JP7490381B2 - Method for calculating the tension of a linear body, the bending rigidity of a linear body, and the characteristics of a damper attached to the linear body - Google Patents

Description

Patent Law Article 30, Paragraph 2 applied Published in "Kanazawa University, Kyoto University, Kobe University, Gifu University, and Fukui National College of Technology Winter Joint Seminar Proceedings Kyoto University" published on January 5, 2019

Application of Article 30, Paragraph 2 of the Patent Act. Presented at the "Kanazawa University, Kyoto University, Kobe University, Gifu University, and Fukui National College of Technology Winter Joint Seminar" held in Gifu University Faculty of Engineering, Room 101 on January 5, 2019.

Applicable to Article 30, Paragraph 2 of the Patent Act. Presented at the "Kyoto University Faculty of Engineering Department of Global Engineering (Civil Engineering, International Course) Special Research Presentation" held on February 5, 2019 at the Global Human Resources Hall (C1-311), C1 Building, C Cluster, Katsura Campus, Kyoto University.

The present invention relates to a method for calculating the tension acting on a linear body, the bending rigidity of the linear body, and the characteristics of a damper attached to the linear body.

Various methods have been devised for calculating the tension acting on cables such as bridge cables, string beams, and electric wires (see Patent Documents 1 and 2). In these methods, the natural frequency of a cable modeled as a one-dimensional beam is calculated. If the calculated value of the natural frequency is close to the natural frequency of the actual cable, it can be said that the tension value used to calculate the natural frequency is close to the tension acting on the actual cable. Therefore, the calculated value of the natural frequency is compared with the actual measured value of the natural frequency appearing in vibration data obtained by applying an impact to an arbitrary point on the actual cable. The calculated value of the natural frequency close to the actual measured value is substituted into the vibration equation of the one-dimensional beam described above, and the tension acting on the cable is calculated.

Japanese Patent Application Laid-Open No. 9-101289 Japanese Patent Application Laid-Open No. 11-271155

The cable for which the tension is calculated may be fitted with a vibration-damping component (hereafter referred to as a "damper") to suppress the vibration of the cable. The damper affects the natural frequency of the cable. However, the calculation method described above calculates the tension acting on the cable without taking the damper into account. Therefore, when a damper is attached to the cable, the calculated tension may differ significantly from the tension actually acting on the cable.

The present invention aims to provide a method for calculating with high accuracy the tension acting on a linear body to which a damper is attached, the bending rigidity of the linear body, and the characteristics of the damper.

A calculation method according to one aspect of the present invention can be used to calculate tension of a linear body to which a damper is attached, comprising: detecting vibration at an arbitrary point on the linear body, obtaining actual measured values of natural frequencies of multiple modes based on the detected vibration, and calculating the tension of the linear body based on a comparison between the calculated values of the natural frequencies of the multiple modes and the actual measured values, using an equation expressing the relationship among the natural frequencies of the multiple modes, tension, bending rigidity, and a phase shift caused by characteristics of the damper, which is set using a boundary condition expressing that the damper is disposed on the linear body.

According to the above configuration, a boundary condition indicating that a damper is disposed on the linear body is used in the tension calculation process, so that a relational equation is established between the natural frequencies of the multiple modes, the tension, the bending rigidity, and the phase shift caused by the characteristics of the damper. Therefore, the calculated values of the natural frequencies of the multiple modes obtained using the above relational equation reflect the influence of the phase shift caused by the characteristics of the damper on the linear body. The calculated values of the natural frequencies of the multiple modes that reflect the influence of the damper are compared with the actual measured values of the natural frequencies of the multiple modes obtained by exciting a linear body with a damper, so that the tension of the linear body is calculated with high accuracy.

With regard to the above configuration, in the process of calculating the tension of the linear body, the tension value at which the value of an objective function set using the tension, the bending rigidity, and the phase shift as variables to represent the deviation between the actual measured value and the calculated value is minimized may be determined as the tension acting on the linear body.

According to the above configuration, a boundary condition indicating that a damper is disposed on the linear body is used in the tension calculation process, so that a relational equation is established between the natural frequencies of multiple modes, the tension, the bending rigidity, and the phase shift caused by the damper characteristics. The calculated value obtained from this equation and an objective function representing the deviation between the actual value and the calculated value of the natural frequency are also set using the tension, the bending rigidity, and the phase shift caused by the damper characteristics as variables. When the value of the objective function is minimized, the deviation between the actual value and the calculated value of the natural frequency is small. The tension value when the value of the objective function is minimized is determined as the tension acting on the linear body, so the tension is calculated with high accuracy.

With respect to the above configuration, the calculation method may further comprise comparing the value of the tension when the objective function is minimized with a predetermined threshold value. Under the condition that the value of the tension when the objective function is minimized is below the threshold value, the value of the tension when the objective function is minimized may be determined as the tension acting on the linear body.

According to the above configuration, the tension of the linear body is determined under the condition that the tension value when the objective function is minimized is below the threshold value. In other words, the tension of the linear body is determined under the condition that the measured and calculated values of the natural frequency are relatively close to each other. Therefore, the tension of the linear body is calculated with a predetermined accuracy. If the tension value when the objective function value is minimized is not below the threshold value, the data of the candidate values of the measured natural frequencies of multiple modes can be reviewed.

In the above configuration, obtaining the actual measured value may include obtaining actual measured values of natural frequencies of multiple modes over the entire length of the linear body. In the step of calculating the tension of the linear body, the tension of the linear body may be calculated using a first objective function as the objective function, the calculated value being set to represent the natural frequencies of multiple modes over the entire length of the linear body.

According to the above configuration, both the actual measured value of the natural frequency and the calculated value of the natural frequency in the first objective function represent the natural frequency of the entire length of the linear body. Since both represent the natural frequency of the entire length of the linear body, the first objective function represents the deviation between the actual measured value and the calculated value of the natural frequency for the entire length of the linear body. By using the first objective function, the value of the tension when the first objective function is minimized accurately represents the tension acting on the linear body.

With regard to the above configuration, in the step of obtaining the actual measured value, the natural frequencies of multiple modes of the longer of the two spans of the linear body separated by the damper may be obtained. In the step of calculating the tension of the linear body, the tension of the linear body may be calculated using, as the objective function, a second objective function set so that the calculated value represents the natural frequencies of multiple modes of the longer span.

The natural frequency of the linear body for a vibration mode that appears only on the short span side cannot be detected from the response value measured on the long span side. With the above configuration, the actual measured value of the natural frequency and the calculated value of the natural frequency in the second objective function both represent the natural frequency of the longer of the two spans of the linear body separated by the damper. The second objective function can appropriately represent the difference between the two natural frequencies in the longer span. The value of the tension when the second objective function is minimized is represented as the tension acting on the linear body.

In the above configuration, in the step of obtaining the actual measured values, the actual measured values of the natural frequencies of the multiple modes of the entire length of the linear body may be obtained, and the actual measured values of the natural frequencies of the multiple modes of the longer of the two spans of the linear body divided by the damper may be obtained. In the step of calculating the tension of the linear body, a first objective function set so that the calculated value represents the natural frequencies of the multiple modes of the entire length of the linear body and a second objective function set so that the calculated value represents the natural frequencies of the multiple modes of the longer span may be used as the objective functions, and the value of the tension when the smaller of the deviations calculated by the first objective function and the second objective function is obtained may be determined as the tension acting on the linear body.

According to the above configuration, the deviation between the actual measured value and the calculated value of the natural frequency for the entire length of the linear body is represented by the first objective function. The deviation between the actual measured value and the calculated value of the natural frequency for the longer span of the linear body is represented by the second objective function. That is, two types of evaluation methods are used to evaluate the deviation between the actual measured value and the calculated value. Depending on the type of damper and the installation conditions, one of these evaluation methods may be more appropriate than the other evaluation method. In this case, the tension value when the smaller of the deviations represented by the first objective function and the second objective function is obtained is determined as the tension acting on the linear body, so that the tension of the linear body is calculated with high accuracy.

A calculation method according to another aspect of the present invention can be used to calculate characteristics of a damper attached to a linear body, the calculation method including: detecting vibration at an arbitrary point on the linear body, obtaining actual measured values of natural frequencies of multiple modes based on the detected vibration, and calculating the characteristics of the damper based on a comparison between the calculated values of the natural frequencies of the multiple modes and the actual measured values, using an equation expressing a relationship among the natural frequencies of the multiple modes, tension, bending rigidity, and a phase shift caused by the characteristics of the damper, which is set using a boundary condition representing that the damper is disposed on the linear body.

According to the above configuration, in the process of calculating the damper characteristics, a boundary condition representing that the damper is disposed on the linear body is used, so that the calculated value of the natural frequency of the linear body is related to the phase shift caused by the characteristics of the damper. Therefore, the calculated value of the natural frequency of the multiple modes reflects the effect of the phase shift caused by the characteristics of the damper on the linear body. The calculated value is compared with the actual measured value of the natural frequency of the multiple modes obtained by applying an impact to the linear body, thereby calculating the characteristics of the damper.

A calculation method according to yet another aspect of the present invention can be used to calculate the bending stiffness of a linear body to which a damper is attached. The calculation method includes detecting vibrations at any point on the linear body and obtaining actual measured values of natural frequencies of multiple modes based on the detected vibrations, and calculating the bending stiffness of the linear body based on a comparison between the calculated values of the natural frequencies of the multiple modes and the actual measured values, using an equation expressing the relationship among the natural frequencies of the multiple modes, tension, bending stiffness, and phase shifts caused by characteristics of the damper, which is set using a boundary condition representing that the damper is disposed on the linear body.

According to the above configuration, a boundary condition indicating that a damper is disposed on the linear body is used in the tension calculation process, so that a relational expression is established between the natural frequencies of the multiple modes, the tension, the bending rigidity, and the phase shift caused by the characteristics of the damper. Therefore, the calculated values of the natural frequencies of the multiple modes obtained using the above relational expression reflect the influence of the phase shift caused by the characteristics of the damper on the linear body. The calculated values of the natural frequencies of the multiple modes, which reflect the influence of the phase shift caused by the characteristics of the damper, are compared with the actual measured values of the natural frequencies of the multiple modes obtained by exciting a linear body with a damper, so that the bending rigidity of the linear body can be calculated with high accuracy.

The techniques described above make it possible to calculate the tension and bending stiffness of the linear body as well as the damper characteristics.

FIG. 1 is a schematic diagram of a measurement device used to measure the natural frequency of a cable equipped with a damper. FIG. 1 is a schematic diagram of a vibration model in which a cable of a measuring device is modeled as a one-dimensional beam. 11 is a schematic flowchart showing a calculation process. 13 is a graph showing vibration data after Fourier transformation processing. 13 is a graph showing vibration data after Fourier transformation processing. 4 is a schematic flowchart showing an exemplary calculation procedure commonly performed in steps S125 and S130 of the process shown in FIG. 3 . 1 is a graph conceptually showing an objective function for finding the minimum value of the deviation between the calculated value and the measured value of the natural frequency. 1 is a graph showing an example of vibration data that is prone to cause an error in setting a mode order; 1 is a table showing test conditions and test results. 1 is a graph showing test results. 10 is a schematic flowchart showing a calculation process using only one of a first calculation formula and a second calculation formula.

First Embodiment
Fig. 1 is a schematic diagram of a measurement device 100 used to measure the natural frequency of a cable 120 to which a damper 110 is attached. Fig. 2 is a schematic diagram of a vibration model in which the cable 120 of the measurement device 100 is modeled as a one-dimensional beam. The calculated value of the natural frequency calculated using the vibration model is compared with the actual value of the natural frequency measured by applying vibration to the cable 120 of the measurement device 100. Based on the comparison of the actual and calculated values of the natural frequency, the tension acting on the cable 120 and the characteristics of the damper 110 are calculated. A method of calculating the tension will be described with reference to Figs. 1 and 2.

The measuring device 100 includes a pair of frames 131, 132 arranged at a distance to support both ends of the cable 120, three support plates 134, a load cell 133, and a pair of fixing devices 139. The two support plates 134 are attached to the left and right sides of the frames 131, 132, respectively. The remaining support plates 134 are arranged at a distance to the left from the support plate 134 attached to the frame 131. The load cell 133 is arranged between these support plates 134. The pair of fixing devices 139 are arranged on the left and right sides of the leftmost and rightmost support plates 134, respectively, and support both ends of the cable 120 so that the cable 120 is in a substantially horizontal position. The total length of the cable 120 is defined as the distance between the pair of fixing devices 139, and is represented by the symbol "L" in the following description. The load cell 133 is configured to detect the tension acting on the cable 120. The tension detected by the load cell 133 is used to verify the calculation accuracy, so the load cell 133 is not required for calculating the tension. The support plate 134 fixes the end of the cable 120 at the position of the frame 131 against the tension of the cable 120.

The measuring device 100 further includes an accelerometer 135 attached to the cable 120 and a measuring instrument 136 that receives a signal output from the accelerometer 135. The accelerometer 135 is configured to detect the acceleration of vibrations occurring in the cable 120. The mounting position of the accelerometer 135 may be any position on the cable 120. Preferably, the accelerometer 135 is mounted at a position where the deflection of the cable 120 is expected to be large. The signal output from the accelerometer 135 represents the acceleration of the vibration. The measuring instrument 136 is configured to record the change in the acceleration of the vibration over time.

The damper 110 is attached to the cable 120 so as to divide the cable 120 into two spans 121, 122. The span 121 extends between the left frame 131 and the damper 110. The span 122 extends between the damper 110 and the right frame 132 on which the damper 133 is located. The damper 110 is attached to the cable 120 closer to the left frame 131 than to the right frame 132. Thus, the left span 121 is shorter than the right span 122. The length of the left span 121 is represented in the following description using the symbol "l ₁ ". The length of the right span 122 is represented in the following description using the symbol "l ₂ ". The right span 122 is attached with the accelerometer 135 described above.

The right span 122 is struck by the hammer 137, giving an impact to the cable 120. As a result, the cable 120 vibrates. The acceleration of the vibration of the cable 120 at the mounting position of the accelerometer 135 is detected by the accelerometer 135. The change in the vibration acceleration over time is recorded by the measuring device 136. The recorded vibration acceleration data is analyzed to obtain actual measurement data of the natural frequency of the cable 120.

For the model shown in FIG. 2, the cable 120 is modeled as a one-dimensional beam simply supported at both ends. The positions of both ends of the cable 120 correspond to the positions of the frames 131 and 132 described above. The following equation is the equation of motion for the deflection y(x,t) of position x at time t for the cable modeled as a one-dimensional beam.

The deflection y(x,t) is separated into variables using the following formula:

The following equation represents the general solution for X(x) obtained by substituting the decomposed deflection y(x,t) into the equation of motion above.

2, spans 121 and 122 are modeled as two beams connected with damper 110 in between. The mode shapes _X1 ( _x1 ) of the beam corresponding to span 121 and _X2 ( _x2 ) of the beam corresponding to span 122 are expressed by the following equations.

The boundary conditions indicating that both ends of the cable are simply supported are expressed by the following "Equation 5" and "Equation 6". "Equation 5" represents the boundary condition at the left end of the cable. "Equation 6" represents the boundary condition at the right end of the cable.

The following formula expresses the boundary condition that the ends of the beams in which spans 121 and 122 are modeled are supported by the damper 110 at the installation position of the damper 110. The following formula is obtained under the condition that the deflection, deflection angle, and curvature of the cable 120 are continuous at the installation position of the damper 110, and the forces are balanced in the vertical direction.

When the characteristics of the damper 110 are expressed by a spring and a dashpod as shown in FIG. 2, "k ^* " in the above formula is expressed by the following formula.

When the mass m of the damper 110 cannot be ignored in consideration of the mass of the cable 120, "k ^* " in the above formula is expressed by the following formula.

When a damper that does not include a spring element (i.e., an oil damper) is used, "k ^* " in the above formula is expressed by the following formula.

When a high-damping rubber damper is used, "Equation 9" may be rewritten as follows, using ku and kv to represent the complex springs.

When the above-mentioned boundary conditions ("Equation 5" to "Equation 7") are substituted into the above-mentioned equation "Equation 4", the following equation is obtained. The following equation is arranged using the relational equation "αl ₂ =α(L-l ₁ )".

The above "Number 12" can be rewritten as the following formula:

In the lower equation of "Number 13" above, the hyperbolic functions have been rewritten as exponential functions.

The relationship between the mode number i and the natural frequency f _i of the i-th mode is derived as follows (i is a natural number).

The symbols "α" and "β" in the above-mentioned "Equation 3" can be rewritten as follows using the natural frequency f _i :

The upper equation in "Equation 13" above holds for each mode, so we obtain the following equation.

When the upper equation of "Number 14" is substituted into "Number 15", the following equation is obtained.

The natural frequency f _i of "Number 16" is a complex number. However, the actual measurement value of the natural frequency obtained from the measurement device 100 described with reference to FIG. 1 corresponds to the real part of the natural frequency f _i of "Number 16". Therefore, the calculated value of the natural frequency to be compared with the actual measurement value of the natural frequency obtained from the measurement device 100 is expressed by the following formula.

The natural frequencies f _i in Equation 17 represent the calculated natural frequencies of the length L of cable 120. Equation 17 shows the relationship between the natural frequencies f _i of the multiple modes, the structural properties of cable 120 (i.e., bending stiffness EI, density ρ, cross-sectional area A and length L), the tension T in cable 120 and the property k ^* of damper 110.

Now, by dividing both sides of "Equation 15" by α _i , the following equation can be obtained.

Considering α _i in "Equation 18" as a wave number, π/α _i can be interpreted as the length of a half wavelength of a wave with wave number α _i . In other words, "Equation 18" can be considered as i half wavelengths π/α _i being included in the apparent cable length (L-φ _i /α _i ). In this respect, the order i in the above-mentioned calculation formula "Equation 17" can be considered as the number of half wavelengths that appear in the entire length of cable 120 at order i. Hereinafter, based on a similar interpretation, the expression "number of half wavelengths" will be used.

If the phase shift φ _i in Equation 15 is divided into a phase shift “φ _i1 ” in span 121 and a phase shift “φ _i2 ” in span 122, the following equation can be obtained.

In addition, from the perspective of how many half wavelengths appear in each mode order for spans 121 and 122, the right-hand side of the above "Equation 15" can be rewritten as follows:

If the phase shift component "φ _i " on the left side of "Equation 15" is considered as being divided into the phase shift components of spans 121 and 122, then "Equation 15" can be rewritten as an equation for each of spans 121 and 122. "Equation 21" below is an equation for span 121. "Equation 22" below is an equation for span 122.

By substituting α _i in “Equation 21” for α _i in “Equation 14”, the natural frequency f _i of the entire length of the cable 120 can be replaced with the natural frequency f _m of the span 121. The following equation expresses the natural frequency f _m of the span 121.

Similarly, by substituting α _i in “Equation 22” for α _i in “Equation 14”, the natural frequency f _i of the entire length of cable 120 can be replaced with the natural frequency f _n of span 122. The following equation expresses the natural frequency f _n of span 121.

The relationship between the phase shift φ _i1 of the span 121 and the phase shift φ _i calculated by "Equation 16" can be obtained as follows. First, by modifying "Equation 21", the phase shift φ _i1 of the span 121 can be expressed as follows.

Here, the tangent of the phase shift φ _i1 is expressed as follows:

Similarly, the tangent of the phase shift φ _i2 is expressed as follows:

Based on "Equation 26" and "Equation 27", the following relational equation is obtained.

The calculated values of the natural frequencies "f _i " and "f _n " calculated based on the above-mentioned "Number 16" and "Number 24" are compared with the natural frequencies based on the vibration data actually obtained from the vibration test using the measurement device 100. "Number 16" representing the natural frequency "f _i " of the entire length of the cable 120 is referred to as the "first calculation formula" in the following description. "Number 24" representing the natural frequency "f _n " of the longer span 122 is referred to as the "second calculation formula" in the following description.

Based on the results of the above comparison, the tension T acting on the cable 120 is calculated. The calculation process from the vibration test to the calculation of the tension T is described with reference to Figures 1 and 3. Figure 3 is a schematic flowchart showing the calculation process.

(Step S105)
Structural data of the cable 120 (total length L of the cable 120, placement position _l2 of the damper 110 (length _l2 of the span 122), density ρ of the cable 120, and cross-sectional area A of the cable 120) are acquired. The structural data of the cable 120 may be an actual measured value or a nominal value. The acquired structural data is used to calculate the natural frequencies "f _i " and "f _n " using the first and second calculation formulas. After acquiring the structural data, step S110 is executed.

(Step S110)
The span 122 of the cable 120 attached to the measuring device 100 is struck by the hammer 137 to apply vibration to the cable 120. Then, step S115 is executed.

(Step S115)
An accelerometer 135 attached to the cable 120 measures the vibration acceleration of the cable 120. The measured vibration acceleration is recorded as a time history response value in a measuring device 136. In this embodiment, since the accelerometer 135 is used to detect the vibration of the cable 120, the time change in the acceleration of the cable 120 is acquired as the time history response value. However, if a device that measures the displacement or velocity of the cable 120 in order to detect the vibration of the cable 120 is attached to the cable 120, the time history response value may be data representing the time change in the displacement or velocity of the cable 120. After the time history response value is acquired, step S120 is executed.

(Step S120)
Based on the time history response values, the actual measured value of the natural frequency of the cable 120 is obtained. In this embodiment, a Fourier transform is performed on the data of the time history response values, and the actual measured value of the natural frequency is obtained from the peak value of the data after the Fourier transform. However, other analysis methods may be used to obtain the natural frequency.

When the measured value of the natural frequency is obtained, the measured value is assigned the mode number i of the vibration of the entire length of the cable 120 (the number i of half wavelengths that appear along the entire length of the cable 120) or the number n of half wavelengths that appear along the span 122. With reference to Figures 1, 3 to 4B, it is explained how the mode number i or the number n of half wavelengths is assigned to the measured value of the natural frequency. Figures 4A and 4B are graphs showing the vibration data after Fourier transformation processing.

The horizontal axis of Fig. 4A and Fig. 4B represents frequency. The vertical axis of Fig. 4A and Fig. 4B represents vibration strength. Fig. 4A and Fig. 4B show a curve representing the vibration strength of a relatively short span 121 and a curve representing the vibration strength of a relatively long span 122.

The frequency at which the peak of the vibration intensity of the short span 121 appears represents the actual measured value of the natural frequency that originates from the short span 121 and appears in the response value obtained from the acceleration sensor 135 on the long span 122 (this natural frequency can also be obtained from the response value data obtained when the accelerometer 135 is attached to the short span 121). The frequency at which the peak of the vibration intensity of the long span 122 appears represents the actual measured value of the natural frequency that originates from the long span 122 and appears in the response value obtained from the acceleration sensor 135 on the long span 122 (this natural frequency does not appear in the response value data obtained when the accelerometer 135 is attached to the short span 121). The actual measured value of the natural frequency that originates from the spans 121 and 122 and appears in the response value obtained from the acceleration sensor 135 on the long span 122 is called the "actual measured value of the natural frequency of the spans 121 and 122". The vibration data in Figures 4A and 4B represent the actual measured values of the natural frequencies of the two spans 121 and 122.

Figure 4A shows how to assign the mode number i (i.e., the number i of half wavelengths appearing over the entire length of cable 120) for the first calculation formula used to calculate the natural frequency of the entire length of cable 120. The mode number i (the number i of half wavelengths appearing over the entire length of cable 120) for the first calculation formula is assigned in ascending order of natural frequencies, regardless of whether it is the natural frequency of span 121 or the natural frequency of span 122.

Figure 4B shows how to assign the number n of half wavelengths for the second calculation formula used to calculate the natural frequency of span 122. The way in which the number n of half wavelengths is assigned in Figure 4B is useful in cases where there is a natural frequency that appears only in the response value obtained when acceleration sensor 135 is attached to short span 121 and does not appear when acceleration sensor 135 is attached to long span 122, and only the response value obtained when acceleration sensor 135 is attached to long span 122 is used.

After obtaining the natural frequency and setting the mode order i and the number of half wavelengths n, steps S125 and S130 are executed. Step S125 is executed to obtain optimal solutions for the tension T of the cable 120, the bending stiffness EI of the cable 120, the spring constant k of the damper 110, the damping constant c of the damper 110, and the mass m of the damper 110 based on the first calculation formula. Step S130 is executed to obtain optimal solutions for the tension T of the cable 120, the bending stiffness EI of the cable 120, the spring constant k of the damper 110, the damping constant c of the damper 110, and the mass m of the damper 110 based on the second calculation formula. Steps S125 and S130 differ only in the calculation formula and are performed using a common procedure.

To obtain the optimal solution, the calculated values of the natural frequencies obtained from the first and second calculation formulas are compared with the actual measured values of the natural frequencies obtained in step S120, and the deviation between the calculated and actual measured values of the natural frequencies is evaluated. To evaluate the deviation between the calculated and actual measured values, an objective function F that represents the deviation between the calculated and actual measured values is used. The objective function F, expressed by the following formula, represents the sum of squared residuals as the deviation between the calculated and actual measured values. However, the objective function F may use other functions to represent the deviation between the calculated and actual measured values (for example, the sum of the absolute values of the deviation between the natural frequencies).

The following "Number 29" indicates an objective function F that represents the sum of squares of the residual between the calculated value of the natural frequency obtained from the first calculation formula and the actual measured value of the natural frequency obtained in step S120. In the following explanation, "Number 29" is referred to as the first objective function. The following "Number 30" indicates an objective function F that represents the sum of squares of the residual between the calculated value of the natural frequency obtained from the second calculation formula and the actual measured value of the natural frequency obtained in step S120. In the following explanation, "Number 30" is referred to as the second objective function. The first objective function and the second objective function are functions whose variables are the tension of the cable 120, the bending rigidity of the cable 120, and the characteristics of the damper 110 (the spring constant k of the damper 110, the damping constant c of the damper 110, and the mass m of the damper 110).

The first objective function and the second objective function are used in steps S125 and S130, respectively. The processes in steps S125 and S130 are described with reference to Figs. 1, 3, 5, and 6. Fig. 5 is a schematic flow chart showing an exemplary calculation procedure commonly performed in steps S125 and S130. Fig. 6 is a graph conceptually showing the above-mentioned objective function F.

The horizontal axis of FIG. 6 represents the tension T acting on the cable 120, the bending stiffness EI of the cable 120, the spring constant k of the damper 110, the damping constant c of the damper 110, or the mass m of the damper 110. Although the graph in FIG. 6 is drawn in two-dimensional coordinates, the calculation using the objective function F is performed on six-dimensional coordinates (six dimensions: sum of squared residuals, tension T, bending stiffness EI, spring constant k, damping constant c, and mass m).

(Step S210)
When steps S125 and S130 are started, step S210 is executed first. In step S210, N initial values are set for the tension T, bending stiffness EI, spring constant k, damping constant c, and mass m. The first to sixth initial values are shown in FIG. 6. After the initial values are set, step S220 is executed.

(Step S220)
When the initial values of the tension T, bending stiffness EI, spring constant k, damping constant c, and mass m are substituted into the first and second calculation formulas, the natural frequency at the initial values is calculated for each mode order i or the number n of half wavelengths. The values of the tension T, bending stiffness EI, spring constant k, damping constant c, and mass m substituted into the objective function F are candidate values of the tension T, bending stiffness EI of the cable 120, and the characteristics k, c, and m of the damper 110 that are estimated to be acting on the cable 120. These candidate values (i.e., substituted values) are referred to as "tension candidate value", "bending stiffness candidate value", "spring constant candidate value", "damping constant candidate value", and "mass candidate value", respectively, in the following description. The "spring constant candidate value", "damping constant candidate value", and "mass candidate value" are candidate values that represent the characteristics of the damper 110, and are collectively referred to as "characteristic candidate value" in the following description. The natural frequency calculated for each mode order i or number n of half wavelengths is a candidate value for the calculated value of the natural frequency of cable 120, and will be referred to as the "candidate value of the natural frequency" in the following description.

As shown in "Equation 29" and "Equation 30" _, the candidate value of the natural frequency calculated for each mode order i or the number n of half wavelengths is compared with the natural frequencies ^fimeasured and _fnmeasured based on actual measurements for each mode ^order i or the number n of half wavelengths. That is, the deviation (i.e., _{residual) of the candidate value of the natural frequency from the natural frequencies fimeasured and fnmeasured is} ^{calculated for} each mode order i or the number n of half wavelengths. Then, the sum of the squared values of the residuals calculated for each mode order i or the number n of half wavelengths is calculated (i.e., the residual sum of squares). The obtained residual sum of squares is used as an evaluation value obtained by evaluating the deviation obtained at the initial value over the mode order i or the number n of half wavelengths.

At least one of the tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value is changed, and a new combination of these calculated values is set. Based on the new combination, the residual sum of squares is recalculated, and it is determined whether the new residual sum of squares is lower than the previous residual sum of squares. Based on the degree of reduction in the residual sum of squares, a new combination of calculated values is repeatedly set, and the minimum value of the objective function F is searched for.

In FIG. 6, four minimum values (hereinafter referred to as the "first minimum value", "second minimum value", "third minimum value", and "fourth minimum value") are shown. With respect to the graph in FIG. 6, when a search is started from the first and second initial values, the first minimum value can be obtained. When a search is started from the third initial value, the second minimum value can be obtained. When a search is started from the fourth and fifth initial values, the third minimum value can be obtained. When a search is started from the sixth initial value, the fourth minimum value can be obtained. After the search process, step S230 is executed.

(Step S230)
The smallest one is found from among the multiple minimum values obtained in step S220. The set of tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value when the smallest residual sum of squares is obtained is extracted as the optimal solution of the objective function F. With respect to the graph in Fig. 6, since the third minimum value is the smallest, the set of tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value used in calculating the third minimum value is extracted as the optimal solution of the objective function F. After the extraction of the optimal solution, step S135 is executed.

(Step S135)
The residual sums of squares (i.e., the value of the objective function F) corresponding to the optimal solutions obtained in steps S125 and S130 are compared with a predetermined threshold value X. If both of the residual sums of squares obtained in steps S125 and S130 exceed the threshold value X, step S140 is executed. Otherwise, step S145 is executed.

(Step S140)
When the first objective function is used, the residual sum of squares exceeds the threshold value X. It is considered that the reason is that the setting of the mode order i performed in step S120 is incorrect. For example, as shown in FIG. 7, when the natural frequency of the short span 121 is close to the natural frequency of the long span 122, the natural frequency of the short span 121 may be overlooked. In this case, as can be seen from a comparison between FIG. 4A and FIG. 7, the mode order i from "6" onwards is shifted by one. In order to correct the shift in the mode order i, in step 140, the mode order i may be assigned while avoiding unclear data parts. Thereafter, step S120 is executed, and the actual measured value of the natural frequency is obtained based on the newly set mode order i. Regarding the setting of the mode order, when there is a part where a peak is not clearly found due to noise in the data of the actual measured natural frequency, the mode order i may be assigned while avoiding the noise.

(Step S145)
If neither of the sums of squared residuals obtained in steps S125 and S130 exceeds the threshold value X, the sum of squared residuals obtained in step S125 is compared with the sum of squared residuals obtained in step S130. The set of tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value that results in the smaller of these sums of squared residuals is determined as the final optimal solution.

If only the residual sum of squares obtained in step S130 exceeds the threshold X, the set of tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value corresponding to the residual sum of squares obtained in step S125 is determined as the final optimal solution. If only the residual sum of squares obtained in step S125 exceeds the threshold X, the set of tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value corresponding to the residual sum of squares obtained in step S130 is determined as the final optimal solution. After the final optimal solution is determined, step S150 is executed.

(Step S150)
The tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value determined as the final optimal solution are output. In addition, the residual sum of squares corresponding to the final optimal solution is also output. The output tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value are calculated values of the tension T of the cable 120, the bending stiffness EI of the cable 120, the spring constant k of the damper 110, the damping constant c of the damper 110, and the mass m of the damper 110. These calculated values can be used to evaluate the state of the cable 120 and the damper 110. The residual sum of squares can be used to evaluate the accuracy of the output tension candidate value, bending stiffness candidate value, spring constant candidate value, damping constant candidate value, and mass candidate value. In the process shown in FIG. 3, the spring constant k of the damper 110, the damping constant c of the damper 110, and the mass m of the damper 110 are output. However, if a high-damping rubber damper is used as the damper 110, then ku, kv (and mass m) representing the complex spring element of the high-damping rubber are output as the characteristics of the damper 110 (see Equation 11).

Test results using the measurement device 100 described with reference to FIG. 1 are described below. FIG. 8A is a table showing the test conditions and test results. FIG. 8B is a graph showing the test results.

Regarding the test conditions, five installation positions of the damper 110 were set. In FIG. 1, the distance from the frame 131 to the damper 110 is indicated by the symbol "l ₁ ". The ratio of the distance l ₁ to the total length L of the cable 120 (i.e., the distance between the frames 131 and 132) is shown in FIG. 8A. Regarding the number of dampers 110, a condition in which two dampers 110 were attached to the set installation positions and a condition in which four dampers 110 were attached were set. Regarding the type of damper 110, anti-vibration rubber and high-damping rubber were prepared. In addition to the condition in which the damper 110 was attached to the cable 120, a test condition (test condition 1) in which the damper 110 was not attached to the cable 120 was also set.

The first and second calculation formulas were used to calculate the tension of the cable 120 (the first and second calculation formulas were derived using the relational equation k ^* =ku+jkv based on "Equation 11"). In addition, the tension was calculated using a calculation method for calculating the tension without considering the presence of the damper 110 (hereinafter referred to as the "conventional calculation formula": see JP 11-271155 A). The tension obtained from the conventional calculation formula is shown in FIG. 8A using the symbol "T ₁ ". The tension obtained from the first calculation formula is shown in FIG. 8A using the symbol "T ₁ ^' ". The tension obtained from the second calculation formula is shown in FIG. 8A using the symbol "T ₂ ^' ".

The tension in the cable 120 was measured using the load cell 133. The tension obtained from the load cell 133 is shown in Figure 8A using the symbol " _T0 ". For all test conditions, the tension in the cable 120, _T0 , is 180.3 kN.

The ratio of the calculated tension values _T1 , _T1 ^' , _T2 ^' to the actual measured tension _T0 was evaluated as the accuracy. The closer this ratio is to 100%, the closer the calculated values _T1 , _T1 ^' , _T2 ^' are to the actual measured tension _T0 . Figure 8B is a graph showing this ratio.

For test condition 1 in which the damper 110 was not attached to the cable 120, all of the calculated values T ₁ , T ₁ ^' , and T ₂ ^' were close to the actual measured value T _0. For test conditions 2 to 16 in which the damper 110 was attached to the cable 120, at least one of the calculated values T ₁ ^' and T ₂ ^' was closer to the actual measured value T ₀ than the calculated value T _1. It was therefore confirmed that the calculation method described with reference to Fig. 3 can calculate the tension T ₀ acting on the cable 120 with higher accuracy than the conventional calculation method.

For test conditions 8, 11, and 14, in which anti-vibration rubber was used as the damper 110, the calculated value _T1 ^' obtained from the first calculation formula is closer to the actual measured value _T0 compared to the other test conditions. On the other hand, the calculated value _T2 ^' obtained from the second calculation formula is farther from the actual measured value _T0 in test conditions 8, 11, and 14. Regarding the test results, it was found that when anti-vibration rubber is used as the damper 110, it is preferable to use the first calculation formula to calculate the tension.

For test conditions 9, 10, 12, 13, 15, and 16 in which high damping rubber was used as damper 110, the calculated value _T2 ^' obtained from the second calculation formula is closer to the actual measured value _T0 compared to the other test conditions. On the other hand, the calculated value _T1 ^' obtained from the first calculation formula is farther from the actual measured value _T0 in test conditions 9, 10, 12, 13, 15, and 16. Regarding the test results, it was found that when high damping rubber is used as damper 110, it is preferable to use the second calculation formula to calculate tension.

In step S145 described with reference to FIG. 3, the optimal solution obtained from the calculation formula that yields the smallest residual sum of squares is determined as the final optimal solution. Therefore, whether the damper 110 is vibration-isolating rubber or high-damping rubber, the calculation method described with reference to FIG. 3 can calculate the tension acting on the cable 120 with high accuracy.

The first calculation formula ("Equation 16") and the second calculation formula ("Equation 24") are derived using the boundary condition ("Equation 7") which indicates that the cable is supported by a damper. Therefore, the tensions " _T1 ^' " and " _T2 ^' " obtained from the first calculation formula and the second calculation formula are calculated taking into account the characteristics of the damper (spring constant, damping constant, mass, etc.). In other words, the first calculation formula and the second calculation formula can be suitably used to calculate the tension acting on a cable to which a damper is attached. As described with reference to Figures 8A and 8B, the tensions " _T1 ^' " and " _T2 ^' " obtained from these calculation formulas have higher accuracy than the tension " _T1 " obtained from the conventional calculation formula.

Regarding the calculation of tensions "T ₁ ^' " and "T ₂ ^' ", a plurality of tension candidate values, a plurality of bending stiffness candidate values, and a plurality of characteristic candidate values (spring constant candidate value, damping constant candidate value, and mass candidate value) are substituted into the first calculation formula and the second calculation formula (steps S125 and S130 in FIG. 3). When these calculated values are substituted into the first calculation formula and the second calculation formula, a plurality of candidate values of natural frequencies are obtained. The residual sum of squares of these candidate values of natural frequencies and the actual measured values of natural frequencies are calculated using the first objective function and the second objective function ("Equation 29" and "Equation 30"). The first objective function and the second objective function include a term expressed using the mode order i or the number n of half wavelengths, and therefore can be suitably used to evaluate the deviation between the candidate value of natural frequency and the actual measured value of natural frequency over the mode order i or the number n of half wavelengths used in the calculation.

Multiple local minima of the first objective function and the second objective function are found using the search method described with reference to FIG. 6 (step S220 in FIG. 5). Then, the tension candidate value, bending stiffness candidate value, and characteristic candidate value that give the smallest local minima among these local minima are extracted as optimal solutions (step S230 in FIG. 5). Therefore, both the optimal solutions obtained from the first calculation formula and the second calculation formula can accurately represent the tension acting on the cable, the bending stiffness of the cable, and the characteristics of the damper.

8A and 8B, depending on the type of damper (or the installation conditions of the damper), the optimal solution obtained from one of the first and second calculation formulas may be less accurate than the other optimal solution. If the sum of squared residuals based on the first calculation formula is smaller than the sum of squared residuals based on the second calculation formula, the optimal solution obtained from the first calculation formula is determined as the final optimal solution in step S145 of FIG. 3. Conversely, if the sum of squared residuals based on the second calculation formula is smaller than the sum of squared residuals based on the first calculation formula, the optimal solution obtained from the second calculation formula is determined as the final optimal solution in step S145 of FIG. 3. Therefore, under various conditions related to the damper, the calculation method described with reference to FIG. 3 can accurately calculate the cable tension, bending stiffness, and damper characteristics.

In step S150 of FIG. 3, the cable tension, bending stiffness, damper characteristics, and residual sum of squares are output. However, the output targets may be determined to suit the purpose of the calculation. If only the cable tension is required, only the cable tension may be output. If the purpose of the calculation is to investigate the damper characteristics, only the damper characteristics may be output.

When it is known from the damper installation conditions that it is preferable to use either the first or second calculation formula, only the preferable one of these calculation formulas may be used for the calculation. FIG. 9 shows a calculation process using only one of the first and second calculation formulas. The calculation process of FIG. 9 differs from the calculation process described with reference to FIG. 3 in that the optimal solution is calculated using only one of the first and second calculation formulas (step S127) instead of steps S125 and S130 in FIG. 3. In addition, the calculation process of FIG. 9 differs from the calculation process described with reference to FIG. 3 in that step S145, in which the minimum sum of squares of the residuals of the first and second calculation formulas is compared, is not executed.

Regarding the calculation process shown in FIG. 9, if the response value obtained from the acceleration sensor 135 attached to the span 122 does not include a response value derived from the short span 121, a calculation process using only the second calculation formula is effective. If the response value obtained from the acceleration sensor 135 attached to the span 122 includes a response value derived from the short span 121, a calculation process using only the first calculation formula is effective.

In the above embodiment, the second calculation formula ("Equation 24") and the second objective function ("Equation 30") are summarized for the relatively long span 122. In the above calculation method, the third calculation formula ("Equation 23") and the third objective function summarized for the relatively short span 121 may be used. In this case, in step S120 described with reference to FIG. 3, the actual measurement value of the natural frequency of the short span 121 is extracted. The extracted actual measurement value is compared with the calculated value of the natural frequency obtained from the third calculation formula, and the deviation between them is evaluated by the third objective function. The tension candidate value when the smallest deviation is obtained is calculated as the tension of the cable 120.

As explained with reference to Figures 8A and 8B, the above calculation method can achieve high accuracy even under conditions without a damper. Therefore, it is applicable to cables that can be modeled as one-dimensional beams, regardless of whether they have a damper or not.

The range of the orders of the natural frequencies used in the calculation may be determined based on the number of peaks of vibration intensity that can be found from the measured vibration data. The range of the orders of the natural frequencies in the test described in relation to Figures 8A and 8B was 1 to 7. In the test, the range of the orders of the natural frequencies was changed to additionally verify the accuracy. It was verified that when the order of the natural frequencies was changed from the range of 1 to 6 to the range of 1 to 10, the tension T was calculated with high accuracy under all test conditions.

In the above-described embodiment, the cable 120 is illustrated as a linear body. However, the above-described techniques are also applicable to other members that can be modeled as one-dimensional beams (e.g., steel bars and solid wires).

In the above-described embodiment, an impact is applied to the cable 120 to obtain an actual measurement of the natural frequency of the cable 120. However, the actual measurement of the natural frequency of the cable 120 may also be obtained based on vibrations caused by other external forces acting on the cable 120 (e.g., wind or vehicle vibrations).

The techniques described in relation to the above embodiments are suitable for use in investigating tensions acting on various linear bodies that can be modeled as one-dimensional beams, and for investigating the characteristics of dampers attached to cables.

110: Damper 120: Cable 121, 122: Span

Claims (8)

A method for calculating tension in a linear body having a damper attached thereto, comprising the steps of:
Detecting vibrations at any point on the linear body, and obtaining actual measured values of natural frequencies of multiple modes based on the detected vibrations;
calculating the tension of the linear body based on a comparison between the calculated values of the natural frequencies of the multiple modes and the actual measured values, using an equation that represents the relationship between the natural frequencies of the multiple modes, tension, bending rigidity, and phase shifts caused by the characteristics of the damper, which is set using a boundary condition that represents that the damper is disposed on the linear body. 2. The calculation method according to claim 1, wherein in the step of calculating the tension of the linear body, the value of the tension when a value of an objective function set using the tension, the bending rigidity, and the phase shift as variables to represent the deviation between the actual measured value and the calculated value is minimized is determined as the tension acting on the linear body. and comparing the value of the tension at which the objective function is minimized to a predetermined threshold.
The calculation method according to claim 2 , further comprising determining, under a condition that the value of the tension when the objective function is minimized is below the threshold value, the value of the tension when the objective function is minimized as the tension acting on the linear body. obtaining the actual measured values includes obtaining actual measured values of natural frequencies of a plurality of modes of the entire length of the linear body,
The calculation method according to claim 2 or 3, wherein in the step of calculating the tension of the linear body, the tension of the linear body is calculated using a first objective function as the objective function, the calculated value being set to represent natural frequencies of multiple modes over the entire length of the linear body. In the step of obtaining the actual measured values, natural frequencies of multiple modes of a longer span of the two spans of the linear body separated by the damper are obtained,
The calculation method according to claim 2 or 3, wherein in the step of calculating the tension of the linear body, the tension of the linear body is calculated using a second objective function as the objective function, the calculated value of which is set to represent natural frequencies of multiple modes of the long span. In the step of obtaining the actual measured values, actual measured values of natural frequencies of multiple modes over the entire length of the linear body are obtained, and actual measured values of natural frequencies of multiple modes of the longer of the two spans of the linear body divided by the damper are obtained,
4. The calculation method according to claim 2 or 3, wherein in the step of calculating the tension of the linear body, a first objective function is used as the objective function, the calculated value of which is set to represent the natural frequencies of multiple modes over the entire length of the linear body, and a second objective function is used as the objective function, the calculated value of which is set to represent the natural frequencies of multiple modes over the long span, and the value of the tension when the smaller of the deviations calculated by the first objective function and the second objective function is obtained is determined to be the tension acting on the linear body. 1. A method for calculating a characteristic of a damper attached to a linear body, comprising the steps of:
Detecting vibrations at any point on the linear body, and obtaining actual measured values of natural frequencies of multiple modes based on the detected vibrations;
and calculating the characteristics of the damper based on a comparison between the calculated values of the natural frequencies of the multiple modes and the actual measured values, using equations that represent relationships among the natural frequencies of the multiple modes, tension, bending rigidity, and phase shifts caused by the characteristics of the damper, which are set using boundary conditions that represent that the damper is disposed on the linear body. A method for calculating the bending stiffness of a linear body to which a damper is attached, comprising the steps of:
Detecting vibrations at any point on the linear body, and obtaining actual measured values of natural frequencies of multiple modes based on the detected vibrations;
and calculating the bending stiffness of the linear body based on a comparison between the calculated values of the natural frequencies of the multiple modes and the actual measured values, using an equation that represents the relationship between the natural frequencies of the multiple modes, tension, bending stiffness, and phase shifts caused by the characteristics of the damper, which is set using a boundary condition that represents that the damper is disposed on the linear body.

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