the second set, the ratio of the mode shape theoretical values should be equal to the ratio of the mode shape measured values, so the following equations (29) and (30) representing this constraint condition hold. Note that equation (30) is an equation considering the case where Φ i1 t 、Φ i1 m in the denominator of equation (29) may become 0.

[0072]

Number

[0073] Since this equation (30) also contains T and EI as unknowns and further contains complex natural frequencies, the attenuation constant of the cable 120 is not exactly known. However, when using the mode order i up to the second order, the number of unknowns is 2 + 2 = 4, and since the number of constraint equations (30) including the first set and the second set is 2 × 2 = 4, T and E1, which are unknowns, can be calculated using the mode order up to the second order. Note that the mode order i of the third order or higher may be used to further improve the accuracy.

[0074] Then, in order to solve the minimization problem of the equation (30), the tension calculation unit 142 calculates the tension of the cable 120 by obtaining T and EI that minimize the value of the objective function by the least squares method, using the following equation (31) based on the equation (30) as the objective function.

[0075] [Number]

[0076] Note that it is not limited to using the equation (31) which is the objective function based on the equation (30), and the objective function based on the equation (29) may be used. However, in that case, it is necessary to exclude the case where the denominator becomes zero. In this case, the Fourier amplitudes Y i1 m , Y i2 m are not used as the mode shape measurement values obtained by normalizing using the equations (22) and (23), but the Fourier amplitudes Y i1 m , Y i2 m are used as the mode shape measurement values as they are. Also, Φ i1 t , Φ i2 t are not used as the mode shape theoretical values, but the absolute values of Y2(p1) and Y2(p2) are used as the mode shape theoretical values.

[0077] (Tension Calculation Process) Here, a method for calculating the tension based on the above tension calculation theorem theory will be described with reference to FIGS. 1 and 4.

[0078] First, the structural data of the cable 120 (the total length L of the cable 120, the installation position l2 of the damper 110 (the length l2 of the second span 122), the length l1 of the first span 121, the unit volume mass (density) ρ of the cable 120, and the cross-sectional area A of the cable 120) is input to the data processing unit 140 (step ST11). Note that the structural data of the cable 120 may be an actual measured value or a nominal value. The input structural data is stored in the data processing unit 140.

[0079] Next, an impact is applied to the second span 122 of the cable 120 by the hammer 145. Accordingly, the cable 120 vibrates. At this time, the accelerometers 135, 136, and 137 measure the vibration acceleration of the cable 120. The measured vibration acceleration is recorded in the data processing unit 140 as a time history response value (step ST12). In the present embodiment, since the accelerometers 135, 136, and 137 are used to detect the vibration of the cable 120, the time change of the acceleration of the cable 120 at three points is acquired as the time history response value. Note that a device for measuring the displacement or velocity of the cable 120 may be attached to the cable 120 to detect the vibration of the cable 120. In that case, the time history response value is data representing the time change of the displacement or velocity of the cable 120.

[0080] In the measurement value processing unit 141 of the data processing unit 140, the data of the time change of the recorded vibration acceleration (time history response value) is used as an acceleration waveform, and Fourier transform is performed (conversion process) (step ST13). Subsequently, from the Fourier-transformed frequency waveform (Fourier amplitude), the Fourier amplitude at the frequency (natural frequency) at which a peak appears is derived as the Fourier amplitude at each mode order (amplitude derivation process). Further, a mode shape measurement value is derived from the Fourier amplitudes at the derived mode orders (measurement value derivation process) (step ST14).

[0081] The tension calculation unit 142 of the data processing unit 140 calculates the tension of the cable 120 by using the data regarding the theoretical values of the mode shapes stored and the data regarding the measured values of the mode shapes derived in the measured value processing unit 141 (step ST15). In this calculation, the objective function of Expression (31) is used. At this time, the tension T and the flexural rigidity EI are obtained by the least squares method so that the value of the objective function becomes the minimum value (or the local minimum value). Thereby, the tension of the cable 120 is calculated.

[0082] (Numerical experiment results) Next, in order to verify the calculation accuracy of the cable tension using the above tension calculation theory, a numerical experiment was conducted, and the results will be described.

[0083] This numerical experiment was conducted for a total of 90 cases combining 10 cables and 9 high-damping rubber dampers, and for a total of 90 cases combining the above 10 cables and 9 viscous shear dampers. MATLAB (registered trademark) was used for the numerical experiment.

[0084] The parameters regarding the cable are shown in Table 1, the parameters regarding the high-damping rubber damper are shown in Table 2, and the parameters regarding the viscous shear damper are shown in Table 3. Since the exact value of the damping coefficient c is unknown for the viscous shear damper, the damping coefficient c is modeled by Expression (32).

[0085] [Number]

[0086] Here, ω1 is the first natural frequency in the cable with the high-damping rubber damper with the same No as the assumed viscous shear damper. k is the spring constant of the viscous shear damper, and γ is the loss coefficient. The attachment position of the viscous shear damper is the same as the attachment position l1 shown for the high-damping rubber damper with the same No.

[0087]

Table 1

[0088]

Table 2

[0089]

Table 3

[0090] The measurement points of the mode shape (i.e., corresponding to the acquisition points of the acceleration waveform) were set as follows: starting from the installation position of the damper, the first point was at a position 0.5 m in the direction of the center of the cable, and the second and third points were at 0.5 m intervals from there. That is, p1 = 0.5, p2 = 1.0, and p3 = 1.5. Regarding the values of the mode function at these measurement points as the values obtained by measurement, the tension was estimated. The search ranges were set as 0.5 to 2 times the true values for tension and bending rigidity, and the imaginary-to-real ratio of the complex natural vibration frequency was set from 0 to 1. In this verification, the cases using up to the 5th mode and the 7th mode were examined. Hereinafter, the results for the case using up to the 5th mode are shown in FIGS. 5 and 6. In FIGS. 5 and 6, the vertical axis represents the ratio of the calculated tension to the true tension, and the horizontal axis represents the model number expressed as the sum of the cable No. and the damper No. Therefore, although the model numbers exist up to 109, the number of data is 90 each.

[0091] As shown in FIG. 5, in the case assuming a high-damping rubber damper, all the errors were less than 0.8%, and it was confirmed that the estimation could be performed with very good accuracy. Also, as shown in FIG. 6, for the case assuming a viscous shear damper, all the errors were less than 0.7%, and it was also confirmed that the estimation could be performed with very good accuracy.

[0092] (Model experiment results) Next, the experimental results using the model will be described. In the experiment, with one end fixed and the other end as the tension end, a tension shown by the set tension in Table 5 was applied to the cable at the tension end. The experiment was conducted for 26 cases. The parameters of the cables used are shown in Table 4, and the cable length, set tension, type of damper, installation position of the damper, relative position (the installation position of the damper divided by the cable length), spring constant, and loss coefficient for each case are shown in Table 5. The installation position of the damper is the distance from the tension end, and the installation positions of the accelerometers p1 + l1, p2 + l1, p3 + l1 are the distances from the tension end. Here, p1, p2, and p3 are the distances from the damper installation position to the accelerometer installation position. Also, the Fourier amplitudes up to the 10th mode for each case are shown in Tables 6 to 8.

[0093]

Table 4

[0094]

Table 5

[0095]

Table 6

[0096]

Table 7

[0097]

Table 8

[0098] For the above 26 cases, the results of calculating the tension using the tension calculation theorem are shown in Fig. 7. In Fig. 7, the vertical axis represents the ratio of the calculated tension to the set tension, and the horizontal axis represents the model number. As shown in Fig. 7, except for model number 18, the calculated tension is within an error of 5%. It is presumed that for model number 18, the error in the mode shape was large.

[0099] Fig. 8 shows a comparison between the method (Example) using the above-mentioned tension calculation theorem and the calculation results (Comparative Example) by a previously proposed method. This Comparative Example is a method using the objective function shown in "Equation 35" in Japanese Patent Application Laid-Open No. 2023-12874 as the calculation formula, and the objective function (Equation (34)) set under the constraint of setting the real part and the imaginary part of the vibration equation of the i-th mode shown in the following Equation (33) to zero. In the previously proposed method, since it is necessary to model the complex stiffness of the damper, the damper was modeled using the design formula of the high-damping rubber damper.

[0100]

Equation

[0101]

Equation

[0102] As shown in Fig. 8, it can be seen that the error is smaller in the method using the above-mentioned tension calculation theorem than in the Comparative Example. Also, when looking at the root mean square error (RMSER), it is 17.59 in the Comparative Example, while it is 3.26 in the method using the above-mentioned tension calculation theorem. The reason why the accuracy is worse in the Comparative Example is that the damper used in this model experiment showed different properties from the high-damping rubber damper. That is, it was affected by the modeling error of the damper. In contrast, in this Example, since no damper model was used, it was not affected by the modeling error of the damper.

[0103] As described above, in the calculation method according to the present embodiment, vibrations at any three or more points on the cable 120 are detected, and from the vibration waveforms at each point, the mode shape measurement values Φ i1 m , Φ i2 m , Φ i3 m ,... Φ ik m at any three or more points are obtained. Also, the theoretical mode shape values Φ i1 t , Φ i2 t , Φ i3 t ,... Φ ik t corresponding to the amplitudes at any three or more points are derived. Then, at least two sets of any two points are selected from three or more points, and in each set, the ratio Φ ik t / Φ i1 t of the mode shape measurement values and the ratio Φ ik m / Φ i1 m of the theoretical mode shape values are made equal for each of the multiple mode orders. Therefore, since at least two constraint conditions are obtained, even if the attenuation coefficient of the cable 120 derived from the complex natural frequency based on the vibration equation of the cable 120 is included as an unknown, it is possible to estimate the unknown. Moreover, since the ratio Φ ik m / Φ i1 m of the theoretical mode shape values corresponding to the arbitrarily selected two points is calculated, even if the general solution of the mode function Y2 obtained from the vibration equation includes integration constants, it is possible to avoid being affected by them. Moreover, the influence of the complex natural frequency based on the vibration equation can also be eliminated. For this reason, since there is no need to model the damper 110 and use its characteristic values as constraint conditions, the tension of the linear body can be calculated without being affected by the modeling error of the damper 110.

[0104] Also, the theoretical mode shape value Φ i1 t, Φ i2 t , Φ i3 t When deriving Φ i1 t , the mode shape theoretical value Φ i1 t is adjusted by adjusting the amplitude so that the maximum value becomes 1. i1 t , Φ i2 t , Φ i3 t is derived, and when obtaining the mode shape measured value Φ i1 m , Φ i2 m , Φ i3 m , the mode shape measured value Φ i1 m , Φ i2 m , Φ i3 m is obtained by adjusting the amplitude so that the maximum value becomes 1. Therefore, considering that the denominator may become zero when calculating the ratio Φ ik m / Φ i1 m of the mode shape theoretical value and the ratio Φ ik t / Φ i1 t of the mode shape measured value, the objective function (suitable when using Equation (31)) is used. i1 m , Φ i2 m , Φ i3 m When obtaining Φ i1 m , Φ i2 m , Φ i3 m , the mode shape measured value Φ i1 m , Φ i2 m , Φ i3 m is obtained by adjusting the amplitude so that the maximum value becomes 1. Therefore, the ratio Φ ik m / Φ i1 m of the mode shape theoretical value and the ratio Φ ik t / Φ i1 t of the mode shape measured value are calculated. i1 m , Φ i2 m , Φ i3 m is obtained. Therefore, considering that the denominator may become zero when calculating the ratio Φ ik m / Φ i1 m of the mode shape theoretical value and the ratio Φ ik t / Φ i1 t of the mode shape measured value, the objective function (suitable when using Equation (31)) is used. ik m / Φ i1 m and the ratio Φ ik t / Φ i1 t of the mode shape measured value are calculated. ik t / Φ i1 t It should be considered that the denominator may become zero when calculating the ratio Φ ik m / Φ i1 m of the mode shape theoretical value and the ratio Φ ik t / Φ i1 t of the mode shape measured value. The objective function (suitable when using Equation (31)) is used.

[0105] It should be noted that the embodiments disclosed this time should be considered as illustrative in all respects and not restrictive. The present invention is not limited to the above embodiments, and various changes, improvements, etc. are possible without departing from the gist thereof.

Explanation of Signs

[0106] 110: Damper 120: Cable 121: First Span 122: Second Span 135: Accelerometer 136: Accelerometer 137: Accelerometer 145: Hammer EI: Bending Rigidity T: Tension

Claims

1. A method for calculating the tension of a linear body to which a damper is attached, comprising: detecting vibrations at three or more arbitrary points on the linear body, and obtaining mode shape measurement values at the three or more points in a plurality of vibration modes based on the detected vibrations; deriving mode shape theoretical values at the three points on the linear body in a plurality of vibration modes based on a vibration equation representing the relationship between the displacement, the tension, and the bending rigidity of the tensioned linear body, and boundary conditions indicating that the damper is disposed on the linear body; selecting at least two sets of any two points from the three or more points; calculating, for each of the two sets, a ratio of the mode shape measurement values in the plurality of vibration modes corresponding to the selected arbitrary two points, and calculating, for each of the two sets, a ratio of the mode shape theoretical values in the plurality of vibration modes corresponding to the selected arbitrary two points; calculating the tension of the linear body using, for each set, a constraint condition that the ratio of the mode shape measurement values is equal to the ratio of the mode shape theoretical values; and a method for calculating the tension of a linear body.

2. When deriving the mode shape theoretical values, the mode shape theoretical values are derived by adjusting the amplitudes at the three or more points in the plurality of vibration modes so that the maximum value becomes 1, and when obtaining the mode shape measurement values, the mode shape measurement values are obtained by adjusting the amplitudes at the three or more points in the plurality of vibration modes so that the maximum value becomes 1. The method for calculating the tension of a linear body according to Claim 1.

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