JP5415393B2 - Tension measuring method and tension measuring device - Google Patents

Description

  The present invention relates to a tension measuring method and a tension measuring apparatus capable of grasping the tension state of a belt-like body such as a copper plate.

  In general, when a belt-like body such as a steel plate is passed, rolls that support the belt-like body are arranged on the front and back of the travel route. The roll is fed through in the longitudinal direction while applying a tension to the strip, thereby passing the strip. By the way, a strip | belt-shaped body may have distortion in the width direction of strip | belt-shaped bodies, such as middle stretch, a half stretch, and an ear wave, for example. Since the belt-like body having such a strain is passed through while a tension is applied, the strain may have an apparently flat shape. As a method for detecting the latent distortion of the belt-like body, a method of measuring the strain in the width direction of the belt-like body as a tension distribution is known.

  Further, in Patent Document 1, any one of the tension distribution patterns that are preliminarily classified by vibrating the strip and measuring the primary resonance mode and the secondary resonance mode at a plurality of measurement positions in the plate width direction. And a method for measuring the tension distribution by allocating the total tension separately measured according to the tension distribution pattern to each position in the plate width direction is disclosed.

  Patent Document 2 discloses a method of measuring the vibration mode in the width direction of the belt and measuring the tension distribution from the vibration mode frequency and the normalized amplitude value of the vibration mode.

JP-A-6-43051 JP 7-218358 A

  However, the methods disclosed in Patent Documents 1 and 2 estimate the tension of the belt-like body based on past tension measurement data, resulting in an error from the actual tension and lowering the measurement accuracy of the tension distribution. There is a problem of doing.

  Accordingly, an object of the present invention is to provide a tension measuring method and a tension measuring apparatus capable of obtaining the tension distribution of a strip with high accuracy using a simple physical model.

  In order to solve the above-described problem, the present invention provides a tension measurement method for measuring displacement characteristics at a plurality of measurement points arranged in the width direction of a strip-like body to which a load is applied. A modeling step for modeling into a two-dimensional multi-mass point system model having a node corresponding to the node and a spring that is coupled to the node and simulates a tension acting on the node, and a displacement amount of the measurement point based on the displacement characteristics And a spring constant calculating step for calculating a spring constant of the spring such that the distribution of the displacement amount of the node coincides, and a tension calculating step for calculating a tension based on the spring constant.

  The present invention also relates to a tension measuring device for measuring displacement characteristics at a plurality of measurement points arranged in the width direction of a band-like body to which a load is applied, the node corresponding to the measurement point and the node for the band-like body. A modeling means for modeling into a two-dimensional multi-mass system model having a spring simulating the tension acting on the node, and a displacement distribution of the measurement point based on the displacement characteristic and the displacement of the node Spring constant calculating means for calculating the spring constant of the spring such that the quantity distribution matches, and tension calculating means for calculating the tension based on the spring constant.

  According to the above configuration, assuming that the change in tension is approximately replaced by the change in rigidity, the belt-like body has a node corresponding to the measurement point and a spring that is joined to the node and simulates the tension acting on the node. Modeled into a two-dimensional multi-mass system model. Then, the tension is calculated from the spring constant of the spring so that the distribution of the displacement of the belt-like body and the displacement amount of the nodes of the multi-mass point system model coincide.

  Thereby, the tension distribution of the strip can be calculated by a two-dimensional multi-mass point system model that physically approximates the strip. As a result, even if it has a complicated tension distribution in the width direction, it is possible to calculate each of the tensions distributed at the measurement points with high accuracy using a simple physical model. Therefore, the total tension obtained by adding the tensions can be calculated with high accuracy.

  The present invention also relates to a tension measurement method for measuring displacement characteristics at a plurality of measurement points arranged in the width direction of a band-like body to which a load is applied, the node corresponding to the measurement point and the node for the band-like body. A modeling step of modeling into a two-dimensional multi-mass system model having a spring simulating the tension acting on the node and bonded to the eigenvalue, an eigenvalue analysis step of analyzing eigenvalues and eigenvectors in the multi-mass system model, A spring constant calculating step of calculating a spring constant of the spring such that the natural frequency and vibration mode vector based on the displacement characteristic and the natural value and the natural vector respectively match, and a tension calculating step of calculating a tension based on the spring constant have.

  The present invention also relates to a tension measuring device for measuring displacement characteristics at a plurality of measurement points arranged in the width direction of a band-like body to which a load is applied, the node corresponding to the measurement point and the node for the band-like body. Modeling means for modeling into a two-dimensional multi-mass system model having a spring simulating the tension acting on the nodal point, and eigenvalue analysis means for analyzing eigenvalues and eigenvectors in the multi-mass system model, A spring constant calculating means for calculating the spring constant of the spring such that the natural frequency and vibration mode vector based on the displacement characteristics and the natural value and the natural vector coincide with each other; and a tension calculating means for calculating the tension based on the spring constant; have.

  According to the above configuration, assuming that the change in tension is approximately replaced by the change in rigidity, the belt-like body has a node corresponding to the measurement point and a spring that is joined to the node and simulates the tension acting on the node. Modeled into a two-dimensional multi-mass system model. Then, the tension is calculated from the spring constant of the spring so that the distribution of the natural frequency and the vibration mode vector coincides with the natural value and the natural vector.

  Thereby, the tension distribution of the belt-like body can be calculated using a two-dimensional multi-mass point system model that physically approximates the belt-like body. As a result, even if it has a complicated tension distribution in the width direction, it is possible to calculate each of the tensions distributed at the measurement points with high accuracy using a simple physical model. Therefore, the total tension obtained by adding the tensions can be calculated with high accuracy.

  In the tension measuring method of the present invention, the spring constant of the spring may be calculated by a least square method in the spring constant calculating step.

  In the tension measuring device of the present invention, the spring constant calculating means may calculate the spring constant of the spring by a least square method.

  According to the above configuration, by using the least square method, the spring constant of the spring can be calculated without causing a problem of convergence in the iterative calculation and a variation in the calculation result due to the initial value. As a result, even if it has a complicated tension distribution in the width direction, it is possible to calculate each of the tensions distributed at the measurement points with high accuracy using a simple physical model.

  By applying a simple physical model to the band, the tension distribution of the band can be obtained with high accuracy.

It is the schematic which shows the tension measuring apparatus of 1st Embodiment. It is the schematic which shows a multi-mass point system model. It is a flowchart which shows the tension | tensile_strength measuring method of 1st Embodiment. It is the schematic which shows the tension measuring apparatus of 2nd Embodiment. It is a flowchart which shows the tension | tensile_strength measuring method of 2nd Embodiment. It is the schematic which shows the modification of a multi-mass point system model. It is a flowchart which shows the tension | tensile_strength measuring method of 3rd Embodiment. It is a graph which shows a tension distribution identification result.

(First embodiment)
Hereinafter, preferred embodiments of the present invention will be described with reference to the drawings.

  FIG. 1 is a schematic diagram showing a tension measuring device 4 for realizing a tension measuring method according to an embodiment of the present invention. The tension measuring device 4 of the present embodiment is applied to a production line for the strip 1 such as a copper plate, an aluminum plate, or a steel plate. As shown in FIG. 1, the band 1 is supported by two rolls 2 and 3 disposed at front and rear (two points) in the traveling direction (longitudinal direction). The rotational speed of the two rolls 2 and 3 is controlled by a control device (not shown), and the belt-like body 1 is fed out while applying a predetermined tension to the belt-like body 1. Hereinafter, the part located between the two rolls 2 and 3 will be described as the band 1.

  The tension measuring device 4 for measuring such a band 1 includes a load applying device 8, a displacement characteristic measuring device 5, and an arithmetic device 7.

(Loading device 8)
As shown in FIG. 1, the load applying device 8 is disposed above the strip 1. The load applying device 8 has a function of applying vibration in the out-of-plane direction to the strip 1. The load applying device 8 is movable in parallel to the strip 1. Thereby, measurement conditions can be changed as appropriate, and for example, the band 1 can be excited to a desired vibration mode. In the present embodiment, the load application device 8 is a vibration device that vibrates the belt-like body 1 by injecting air onto the belt-like body 1, but is not limited thereto. For example, a vibration device that injects fluid such as water and oil onto the belt 1, a vibration device that vibrates the belt 1 with magnetic force, electrostatic force, eddy current due to electromagnetic induction, sound waves, and the like. Also good. Further, the load applying device 8 is preferably a non-contact type, but is not limited thereto. For example, the load applying device 8 may be a device that strikes a point on the belt or a device that vibrates a support roll.

(Displacement characteristic measuring device 5)
The displacement characteristic measuring device 5 measures the displacement characteristics at the plurality of measurement points 30 arranged in the width direction with respect to the belt-like body 1 to which the load is applied in the out-of-plane direction by the load applying device 8 as described above. The displacement characteristic measuring device 5 is disposed so as to face the upper surface of the central portion in the longitudinal direction of the strip 1. Further, the displacement characteristic measuring device 5 has five non-contact displacement meters 5 a, 5 b, 5 c, 5 d, and 5 e arranged in the width direction of the strip 1. The non-contact displacement gauges 5a to 5e measure the displacement amount of vibration at the five measurement points 30 (measurement points 30a, 30b, 30c, 30d, and 30e) with respect to the opposed strip 1. Examples of the non-contact displacement meters 5a to 5e include a light reflection type laser displacement meter and an overcurrent displacement meter. The non-contact displacement meters 5 a to 5 e are arranged at equal intervals from end to end in the width direction of the band 1. The displacement characteristic measuring device 5 is connected to the arithmetic device 7 so that data communication is possible. That is, the displacement characteristic measuring apparatus 5 measures how much the vibration in the out-of-plane direction has at the plurality of measurement points 30 a to 30 e arranged in the width direction of the strip 1. In the present embodiment, an example in which five non-contact displacement meters are provided is shown, but the number of non-contact displacement meters may not be five.

  In the present embodiment, the displacement characteristic indicates the amount of vibration displacement at each measurement point 30 of the strip 1, but is not limited thereto. For example, the velocity at each measurement point 30 measured by the displacement characteristic measurement device 5 It may be acceleration or the like. Further, when the strip 1 has a static displacement by applying a load, the displacement characteristics may indicate a static displacement at each measurement point 30.

  Further, the displacement characteristic measuring device 5 outputs signals corresponding to the vibration displacement characteristics measured by the non-contact displacement meters 5a to 5e to the arithmetic device 7 in association with the respective measurement points 30a to 30e. Note that the measurement points are preferably measured for the order of the vibration mode to be measured. For example, when measuring vibration modes from the first to the fourth order, it is preferable to arrange four or more measurement points.

  In the present embodiment, the same number (five) of non-contact displacement meters as the measurement points are arranged, but the present invention is not limited to this. For example, the structure which has one fixed non-contact displacement meter fixed and one movement non-contact displacement meter which can move to the width direction of a strip | belt-shaped body may be sufficient. As a result, even if there are more than two measurement points, the displacement amount of the fixed non-contact displacement meter is fixed to one measurement point, and the movement non-contact displacement meter is moved for the other measurement points. The vibration mode can be measured by measuring the displacement amount of each measurement point and calculating the displacement amount of each measurement point relative to the reference displacement amount.

(Calculation device 7)
The arithmetic device 7 includes a vibration characteristic calculation unit 11, a modeling unit 12, an eigenvalue analysis unit 13, a spring constant calculation unit 14, a tension calculation unit 15, and a temporary storage unit 16. The arithmetic unit 7 includes, for example, a CPU (Central Processing Unit), a program executed by the CPU, a storage device that stores data used for these programs in a rewritable manner, and a program execution, like a general personal computer Any device may be used as long as it includes a temporary storage device such as a RAM (Random Access Memory) that temporarily stores data. The functional units 11 to 16 included in the arithmetic device 7 are constructed by cooperation of these hardware and software in the storage device. For example, the software includes a program that analyzes the natural frequency and the vibration mode vector from the measured displacement. The arithmetic unit 7 is not limited to a single unit, and the functions of the functional units 11 to 16 are distributed to a plurality of devices each having the hardware and software in the storage device. It may be a thing.

(Temporary storage unit 16)
The temporary storage unit 16 stores temporary data used by the functional units 11 to 15. For example, the temporary storage unit 16 stores a signal indicating the displacement characteristic output from the displacement characteristic measuring device 5 in association with each of the measurement points 30a to 30e. In addition, the temporary storage unit 16 includes the Young's modulus of the strip 1, the cross-sectional secondary moment of the strip 1, the distance between the rolls 2 and 3 (hereinafter referred to as the inter-roll distance), and adjacent measurement points. The length between 30a to 30e (hereinafter referred to as the link length), the density of the strip 1 and the cross sectional area of the cross section perpendicular to the longitudinal direction of the strip 1 (hereinafter referred to as the strip cross section) are stored in advance. ing.

(Vibration characteristic calculation unit 11)
The vibration characteristic calculation unit 11 analyzes the natural frequency and the vibration mode vector based on the displacement characteristic from the displacement characteristic measuring device 5. Here, the natural frequency is a frequency of each vibration mode. The vibration mode vector is a vector amount indicating the shape of each vibration mode. The vibration characteristic calculation unit 11 performs a fast Fourier transform (FFT) or the like based on the displacement characteristic stored in the temporary storage unit 16, and mode-separates the result to perform natural frequency (natural angular frequency). ) And the vibration mode vector, respectively.

(Modeling unit 12)
The modeling unit 12 has a node corresponding to the measurement point 30 and a spring that simulates the tension acting on the node, which is joined to the node, with respect to the band 1 in a state where vibration is applied by the load applying device 8. Model into a multi-dimensional multi-point system model. In the present embodiment, the modeling unit 12 models a multi-mass point system model 40 as shown in FIG.

  Here, each configuration of the multi-mass point system model 40 obtained by modeling the band 1 will be described with reference to FIG. The multi-mass point system model 40 is simplified to a two-dimensional model of the width direction of the strip 1 and the vibration direction of the measurement point 30. As shown in FIG. 2, the multi-mass point system model 40 includes a node 41 (nodes 41a, 41b, 41c, 41d, and 41e), a connecting member 42, and a distributed spring 43 (distributed springs 43a, 43b, 43c, 43d, and 43e). ), A rotation spring 44, and a fixed surface 45.

  The fixing surface 45 is disposed so that the extending direction coincides with the width direction of the band 1. The nodes 41 are distributed corresponding to the measurement points 30. That is, similarly to the measurement points 30a to 30e distributed at equal intervals in the width direction of the strip 1, the nodes 41 are distributed at equal intervals in the width direction. Specifically, the node 41a corresponds to the measurement point 30a, the node 41b corresponds to the measurement point 30b, the node 41c corresponds to the measurement point 30c, the node 41d corresponds to the measurement point 30d, and the node 41e corresponds to the measurement point 30e. It corresponds to.

  The connecting member 42 rotatably connects adjacent nodes 41 distributed in this way. The sum total of the mass of each connection member 42 shall correspond with the mass of the strip | belt-shaped body 1. FIG.

  The distribution spring 43 is disposed perpendicular to the fixed surface 45. One end of each of the distribution springs 43 a to 43 e is joined to each of the nodes 41 a to 41 e, and the other end is fixed to the fixed surface 45. The distribution springs 43a to 43e simulate the tension acting on each of the nodes 41a to 41e.

  The rotation spring 44 connects adjacent connecting members 42 to each other. That is, the spring constant of the rotary spring 44 indicates the bending rigidity at each node 41 of the strip 1.

  As described above, the multi-mass point system model 40 is modeled on the assumption that the displacement amount at each measurement point 30 of the belt-like body 1 vibrated by the load applying device 8 is equal to the displacement amount at each node 41 of the distribution spring 43. It has been done. That is, the multi-mass point system model 40 pays attention to the fact that there is a correlation between the level of the tension of the band 1 and the level of the natural frequency of the band 1, and the intrinsic mass determined by the mass, rigidity, and tension of the band 1 The change in the frequency was modeled on the assumption that the change in the spring constant of the virtual distributed spring 43 corresponding to the change in the tension can be approximated because the mass and rigidity, which are the physical properties of the strip 1, remain unchanged. Is.

(Eigenvalue analysis unit 13)
The eigenvalue analysis unit 13 analyzes eigenvalues and eigenvectors in the multi-mass point system model 40. Specifically, the eigenvalue analysis unit 13 calculates an equation of motion of the multi-mass point system model 40. In addition, the eigenvalue analysis unit 13 performs eigenvalue analysis using the mass matrix obtained from this equation of motion and the stiffness matrix, and calculates the eigenvalue and eigenvector of the node 41. Here, the stiffness matrix is expressed by an unknown spring constant of the distributed spring 43 and a spring constant of the rotary spring 44 showing the known bending stiffness.

(Spring constant calculation unit 14)
The spring constant calculation unit 14 calculates the spring constant of the distributed spring 43 such that the natural frequency and vibration mode vector based on the displacement characteristics of the measurement point 30 coincide with the natural value and natural vector of the multi-mass point system model 40. Here, “matching” is not limited to complete matching. In this embodiment, the natural frequency and vibration mode vector of the measurement point 30 obtained by measurement are compared with the natural value and natural vector of the node 41, and the spring constants of the distributed springs 43 that minimize the difference are respectively determined. It comes to calculate.

  As described above, in this embodiment, the natural frequency and vibration mode vector obtained from the measured displacement amount are compared with the eigenvalue and eigenvector obtained from the multi-mass point system model 40, so that the distributed springs such that both coincide with each other. 43 spring constants are calculated. Note that the present invention is not limited to such a calculation mode. For example, the spring constant of the distributed spring 43 is calculated by comparing the displacement amount at the measurement point 30 with the displacement amount of the node 41 in the multi-mass system model 40. Also good.

(Tension calculation unit 15)
The tension calculator 15 calculates the tension from the natural frequency based on the calculated spring constant of the distributed spring 43. Specifically, the strip 1 is supported by rolls 2 and 3 at both ends in the longitudinal direction. Therefore, the belt-like body 1 vibrates freely with a vibration antinode between the rolls 2 and 3. In this case, the tension of the strip 1 is indicated by the distance between the rolls, the natural frequency, and the mass of the strip 1. The tension calculation unit 15 calculates the tension at each measurement point 30 of the strip 1 assuming that the natural frequency matches the natural frequency obtained from the spring constant of the distributed spring 43 calculated by the spring constant calculation unit 14. To do. In addition, the tension calculating unit 15 calculates the total tension of the strip 1 by adding the tensions at the respective measurement points 30.

  A tension measuring method executed by the tension measuring device 4 configured as described above will be described with reference to the flowchart of FIG.

(Load application process)
First, the strip 1 is vibrated in the out-of-plane direction (S10). Specifically, the belt-like body 1 is vibrated when compressed air is injected from the load applying device 8. The load applying device 8 may be one whose driving is controlled by the arithmetic device 7.

(Displacement characteristic measurement process)
Next, the displacement characteristics in the out-of-plane direction at the plurality of measurement points 30 arranged in the width direction of the strip 1 are measured (S20). Specifically, the displacement amounts of the measurement points 30 a to 30 e of the strip 1 to be excited are measured by the non-contact displacement meters 5 a to 5 e of the displacement characteristic measuring device 5. The measured displacement amount is output to the computing device 7 and stored in the temporary storage unit 16 in association with each of the measurement points 30a to 30e.

(Vibration characteristic calculation process)
Next, the natural frequency ω and the vibration mode vector ν are calculated based on the displacement amounts at the plurality of measurement points 30 stored in the temporary storage unit 16 (S30). Specifically, the i-th vibration mode vector ν _i is represented by a vector quantity as shown in the following formula (1). As a method for calculating the natural frequency and the vibration mode vector, there are an MDOF (Multiple Degrees Of Freedom method) method and an ERA (Eigensystem Realization Algorithm) method.

ν _i = {ν _{i, 1} , ν _{i, 2} ,..., ν _{i, n} } ^T (1)
Here, n shows the measurement point 30, and shows the measurement points 30a-30e in the order of 1-5. That is, ν _i consists of each element indicated by the measurement point 30 and the mode order. In the present embodiment, the vibration mode that matches the number of measurement points 30 is measured. The natural frequency and vibration mode vector calculated in this way are stored in the temporary storage unit 16.

(Modeling process)
Next, the strip 1 is modeled into a multi-mass point system model 40 as shown in FIG. 2 (S40). Specifically, the band 1 includes the Young's modulus of the band 1 stored in the temporary storage unit 16, the cross-sectional secondary moment of the band 1, the link length (the length of the connecting member 42), the density of the band 1, A belt-like body cross-sectional area or the like is used and modeled into a multi-mass system model 40.

(Eigenvalue analysis process)
Next, with respect to the multi-mass point system model 40 as described above, eigenvalues are analyzed, and eigenvalues and eigenvectors are calculated (S50). Specifically, first, the spring constant τ of the rotary spring 44 that connects the connecting member 42 is calculated as in the following equation (2). The spring constant τ of the rotary spring 44 indicates the bending rigidity at the measurement point 30.

τ = EI / l (2)
Here, E: Young's modulus of the strip 1, I: sectional moment of the strip 1, and l: link length.

  Then, a mass matrix and a stiffness matrix are calculated from an equation of motion such as Equation (3) based on the multi-mass point system model 40. The stiffness matrix is represented by an unknown spring constant of the distributed spring 43 and a spring constant τ of the rotary spring 44.

Ma + Kx = 0 (3)
Here, M: mass matrix, K: stiffness matrix, a: acceleration vector at each node 41, and x: displacement vector at each node 41. The displacement vector x is represented by a vector whose component is the displacement amount at each node 41 as in the following equation (4).

x = {x ₁ , x ₂ ,..., x _n } ^T (4)

  Then, eigenvalues and eigenvectors are calculated based on the mass matrix and stiffness matrix obtained from the equation of motion of Equation (3). Specifically, an eigenvalue Λ and an eigenvector Φ that satisfy the following equation (5) are calculated.

M ⁻¹ K = ΦΛΦ ^T (5)
That is, the eigenvalue analysis is performed so that the eigenvalue λ and the eigenvector φ where M ⁻¹ Kφ = λφ are calculated. The eigenvalue Λ and the eigenvector Φ calculated in this way are stored in the temporary storage unit 16. The calculated eigenvalues Λ and eigenvectors Φ are represented by matrices such as the following formulas (6) and (7). That is, the i-th order eigenvector φ _i is represented by a vector quantity as shown in the following equation (8).

Φ = [φ ₁ , φ ₂ ,..., Φ _n ] (7)
φ _i = {φ _{i, 1} , φ _{i, 2} ,..., φ _{i, n} } (8)
Here, λ _{n is} an eigenvalue for each vibration mode, and φ _n is an eigenvector for each vibration mode.

  In this way, the eigenvalue Λ and the eigenvector Φ are calculated. Since the spring constant of the distributed spring 43 is unknown, the eigenvalue Λ and eigenvector Φ are calculated by setting the initial value to the spring constant of the distributed spring 43, and the distributed spring 43 that minimizes the error by repeated calculation. Is calculated.

(Spring constant calculation process)
Next, a spring constant is calculated (S60). That is, the spring constant of the distributed spring 43 is calculated by repeated calculation so that the natural frequency and vibration mode vector based on the displacement characteristics obtained by the measurement match the natural value and natural vector obtained from the multi-mass system model 40, respectively. Is done.

Specifically, the difference between each component of the natural vector ν _i and the vibration mode vector φ _i and the square of the natural frequency ω _i and the difference between the natural value λ _i and the square of the natural frequency ω _i are calculated. A function represented by the following formula (9) that sums these squares is defined as an evaluation function J.

This evaluation function J has the spring constant k of the distributed spring 43 as a parameter because the vibration mode vector φ _i and the eigenvalue λ _i are calculated from the distributed spring constant k. As shown in the multi-mass point system model 40, the spring constant k _j is physically a positive value. Therefore, the optimization problem is solved so that the evaluation function J is minimized with k _j > 0 as a constraint. J indicates each node 41 in order. For example, the steepest descent method and the quasi-Newton method are used as the optimization method. Thereby, the spring constant k _j for each distributed spring 43 is calculated.

  Thus, in this embodiment, in order to calculate the spring constant for each distributed spring 43, the natural frequency and vibration mode vector based on the displacement obtained by measurement, the eigenvalue and eigenvector in the multi-mass system model 40, Is compared with the amount of displacement obtained by measurement and the amount of displacement of the node 41. Note that the comparison mode of the evaluation function is not limited to this, and for example, the displacement amount obtained by measurement and the displacement amount of the node 41 may be compared.

(Tension calculation process)
Next, the tension is calculated from the spring constant for each distributed spring 43 (S70). Specifically, the tension T _n applied to the belt-like body 1 and the natural frequency f _n have the relationship of the expression (10).
T _n = 4L ² f _n ² ρA (10)
Here, T _n is the tension value at the measurement point 30, L is the distance between the rolls, f _{n is the} natural frequency, ρ is the density of the strip 1, and A is the cross-sectional area of the strip.

Assuming that this natural frequency f _n coincides with the natural frequency represented by the spring constant k _j of the distributed spring 43, the tension at each measurement point 30 of the strip 1 is calculated. That is, the natural frequency f _n is _expressed by the following equation (11) by the spring constant k _j .
f _n = (√ (k _j / m _eq )) / 2π (11)
Where m _eq is the equivalent mass of the strip 1 at each node 41 at the mode order i. This equivalent mass m _eq is represented by the following formula (12).

m _eq = m _modal / ν _{i, n} ² (12)
Here, m _modal is the mode mass of the plate, and is calculated from [Φ] ^T [M] [Φ]. That is, ν _{i, n} is a component of the i-th order vibration mode vector obtained from the measurement.

Further, the mass m of the plate is represented by the following formula (13).
m = ρAL (13)

Thus, the tension value is given by the following equation (14), and the tension value is calculated for each measurement point 30 using this.
T _j = (k _j × L × m) / (π ² × m _{eq, j} ) (14)

Further, the total tension T _total in the width direction of the belt-like body 1 is given by the total tension value at each measurement point 30 as shown in the following equation (15).
T _total = ΣT _n (15)

  As described above, assuming that the change in tension is approximately replaced by the change in stiffness, the band 1 has a node 41 (nodes 41a to 41e) corresponding to the measurement point 30 (measurement points 30a to 30e) and the node 41. This is modeled as a two-dimensional multi-mass point system model 40 having distributed springs 43 (distributed springs 43 a to 43 e) that simulate the tension acting on the joints 41. Then, the tension is calculated from the spring constant of the distributed spring 43 such that the natural frequency and vibration mode vector distribution at the measurement point 30 matches the natural value and natural vector of the node 41.

  Thereby, the tension distribution of the strip 1 can be calculated using the two-dimensional multi-mass system model 40 that physically approximates the strip 1. As a result, even when the tension distribution is complicated in the width direction, it is possible to calculate each of the tensions distributed at the measurement points 30 with high accuracy using a simple physical model. Therefore, the total tension obtained by adding the tensions can be calculated with high accuracy.

(Second Embodiment)
Next, a tension measuring device according to the second embodiment will be described. However, about the thing which has the structure similar to 1st Embodiment, the same code | symbol is attached | subjected and the description is abbreviate | omitted suitably.

  The tension measuring device 204 of the present embodiment includes a load applying device 208, a displacement characteristic measuring device 5, and a computing device 207.

(Loading device 208)
As shown in FIG. 4, the load applying device 208 is disposed below the strip 1. The load applying device 208 extends in the width direction in which the measurement points 30a to 30e are distributed. The load applying device 208 has a function of applying a static load in the out-of-plane direction to the strip 1. The load applying device 208 is movable in parallel with the strip 1. Thereby, measurement conditions can be changed as appropriate. In the present embodiment, the load applying device 208 has a plurality of holes in the width direction facing the strip 1. The load applying device 208 continuously injects the fluid from the plurality of holes to the strip 1 so that the strip 1 has a static deflection at the injection location.

(Calculation device 207)
The arithmetic device 207 includes a vibration characteristic calculation unit 11, a modeling unit 12, a spring constant calculation unit 214, a tension calculation unit 15, and a temporary storage unit 16.

(Spring constant calculation unit 214)
The spring constant calculation unit 214 calculates the spring constant of the distributed spring 43 such that the distribution of the displacement amount at the measurement points 30a to 30e matches the distribution of the displacement amount at the nodes 41a to 41e.

  The tension measuring method of the present embodiment executed by the tension measuring device 204 configured as described above will be described with reference to the flowchart of FIG.

(Load application process)
First, a static load is applied to the strip 1 (S210). In the present embodiment, unlike the first embodiment, the belt-like body 1 is statically bent by the load applying device 208.

(Displacement characteristic measurement process)
Next, the displacement characteristics in the out-of-plane direction at the plurality of measurement points 30 arranged in the width direction of the strip 1 are measured (S220). In the present embodiment, unlike the first embodiment, the belt-like body 1 is statically loaded. Therefore, the displacement vector x _exp indicating the static displacement amount at the measurement point 30 of the strip 1 is output to the computing device 207 as the displacement characteristic.

(Modeling process)
Next, the strip 1 is modeled into a multi-mass point system model 40 as shown in FIG. 2 (S240). Specifically, the band 1 uses the Young's modulus of the band 1 stored in the temporary storage unit 16, the cross-sectional secondary moment of the band 1, the link length, the density of the band 1, the band cross-sectional area, and the like. The multi-mass point model 40 is modeled.

(Spring constant calculation process)
Next, the spring constant is calculated (S260). In this embodiment, the spring constant of the distributed spring 43 is calculated such that the distribution of the displacement amount at the measurement point 30 and the distribution of the displacement amount at the node 41 match based on the displacement characteristics.

Specifically, for the modeled multi-mass point system model 40, an equation of motion with a static load F as an external force is given by the following equation (16).
Kx _ana = F ···· (16)
Here, x _ana is a displacement vector indicating the displacement of each node 41.

Then, the displacement vector _xana is calculated by multiplying both sides of the equation (16) by the inverse matrix of K. The displacement vector _xana has the spring constant of the distributed spring 43 as a parameter for each component.

Then, for each measurement point 30 and each node 41, the following equation (2) squares the difference between each component of the displacement vector x _ana obtained from the multi-mass system model 40 and the displacement vector x _exp obtained by the measurement ( The function shown in 17) is defined as an evaluation function J.

This evaluation function J has a spring constant k _j of the distributed spring 43 as a parameter. As shown in the multi-mass point system model 40, the spring constant k _j is physically a positive value. Therefore, with the constraint that k _j > 0, the optimization is performed for each distributed spring 43 so that the evaluation function J is minimized. Note that j is 1, 2,..., N, and indicates each node 41 in order. For example, the steepest descent method and the quasi-Newton method are used as the optimization method. Thereby, the spring constant k _j for each distributed spring 43 is calculated.

(Tension calculation process)
Next, similarly to the tension calculation step of the first embodiment, the tension value and the total tension value at each measurement point 30 are calculated based on the spring constant of the distributed spring 43 calculated by the spring constant calculation step.

  As described above, assuming that the change in tension is approximately replaced by the change in stiffness, the band 1 has a node 41 (nodes 41a to 41e) corresponding to the measurement point 30 (measurement points 30a to 30e) and the node 41. This is modeled as a two-dimensional multi-mass point system model 40 having distributed springs 43 (distributed springs 43 a to 43 e) that simulate the tension acting on the joints 41. Then, the tension is calculated from the spring constant of the distributed spring 43 so that the distribution of the displacement of the strip 1 matches the amount of displacement of the node 41 of the multi-mass point system model.

  Thereby, the tension distribution of the strip 1 can be calculated by the two-dimensional multi-mass point system model 40 that physically approximates the strip 1. As a result, even when the tension distribution is complicated in the width direction, it is possible to calculate each of the tensions distributed at the measurement points 30 with high accuracy using a simple physical model. Therefore, the total tension obtained by adding the tensions can be calculated with high accuracy.

(Third embodiment)
Next, a tension measuring device according to a third embodiment will be described. However, about the thing which has the structure similar to 1st Embodiment, the same code | symbol is attached | subjected and the description is abbreviate | omitted suitably.

  The tension measuring device of the present embodiment has the same configuration as the tension measuring device 4 (see FIG. 1) of the first embodiment. A tension measuring method executed by the tension measuring device of the present embodiment will be described with reference to the flowchart of FIG.

  In the flowchart of FIG. 7, the load adding step (S10), the displacement characteristic measuring step (S20), the vibration characteristic calculating step (S30), the modeling step (S40), and the tension calculating step (S70) are the first embodiment. The description is omitted because it is the same.

(Spring constant calculation process)
After the modeling step (S40) similar to that of the first embodiment, the spring constant is calculated (S360). That is, the spring constant of the distributed spring 43 such that the natural frequency and vibration mode vector based on the displacement characteristics obtained by the measurement and the eigenvalue and eigenvector obtained from the multi-mass system model 40 coincide with each other by the least square method. Calculated.

Specifically, assuming that the equation of motion of the multi-mass point system model 40 in FIG. 2 is expressed by equation (3), the stiffness matrix K is a rotary spring that represents the bending stiffness of the plate as shown in equation (18). and component K _R, is divided into a distribution spring component K _T simulating the tension. The rotational spring component K _R in a known matrix obtained from the physical properties and dimensions of the plate, the distribution spring component K _T is a matrix of unknown parameters to be determined.

K = K _T + K _R (18)

  Based on the experimental value of the displacement characteristic obtained in the displacement characteristic measurement step (S20), the natural frequency Λ and the vibration mode Φ are calculated in the vibration characteristic calculation step (S30). The mass matrix and stiffness matrix obtained from the equation of motion of equation (3) and the natural frequency and vibration mode obtained by experiment have the relationship of the following equation (19).

(Φ ^T MΦ) ⁻¹ Φ ^T (K _T + K _R ) Φ = Λ (19)
Here, Λ and Φ is a known parameter obtained by calculation experimental values, M and K _R. The stiffness matrix K _T of the spring distribution representing the tension distribution and the unknown parameter, by focusing on the components of each matrix, organize equation (19). In the calculation, any number of natural frequencies and vibration mode vectors of Λ and Φ may be used. Here, a case where the first to fourth orders are used will be described. That is, the eigenvector Φ is as follows.
Φ = [φ ₁ , φ ₂ , φ ₃ , φ ₄ ] (20)

  When formula (19) is expanded for each component, the following formula is obtained.

Since both sides of the equation (21) is a symmetric matrix, ten independent equations shown in the following equation is obtained, and arranging for the unknowns K _T, as follows.

Further, since the stiffness matrix _KT is a diagonal matrix, it is expressed by the following equation (23). Here, k _Tn in the equation (23) corresponds to the spring constant of each distributed spring in FIG.

When the left side of Expression (22) is expanded and the unknown _KT is summarized, it is expressed by the following Expression (24).

If the equation (24) is solved for the unknown vector k _T = {k _T1 , k _T2 ,..., K _Tn } ^{T by the} least square method, the spring distribution can be obtained.

  Here, with respect to the equation (24), the equation when the number of modes is generalized as m is shown below. By using the following equation, a least square method adapted to the number of modes obtained by experiments can be applied.

Least squares method, the coefficient matrix of the k _T in equation (24) is V, the vector of the right side as k _x, solving the following equation.

Vk _T = k _x (26)
k _T = V ^† k _x (27)
However,
V ^† = (V ^T V) ⁻¹ V ^T (28)

Here, when the rank of the matrix V is less than the number of columns, the solution of Expression (27) diverges. Therefore, the least square method is solved using the singular value decomposition shown in Equation (29). Where P and Q are orthogonal matrices having a singular vector of V as a column vector, D is a diagonal matrix having a singular value of V as a diagonal component, and Q ^* is a conjugate transpose of Q.
V = PDQ ^* (29)

In order to improve the identification accuracy, only a singular value having a large absolute value is adopted in the matrix D, and a matrix obtained by replacing a singular value having a small absolute value with “0” is defined as D ′. Further, a pseudo inverse matrix of V is obtained as shown in the following equation, where D ′ ^† is a transposed matrix of a matrix whose diagonal component is the reciprocal of a singular value that is not “0” of D ′.
V ^† = QD ' ^† P ^* (30)

  As described above, the least square method is solved and the spring distribution is obtained.

(Tension calculation process)
Next, the tension distribution and the total tension are calculated by converting the value of the spring distribution into the tension by the same tension calculating step (S70) as in the first embodiment (S70).

  FIG. 8 shows the calculation results of the tension distribution according to Equation (28) using all singular values and Equation (30) from which small singular values are removed. It can be seen that better results are obtained by removing smaller singular values than using all singular values. Note that the number of singular values having a large absolute value employed in the equation (30) is arbitrary, but a relatively good result can be obtained by making it coincide with the number of modes obtained through experiments, for example.

  In this way, by using the least square method, the spring constant of the distributed spring 43 can be calculated without causing a convergence problem in repeated calculations and variations in calculation results due to initial values. As a result, even when the tension distribution is complicated in the width direction, it is possible to calculate each of the tensions distributed at the measurement points 30 with high accuracy using a simple physical model.

(Modification)
The embodiment of the present invention has been described above, but only specific examples are illustrated, and the present invention is not particularly limited, and the specific configuration and the like can be appropriately changed in design. In addition, the actions and effects described in the embodiments of the present invention only list the most preferable actions and effects resulting from the present invention, and the actions and effects according to the present invention are described in the embodiments of the present invention. It is not limited to things.

  For example, in the present embodiment, the belt-like body 1 is modeled as a two-dimensional model such as the multi-mass point system model 40 shown in FIG. 2, but the present invention is not limited to this. For example, a multi-mass point model 340 as shown in FIG. 6 may be used.

  Specifically, the multi-mass point system model 340 includes a node 341, a distribution spring 43 (distribution springs 43a, 43b, 43c, 43d, and 43e), a shear spring 344, and a fixed surface 45. As shown in FIG. 6, the node 341 has the mass of the band 1. The shear spring 344 connects adjacent nodes 341 and a restoring force proportional to the relative displacement between the connected nodes 341 acts.

DESCRIPTION OF SYMBOLS 1 Band-like body 2/3 Roll 4 Tension measuring device 5 Displacement characteristic measuring device 5a * 5b * 5c * 5d * 5e Non-contact displacement meter 7 Arithmetic device 8 Load application apparatus 11 Vibration characteristic calculation part 12 Modeling part 13 Eigenvalue analysis part 14 Spring constant calculation unit 15 Tension calculation unit 16 Temporary storage unit 30 (30a / 30b / 30c / 30d / 30e) Measurement point 40 Multi-mass point system model 41 (41a / 41b / 41c / 41d / 41e) Node 42 Connecting member 43 (43a · 43b · 43c · 43d · 43e) Distributed spring 44 Rotating spring 45 Fixed surface 204 Tension measuring device 207 Arithmetic device 208 Load applying device 214 Spring constant calculation unit

Claims (6)

A tension measurement method for measuring displacement characteristics at a plurality of measurement points arranged in the width direction of a belt-like body to which a load is applied,
A modeling step for modeling the band-like body into a two-dimensional multi-mass system model having a node corresponding to the measurement point and a spring that is joined to the node and simulates a tension acting on the node;
A spring constant calculation step of calculating a spring constant of the spring such that a distribution of the displacement amount of the measurement point based on the displacement characteristic and a distribution of the displacement amount of the node coincide with each other;
And a tension calculating step of calculating a tension based on the spring constant. A tension measurement method for measuring displacement characteristics at a plurality of measurement points arranged in the width direction of a belt-like body to which a load is applied,
A modeling step for modeling the band-like body into a two-dimensional multi-mass system model having a node corresponding to the measurement point and a spring that is joined to the node and simulates a tension acting on the node;
An eigenvalue analysis step of analyzing eigenvalues and eigenvectors in the multi-mass point system model;
A spring constant calculating step of calculating a spring constant of the spring such that the natural frequency and vibration mode vector based on the displacement characteristics and the natural value and natural vector respectively match;
And a tension calculating step of calculating a tension based on the spring constant.   The tension measuring method according to claim 2, wherein, in the spring constant calculating step, a spring constant of the spring is calculated by a least square method. A tension measuring device that measures displacement characteristics at a plurality of measurement points arranged in the width direction of a belt-like body to which a load is applied,
Modeling means for modeling the belt-like body into a two-dimensional multi-mass point system model having a node corresponding to the measurement point and a spring that is joined to the node and simulates a tension acting on the node;
A spring constant calculating means for calculating a spring constant of the spring such that a distribution of the displacement amount of the measurement point based on the displacement characteristic and a distribution of the displacement amount of the node coincide with each other;
A tension measuring device comprising tension calculating means for calculating a tension based on the spring constant. A tension measuring device that measures displacement characteristics at a plurality of measurement points arranged in the width direction of a belt-like body to which a load is applied,
Modeling means for modeling the belt-like body into a two-dimensional multi-mass point system model having a node corresponding to the measurement point and a spring that is joined to the node and simulates a tension acting on the node;
Eigenvalue analysis means for analyzing eigenvalues and eigenvectors in the multi-mass point system model;
A spring constant calculating means for calculating a spring constant of the spring such that the natural frequency and vibration mode vector based on the displacement characteristic and the natural value and natural vector respectively match;
A tension measuring device comprising tension calculating means for calculating a tension based on the spring constant.   6. The tension measuring apparatus according to claim 5, wherein the spring constant calculating means calculates a spring constant of the spring by a least square method.