JP7680915B2 - Calculation method for tension, bending rigidity and rotational rigidity at both ends of a linear body - Google Patents

Description

Article 30, paragraph 2 of the Patent Act applies. 1. Date of website posting: May 23, 2021. 2. Website address: Home page of the 2021 Annual Academic Conference of the Kansai Branch of the Japan Society of Civil Engineers https://www.ac-research.jp/jsce/js-kansai/2021/precede/login.php 3. Publisher: Norimasa Takatsuru

Article 30, paragraph 2 of the Patent Act applies. 1. Date of website posting: August 2, 2021. 2. Website address: 2021 National Conference of the Japan Society of Civil Engineers in Kanto Online Homepage https://confit.atlas.jp/guide/event/jsce2021/top 3. Publisher: Norimasa Takatsuru

Article 30, paragraph 2 of the Patent Act applies. 1. Date of website posting: May 15, 2021. 2. Website address: The 24th (2021) Symposium on Applied Mechanics website https://confit.atlas.jp/guide/event/jsceam2021/proceedings/list. 3. Publisher: Norimasa Takatsuru.

The present invention relates to a method for calculating the tension, bending stiffness, and rotational stiffness at both ends of two linear bodies held by a holding device so as to cross each other.

Various methods have been devised for calculating the tension acting on linear objects 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, and the calculated natural frequency is compared with the actual measured natural frequency appearing in vibration data obtained by applying an impact to an actual cable. The cable tension is calculated based on the results of this comparison.

Japanese Patent Application Publication No. 9-101289 Japanese Patent Application Publication No. 11-271155

The above-mentioned calculation method, which models a one-dimensional beam, can be suitably used to calculate the tension of a single linear body. However, even if the above-mentioned calculation method is applied to a structure in which two linear bodies are arranged crossing each other and the intersection of these linear bodies is grasped by a grasping device, only the tension is calculated under a condition in which these linear bodies move independently at the intersection. In other words, with the conventional tension calculation method, it is not possible to obtain a highly accurate calculated value of tension due to the influence of the linear bodies being mutually constrained at the intersection.

The present invention aims to provide a method for calculating the tension, bending stiffness, and rotational stiffness at both ends of two linear bodies grasped by a grasping device so that they cross each other.

A calculation method according to one aspect of the present invention can be used to calculate the tension, bending stiffness, and rotation stiffness at both ends of two linear bodies held by a holding device so as to intersect with each other. The calculation method includes a measurement step of obtaining actual measured values of natural frequencies of multiple modes for each of the two linear bodies based on the vibration of the two linear bodies, and a calculation step of calculating the tension, bending stiffness, and rotation stiffness using a calculation standard formula set using boundary conditions that represent that the intersection of the two linear bodies is held by the holding device and that both ends of each of the two linear bodies have the rotation stiffness, and the actual measured values of the natural frequencies of the multiple modes. The calculation standard formula is expressed as an equation in which a function including variables of the tension, bending stiffness, rotation stiffness, and the natural frequencies of the multiple modes is equal to a predetermined value. In the calculation process, the candidate values set for the tension, bending stiffness, and rotation stiffness, and the actual measured values of the natural frequencies of the multiple modes are substituted into the corresponding variables of the function to obtain the calculated values of the function, and the tension, bending stiffness, and rotation stiffness are calculated based on a comparison between the obtained calculated values and the predetermined values.

In the above-described configuration, the calculation standard equation used in the calculation step for calculating tension and the like is set using the following two boundary conditions.
The intersecting portion of the two linear bodies is held by a holding device.
- Both ends of each of the two linear bodies have rotational rigidity.

Because the boundary condition is that the intersection of the two linear bodies is gripped by a gripping device, the calculation standard formula can represent the vibration mode in which the two linear bodies vibrate together at the intersection. In addition, because rotational rigidity represents the resistance of the linear body to bending at both ends, the calculation standard formula set using this boundary condition can represent the degree of constraint against bending at both ends of the linear body in an actual building.

The calculation standard equation set using the above-mentioned boundary conditions is expressed as an equation in which a function including variables of tension, bending stiffness, rotation stiffness, and natural frequencies of multiple modes is equal to a predetermined value. The closer the calculated value obtained by substituting numerical values for these variables is to the above-mentioned predetermined value, the closer the substituted numerical value is to the true value that satisfies the calculation standard equation. For this reason, in the calculation process, candidate values of tension, bending stiffness, and rotation stiffness, and the measured values of natural frequencies of multiple modes are respectively substituted for the corresponding variables. By comparing the calculated value obtained by this substitution process with the predetermined value taken by the function of the calculation standard equation, it is possible to know whether the substituted candidate value is close to the true value that satisfies the calculation standard equation. In particular, in the above-mentioned calculation method, the substitution process is performed using rotation stiffness as a variable, and tension, etc. can be calculated taking into account the degree of constraint against bending at both ends of the linear body. Therefore, the calculated tension, etc. has high accuracy.

With respect to the above-mentioned configuration, the calculation formula may express the relationship between the natural frequencies, the tension, the bending stiffness, and the rotational stiffness of the multiple modes without including the modal order.

In the calculation methods of tension and the like in Patent Documents 1 and 2, it is necessary to assign the measured values of the natural frequencies of multiple modes to the mode orders in correspondence with each other. However, it is expected that the correspondence between the natural frequencies and the mode orders will be performed incorrectly. For example, if a natural frequency with a mode order of 3 is overlooked, a natural frequency with a mode order of 4 may actually be obtained as a natural frequency with a mode order of 3. In this case, 3 may be assigned to the variable for the mode order in the calculation formula in Patent Documents 1 and 2, while a natural frequency with a mode order of 4 may be assigned to the variable for the natural frequency in the calculation formula. If such an incorrect assignment process is performed, the tension and the like of the linear body cannot be calculated with high accuracy.

On the other hand, the above-mentioned calculation method uses a calculation standard formula that does not include the mode order. Therefore, the actual measured value of the natural frequency of the linear body can be substituted into the function of the calculation standard formula without corresponding to the mode order, preventing the above-mentioned erroneous substitution process.

With respect to the above-mentioned configuration, the boundary conditions may include a condition indicating that the displacement directions and displacement amounts of the two linear bodies at the intersection are equal.

When two linear bodies impact a structure held by a holding device at an intersection, the displacements (i.e., the direction and amount of displacement) of the two linear bodies at the intersection are equal. In the above-described configuration, the boundary condition that the displacements (i.e., the direction and amount of displacement) of the two linear bodies are equal is used, making it possible to calculate the tensions of the two linear bodies held by the holding device at the intersection.

Regarding the above-mentioned configuration, the function of the calculation standard formula may express the sum of the force acting on the intersection of one of the two linear bodies and the force acting on the intersection of the other linear body using variables of the natural frequency of the multiple modes, the tension, the bending stiffness, and the rotation stiffness. The calculation standard formula may be expressed as an equation in which the function is equal to a value of zero. In the calculation step, the candidate values of the tension, the bending stiffness, and the rotation stiffness, and the actual measured value of the natural frequency of the multiple modes may be substituted for the corresponding variables of the function to obtain the calculated value. The tension, the bending stiffness, and the rotation stiffness may be calculated based on a comparison between the obtained calculated value and the value of zero.

In the above-mentioned configuration, the two linear bodies are held by the holding parts at their intersections, so that the forces acting on these linear bodies at the intersections are balanced. In other words, the sum of the force acting on one linear body at the intersection and the force acting on the other linear body at the intersection is equal to zero, and this balance of forces is expressed by the calculation formula. The force acting on the intersections can be expressed as a function including variables for tension, bending rigidity, rotational rigidity, and natural vibration frequencies of multiple modes. The closer the calculated value obtained from the function by substituting numerical values for these variables is to zero, the closer the substituted numerical values are to the conditions under which the balance of forces at the intersections is established. For this reason, in the calculation process, tension, etc. are calculated by comparing the calculated value obtained by the substitution process with the value of zero.

Regarding the above-mentioned configuration, in the actual measurement step, the Fourier amplitudes at the natural frequencies of the multiple modes corresponding to the actual values of the natural frequencies of the multiple modes are measured at two different positions on each of the two linear bodies, and an actual value of a Fourier amplitude ratio, which is the ratio of the actual values of the Fourier amplitudes obtained at the two positions, is obtained. The function of the calculation standard formula is set to represent the difference between the theoretical value obtained from the theoretical formula of the Fourier amplitude ratio expressed as a function of the tension, the bending rigidity, the rotational rigidity, and the natural frequencies of the multiple modes, and the actual value of the Fourier amplitude ratio. The calculation standard formula is set to represent the relationship in which the difference between the theoretical value of the Fourier amplitude ratio and the actual value of the Fourier amplitude ratio is equal to zero. In the calculation process, the actual value of the Fourier amplitude ratio acquired in the actual measurement process is substituted into the term of the actual value of the Fourier amplitude ratio in the function, and the candidate values of the tension, bending stiffness, and rotational stiffness, and the actual values of the natural frequencies of the multiple modes are substituted into the theoretical formula to obtain the difference between the actual value of the Fourier amplitude ratio and the theoretical value of the Fourier amplitude ratio as the calculated value, and the tension, bending stiffness, and rotational stiffness are calculated based on a comparison between the calculated value of the difference and the value of zero.

The theoretical formula for the Fourier amplitude ratio can be expressed as a function of tension, bending stiffness, rotational stiffness, and the natural frequencies of multiple modes. In the above configuration, the theoretical value of the Fourier amplitude ratio is calculated by substituting the candidate values of tension, bending stiffness, and rotational stiffness and the actual measured values of the natural frequencies of multiple modes into this theoretical formula. If this theoretical value is close to the actual measured value of the Fourier amplitude ratio, the calculated value, which is the difference between the actual and theoretical values of the Fourier amplitude ratio, is close to zero. Therefore, by comparing the calculated value with zero, it is possible to determine whether the candidate values of tension, bending stiffness, and rotational stiffness substituted into the theoretical formula are close to the true values that establish the calculation standard formula.

In the above-mentioned configuration, in the calculation step, the calculation process may repeatedly obtain other calculated values by substituting another candidate value set for at least one of the tension, the bending stiffness, and the rotation stiffness, and the actual measured value of the natural frequency of the multiple modes into the function, thereby obtaining multiple calculated values, and the candidate value that is closest to the predetermined value among the multiple calculated values may be determined as the calculated value of the tension, the bending stiffness, and the rotation stiffness.

In the above-mentioned calculation method, multiple candidate values are substituted into the function of the calculation standard formula, resulting in multiple calculated values. Among these calculated values, the candidate value obtained is close to the predetermined value when the calculation standard formula holds, and is relatively close to the true value that holds for the calculation standard formula. Since such candidate values are determined as the calculated values of tension, etc., the accuracy of the calculated values of tension, etc. is improved.

Regarding the above-mentioned configuration, in the calculation step, a substitution process may be repeated in which another candidate value set for at least one of the tension, the bending stiffness, and the rotational stiffness, and the actual measured values of the natural frequencies of the multiple modes are substituted into the function until the difference between the calculated value and the predetermined value becomes less than a predetermined threshold value.

In the above-mentioned calculation method, in order to guarantee the accuracy of the calculated values of tension, etc., a threshold is set for the difference between the calculated value and a predetermined value when the calculation standard formula is satisfied. The substitution process is repeated until the difference between these values becomes less than the threshold, so that the calculated values of tension, etc. have the accuracy guaranteed by the threshold.

The above-described technique makes it possible to calculate the tension and bending stiffness of each of two linear bodies grasped by a grasping device so that they cross each other.

FIG. 1 is a schematic diagram of a structure having two crossed cables. FIG. 1 is a schematic diagram of a vibration model in which a cable is modeled as a one-dimensional beam. FIG. 1 is a schematic diagram of forces acting around a cable intersection. 10 is a schematic flowchart of a method for calculating tension and the like. This data was obtained by performing a Fourier transform on the time history response values of the cable vibration. 1 is a graph showing a method for searching for an optimal solution based on the MultiStart method. This data was obtained by performing a Fourier transform on the time history response values of the cable vibration. 10 is a schematic flowchart of a method for calculating tension and the like. FIG. 1 is a schematic diagram of a structure having two crossed cables. 10 is a schematic flowchart of a method for calculating tension and the like. This data was obtained by performing a Fourier transform on the time history response values of the cable vibration.

Figure 1 is a schematic plan view of a structure 100 in which two cables 121, 122 (linear bodies) cross each other. The cables 121, 122 are fixed in a tensioned state. Both ends of the cables 121, 122 are supported with a predetermined rotational rigidity.

A gripping device 130 is attached to the intersection of the cables 121 and 122. The gripping device 130 is configured to grip the cables 121 and 122 at their intersection. Therefore, the vibration amplitude and direction of both cables 121 and 122 are equal at the intersection.

The cables 121 and 122 are each divided into two spans by the gripping device 130 attached to the intersection. In the following description, the length of the cable 121 from the left end of the cable 121 to the gripping device 130 in FIG. 1 is referred to as "span 171." In the following description, the length of the cable 121 from the right end of the cable 121 to the gripping device 130 in FIG. 1 is referred to as "span 172." In the following description, the length of the cable 121 from the left end of the cable 122 to the gripping device 130 in FIG. 1 is referred to as "span 173." In the following description, the length of the cable 122 from the right end of the cable 121 to the gripping device 130 in FIG. 1 is referred to as "span 174."

An acceleration sensor 141 is attached to span 171 of cable 121, and an acceleration sensor 143 is attached to span 173 of cable 122. Acceleration sensors 141, 143 are configured to detect the acceleration of vibrations generated in cables 121, 122 due to an impact given to at least one of cables 121, 122. In this embodiment, cables 121, 122 are vibrated in a direction perpendicular to the plane including cables 121, 122 (direction perpendicular to the paper surface of FIG. 1). In the following description, the direction perpendicular to the plane including cables 121, 122 is referred to as the "out-of-plane direction."

The acceleration sensors 141, 143 are electrically connected to a data collection device 160 (e.g., a personal computer), and the acceleration data acquired by the acceleration sensors 141, 143 is stored in the data collection device 160 as a time history response value. The data collection device 160 is configured to perform a predetermined analysis process on the acceleration data, and through this analysis process, data on the actual measured values of the natural frequencies (natural frequencies in the out-of-plane direction) of the cables 121, 122 is obtained. Based on the obtained natural frequency data, the tensions, etc. of the cables 121, 122 are calculated.

<Derivation of the calculation formula>
In order to calculate the tensions, etc. of the cables 121, 122, a calculation standard formula based on the vibration equation for the cables 121, 122 is derived. The calculation standard formula is expressed as a formula in which a function including variables such as tensions is equal to a predetermined value. Using the predetermined value taken by the function of the calculation standard formula as a standard, it is determined whether the calculated values of tensions, etc. obtained by the calculation processing described below are close to true values.

The calculation formula can be set by modeling each of cables 121 and 122 as a one-dimensional beam supported at both ends with a predetermined rotational stiffness, as shown in FIG. 2. In FIG. 2, subscript k takes the value of 1 or 2. When subscript k takes the value of 1, the model in FIG. 2 refers to cable 121 in FIG. 1. When subscript k takes the value of 2, the model in FIG. 2 refers to cable 122 in FIG. 1. In the following description, the term "cable k" refers to one of cables 121 and 122. That is, when "k" takes the value of 1, the term "cable k" refers to cable 121, and when "k" takes the value of 2, the term "cable k" refers to cable 122.

In the model of FIG. 2, a rotational spring is inserted into both ends of cable k to indicate that both ends of cable k have a predetermined rotational stiffness, and the rotational stiffness of both ends of cable k is represented by the stiffness of the rotational spring (i.e., rotational spring stiffness K _k ).

In Fig. 2, an intersection 131 is shown on cable k. The intersection 131 refers to the portion where the cables 121 and 122 intersect (i.e., the gripping position of the gripping device 130 in Fig. 1). In the following description, the length of the span from the left end of cable k to the intersection 131 (i.e., spans 171 and 173 of cables 121 and 122) is defined as _Lk1 . The length of the span from the other end of cable k to the intersection 131 (i.e., spans 172 and 174 of cables 121 and 122) is defined as _Lk2 . The total length of cable k ( _Lk1 + _Lk2 ) is defined as _Lk .

In Fig. 2, an _xk1 axis extends rightward from the left end of cable k as the origin, and an _xk2 axis extends leftward from the right end of cable k as the origin. The direction perpendicular to the paper surface of Fig. 2 is the out-of-plane direction, and the displacement of cable k in this direction is indicated by _wkd . Subscript d is used to indicate the span of cable k and takes a value of 1 or 2. Subscript d with a value of 1 indicates the left span of cable k. Subscript d with a value of 2 indicates the right span of cable k. That is, displacement amount " _w11 " indicates the displacement amount of span 171 of cable 121 in the out-of-plane direction. Displacement amount " _w12 " indicates the displacement amount of span 172 of cable 121 in the out-of-plane direction. Displacement amount " _w21 " indicates the displacement amount of span 173 of cable 122 in the out-of-plane direction. Displacement "w ₂₂ " represents the amount of displacement of span 174 of cable 122 in the out-of-plane direction.

The vibration equation of cable k at time t is given as follows. Note that the mass of the gripping device 130 is not taken into account.

When the above vibration equation (equation 1) is solved using the variable separation method, the following relationship is obtained.

Using this relationship (equation 2), the vibration equation (equation 1) can be rewritten as follows:

The general solution to the vibration equation in "Equation 3" is expressed as follows.

At both ends of cable k (x _kd =0), no displacement of cable k occurs, and since rotational springs having a rotational spring stiffness K _k are inserted at both ends of cable k, the following boundary conditions are established regarding the displacement and bending moment at both ends of cable k. The following boundary conditions regarding the bending moment indicate that both ends of cables 121 and 122 have rotational stiffness.

At the intersection 131 (x _k1 =L _k1 , x _k2 =L _k2 ) of cable k, the displacement, inclination, and bending moment of the left and right spans of the intersection 131 are equal, and the following boundary condition is established.

Here, we consider the effect of restraint by the gripping device 130 on the cables 121 (k=1) and 122 (k=2) at the intersection 131 ( _xk1 = _Lk1 , _xk2 = _Lk2 ). First, at the intersection 131 ( _xk1 = _Lk1 , _xk2 = _Lk2 ), the gripping device 130 grips the cables 121 (k=1) and 122 (k=2), and the displacement directions and amounts of displacement of these cables 121 and 122 are equal. Furthermore, at the intersection 131 ( _xk1 = _Lk1 , _xk2 = _Lk2 ), the forces acting on the cables 121 (k=1) and 122 (k=2) are balanced.

First, the boundary condition that the displacement directions and displacement amounts of the cables 121 and 122 are equal at the intersection 131 (x _k1 =L _k1 , x _k2 =L _k2 ) is expressed as follows.

Next, the balance of forces at the intersection 131 (x _k1 =L _k1 , x _k2 =L _k2 ) when the cables 121 and 122 are held by the holding device 130 at the intersection 131 will be described with reference to FIG.

At the intersection 131, a shear force _Qk1 ( _Lk1 ) acts on the left span of cable k, and a shear force _Qk2 ( _Lk2 ) acts on the right span of cable k. These shear forces are balanced at the intersection 131. Therefore, the following boundary conditions are obtained for the balance of forces at the intersection 131 ( _xk1 = _Lk1 , _xk2 = _Lk2 ) when the cables 121, 122 are gripped by the gripping device 130 at the intersection 131.

The first term on the left side of "Equation 8" represents the force acting on intersection 131 on cable 121, and the second term represents the force acting on intersection 131 on cable 122. The left side of "Equation 8" represents the sum of the forces acting on intersection 131 of cables 121, 122. "Equation 8" represents the state in which the sum of the forces acting on intersection 131 of cables 121, 122 is equal to zero, in other words, the forces are balanced.

The determinant obtained by applying the boundary condition equation in "Equation 5" above to the general solution equation in "Equation 4" is shown below.

The boundary condition equations for the displacement and bending moment in the above "Equation 6" are expressed by the following determinant:

This determinant (number 10) can be expressed as follows using the general solution equation of "number 4" and the second derivative of this general solution equation:

By substituting the above "Number 9" into "Number 11", we obtain the following relational equation.

Using the matrix F _kd of Equation 12 above, Equation 10 can be rewritten as follows:

Next, the first derivative of the general solution equation of "Mathematical formula 4" is expressed as follows.

By substituting the determinant of "Number 9" into "Number 14", we obtain the following relationship.

Using the matrix H _kd in "Equation 15", the boundary condition for the slope in "Equation 6" is expressed as follows:

By substituting the equation in "Number 13" into "Number 16", we obtain the following equation.

Based on "Equation 17", the following equation is obtained:

Next, the general solution of "Equation 4" can be expressed using a matrix as follows:

By substituting "Number 9" into "Number 19", we get the following equation.

Using "Number 20", the above "Number 7" (the equation for the boundary condition related to the displacement at the intersection 131) can be expressed as follows.

By substituting "Number 18" into "Number 21", we obtain the following equation.

Under the assumption that the displacement at intersection 131 is not zero, the above "equation 22" can be rewritten as follows:

Next, the third derivative of the general solution equation of "Mathematical formula 4" is expressed as follows.

By substituting "Number 9" into "Number 24", we get the following equation.

Using the matrix J _kd in "Equation 25", the equation in the curly brackets of "Equation 8" (the equation for the boundary condition related to the balance of forces) can be rewritten as follows:

By substituting "Number 13" into "Number 26," "Number 26" can be transformed as follows.

By substituting "Number 18" into "Number 27", "Number 27" can be transformed as follows.

Based on "Number 28," "Number 8" (the equation for the boundary condition related to the balance of forces) can be rewritten as follows.

By substituting "Number 29" into "Number 23", we get the following equation.

For "Number 30" to have a solution other than the trivial one, the value in the curly brackets must be zero. This gives us the following relationship:

Since "Number 31" contains a hyperbolic function, in order to prevent problems such as cancellation of digits, the numerator and denominator of the first term of "Number 31" are divided by "m ₁₁ ", and the numerator and denominator of the second term are divided by "m ₂₁ ". As a result of this process, "Number 31" is transformed as follows:

The first term of "number 32" is represented as "fun1" and the second term is represented as "fun2".

When each term in fun1 is simplified and processed to prevent problems such as truncation, each term in fun1 is expanded as follows:

In addition, "Na _nm11 ", "Na _nm12 ", "Na _Fm11 ", and "Na _Fm12 " in "Equation 34" are expressed as follows.

Using "Na _nm11 ", "Na _nm12 ", "Na _Fm11 ", and "Na _Fm12 ", the above-mentioned "fun1" is expressed as follows.

Here, the rotational spring stiffness r _k at both ends of the cable k is defined as follows:

In the above-mentioned "Equation 44", the closer the rotational spring stiffness r _k is to zero, the closer both ends of the cable k are to being supported as free ends. On the other hand, the closer the rotational spring stiffness r _k is to 1, the closer both ends of the cable k are to being supported as fixed ends.

Regarding "fun1" in "Equation 43", if the case is divided into when the rotational spring stiffness _r1 is less than 1 and when the rotational spring stiffness _r1 is equal to 1, "fun1" is expressed as follows. Note that when the rotational spring stiffness _r1 is equal to 1, _P1 becomes infinity. In order to avoid calculations that include infinity, the case is divided according to the value of the rotational spring stiffness _r1 .

Regarding “fun2” in “Equation 43”, when the rotational spring stiffness _r2 is less than 1 and when the rotational spring stiffness _r2 is equal to 1, “fun2” can be expressed as follows:

"q _k0 ", "q _k1 ", "q _k2 ", "q _k2 ", "s _k0 ", "s k1 ", "s _k2 ", and "s _k2 " in "Equation 45" and "Equation ₄₆ " are expressed as follows.

Using "fun1" and "fun2" defined as above, the above "Number 31" can be rewritten as follows:

The denominators of "fun1" and "fun2" (see "Equation 45") are represented as "fun3" and "fun4", respectively, as shown in the following equation.

When "fun3" and "fun4" become zero, the above-mentioned "Number 58" no longer holds, so "Number 58" is multiplied by the product of "fun3" and "fun4" as follows. Note that "fun3" takes a value of zero when the amplitude of cable 121 becomes zero at intersection 131, and "fun4" takes a value of zero when the amplitude of cable 122 becomes zero at intersection 131. By multiplying "Number 58" by the product of "fun3" and "fun4", it is possible to include the case where the amplitude at intersection 131 becomes zero when one of cables 121, 122 does not vibrate at all while the other cable only vibrates.

The formula "60" can be expressed as the formula for the i-th mode as follows. In the formula below, the mode number i is expressed as a superscript. The formula below is normalized by dividing by the values of "fun3" and "fun4" of the first mode. In this embodiment, the formula below is used as the calculation standard formula.

"Number 61" is shown as an equation in which the function on the left side is equal to zero. As described above, "Number 61" is set based on "Number 8" which represents the state of balance of forces at intersection 131, and when "Number 61" is true, the forces are balanced at intersection 131.

The function on the left side of "Equation 61" includes the following constants and variables (see "Equation 4", "Equation 5", "Equation 45" to "Equation 57" and "Equation 59"), and the calculation standard formula of "Equation 61" expresses the relationship between the following constants and variables. Note that the function on the left side of "Equation 61" does not include the mode order i as a variable.
Total length of cable k: L _k
Position of intersection 131 (i.e., length of span of cable k): L _k1 , L _k2
Density of cable k: ρ _k
Cross-sectional area of cable k: A _k
Measured value of the natural frequency of cable k: f _k ⁱ
Tension of cable k: T _k
Bending rigidity of cable k: E _k I _k
Rotational spring stiffness: _K

(Equation for calculating tension _Tk etc.)
The function on the left side of the calculation standard equation of "Equation 61" is used as an arithmetic expression G ₁ ⁱ for calculating the tension T _k and the like.

The calculated value of the arithmetic expression G ₁ ⁱ is calculated by substituting numerical values into the constants and variables of the arithmetic expression G ₁ ⁱ of "Equation 62". The closer the calculated value calculated from the arithmetic expression G ₁ ⁱ is to the value of the calculation standard formula of "Equation 61" (i.e., zero), the closer the numerical value substituted into the arithmetic expression G ₁ ⁱ is to the true value when the calculation standard formula of "Equation 61" is satisfied. Therefore, the tension T _k and the like can be calculated by comparing the calculated value calculated from the arithmetic expression G ₁ ⁱ with the value of the calculation standard formula of "Equation 61" (i.e., zero). For this comparison process, the sum of squares of the arithmetic expression G ₁ ⁱ is set as shown below.

If the sum of squares of the calculation formula G ₁ ⁱ is close to zero, it can be seen that the numerical value substituted into the calculation formula G ₁ ⁱ across a plurality of vibration modes is close to the true value when the calculation standard formula of "Equation 61" holds. A specific method for calculating the tension T _k etc. using the sum of squares of the calculation formula G ₁ ⁱ will be described below with reference to FIG.

(Calculation method of tension _Tk, etc.)
The following structural data regarding cable k is acquired (step S105): The following structural data is substituted as fixed values into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63).
Total length of cable k: L _k
Position of intersection 131 (i.e., length of span of cable k): L _k1 , L _k2
Density of cable k: ρ _k
Cross-sectional area of cable k: A _k

After acquiring the structural data, as shown in Fig. 1, acceleration sensors 141, 143 are attached to the cables 121, 122, and a data collecting device 160 is connected to the acceleration sensors 141, 143. Then, a measurement process (steps S110 to S120 in Fig. 4) is performed to acquire actual measured values _fk ⁱ of the natural frequencies of the cables 121, 122 for a plurality of vibration modes.

In the actual measurement process, the cables 121 and 122 are vibrated in the out-of-plane direction (step S110). The acceleration of the vibration generated in the cables 121 and 122 is measured by the acceleration sensors 141 and 143. The measured acceleration of the vibration is recorded as a time history response value in the data collection device 160 (step S115). The data collection device 160 performs a Fourier transform on the time history response value, and the actual measured value of the natural frequency _fk ⁱ is obtained from the peak value of the data after the Fourier transform (step S120).

As a result of the above-mentioned Fourier transform, data such as that shown in Fig. 5 is obtained. The frequencies at which the vibration intensity peaks appear are acquired as the measured values of the natural frequencies _fk1 ^to _fk6 ^. Since the arithmetic expression _G1i does not include a variable for the mode order ⁱ , it is not ^necessary to acquire the measured values of the natural frequencies _fk1 ^to _fk6 in association with the mode order _i for the purpose ^of the substitution process ⁽ described later) into the arithmetic expression G1i. Therefore, for example, as shown in Fig. 5, the frequencies at which the peaks appear may be acquired as the measured values of the natural frequencies _fk1 ^to _fk6 in ascending order.

The structural data acquired in step S105 and the data of the measured values of the natural frequencies f _k ¹ to f _k ⁶ acquired in step S120 are substituted into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) (calculation step). As a result, the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) can be handled as a function including the tension T _k of the cable k, the bending stiffness E _k I _k of the cable k, and the rotational spring stiffness K _k as variables. After the structural data and the data of the measured values of the natural frequencies f _k ¹ to f _k ⁶ are substituted into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63), a search process (calculation step) is performed to find the optimal solution for the tension T _k of the cable k, the bending stiffness E _k I _k , and the rotational spring stiffness K _k (step S125).

The search process for obtaining the optimal solution of the tension T _k , bending stiffness E _k I _k and rotational spring stiffness K _k will be described with reference to Fig. 6. Fig. 6 is a conceptual graph of the sum of squares of the calculation formula G ₁ ⁱ (Equation 63). In this embodiment, the optimal solution of the tension T _k of cable k, bending stiffness E _k I _k of cable k and rotational spring stiffness K _k is calculated based on the MultiStart method.

The horizontal axis in Fig. 6 represents tension _Tk , bending stiffness _{EkIk ,} or rotational spring stiffness _Kk . Although the graph in Fig. 6 is drawn in two ^- dimensional coordinates, the calculation in the search process is performed on four-dimensional coordinates (four axes: the sum of squares of the calculation formula _G1i , tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk ).

N initial values are set for the tension T _k , bending stiffness E _k I _k and rotational spring stiffness K _k . FIG. 6 shows the first to sixth initial values. The set initial values are substituted into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63). The value of the tension T _k substituted into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) is a candidate value for the tension estimated to be acting on the cable k. The value of the bending stiffness E _k I _k substituted into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) is a candidate value for the bending stiffness of the cable k. The value of the rotational spring stiffness K _k substituted into the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) is a candidate value for the rotational spring stiffness of the rotational springs inserted at both ends of the cable k. These candidate values (i.e., substituted values) are referred to as "tension candidate value", "bending stiffness candidate value" and "spring stiffness candidate value" in the following description.

When an initial value is input and the calculated value of the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) is obtained, at least one of the tension candidate value, bending stiffness candidate value, and spring stiffness candidate value is changed, and the calculated value of the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) is calculated based on the new combination. It is determined whether or not the value of the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) obtained from the new combination is lower than the value of the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) obtained from the previous combination. Based on the degree of decrease in the calculated value of the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63), a new combination of the tension candidate value, bending stiffness candidate value, and spring stiffness candidate value is repeatedly set, and the minimum value of the sum of squares of the arithmetic expression G ₁ ⁱ (Equation 63) is searched for.

Four minimum values (hereinafter referred to as the "first minimum value", "second minimum value", "third minimum value" and "fourth minimum value") of the sum of squares (Equation 63) of the arithmetic expression G ₁ ⁱ are shown in Fig. 6. With respect to the graph in Fig. 6, when the search is started from the first and second initial values, the first minimum value can be obtained. When the search is started from the third initial value, the second minimum value can be obtained. When the search is started from the fourth and fifth initial values, the third minimum value can be obtained. When the search is started from the sixth initial value, the fourth minimum value can be obtained.

The smallest one is found from among the multiple minimum values obtained by the above-mentioned search process. The tension candidate value, bending stiffness candidate value, and spring stiffness candidate value when the smallest minimum value is obtained are determined as the calculated values of tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk . With respect to the graph in Figure 6, since the third minimum value is the smallest, the set of tension candidate value, bending stiffness candidate value, and spring stiffness candidate value used in calculating the third minimum value is determined as the calculated values of tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk .

It is assumed that the peak in the data after Fourier transform shown in FIG. 5 is overlooked due to noise. For example, as shown in FIG. 7, if the peak corresponding to the measured value f _k ⁴ of the natural frequency is overlooked, the natural frequency acquired as the measured values f _k ⁵ and f _k ⁶ of the natural frequency in FIG. 5 can be acquired as the measured values f _k ⁴ and f _k ⁵ of the natural frequency. In this case, since the calculation formula G ₁ ⁱ does not include the mode order i, it is not necessary to assign the measured value of the natural frequency acquired in step S120 to the mode order i. Therefore, even if the peak corresponding to the measured value f _k ⁴ of the natural frequency is overlooked, no erroneous calculation occurs due to an error in the correspondence between the mode order i and the measured value f _k ⁱ of the natural frequency. Therefore, the tension T _k , the bending stiffness E _k I _k and the rotational spring stiffness K _k can be calculated with high accuracy.

The calculation formula G ₁ ⁱ is based on the model shown in FIG. 2, in which a rotational spring is inserted at both ends of the cable k. The characteristics of the rotational spring are represented by the rotational spring stiffness K _k . If the rotational spring stiffness K _k is large, both ends of the cable k are close to the fixed ends, and conversely, if the rotational spring stiffness K _k is small, both ends of the cable k are close to the free ends. The rotational spring stiffness K _k is treated as a variable together with the tension T _k and the bending stiffness E _k I _k in the search process (step S125 in FIG. 4, FIG. 6). If such a search process is performed, the tension T _k , etc. can be calculated while taking into account the state of both ends of the cable k. This increases the accuracy of the calculated value of the tension T _k , etc.

In the process shown in Fig. 4, a set of tension candidate value, bending stiffness candidate value, and spring stiffness candidate value when the minimum value obtained in the search process (step S125) is obtained is determined as the calculated values of tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk . In addition to this process, a process for ensuring the calculation accuracy of tension _Tk, etc. may be performed. For example, as shown in Fig. 8, the minimum value of the sum of squares of the arithmetic expression _G1i obtained by the search process in step S125 ^may be compared with a predetermined threshold value (step S130). If the minimum value of the sum ^of squares of the arithmetic expression _G1i is less than the threshold value (step S130: Yes), the set of tension candidate value, bending stiffness candidate value, and spring stiffness candidate value at this time is determined as the calculated values of tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk . If the minimum value of the sum of squares of the arithmetic expression G ₁ ⁱ is equal to or greater than the threshold value (step S130: No), the search range may be changed by changing the initial value shown in Fig. ⁶ (step S135), and the search process (step S125) may be performed again. By repeating the change of the search range (step S135) and the search process (step S125) until the minimum value of the sum of squares of the arithmetic expression _G 1 i becomes less than the threshold value, the tension T _k etc. close to the true value that satisfies the calculation standard equation of "Equation 61" is calculated.

In the process shown in FIG. 8, if you want to obtain high calculation accuracy, you can set a small threshold value. Conversely, if you do not need high calculation accuracy, you can set a large threshold value.

Second Embodiment
A calculation standard equation used for calculating the tension T _k etc. may be set based on the vibration shape of the cable k in each vibration mode. In order to measure the vibration shape of the cable k in each vibration mode, for example, as shown in Fig. 9, two acceleration sensors 141, 142 may be attached to a span 171 of the cable 121. In this case, the data collection device 160 is connected to these acceleration sensors 141, 142 so as to receive data from these acceleration sensors 141, 142. In the following description, the attachment position of the acceleration sensor 141 is represented by "x ₁₁ =u ₁₁ ", and the attachment position of the acceleration sensor 142 is represented by "x ₁₁ =u ₁₂ ".

The general solution of the vibration equation (Equation 3) for cable 121 is expressed as follows. Note that the general solution below is the same as the above Equation 4, except that "k" is expressed as "1".

Furthermore, the boundary condition equations of "Equation 5" and "Equation 6" above can be rewritten as follows for cable 121.

Based on the boundary conditions of "Equation 65", the relationship among the integral constants C _1d1 to C _1d4 in the general solution of "Equation 64" is expressed as follows.

In this embodiment, since the vibration shape of the cable 121 is the issue, the magnitudes of the integral constants C _1d1 to C _1d4 do not matter. Therefore, in order to simplify the above-mentioned "Equation 66", "C ₁₁₂ =m ₁₂ " is set. As a result, "Equation 66" can be rewritten as follows.

Based on "number 67", "number 64" can be rewritten as follows.

If we divide "Equation 68" into the cases "r ₁ <1" and "r ₁ =1" following the above-mentioned "Equation 45", "Equation 68" can be rewritten as follows. Since we solve the constraint equation that the Fourier amplitude ratio, not the Fourier amplitude, is equal between the actual value and the theoretical value, not the Fourier amplitude itself but the Fourier amplitude ratio is meaningful. For this reason, the term P ₁ ² , which becomes infinite when "r ₁ =1", is dropped from the equation when "r ₁ =1" in the following "Equation 69". Note that X ₀ (x ₁₁ ), X ₁ (x ₁₁ ), and X ₂ (x ₁₁ ) in "Equation 68" are defined in "Equation 70" to "Equation 72".

"W ₁₁ (x ₁₁ )" in "Equation 69" corresponds to the Fourier amplitude obtained by applying a Fourier transform process to the data obtained from the acceleration sensors 141, 142. To make the data easier to handle, the absolute value of the Fourier amplitude, i.e., the absolute value of "W ₁₁ (x ₁₁ )", is set to φ _i ^t in the following explanation. The theoretical formula for the absolute value of the Fourier amplitude at the mounting positions of the acceleration sensors 141, 142 is shown below.

The theoretical formula of "Equation 73" is a function _of tension _Tk , bending stiffness _EkIk , rotational spring stiffness _Kk and ^natural frequencies _fki of multiple modes (see Equations 4, 5, 68 and 69).

The constraint equation below specifies that the difference between the ratio of the absolute values of the theoretical values obtained from the theoretical formula of "Equation 73" and the ratio of the absolute values of the measured values of the Fourier amplitudes obtained by performing a Fourier transform process on the data obtained from the acceleration sensors 141 and 142 (i.e., the Fourier amplitude ratio) is equal to zero. Note that the constraint equation below distinguishes between cases in which the ratio of the absolute values of the Fourier amplitudes is 1 or less.

By carrying out the same calculations as in "Equation 60" and "Equation 61" taking into account the case where the deflection at the intersection 131 becomes zero, "Equation 74" can be rewritten as follows.

Assuming that j acceleration sensors (j≧2) are attached to span 171, the attachment positions of these acceleration sensors can be represented as “u _1j ”, and “Equation 75” can be rewritten as follows and used as the calculation standard equation.

"Equation 76" is shown as an equation where the function on the left side is equal to zero. The function on the left side of "Equation 76" includes the following constants and variables: Note that the function on the left side of "Equation 76" does not include the mode number i as a variable.
Total length of cable 121: _L1
Position of intersection 131 (i.e., lengths of spans 171, 172 of cable 121): L ₁₁ , L ₁₂
Density of cable 121: ρ ₁
Cross-sectional area of cable 121: A ₁
Actual measured value of the natural frequency of the cable 121: f ₁ ⁱ
Tension of cable 121: _T1
Bending rigidity of cable 121: E ₁ I ₁
Rotational spring stiffness: _K1

(Equation for calculating tension T1 _etc. )
The function on the left side of the calculation standard equation of "Equation 76" is used as an arithmetic expression G ₂ ^ij for calculating the tension T ₁ and the like.

The value of the arithmetic expression G ₂ ^ij is calculated by substituting numerical values into the constants and variables of the arithmetic expression G ₂ ^ij of "Equation 77". The closer the value calculated from the arithmetic expression G ₂ ^ij is to the value of the calculation standard equation of "Equation 76" (i.e., zero), the closer the numerical value substituted into the arithmetic expression G ₂ ^ij is to the true value when the calculation standard equation of "Equation 76" is satisfied. Therefore, the tension T ₁ , etc. can be calculated by comparing the value calculated from the arithmetic expression G ₂ ^ij with the value of the calculation standard equation of "Equation 76" (i.e., zero). For this comparison process, the sum of squares of the arithmetic expression G ₂ ^ij is set as shown below.

If the sum of squares of the calculation formula G ₂ ^ij is close to zero, it can be seen that the numerical value substituted into the calculation formula G ₂ ^ij is close to the true value when the calculation standard formula of "Equation 75" holds across a plurality of vibration modes. A specific method for calculating the tension T1 _etc. using the sum of squares of the calculation formula G ₂ ^ij will be described below with reference to FIG. 10.

(Calculation method for tension _T1 , etc.)
First, the following structural data regarding the cable 121 is acquired (step S106): The following structural data is substituted as fixed values into the sum of squares of the arithmetic expression G ₂ ^ij (Equation 77).
Total length of cable 121: _L1
Position of intersection 131 (i.e., lengths of spans 171, 172 of cable 121): L ₁₁ , L ₁₂
Density of cable 121: ρ ₁
Cross-sectional area of cable 121: A ₁

After acquiring the structural data, as shown in Fig. 9, acceleration sensors 141, 142, etc. are attached to cable 121, and data collection device 160 is connected to acceleration sensors 141, 142, etc. Then, a measurement process (steps S111 to S121 in Fig. 9) is performed to acquire actual measured values f ₁ ⁱ of natural frequencies of cables 121, 122, etc. for a plurality of vibration modes.

In the actual measurement step, similarly to the calculation method of tension _Tk etc. in the first embodiment, the cables 121, 122 are vibrated in the out-of-plane direction (step S111), and the acceleration of the vibration generated in the cable 121 is measured by the acceleration sensors 141, 142 etc. The measured acceleration of the vibration is recorded as a time history response value in the data collection device 160 (step S116). The data collection device 160 performs a Fourier transform on the time history response value, and the actual measured value of the natural frequency f ₁ ⁱ and the actual measured value of the Fourier amplitude φ _i ^m (u _1j ) are obtained from the peak value of the data after the Fourier transform (step S121).

As a result of the above-mentioned Fourier transform, for example, data as shown in Fig. 11 is obtained. Fig. 11(a) shows data after Fourier transform of the time history response value obtained from acceleration sensor 141 ( _x11 = _u11 ). Fig. 11(b) shows data after Fourier transform of the time history response value obtained from acceleration sensor 142 etc. attached at the attachment position where " _x11 = _u1j ".

In the data in Fig. 11, the measured Fourier amplitude _φim ⁽ _u1j ) is smaller than the measured Fourier ^amplitude _φim ( _u11 ). Therefore, the ratio of the measured Fourier amplitudes is calculated for each mode order i by using the Fourier ^amplitude _φim ( _u1j ) as the denominator and the Fourier ^{amplitude φim} _{( u11} ) as the numerator.

The structural data acquired in step S106 and the data of the ratio of the measured value of the natural frequency f ₁ ⁱ and the measured value of the Fourier amplitude (=φ _i ^m (u ₁₁ )/φ _i ^m (u _1j )) acquired in step S121 are substituted into the sum of squares of the arithmetic expression G ₂ ^ij (Equation 77). As a result, the sum of squares of the arithmetic expression G ₂ ^ij (Equation 77) can be treated as a function including the tension T ₁ of the cable 121, the bending rigidity E ₁ I ₁ of the cable 121, and the rotational spring rigidity K ₁ as variables. After the structural data, the data of the ratio of the measured value of the natural frequency f ₁ ⁱ and the measured value of the Fourier amplitude are input into the sum of squares of the arithmetic expression G ₂ ^ij (Equation 77), a search process is performed to find an optimal solution for the tension T ₁ of the cable 121, etc. (step S126). This search process is executed based on the MultiStart method as in the first embodiment (see FIG. 6). That is, in the search process, the sum of squares of the calculation formula _G2ij is calculated while changing the candidate values of the tension _T1 , bending stiffness _{E1I1 ,} and rotational spring stiffness _K1 . The candidate values of the tension _T1 , bending stiffness _{E1I1 ,} and rotational spring stiffness _K1 that minimize the sum ^of squares within the _search range are determined as the tension _T1 , bending stiffness _E1I1 , and rotational spring stiffness _K1 .

As in the first embodiment, ^the calculation formula _G2ij does not include the mode order i, so there is no need to assign the measured value of the natural frequency acquired in step S121 to the mode order i. Therefore, no calculation error occurs due to an incorrect association ^between the mode order i and the measured value of the natural frequency _f1i . Therefore, the tension _T1 , bending stiffness _{E1I1 ,} and rotational spring stiffness _K1 can be calculated with high accuracy.

^In the calculation formula _G2ij , the rotational spring stiffness _K1 is _also treated as a variable in the search process (step S126). Therefore, the tension _T1 and the bending stiffness _E1I1 are calculated taking into consideration the support state of both ends of the cable 121 from a state close to the fixed end to a state close to the free end. This increases the accuracy of the calculated values of the tension _T1, etc.

9, the acceleration sensors 141 and 142 are attached to span 171 (the left span) of the cable 121. Alternatively, the acceleration sensors 141 and 142 may be attached to span 172 (the right span) of the cable 121. If the acceleration sensors 141 and 142 are attached to either span 171 or 172 of the cable 121, the tension _T1, etc. of the cable 121 can be calculated.

In order to calculate the tension _T2 , bending stiffness _E2I2 , and rotational spring stiffness _K2 of cable 122, multiple acceleration sensors may be attached to span 173 or span 174 of cable _122. In this case, it becomes possible to calculate the tension T2, _etc. of cable 122 based on data of time history response values obtained from the multiple acceleration sensors attached to cable 122.

In the calculation method of the second embodiment, additional processing (similar to steps S130 and S135 in FIG. 8) may be performed to ensure calculation accuracy. That is, a threshold may be set for the minimum value of the sum of squares of the arithmetic expression G ₂ ^ij , and whether or not to continue the search process may be determined based on a comparison between the minimum value of the sum of squares of the arithmetic expression G ₂ ^ij and the threshold.

The tension _Tk and the like may be calculated by combining the calculation methods of the first and second embodiments. In this case, in the search process, the candidate values of the tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk that minimize the calculated value obtained from the following sum-of-squares formula are determined as the tension _Tk , bending stiffness _{EkIk ,} and rotational spring stiffness _Kk .

In the above-described embodiment, cables 121 and 122 are used as the linear body. Alternatively, a steel rod or other linear member may be used as the linear body.

The techniques described in relation to the above embodiments are well suited for investigating the tensions and stiffness acting on various linear bodies that can be modeled as one-dimensional beams.

121, 122: Cable (linear body)
130: Grip device 131: Intersection

Claims (7)

A method for calculating tension, bending rigidity, and rotational rigidity at both ends of two linear bodies held by a holding device so as to cross each other, comprising the steps of:
a measuring step of obtaining actual measured values of natural frequencies of multiple modes for each of the two linear bodies based on vibration of the two linear bodies;
a calculation step of calculating the tension, the bending stiffness, and the rotation stiffness using a calculation standard equation set using boundary conditions that represent that an intersection of the two linear bodies is gripped by the gripping device and that both ends of each of the two linear bodies have the rotation stiffness, and the actual measured values of the natural frequencies of the multiple modes,
the calculation standard formula is expressed as a formula in which a function including variables of the tension, the bending stiffness, the rotation stiffness, and the natural frequencies of the multiple modes is equal to a predetermined value,
In the calculation step,
Substituting the candidate values set for each of the tension, the bending stiffness, and the rotational stiffness, and the actual measured values of the natural frequencies of the multiple modes into corresponding variables of the function, respectively, to obtain a calculated value of the function;
A calculation method, comprising: calculating the tension, the bending stiffness, and the rotational stiffness based on a comparison between the acquired calculated value and the predetermined value. The calculation method according to claim 1, wherein the calculation formula expresses the relationship between the natural frequencies, the tension, the bending stiffness, and the rotational stiffness of the multiple modes without including modal orders. The calculation method according to claim 1 or 2, wherein the boundary conditions include a condition indicating that the displacement directions and displacement amounts of the two linear bodies at the intersection are equal. the function of the calculation standard equation expresses a sum of a force acting on the intersection of one of the two linear bodies and a force acting on the intersection of the other linear body, using variables of the natural frequencies of the multiple modes, the tension, the bending rigidity, and the rotational rigidity,
The calculation criterion formula is expressed as a formula for which the function is equal to a value of zero,
In the calculation step,
Substituting the candidate values of the tension, the bending stiffness, and the rotational stiffness, and the actual measured values of the natural frequencies of the multiple modes into corresponding variables of the functions, respectively, to obtain the calculated values;
The method according to claim 1 , further comprising the steps of: calculating the tension, the bending stiffness and the rotational stiffness based on a comparison between the obtained calculated value and a value of zero. In the actual measurement step, Fourier amplitudes at the natural frequencies of the multiple modes corresponding to the actual measured values of the natural frequencies of the multiple modes are measured at two mutually different positions on each of the two linear bodies, and an actual measured value of a Fourier amplitude ratio, which is a ratio of the actual measured values of the Fourier amplitudes obtained at the two positions, is obtained;
the function of the calculation standard equation is set to represent a difference between a theoretical value obtained from a theoretical equation of a Fourier amplitude ratio expressed as a function of the tension, the bending rigidity, the rotational rigidity, and the natural frequencies of the multiple modes, and the actual measured value of the Fourier amplitude ratio;
The calculation standard formula is set to express a relationship in which the difference between the theoretical value of the Fourier amplitude ratio and the actual measured value of the Fourier amplitude ratio is equal to a value of zero,
In the calculation step,
the actual value of the Fourier amplitude ratio acquired in the actual measurement step is substituted into the term of the actual value of the Fourier amplitude ratio in the function, and the candidate values of the tension, the bending rigidity, and the rotational rigidity, and the actual values of the natural frequencies of the multiple modes are substituted into the theoretical formula, thereby obtaining the difference between the actual value of the Fourier amplitude ratio and the theoretical value of the Fourier amplitude ratio as the calculated value;
The method according to claim 1 , further comprising the step of calculating the tension, the bending stiffness and the rotational stiffness based on a comparison of the calculated value of the difference with the value of zero. In the calculation step,
Repeating the process of substituting another candidate value set for at least one of the tension, the bending stiffness, and the rotational stiffness, and the actual measured values of the natural frequencies of the multiple modes into the function to obtain another calculated value, thereby obtaining a plurality of calculated values;
The calculation method according to claim 1 , wherein a candidate value that is closest to the predetermined value among the plurality of calculated values is determined as the calculated values of the tension, the bending stiffness, and the rotational stiffness. In the calculation step,
6. The calculation method according to claim 1, further comprising repeating an assignment process of assigning another candidate value set for at least one of the tension, the bending stiffness, and the rotational stiffness, and the actual measured values of the natural frequencies of the multiple modes to the function, until a difference between the calculated value and the predetermined value becomes less than a predetermined threshold value.

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