Method for calculating tension and bending rigidity of linear bodies of structure with intersecting linear bodies

Abstract

To provide a method for calculating tension of two linear bodies in a structure having the linear bodies held at an intersect part by a holding device.SOLUTION: The present application discloses a calculation method for calculating tension, etc., of two linear bodies in a structure having two intersecting linear bodies. The calculation method has a measurement step of obtaining n (n is a natural number) actual measurement values for a peak vibration frequency of each linear body based on the vibration of each linear body. Then, m (m is a natural number smaller than n) actual measurement values are selected from these actual measurement values, and the tension, etc., of the linear body are calculated.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to a method for calculating the tension and bending rigidity of a plurality of linear bodies extending so as to intersect each other in a structure in which an intersection portion of the linear bodies is gripped by a gripping device.

Background Art

[0002] In Patent Document 1, in a structure in which an intersection portion of a plurality of linear bodies extending so as to intersect each other is gripped by a gripping device, the tension of these linear bodies is calculated. In this calculation of the tension, an acceleration sensor is installed on these linear bodies, and the vibration frequency at which the vibration of these linear bodies is prominent is measured. Then, the measured vibration frequency is used as the natural vibration frequency of these linear bodies.

Prior Art Document

Patent Document

[0003]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0004] In a structure having only a single linear body, the natural frequency in the direction perpendicular to the axis of this linear body, in the direction perpendicular to the axis (the axis-perpendicular direction), can theoretically measure the same natural frequency regardless of the direction in which the acceleration sensor is installed as long as it is in the axis-perpendicular direction. However, a structure in which the intersection of a plurality of linear bodies extending so as to cross each other is gripped by a gripping device is more complex than a structure having only a single linear body, and the natural frequency can vary depending on the direction of vibration. That is, the in-plane direction within the plane including the plurality of linear bodies and the out-of-plane direction perpendicular to this plane can have different natural frequencies. Even if an acceleration sensor is installed for the purpose of measuring the out-of-plane natural frequency, due to an error in the installation direction of the acceleration sensor, the measured vibration waveform can include vibrations not only in the out-of-plane direction but also in the in-plane direction. In this case, the tension calculated by a calculation method premised on the use of the out-of-plane natural frequency can deviate significantly from the true value.

[0005] An object of the present invention is to provide a method for calculating the tension and bending rigidity of the linear bodies of a structure in which the intersection of a plurality of linear bodies extending so as to cross each other is gripped by a gripping device.

Means for Solving the Problems

[0006] The calculation method according to one aspect of the present invention can be used to calculate the tension and bending rigidity of the first linear body and the second linear body based on the natural frequency of vibration in the out-of-plane direction perpendicular to the plane including the first linear body and the second linear body in a structure in which the intersection of the first linear body and the second linear body extending so as to intersect each other is gripped by a gripping device. This calculation method includes a measurement step of obtaining n (n is a natural number) measured values for the frequency at which the out-of-plane vibration of at least one of the first linear body and the second linear body is dominant, a calculation reference formula set using boundary conditions indicating that the intersection is gripped by the gripping device, the n measured values, and a calculation step of selecting the natural frequency of the structure from the n measured values and calculating the tension and bending rigidity of each linear body. The calculation reference formula is expressed as an equation in which a function including variables of tension, bending rigidity, and natural frequency becomes equal to a predetermined value so as to represent the relationship between the tension, bending rigidity, and natural frequency of each linear body and the natural frequency of an arbitrary mode number. The calculation step includes: (i) substituting each measured value into the variable of the natural frequency of the function to obtain n calculation formulas including variables of tension and bending rigidity; (ii) for each calculation formula, bringing the value of each calculation formula closer to a predetermined value while changing the substitution values substituted into the variable of tension and the variable of bending rigidity; (iii) determining the smallest measured value among the measured values as one of the natural frequencies of the structure; and (iv) the difference between the value of the calculation formula corresponding to the smallest measured value and the predetermined value, and the sum of the squares of the differences between the value of each calculation formula for (n - 1) measured values excluding the smallest measured value and the predetermined value, and the predetermined value, and calculating the substitution values into the variable of tension and the variable of bending rigidity as the tension and bending rigidity when the sum of the squares of the differences becomes small.

[0007] In a structure having only a single linear body, the measured value of the frequency at which the vibration of this linear body is prominent can be treated as the natural frequency of this linear body. However, in a structure where the gripping device grips the intersection of the first linear body and the second linear body, the in-plane natural frequency in the in-plane direction within the plane including the first linear body and the second linear body and the out-of-plane natural frequency perpendicular to this plane can be different. And, even if the frequency at which the out-of-plane vibration of at least one of the first linear body and the second linear body is prominent is measured so as to obtain the out-of-plane natural frequency of the structure, this measured value is affected by the in-plane vibration and does not necessarily accurately represent the out-of-plane natural frequency. That is, among these measured values, there may be those that deviate greatly from the true value of the natural frequency of the structure. In the above-described configuration, a calculation process is performed so as to exclude measured values that deviate greatly from the true value of the natural frequency.

[0008] In the calculation process, a calculation reference formula is used, and this calculation reference formula is set using a boundary condition indicating that the intersection of the first linear body and the second linear body is gripped by the gripping device. For this reason, the calculation reference formula can represent a vibration mode in which the first linear body and the second linear body vibrate integrally at the intersection.

[0009] This calculation reference formula is expressed as an equation in which a function including variables of tension, bending rigidity, and natural frequency becomes equal to a predetermined value. By substituting each of the n measured values of the frequency at which the out-of-plane vibration is prominent into the variable of the natural frequency, n calculation formulas can be obtained. These calculation formulas include variables of tension and variables of bending rigidity. By changing the substitution values substituted into these variables, the value of the calculation formula can be varied. Note that the smallest of the n measured values is selected as one of the natural frequencies of the structure because it has a large influence on the calculation of tension and the like.

[0010] If the measured value of the frequency substituted into the calculation formula is close to the true value of the natural frequency of the structure, the value of the calculation formula can approach a predetermined value taken by the calculation reference formula. On the other hand, if the measured value substituted into the calculation formula deviates greatly from the true value of the natural frequency of the structure, the value taken by this calculation formula does not approach the predetermined value as much as the value taken by the calculation formula obtained by substituting a measured value close to the true value of the natural frequency. Therefore, the measured values corresponding to (m - 1) calculation formulas (where m is a natural number smaller than n) for which values close to the predetermined value taken by the calculation reference formula are obtained are selected as the natural frequency of the structure. In other words, the measured values corresponding to the remaining calculation formulas (i.e., (n - m) calculation formulas) are likely to deviate greatly from the true value of the natural frequency. Therefore, after excluding these measured values, the tension and bending rigidity are calculated.

[0011] In the calculation of the tension and bending rigidity, the smaller the sum of the squares of the differences between the value of the calculation formula corresponding to the selected natural frequency and the predetermined value, the more likely it is that the values substituted into the tension and bending rigidity approximate the true values. Therefore, the substitution values into the variables of the tension and the bending rigidity when this sum of the squares becomes small are calculated as the tension and the bending rigidity.

[0012] Regarding the above configuration, the boundary condition may be set using a balance equation including a variable of the mass of the gripping device so as to represent that the forces acting on the first linear body and the second linear body at the intersection are balanced. The calculation reference formula may be expressed as an equation in which a function including variables of the tension, bending rigidity, natural frequency, and mass of the gripping device is equal to a predetermined value so as to represent the relationship between the tension, bending rigidity, natural frequency, and mass of the gripping device when each linear body vibrates in the i-th mode. In the stage of obtaining n calculation formulas, the value of the mass of the gripping device may be substituted into the variable of the mass of the gripping device.

[0013] In the above configuration, the balance equation represents that the forces acting on the first linear body and the second linear body at the intersection are balanced. Therefore, the calculation reference formula set using this balance equation can represent the vibration mode in which the first linear body and the second linear body vibrate integrally at the intersection.

[0014] Since the balance formula includes the variable of the mass of the gripping device, the calculation reference formula set using this balance formula is also expressed in a form including the variable of the mass of the gripping device. Since the value of the mass of the gripping device is substituted into this variable, the tension and the flexural rigidity are accurately calculated in consideration of the mass of the gripping device.

[0015] Regarding the above configuration, the structure may include a third linear body extending in a plane including the first linear body and the second linear body and intersecting the first linear body at a position different from the second linear body, and another gripping device for gripping the intersection of the first linear body and the third linear body. The calculation reference formula may be set using a boundary condition indicating that the intersection of the first linear body and the second linear body is gripped by the gripping device and a boundary condition indicating that the intersection of the first linear body and the third linear body is gripped by another gripping device. In the actual measurement process, n (n is a natural number) actual measurement values may be obtained for the frequency at which the out-of-plane vibration of at least one of the first to third linear bodies is dominant. The calculation process includes: (i) substituting each actual measurement value into the variable of the natural vibration frequency of the function to obtain n calculation formulas including the variables of the tension and the flexural rigidity; (ii) for each calculation formula, approaching the value of each calculation formula to a predetermined value while changing the substitution values substituted into the variable of the tension and the variable of the flexural rigidity; (iii) determining the smallest actual measurement value among the actual measurement values as one of the natural vibration frequencies of the structure; and (iv) the difference between the value of the calculation formula corresponding to the smallest actual measurement value and the predetermined value, and the sum of the squares of the differences between the predetermined value and the values of (m - 1) (m is a natural number smaller than n) calculation formulas for which a value close to the predetermined value is obtained among the calculation formulas for the (n - 1) actual measurement values excluding the smallest actual measurement value is small, calculating the substitution values into the variable of the tension and the variable of the flexural rigidity as the tension and the flexural rigidity of each linear body.

[0016] In the above configuration, the tension and the flexural rigidity can also be calculated in a structure in which not only the second linear body but also the third linear body intersects the first linear body.

[0017] Regarding the above configuration, the calculation reference formula may represent the relationship between the tension, bending rigidity, and natural vibration frequency of any mode order without including the variable of the mode order.

[0018] In the above configuration, since the calculation reference formula does not include the variable of the mode order, the measured value of the vibration frequency can be substituted into the function in the calculation reference formula without associating the measured value of the peak vibration frequency with the mode order, and n calculation formulas can be obtained.

Effect of the Invention

[0019] The above technology enables accurate calculation of the tension and bending rigidity of the linear bodies of the structure in a structure in which the intersection portions of a plurality of linear bodies extending so as to cross each other are gripped by a gripping device.

Brief Description of the Drawings

[0020]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Mode for Carrying Out the Invention

[0021] <First Embodiment> FIG. 1 is a schematic plan view of a structure 100 having a structure in which two linear bodies 121 and 122 (a first linear body and a second linear body) intersect each other. As the linear bodies 121 and 122, cables for bridges are used. The linear bodies 121 and 122 are fixed in a tensioned state. Both ends of each of the linear bodies 121 and 122 are supported so as to be fixed ends.

[0022] A gripping device 130 is attached to the intersection of the linear bodies 121 and 122. The gripping device 130 is configured to grip these linear bodies 121 and 122 at the intersection of the linear bodies 121 and 122. Therefore, at the intersection, the vibration amplitudes and vibration directions of both linear bodies 121 and 122 are equal.

[0023] By the gripping device 130 attached to the intersection, the linear bodies 121 and 122 are each divided into two spans. The length portion of the linear body 121 from the left end of the linear body 121 to the gripping device 130 in FIG. 1 is referred to as "span 171" in the following description. The length portion of the linear body 121 from the right end of the linear body 121 to the gripping device 130 in FIG. 1 is referred to as "span 172" in the following description. The length portion of the linear body 121 from the left end of the linear body 122 to the gripping device 130 in FIG. 1 is referred to as "span 173" in the following description. The length portion of the linear body 122 from the right end of the linear body 121 to the gripping device 130 in FIG. 1 is referred to as "span 174" in the following description.

[0024] An acceleration sensor 141 is attached to the span 171 of the linear body 121, and an acceleration sensor 143 is attached to the span 173 of the linear body 122. The acceleration sensors 141 and 143 are configured to detect the acceleration of the vibration generated in the linear bodies 121 and 122 by an impact applied to at least one of the linear bodies 121 and 122. In the present embodiment, the linear bodies 121 and 122 are vibrated in a direction perpendicular to the plane including the linear bodies 121 and 122 (a direction perpendicular to the paper surface of FIG. 1). In the following description, the direction perpendicular to the plane including the linear bodies 121 and 122 is referred to as the "out-of-plane direction".

[0025] Acceleration sensors 141 and 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 and 143 is stored in the data collection device 160 as time history response values. The data collection device 160 is configured to perform a predetermined analysis process on the acceleration data, and through this analysis process, data of the measured values of the natural frequencies (out-of-plane natural frequencies) of the linear bodies 121 and 122 can be obtained. Based on the obtained natural frequency data, the tension and the like of each of the linear bodies 121 and 122 are calculated.

[0026] To obtain the measured data of the natural frequency, the data collection device 160 may perform a Fourier analysis process on the acceleration data. In this case, data of a plurality of frequencies (i.e., a plurality of peak frequencies at which the vibration intensity appears as a peak) at which the vibration of the linear bodies 121 and 122 in the out-of-plane direction is prominent can be acquired. These frequencies can be treated as natural frequencies in the calculation of the tension and the like. However, the structure 100 shown in FIG. 1 is more complex than a structure in which a single linear body is stretched, and the natural frequency in the out-of-plane direction may be different from the natural frequency in the in-plane direction in which the linear bodies 121 and 122 are included. For this reason, if the acceleration sensors 141 and 143 are not accurately attached to the linear bodies 121 and 122, the measured value of the obtained peak frequency may not represent the out-of-plane natural frequency of the linear bodies 121 and 122. Therefore, a calculation step of calculating the tension and the like of the linear bodies 121 and 122 is performed after selecting the natural frequency of the linear bodies 121 and 122 from among the obtained peak frequencies.

[0027] <Derivation of the calculation reference formula> For the above-described calculation step, a calculation reference formula based on the vibration equation for the linear bodies 121 and 122 is derived. The calculation reference formula is expressed as an equation in which a function including variables such as the natural frequency, the tension, and the flexural rigidity is equal to a predetermined value. Based on the predetermined value taken by the function of the calculation reference formula, a natural frequency close to the true value is selected from among the measured values of the peak frequencies.

[0028] As shown in FIG. 2, the calculation reference formula can be set by modeling each of the linear bodies 121 and 122 as a one-dimensional beam with both ends fixed. In FIG. 2, the subscript k takes a value of 1 or 2. In the following description, the term "linear body k" means one of the linear bodies 121 and 122. When "k" takes a value of 1, the term "linear body k" means the linear body 121. When "k" takes a value of 2, the term "linear body k" means the linear body 122.

[0029] In FIG. 2, an intersection 131 is shown on the linear body k. The intersection 131 means the portion where the linear bodies 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 the linear body k to the intersection 131 (i.e., the spans 171 and 173 of the linear bodies 121 and 122) is denoted as L k1 Let it be. The length of the span from the intersection 131 to the other end of the linear body k (i.e., the spans 172 and 174 of the linear bodies 121 and 122) is denoted as L k2 Let it be. The total length of the linear body k (L k1 + L k2 ) is denoted as L k Let it be.

[0030] In FIG. 2, an x k1 axis extending rightward with the left end of the linear body k as the origin and an x k2 axis extending rightward with the intersection 131 as the origin are provided. Also, taking the direction perpendicular to the plane of FIG. 2 as the out-of-plane direction, the displacement amount of the linear body k in this direction is denoted as w kd . The subscript d is used to indicate the span of the linear body k and takes a value of 1 or 2. The subscript d taking a value of 1 indicates the left span of the linear body k. The subscript d taking a value of 2 indicates the right span of the linear body k. That is, the displacement amount "w 11 " indicates the displacement amount in the out-of-plane direction of the span 171 of the linear body 121. The displacement amount "w 12 " indicates the displacement amount in the out-of-plane direction of the span 172 of the linear body 121. The displacement amount "w 21"w" indicates the displacement amount of the span 173 of the linear body 122 in the out-of-plane direction. The displacement amount "w" 22 "w'" indicates the displacement amount of the span 174 of the linear body 122 in the out-of-plane direction.

[0031] When the vibration equation of the linear body k at time t is considered as an Euler-Bernoulli beam under tension, it is given as follows.

[0032]

Equation

[0033] When solving the vibration equation of Equation 1 by the method of separation of variables, the following relational expressions are obtained.

[0034]

Equation

[0035] Substituting Equation 2 into the above vibration equation (Equation 1), the general solution of the mode function W kd (x kd ) can be expressed as follows.

[0036]

Equation

[0037] Since both ends of the linear body k are fixed ends, the displacement and deflection angle at both ends of the linear body k are 0. Therefore, the boundary conditions at both ends of the linear body k are given as follows.

[0038]

Equation

[0039] At the intersection 131, the displacements, deflection angles, and bending moments of the left span and the right span are equal to each other. Therefore, the following boundary condition equations are obtained for the displacements, deflection angles, and bending moments at the intersection 131.

[0040] [Number]

[0041] When the crossing part 131 is gripped by the gripping device 130, at the crossing part 131, the displacements of the linear bodies 1 and 2 are equal, and the forces acting on them are also balanced. That is, these relationships represent the boundary conditions indicating that the crossing part 131 is gripped by the gripping device 130. Therefore, the following boundary condition equations can be obtained for the linear bodies 1 and 2. Note that the following force balance equation includes the variable of the mass m of the gripping device 130, so the mass m of the gripping device 130 is considered. c of the gripping device 130, so the mass m c of the gripping device 130 is considered.

[0042] [Number]

[0043] Numbers 4 to 6 are represented by the matrix of Number 7. Note that the matrix D in Number 7 is a 16-row and 16-column matrix. The first row to the seventh row of this matrix are the coefficients of the equation regarding the linear body 1, and the eighth row to the fourteenth row are the coefficients of the equation regarding the linear body 2. The fifteenth row is the coefficient of the boundary condition equation regarding the displacement at the crossing part 131 of Number 6, and the sixteenth row is the coefficient of the boundary condition equation regarding the force balance at the crossing part 131 of Number 6. These coefficients are shown in Numbers 9 to 24. Note that all matrix components not shown in Numbers 9 to 24 are 0.

[0044] Also, the following matrix Z is a vector composed of unknowns, and this vector is shown in Number 8.

[0045] [Number]

[0046] [Number]

[0047] The component in the \(i\)-th row and \(k\)-th column of matrix \(D\) is denoted as \(D(i, j)\). Using this notation, the components of matrix \(D\) are shown below.

[0048]

Number

[0049]

Number

[0050]

Number

[0051]

Number

[0052]

Number

[0053]

Number

[0054]

Number

[0055]

Number

[0056]

Number

[0057]

Number

[0058]

Number

[0059]

Number

[0060]

Number

[0061]

Number

[0062]

Number

[0063]

Number

[0064] The operations for reducing the unknowns in the above seven numbers are described below. The matrix obtained by extracting the i-th to j-th rows and k-th to l-th columns of matrix D is hereinafter denoted as matrix D(i:j,k:l).

[0065] For the seven equations regarding the linear body 1, the product of matrix D(1:7,1:8) and matrix Z(1:8) is the zero vector. The expression representing this product can be transformed as shown in Equation 25 below.

[0066]

Number

[0067] According to Equation 25, the seven unknowns can be reduced to one unknown.

[0068] For the linear body 2, similar deformations can be performed to reduce the unknowns. That is, for the seven equations regarding the linear body 2, the product of the matrix D(8:14,9:16) and the matrix Z(9:16) is the zero vector. The expression representing this product can be deformed as shown in Equation 26 below.

[0069]

Equation

[0070] Furthermore, the product of the matrix D(15:16,1:16) and the matrix Z(1:16) is the zero vector. The expression representing this product can be deformed as shown in Equation 27 below.

[0071]

Equation

[0072] The coefficient multiplying Z(8) regarding the linear body 1 can be rewritten as shown in Equation 28 below. Also, the coefficient multiplying Z(16) regarding the linear body 2 can be rewritten as shown in Equation 29 below.

[0073]

Equation

[0074]

Equation

[0075] Note that the functions fun 1a , fun 1b , fun 2a and fun 2bRegarding this, arithmetic processing may be performed to prevent information loss, digit loss, or division by infinity. For example, by combining the denominators and numerators of these functions with the exponential function to the base e, and then dividing the denominator and numerator by the term with the largest exponent of the base e in the exponential function of the denominator, the inconvenience in the above arithmetic processing can be avoided.

[0076] Using numbers 25, 26, 28, and 29, number 27 can be rewritten as the following number 30.

[0077]

Number

[0078] For this number 30 to have a solution other than 0, the following number 31 must hold. In this embodiment, the bottommost equation of number 31 is used as the calculation reference equation. In this calculation reference equation, fun 1a , fun 1b , fun 2a , fun 2b is the natural vibration frequency f of the linear body k when the linear body k is vibrating in the i-th mode k i , the tension T of the linear body k k , the bending rigidity E k I k and the mass m of the gripping device 130 c and other variables. The following calculation reference equation is expressed as an equation in which the function containing these variables is equal to a predetermined value (in this embodiment, 0), and represents the relationship between the natural vibration frequency f of the linear body k k i , the tension T of the linear body k k , the bending rigidity E k I k and the mass m of the gripping device 130 c and so on. Note that fun in the following calculation reference equation 1a , fun 1b , fun 2a , fun 2b does not contain variables of the vibration mode of the linear body k.

[0079] [Number]

[0080] (Derivation of objective function for calculating tension etc. of linear body) The calculation reference formula of Equation 31 is for the tension T of the linear body k k and the bending rigidity E k I k is used for calculation. That is, the formula used for this calculation can be defined as follows.

[0081] [Number]

[0082] In the formula of Equation 32, if all the values substituted into the left side are true values, the value of the left side will be 0. And the more the values substituted into the left side deviate from the true value, the larger the value of the left side will be. Utilizing such a relationship, the objective function J for calculating the tension T k and the bending rigidity E k I k can be set as follows. The value of this objective function J represents the sum of the squares of the difference between zero, which is the value taken by the left side of the calculation reference formula of Equation 31, and the value obtained by substituting numerical values into the variables on the left side of the calculation reference formula.

[0083] [Number]

[0084] [Calculation process for calculating natural frequency, tension and bending rigidity] The function fun on the left side of the objective function of Equation 33 1a , fun 1b , fun 2a , fun 2b contains the following variables, but the mode order i is not included as a variable. · Total length of the linear body k: L k · Position of the intersection part 131 (i.e., the length of the span of the linear body k): L k1 , Lk2 · Mass of the gripping device 130: m c · Density of the linear body k: ρ k · Cross-sectional area of the linear body k: A k · Natural frequency of the linear body k: f k i · Tension of the linear body k: T k · Bending rigidity of the linear body k: E k I k Here, the expression inside the parentheses of the objective function J of Equation 33 is hereinafter referred to as calculation formula G i and this calculation formula G i can be defined as in Equation 34 below.

[0085]

Equation

[0086] Calculation formula G i Among the variables in, the total length L of the linear body k k , the position L of the intersection 131 k1 , L k2 , the mass m of the gripping device 130 c , the density ρ of the linear body k k and the cross-sectional area A of the linear body k k are numerical values obtained from the design drawing etc. of the structure 100 shown in FIG. 1. Since these numerical values are considered to have high accuracy, they are substituted into the calculation formula G i as fixed values in the calculation process.

[0087] The variable f of the natural frequency of the linear body k k is substituted with the peak frequency obtained by actually vibrating the linear bodies 121, 122 of the structure 100 shown in FIG. 1 (actual measurement process). The peak frequency is obtained, for example, as follows.

[0088] That is, the accelerations of the linear bodies 121 and 122 are acquired by the acceleration sensors 141 and 143, and the time history response values of this acceleration are stored in the data collection device 160. When Fourier analysis is performed on the data of the time history response values stored in the data collection device 160, for example, analysis results as shown in FIG. 3 are obtained. From the analysis results shown in FIG. 3, the peak vibration frequency (the measured value of the natural vibration frequency of the linear body k) at which the vibration intensity peaks can be known. In a simple structure having only one linear body, the peak vibration frequency can be directly treated as the natural vibration frequency of this linear body. However, in the structure 100 shown in FIG. 1, the two linear bodies 121 and 122 intersect, and the vibration modes of these linear bodies 121 and 122 are complex. For this reason, the peak vibration frequency does not necessarily become a value close to the natural vibration frequencies of these linear bodies 121 and 122.

[0089] The user acquires n (for example, 12) peak vibration frequencies from the analysis results shown in FIG. 3. Preferably, in the analysis results shown in FIG. 3, the smallest peak vibration frequency and (n - 1) peak vibration frequencies are obtained from the remaining small peak vibration frequencies. This is because the small peak vibration frequencies are likely to represent the lower-order natural vibration frequencies, and the lower-order natural vibration frequencies have a greater impact on the calculation accuracy of the tension T of the linear bodies 121 and 122. k This is because the influence on the calculation accuracy is large.

[0090] Each of the n peak frequencies obtained from the analysis results shown in FIG. 3 is substituted into the variable f of the natural vibration frequency, and n calculation formulas G k ~G 1 ~G n are obtained. As described above, since the total length L of the linear body k k and the like are fixed values, the variables of the calculation formulas G 1 ~G n , that is, the variables of the objective function J, are only the tension T of the linear body k k and the flexural rigidity E k I k . The tension T k and the flexural rigidity E k I kA search process is performed to search for the minimum value of the objective function J while changing the numerical value to be substituted. For example, the MultiStart method can be used for this search process.

[0091] For example, the value of the objective function J varies as shown in FIG. 4 depending on the substitution values for the tension T k and the bending rigidity E k I k As shown in FIG. 4, and in FIG. 4, six initial values are set for these substitution values. Then, from each initial value, by increasing and decreasing the substitution values for the tension T k and the bending rigidity E k I k of the linear body k, four minimum values of the objective function J are obtained. The smallest value among these minimum values is recorded as the minimum value of the objective function J.

[0092] In performing the search process for the objective function J, m arithmetic expressions that satisfy the following condition 1 or 2 are used.

[0093] (Condition 1) An arithmetic expression obtained by substituting the minimum value among the peak frequencies obtained in the actual measurement process (that is, arithmetic expression G 1 ).

[0094] (Condition 2) (m - 1) arithmetic expressions that do not satisfy Condition 1 but obtain values relatively close to zero in the search process.

[0095] The reason for using the arithmetic expression that satisfies Condition 1 is that small frequencies are greatly affected by the calculation of the tension T k and the bending rigidity E k I k . Preferably, the peak frequency representing the vibration of mode order 1 is used. However, if such a frequency cannot be obtained in the actual measurement process, the smallest one among the obtained peak frequencies can be used.

[0096] The reason for using the arithmetic expression that satisfies Condition 2 is that it is considered that the peak frequency substituted to obtain the arithmetic expression is close to the true value of the natural frequency.

[0097] Note that in the objective function J of Equation 33, "fun 1a ·fun 2b " is in the denominator, so this objective function J does not hold in the vibration mode where the value of "fun 1a ·fun 2b " becomes 0. However, when the peak frequency corresponding to the vibration mode where the value of "fun 1a ·fun 2b " becomes 0 is substituted into the objective function J, the minimum value obtained by the above search process becomes larger than the minimum value when other peak frequencies are substituted. Therefore, by excluding the ones with relatively large minimum values among the n peak frequencies, it is allowed to handle only the peak frequencies under the conditions where the objective function J holds as the natural frequencies.

[0098] In the calculation reference formula of Equation 31, the boundary condition formula of Equation 6 (the boundary condition formula regarding the balance of forces at the intersection 131) is used. This boundary condition formula includes the variable of the mass m c of the gripping device 130. Therefore, the calculation of the natural frequency f k i and the tension T k etc. is performed considering the mass m c of the gripping device 130. Accordingly, the accuracy of the value of the natural frequency f k i and the value of the tension T k etc. can be improved. Note that in order to reduce the calculation load, the mass m c may be ignored. In this case, the left side of the boundary condition formula (Equation 6) regarding the balance of forces at the intersection 131 may be set to a value of 0.

[0099] In FIG. 1, the peak frequencies of the linear bodies 121 and 122 are measured using the acceleration sensors 141 and 143. However, since these linear bodies 121 and 122 vibrate integrally by the gripping device 130, there may be no significant difference in the peak frequencies between the acceleration sensors 141 and 143. Therefore, an acceleration sensor may be installed on only one of the linear bodies 121 and 122. And the variable f of the natural frequency in the objective function J of Equation 33k The same measured values may be substituted.

[0100] <Second Embodiment> In the first embodiment, the structure 100 has a structure in which two linear bodies 121 and 122 intersect. However, in the Nielsen bridge, as shown in FIG. 5, there may be a case where three linear bodies 121 to 123 intersect. In the structure 100 of FIG. 5, the linear bodies 122 and 123 intersect the linear body 121 at different positions. And the intersection of the linear bodies 121 and 122 is held by the holding device 130, and the intersection of the linear bodies 121 and 123 is held by the holding device 132.

[0101] In the following description, these linear bodies 121 to 123 are represented by the symbol k. When "k = 1", the linear body 121 is represented. When "k = 2", the linear body 122 is represented. When "k = 3", the linear body 123 is represented.

[0102] The linear body 121 (k = 1) is modeled as shown in FIG. 6. Also, the linear bodies 122 and 123 (k = 2, 3) are modeled as shown in FIG. 7. The intersection of the linear bodies 121 and 122 is indicated by the symbol "131" in FIGS. 6 and 7, and the intersection of the linear bodies 121 and 123 is indicated by the symbol "133" in FIGS. 6 and 7.

[0103] In FIGS. 6 and 7, both ends of the linear bodies 121 to 123 are fixed ends. Therefore, for the linear body 121 (k = 1), the following boundary condition equations hold.

[0104]

Equation

[0105] Also, at the intersection 131, the following boundary condition equations hold. The following boundary condition equations represent that the holding device 130 holds the intersection 131.

[0106]

Equation

[0107] Furthermore, at the intersection 133, the following boundary condition equations hold. The following boundary condition equations represent that the gripping device 132 is gripping the intersection 133.

[0108]

Equation

[0109] Considering the relationship between the linear bodies 121 and 122 (k = 1, 2) at the intersection 131, the boundary condition equations shown in Equation 38 hold. In the boundary condition equations shown in Equation 38, the mass of the gripping device 130 is represented by the variable m c1 and, except for this point, Equation 38 is the same as Equation 6 described above.

[0110]

Equation

[0111] Similar to Equation 38, at the intersection 133, the boundary condition equations shown in Equation 39 hold. In the boundary condition equations shown in Equation 39, the mass m c2 of the gripping device 132 is represented by a variable.

[0112]

Equation

[0113] The boundary condition equations for Numbers 36 to 39 are represented by the following matrix in the same manner as in the first embodiment. In the following matrix, matrix D is a 28-row and 28-column matrix, and the coefficients of the equations related to the linear body 121 (k = 1) are described in rows 1 to 10. Also, the coefficients of the equations related to the linear body 122 (k = 2) are described in rows 11 to 17. And the coefficients of the equations related to the linear body 123 (k = 3) are described in rows 18 to 24. Further, in rows 25 and 26, the coefficients of the "boundary condition equations regarding displacements in the intersection parts 131, 133" shown in Numbers 38 and 39 are described. In the remaining rows 27 and 28, the coefficients of the "boundary condition equations regarding the balance of forces in the intersection parts 131, 133" shown in Numbers 38 and 39 are described.

[0114]

Number

[0115] Since the matrix components shown in Numbers 9 to 15 can be directly applied to matrix D of Number 40, the remaining coefficients are shown in the following Numbers 41 to 63. Note that all matrix components not shown in Numbers 9 to 15 and Numbers 41 to 63 are 0.

[0116]

Number

[0117]

Number

[0118]

Number

[0119]

Number

[0120]

Number

[0121]

Number

[0122]

Number

[0123]

Number

[0124]

Number

[0125]

Number

[0126]

Number

[0127]

Number

[0128]

Number

[0129]

Number

[0130]

Number

[0131] [Number]

[0132] [Number]

[0133] [Number]

[0134] [Number]

[0135] [Number]

[0136] [Number]

[0137] [Number]

[0138] [Number]

[0139] The operations for reducing the unknowns in the above matrix D will be described below. The matrix obtained by extracting the i-th to j-th rows and k-th to l-th columns of matrix D is hereinafter denoted as matrix D(i:j,k:l).

[0140] For the linear body 121 (k = 1), by using the fact that the product of matrix D(1:10,1:12) and matrix Z(1:12) is the zero vector, Z(1:10) = {C 111 ···C 132} T is, Z(11:12) = {C 133 C134} T It can be represented by this. As a result, the number of unknowns can be reduced to two. For this reduction process of the unknowns, it is utilized that the product of matrix D(1:5,1:8) and matrix Z(1:8) is a zero vector. The expression representing this product can be transformed as shown in Equation 64 below.

[0141]

Equation

[0142] Also, since the product of matrix D(6:10,1:12) and matrix Z(1:12) is a zero vector, the expression representing this product can be transformed as shown in Equation 65 below.

[0143]

Equation

[0144] The matrix Z1 in Equation 65 is a 5×5 matrix. When Equation 64 is applied, Equation 66 below is obtained.

[0145]

Equation

[0146] Here, when solving Equation 65 for matrix Z(6:10), Equation 67 below is obtained.

[0147]

Equation

[0148] Furthermore, the coefficients of matrix Z(11:12) are represented by matrix Z2 as follows.

[0149]

Equation

[0150] Using the matrices Z1 and Z2, the matrices Z(1:5) and Z(6:10) are expressed as follows.

[0151]

Number

[0152] For the linear body 122 (k = 2) as well, the unknowns can be reduced as follows. That is, since the product of the matrix D(11:17, 13:20) and the matrix Z(13:20) is the zero vector, Z(13:19) = {C 211 ···C 223} T can be represented as Z(20) = C 224 The arithmetic processing for reducing the unknowns for the linear body 122 (k = 2) is shown below. The expression representing the product of the matrix D(11:17, 13:20) and the matrix Z(13:20) can be transformed as shown in Equation 70 below.

[0153]

Number

[0154] Here, the coefficients of the matrix Z(20) are represented by the matrix Z3 as follows.

[0155]

Number

[0156] And Equation 69 is rewritten as follows using the matrix Z3.

[0157]

Number

[0158] For the linear body 123 (k = 3) as well, the unknowns can be reduced as follows. That is, since the product of the matrix D(18:24, 21:28) and the matrix Z(21:28) is the zero vector, Z(21:27) = {C311 ···C 323} T is Z(28)=C 324 and can be represented as follows. The arithmetic processing for reducing the unknowns for the linear body 123 (k = 3) is shown below. The expression representing the product of the matrix D(18:24,21:28) and the matrix Z(21:28) can be transformed as shown in Equation 73 below.

[0159]

Equation

[0160] Here, the coefficients of the matrix Z(28) are represented by the matrix Z4 as follows.

[0161]

Equation

[0162] And Equation 72 is rewritten as follows using the matrix Z4.

[0163]

Equation

[0164] Furthermore, the product of the matrix D(25:28,1:28) and the matrix Z(1:28) is the zero vector. The expression representing this product can be transformed as shown in Equation 76 below.

[0165]

Equation

[0166] In Equation 76 above, the matrix Z(11:12) relates to the linear body 121 (k = 1). Also, the matrix Z(20) relates to the linear body 122 (k = 2). And the matrix Z(28) relates to the linear body 123 (k = 3). Replace the coefficients multiplied by these matrices as shown in Equation 77 below.

[0167] [Number]

[0168] And the number 76 can be rewritten as follows using the numbers 69, 72, 75, and 77.

[0169] [Number]

[0170] Note that for the functions fun in the number 78 1a ~fun 1h , fun 2a , fun 2c , fun 3b and fun 3d , arithmetic processing may be performed to prevent information loss, digit loss, or division by infinity. For example, by combining the denominators and numerators of these functions with the exponential function of base e and then dividing the numerator and denominator by the term with the largest exponent of base e in the exponential function of the denominator, the inconvenience in the above arithmetic processing can be avoided.

[0171] For the above number 78 to have a solution other than 0, the following number 79 must hold.

[0172] [Number]

[0173] The number 79 can be transformed as follows to obtain the following calculation reference formula. Note that the matrix components in the number 79 do not include the variable of the order of the vibration mode. Therefore, the following calculation reference formula also does not include the variable of the order of the vibration mode. This calculation reference formula is used for the selection of the natural vibration frequency and the calculation of the tension and bending rigidity.

[0174] [Number]

[0175] When the natural frequency is selected using the calculation reference formula of Equation 80, it is possible to prevent a peak frequency that greatly deviates from the true value of the natural frequency from being used as the natural frequency. Also, it is possible to prevent a peak frequency corresponding to the vibration mode under the condition that the denominator in Equation 80 becomes 0 from being selected as the natural frequency.

[0176] Also, an objective function J for calculating the tension or the like of the linear bodies 121 to 123 can be set as follows using the left side of the calculation reference formula of Equation 80. Note that the calculation process for calculating the tension or the like is the same as the calculation process of the first embodiment. Since a calculation formula that satisfies the above-described condition 1 or 2 is used in the calculation process, the tension or the like can be accurately calculated.

[0177]

Equation

[0178] Note that instead of Equation 80, it is also possible to obtain the following calculation reference formula from Equation 79. Using this calculation reference formula, the tension and the bending rigidity may be calculated by the same method as the calculation process of the first embodiment.

[0179]

Equation

[0180] Even when the natural frequency is selected using the calculation reference formula of Equation 82, it is possible to prevent a peak frequency that greatly deviates from the true value of the natural frequency from being selected as the natural frequency. Also, it is possible to prevent a peak frequency corresponding to the vibration mode under the condition that the denominator in Equation 82 becomes 0 from being selected as the natural frequency.

[0181] Further, an objective function J for calculating the tension and the like of the linear bodies 121 to 123 can be set as follows by using the left side of the calculation reference formula of Equation 82. Note that the calculation process for calculating the tension and the like is the same as the calculation process of the first embodiment. Since a calculation formula that satisfies the above-mentioned condition 1 or 2 is used in the calculation process, the tension and the like can be accurately calculated.

[0182]

Equation

[0183] In the second embodiment, the peak frequency may be measured for each of the linear bodies 121 to 123, or the peak frequency may be measured for one of these linear bodies 121 to 123. When the peak frequency is measured for one of these linear bodies 121 to 123, the variable f of the natural frequency k may be substituted with the same measured value.

Industrial Applicability

[0184] The technology described in relation to the above embodiments is suitably used for investigating the tension and rigidity acting on various linear bodies that can be modeled as one-dimensional beams.

Explanation of Reference Numerals

[0185] 100 ····················· Structure 121 to 123 ················· Linear bodies 130, 132 ················· Gripping devices 131, 133 ················· Intersection parts

Claims

1. In a structure in which an intersection of a first linear body and a second linear body extending so as to intersect each other is gripped by a gripping device, based on a natural frequency of vibration in a direction perpendicular to the plane including the first linear body and the second linear body, a method for calculating the tension and bending rigidity of the first linear body and the second linear body, comprising: a measurement step of obtaining n (n is a natural number) measured values for a frequency at which vibration in the out-of-plane direction of at least one of the first linear body and the second linear body is dominant; a calculation step of selecting a natural frequency of the structure from the n measured values and calculating the tension and bending rigidity of each linear body, using a calculation reference formula set using a boundary condition indicating that the intersection is gripped by the gripping device and the n measured values; the calculation reference formula is expressed as an equation in which a function including variables of the tension, the bending rigidity, and the natural frequency is equal to a predetermined value so as to represent a relationship between the tension, the bending rigidity, and the natural frequency of each linear body for an arbitrary mode number; the calculation step includes: (i) substituting each measured value into the variable of the natural frequency of the function to obtain n calculation formulas including variables of the tension and the bending rigidity; (ii) for each calculation formula, bringing the value of each calculation formula closer to the predetermined value while changing substitution values substituted into the variable of the tension and the variable of the bending rigidity; (iii) determining, as one of the natural frequencies of the structure, the smallest measured value among the measured values; (iv) calculating, as the tension and the bending rigidity, substitution values into the variable of the tension and the variable of the bending rigidity when the sum of squares of differences between the value of the calculation formula corresponding to the smallest measured value and the predetermined value and the values of (m - 1) (m is a natural number smaller than n) calculation formulas, among the calculation formulas for the (n - 1) measured values excluding the smallest measured value, that are close to the predetermined value and the predetermined value is minimized. A method for calculating the tension and bending rigidity of a linear body.

2. The boundary condition is set using a balance formula including a variable of the mass of the gripping device so as to represent that forces acting on the first linear body and the second linear body at the intersection are balanced. The calculation reference formula is expressed as an equation in which a function including variables of the tension, bending rigidity, natural frequency, and the mass of the gripping device is equal to a predetermined value so as to represent the relationship among the tension, bending rigidity, natural frequency, and the mass of the gripping device when each linear body vibrates in the i-th mode. The method for calculating the tension and bending rigidity of a linear body according to claim 1, wherein in the step of obtaining the n calculation formulas, a value of the mass of the gripping device is substituted into the variable of the mass of the gripping device.

3. The structure includes a third linear body extending in the plane including the first linear body and the second linear body and intersecting the first linear body at a position different from the second linear body, and another gripping device for gripping the intersection of the first linear body and the third linear body. The calculation reference formula is set using the boundary condition indicating that the intersection of the first linear body and the second linear body is gripped by the gripping device and the boundary condition indicating that the intersection of the first linear body and the third linear body is gripped by the other gripping device. In the actual measurement step, n (n is a natural number) actual measurement values are obtained for the frequency at which the out-of-plane vibration of at least one of the first to third linear bodies is dominant. The calculation step includes: (i) substituting each actual measurement value into the variable of the natural frequency of the calculation reference formula to obtain n calculation formulas; (ii) for each calculation formula, obtaining n values of the calculation formula when it is closest to the predetermined value while changing the substitution values substituted into the variable of the tension and the variable of the bending rigidity; (iii) determining the smallest actual measurement value among the actual measurement values as one of the natural frequencies of the structure; (iv) calculating the substitution values into the variable of the tension and the variable of the bending rigidity of each linear body as the tension and the bending rigidity of each linear body when the sum of the squares of the differences between the value of the calculation formula corresponding to the smallest actual measurement value and the predetermined value and the values of (m - 1) (m is a natural number smaller than n) calculation formulas among the calculation formulas for the (n - 1) actual measurement values excluding the smallest actual measurement value, which are close to the predetermined value, and the predetermined value is minimized. The method for calculating the tension and bending rigidity of a linear body according to claim 1.

4. The calculation reference formula according to claim 1 or 2, which represents the relationship between the tension, the bending rigidity, and the natural vibration frequency of an arbitrary mode order without including a variable of the mode order, for calculating the tension and the bending rigidity of a linear body.

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