JP5531350B2

JP5531350B2 - Tension measuring device

Info

Publication number: JP5531350B2
Application number: JP2011277763A
Authority: JP
Inventors: 広敏在原; 徹岡田
Original assignee: Kobe Steel Ltd
Current assignee: Kobe Steel Ltd
Priority date: 2011-12-20
Filing date: 2011-12-20
Publication date: 2014-06-25
Anticipated expiration: 2031-12-20
Also published as: JP2013130393A

Description

本発明は、帯状体の幅方向の張力分布を測定する張力測定装置に関する。 The present invention relates to a tension measuring device that measures a tension distribution in the width direction of a belt-like body.

薄鋼板等の帯状体を通板して、圧延、矯正、焼鈍、表面処理、脱脂、スリッティング等の各種処理を連続的に行うプロセスラインでは、帯状体の走行時の蛇行を防止するためや、各種処理を良好に行うために、帯状体が走行する長手方向に張力を付与することが多い。また、これらのプロセスラインに通板される帯状体には、耳波、中伸び、片伸びや、これらが複合した複合伸び等の幅方向での不均一歪みが存在することがある。これらの幅方向の不均一歪みは、板製品の平坦度不良となるのみでなく、スムーズな通板を阻害することもあるので、ロールベンディング装置やクーラントのゾーンコントロール装置等によって、不均一歪みを除去または低減するように平坦度制御が行われている。 In a process line that continuously passes various strips such as rolling, straightening, annealing, surface treatment, degreasing, slitting, etc., by passing strips such as thin steel plates, to prevent meandering during strip travel In order to perform various treatments satisfactorily, tension is often applied in the longitudinal direction in which the strip travels. In addition, the band-shaped body passed through these process lines may have non-uniform distortion in the width direction such as ear waves, medium elongation, single elongation, or composite elongation in which these are combined. These non-uniform distortions in the width direction not only cause poor flatness of the plate product, but also hinder smooth plate-through, so non-uniform distortion may be caused by roll bending equipment or coolant zone control equipment. Flatness control is performed so as to eliminate or reduce.

上述した帯状体の長手方向に張力が付与されるプロセスラインでは、幅方向の不均一歪みが潜在化して、見かけ上平坦になることがある。このため、平坦度制御を行うために、帯状体がロール等によって支持される長手方向の２箇所の支持部位間で、帯状体に振動荷重や静荷重を負荷して、そのときの帯状体の幅方向の変位分布を計測することにより、幅方向の張力分布を測定し、不均一歪みを間接的に検出する技術が開発されている（例えば、特許文献１、２参照）。 In the process line in which tension is applied in the longitudinal direction of the band-shaped body described above, uneven strain in the width direction may become latent and appear flat. For this reason, in order to control the flatness, a vibration load or a static load is applied to the belt-like body between two longitudinal support portions where the belt-like body is supported by a roll or the like, and the belt-like body at that time A technique has been developed in which a tension distribution in the width direction is measured by measuring a displacement distribution in the width direction to indirectly detect non-uniform strain (for example, see Patent Documents 1 and 2).

特許文献１、２に記載されたものは、いずれも幅方向の不均一歪みが全て潜在化する場合を想定しており、特許文献１では、振動荷重による変位分布として、一次共振モードと二次共振モードを板幅方向の複数点で計測し、計測した変位分布の形状を、予め区分した張力分布パターンのいずれかに決定し、決定した張力分布パターンに従って、別途に計測した全張力を板幅方向の各位置に割り振ることで、張力分布を測定するようにしている。また、特許文献２では、幅方向の複数点で振動モードを計測し、各振動モードの周波数と振幅の正規化値から、幅方向の張力分布を測定するようにしている。 Both of Patent Documents 1 and 2 assume the case where all of the non-uniform distortion in the width direction is latent. In Patent Document 1, as a displacement distribution due to a vibration load, primary resonance mode and secondary The resonance mode is measured at a plurality of points in the plate width direction, and the shape of the measured displacement distribution is determined as one of the tension distribution patterns divided in advance, and the total tension separately measured according to the determined tension distribution pattern The tension distribution is measured by assigning each position in the direction. In Patent Document 2, vibration modes are measured at a plurality of points in the width direction, and the tension distribution in the width direction is measured from the normalized values of the frequency and amplitude of each vibration mode.

特許文献１に記載されたものは、張力分布を予め設定したいずれかの張力分布パターンに当てはめるので、実際に生じる複雑な張力分布を精度よく測定できない問題があった。また、特許文献２に記載されたものは、加振方法や加振位置によって振動モード形状がばらつくので、測定精度が悪化する問題があった。 Since the tension distribution is applied to any preset tension distribution pattern, the one disclosed in Patent Document 1 has a problem that a complex tension distribution actually generated cannot be measured with high accuracy. Further, the device described in Patent Document 2 has a problem that the measurement accuracy deteriorates because the vibration mode shape varies depending on the vibration method and the vibration position.

本発明者らは、これらの問題に対して、どのような張力分布であっても精度よく測定できるように、荷重が負荷された帯状体の幅方向の複数の測定点における変位を計測する変位計測手段と、帯状体について、複数の測定点に対応する節点と、該節点に接合された節点に作用する張力を模擬したばねとを有する２次元の多質点系モデルにモデル化するモデル化手段と、計測された測定点の変位量分布と、モデルの節点の変位量分布とが一致するようなばねのばね定数を算出するばね定数算出手段と、算出されたばね定数に基づいて張力を算出する張力算出手段とを有する張力測定装置を先に提案している（特願２０１０−２７４３４５）。この張力測定装置も、帯状体の幅方向の不均一歪みが全て潜在化する場合を想定している。 In order to solve these problems, the present inventors can measure the displacement at a plurality of measurement points in the width direction of the band-like body loaded with a load so that any tension distribution can be accurately measured. Modeling means for modeling a two-dimensional multi-mass point system model having measuring means and a strip corresponding to a plurality of measurement points and a spring simulating a tension acting on the nodes joined to the nodes. And a spring constant calculation means for calculating a spring constant of the spring so that the measured displacement distribution of the measurement point and the displacement distribution of the model node coincide with each other, and a tension is calculated based on the calculated spring constant. A tension measuring device having a tension calculating means has been previously proposed (Japanese Patent Application No. 2010-274345). This tension measuring device also assumes a case in which all non-uniform distortion in the width direction of the belt-like body is latent.

特開平６−４３０５１号公報JP-A-6-43051 特開平７−２１８３５８号公報JP 7-218358 A

上述した薄鋼板等のプロセスラインには、帯状体に付与できる張力が低い値に制約されるものや、例えば、矯正ラインの上流側のように、帯状体に大きな幅方向の不均一歪みが存在するものがあり、不均一歪みが全て潜在化せずに一部が顕在することがある。このように幅方向の不均一歪みが一部顕在する場合は、特許文献１、２に記載された張力分布測定方法や特願２０１０−２７４３４５で提案した張力測定装置は、いずれも幅方向の不均一歪みが全て潜在化することを前提としているので、張力分布を精度よく測定できない問題がある。 In the process line such as the above-mentioned thin steel plate, there is a large tension unevenness in the band direction, such as those where the tension that can be applied to the band is limited to a low value, for example, upstream of the correction line In some cases, some non-uniform distortion does not become latent and part of it appears. In this way, when some uneven strain in the width direction is manifested, the tension distribution measuring method described in Patent Documents 1 and 2 and the tension measuring device proposed in Japanese Patent Application No. 2010-274345 are both non-uniform in the width direction. Since it is assumed that all uniform strains are latent, there is a problem that the tension distribution cannot be measured accurately.

そこで、本発明の課題は、帯状体に幅方向の不均一歪みが一部顕在しても、張力分布を精度よく測定できるようにすることである。 Therefore, an object of the present invention is to make it possible to accurately measure the tension distribution even if a non-uniform strain in the width direction appears partially in the belt-like body.

上記の課題を解決するために、本発明は、長手方向に張力を付与された帯状体の幅方向の張力分布を、長手方向の２箇所の部位で支持された支持部位間で測定する張力測定装置において、前記２箇所の支持部位間で前記帯状体に振動荷重を負荷する振動荷重負荷手段と、この振動荷重負荷手段によって生じる振動変位を帯状体の幅方向の複数の測定点で計測する変位計測手段とを備え、前記帯状体について、前記幅方向の複数の各測定点に対応する各節点に作用する張力を模擬する直線ばねと、該各節点における幅方向の曲げ剛性を模擬する回転ばねとを含む２次元の多質点系モデルにモデル化するモデル化手段と、前記多質点系モデルの固有値解析から得られる前記各節点での固有振動数および振動モードが、前記計測された各測定点での振動変位から得られる固有振動数および振動モードと一致するような前記直線ばねおよび回転ばねの各ばね定数を算出するばね定数算出手段と、算出された前記直線ばねおよび回転ばねの各ばね定数を、それぞれ前記各測定点における張力値および曲げ剛性値に換算するばね定数換算手段とを有し、換算された前記各測定点における張力値から前記帯状体の幅方向の張力分布を求める構成を採用した。 In order to solve the above-described problems, the present invention measures the tension distribution in the width direction of the belt-like body that is tensioned in the longitudinal direction between the support parts supported by the two parts in the longitudinal direction. In the apparatus, a vibration load loading means for applying a vibration load to the belt-like body between the two support parts, and a displacement for measuring the vibration displacement caused by the vibration load loading means at a plurality of measurement points in the width direction of the belt-like body A linear spring that simulates the tension acting on each node corresponding to each of the plurality of measurement points in the width direction, and a rotary spring that simulates the bending rigidity in the width direction at each node. Modeling means for modeling into a two-dimensional multi-mass point system model including the natural frequency and vibration mode at each node obtained from the eigenvalue analysis of the multi-mass point system model. In A spring constant calculating means for calculating each spring constant of the linear spring and the rotary spring so as to coincide with a natural frequency and a vibration mode obtained from a dynamic displacement, and each spring constant of the calculated linear spring and the rotary spring, Each has a spring constant conversion means for converting into a tension value and a bending stiffness value at each measurement point, and adopts a configuration for obtaining a tension distribution in the width direction of the strip from the converted tension value at each measurement point. .

すなわち、２箇所の支持部位間で帯状体に振動荷重を負荷する振動荷重負荷手段と、振動荷重負荷手段によって生じる振動変位を帯状体の幅方向の複数の測定点で計測する変位計測手段とを備え、帯状体について、幅方向の複数の各測定点に対応する各節点に作用する張力を模擬する直線ばねと、該各節点における幅方向の曲げ剛性を模擬する回転ばねとを含む２次元の多質点系モデルにモデル化するモデル化手段と、多質点系モデルの固有値解析から得られる各節点での固有振動数および振動モードが、計測された各測定点での振動変位から得られる固有振動数および振動モードと一致するような直線ばねおよび回転ばねの各ばね定数を算出するばね定数算出手段と、算出された直線ばねおよび回転ばねの各ばね定数を、それぞれ各測定点における張力値および曲げ剛性値に換算するばね定数換算手段とを有し、換算された各測定点における張力値から帯状体の幅方向の張力分布を求めることにより、不均一歪みの一部顕在によって幅方向の曲げ剛性が変化する帯状体であっても、張力を模擬する直線ばねのばね定数と、幅方向の曲げ剛性を模擬する回転ばねのばね定数とを未知数として算出し、張力分布を精度よく測定できるようにした。また、この張力測定装置は、幅方向の曲げ剛性分布も併せて求めることができるので、不均一歪みの顕在形態の推定にも寄与することができる。 That is, vibration load loading means for applying a vibration load to the band between two support parts, and displacement measuring means for measuring vibration displacement caused by the vibration load loading means at a plurality of measurement points in the width direction of the band. A two-dimensional including a linear spring that simulates a tension acting on each node corresponding to each of a plurality of measurement points in the width direction, and a rotary spring that simulates a bending rigidity in the width direction at each node. Modeling means for modeling into a multi-mass system model, and the natural frequency and vibration mode at each node obtained from eigenvalue analysis of the multi-mass system model are obtained from the vibration displacement at each measured measurement point. The spring constant calculating means for calculating the spring constants of the linear spring and the rotary spring that match the number and the vibration mode, and the calculated spring constants of the linear spring and the rotary spring, respectively, Spring constant conversion means for converting the tension value and the bending stiffness value in the case, and by obtaining the tension distribution in the width direction of the strip from the converted tension value at each measurement point, Even for strips with varying bending rigidity in the width direction, the spring constant of the linear spring that simulates the tension and the spring constant of the rotary spring that simulates the bending rigidity in the width direction are calculated as unknowns, and the tension distribution is accurate. I was able to measure well. Moreover, since this tension measuring apparatus can also obtain | require the bending rigidity distribution of the width direction collectively, it can also contribute to estimation of the manifestation form of a nonuniform distortion.

前記ばね定数算出手段は、前記固有値解析から得られる前記各節点での固有振動数および振動モードと、前記各測定点での振動変位から得られる固有振動数および振動モードとの各差の二乗和が最小となるような前記直線ばねおよび回転ばねの各ばね定数を、繰り返し計算を用いた最小二乗法によって算出するものとすることができる。 The spring constant calculating means is a sum of squares of differences between the natural frequency and vibration mode at each node obtained from the eigenvalue analysis and the natural frequency and vibration mode obtained from vibration displacement at each measurement point. It is possible to calculate the spring constants of the linear spring and the rotary spring that minimize the value by the least square method using iterative calculation.

前記ばね定数算出手段が、前記帯状体の振動モードに対する前記直線ばねおよび回転ばねの各ばね定数の係数行列の擬似逆行列を計算し、前記固有値解析から得られる前記各節点での固有振動数および振動モードと、前記各測定点での振動変位から得られる固有振動数および振動モードとが一致するような前記直線ばねおよび回転ばねの各ばね定数を、前記擬似逆行列を用いた最小二乗法によって算出するものとすることもできる。 The spring constant calculation means calculates a pseudo inverse matrix of a coefficient matrix of each spring constant of the linear spring and the rotary spring with respect to the vibration mode of the strip, and the natural frequency at each node obtained from the eigenvalue analysis and The spring constants of the linear spring and the rotary spring, in which the vibration mode matches the natural frequency and vibration mode obtained from the vibration displacement at each measurement point, are obtained by the least square method using the pseudo inverse matrix. It can also be calculated.

前記計算された擬似逆行列から特異値分解により誤差成分を除去し、この誤差成分を除去した擬似逆行列を前記最小二乗法による各ばね定数の算出に用いることにより、直線ばねおよび回転ばねの各ばね定数を求める演算を簡単にすることができるとともに、より精度よく張力分布を測定することができる。 An error component is removed from the calculated pseudo inverse matrix by singular value decomposition, and the pseudo inverse matrix from which the error component is removed is used for calculation of each spring constant by the least square method. The calculation for obtaining the spring constant can be simplified, and the tension distribution can be measured with higher accuracy.

本発明に係る張力測定装置は、２箇所の支持部位間で帯状体に振動荷重を負荷する振動荷重負荷手段と、振動荷重負荷手段によって生じる振動変位を帯状体の幅方向の複数の測定点で計測する変位計測手段とを備え、帯状体について、幅方向の複数の各測定点に対応する各節点に作用する張力を模擬する直線ばねと、該各節点における幅方向の曲げ剛性を模擬する回転ばねとを含む２次元の多質点系モデルにモデル化するモデル化手段と、多質点系モデルの固有値解析から得られる各節点での固有振動数および振動モードが、計測された各測定点での振動変位から得られる固有振動数および振動モードと一致するような直線ばねおよび回転ばねの各ばね定数を算出するばね定数算出手段と、算出された直線ばねおよび回転ばねの各ばね定数を、それぞれ各測定点における張力値および曲げ剛性値に換算するばね定数換算手段とを有し、換算された各測定点における張力値から帯状体の幅方向の張力分布を求めるようにしたので、不均一歪みの一部顕在によって幅方向の曲げ剛性が変化する帯状体であっても、張力分布を精度よく測定することができる。 The tension measuring device according to the present invention includes a vibration load loading means for applying a vibration load to the band between two support sites, and vibration displacement caused by the vibration load loading means at a plurality of measurement points in the width direction of the band. A linear spring that simulates the tension acting on each node corresponding to each of a plurality of measurement points in the width direction, and a rotation that simulates the bending rigidity in the width direction at each node. Modeling means for modeling into a two-dimensional multi-mass system model including a spring, and the natural frequency and vibration mode at each node obtained from the eigenvalue analysis of the multi-mass system model are as follows. Spring constant calculating means for calculating the spring constants of the linear spring and the rotary spring that match the natural frequency and vibration mode obtained from the vibration displacement, and the calculated spring constants of the linear spring and the rotary spring It has a spring constant conversion means for converting to a tension value and a bending stiffness value at each measurement point, and the tension distribution in the width direction of the strip is obtained from the converted tension value at each measurement point. Even in the case of a belt-like body whose bending rigidity in the width direction changes due to partial manifestation of the strain, the tension distribution can be accurately measured.

帯状体の張力分布測定装置の実施形態を示す構成図The block diagram which shows embodiment of the tension | tensile_strength distribution measuring apparatus of a strip | belt-shaped body 図１のモデル化部で帯状体をモデル化する２次元多質点系モデルの概念図Conceptual diagram of a two-dimensional multi-mass point system model that models a band in the modeling unit of FIG. 図１の張力分布測定装置で張力分布を測定する第１の実施形態を示すフローチャートThe flowchart which shows 1st Embodiment which measures tension distribution with the tension distribution measuring apparatus of FIG. 図１の張力分布測定装置で張力分布を測定する第２の実施形態を示すフローチャートThe flowchart which shows 2nd Embodiment which measures tension distribution with the tension distribution measuring apparatus of FIG. 図４での特異値分解結果を示すグラフThe graph which shows the singular value decomposition result in FIG. 図４での特異値分解の張力分布測定精度に対する効果を示すグラフThe graph which shows the effect with respect to the tension distribution measurement precision of singular value decomposition in FIG. （ａ）、（ｂ）、（ｃ）は、それぞれ実施例での薄鋼板の張力分布測定結果をＦＥＭ解析結果と対比して示すグラフ(A), (b), (c) is the graph which each shows the tension distribution measurement result of the thin steel plate in an Example with the FEM analysis result. （（ａ）、（ｂ）、（ｃ）は、それぞれ実施例でのアルミニウム薄板の張力分布測定結果をＦＥＭ解析結果と対比して示すグラフ((A), (b), (c) is a graph showing the tension distribution measurement result of the aluminum thin plate in the example in comparison with the FEM analysis result, respectively. （ａ）、（ｂ）、（ｃ）は、それぞれ実施例での薄鋼板の断面二次モーメント分布の算出結果を示すグラフ(A), (b), (c) is a graph which shows the calculation result of the cross-sectional secondary moment distribution of the thin steel plate in an Example, respectively. （ａ）、（ｂ）、（ｃ）は、それぞれ実施例でのアルミニウム薄板の断面二次モーメント分布の算出結果を示すグラフ(A), (b), (c) is a graph which respectively shows the calculation result of the cross-sectional secondary moment distribution of the aluminum thin plate in an Example.

以下、図面に基づき、本発明の実施形態を説明する。この張力測定装置は、図１に示すように、走行する長手方向に張力を付与された帯状体１の張力分布を、長手方向の２箇所の部位で支持ロール２ａ、２ｂによって支持された支持部位間で測定するものであり、支持部位間の帯状体１に振動荷重を負荷する振動荷重負荷装置３と、振動荷重負荷装置３によって生じる振動変位を、帯状体１の幅方向の複数の測定点１ａで計測する非接触式の変位計４と、各変位計４の出力に基づいて、帯状体１の張力分布を演算する演算装置５とからなる。帯状体１の各測定点１ａは、支持部位間の長手方向の中間位置に等間隔で設定されている。 Hereinafter, embodiments of the present invention will be described with reference to the drawings. As shown in FIG. 1, this tension measuring device is configured to support the tension distribution of the belt-like body 1 provided with tension in the traveling longitudinal direction by supporting rolls 2 a and 2 b at two longitudinal portions. The vibration load device 3 that applies a vibration load to the belt 1 between the support parts, and the vibration displacement caused by the vibration load device 3 are measured at a plurality of measurement points in the width direction of the belt 1. It comprises a non-contact displacement meter 4 that is measured at 1a and an arithmetic device 5 that calculates the tension distribution of the strip 1 based on the output of each displacement meter 4. Each measurement point 1a of the belt-like body 1 is set at equal intervals at intermediate positions in the longitudinal direction between the support portions.

前記振動荷重負荷装置３は、帯状体１に空気を間歇的に噴射して振動荷重を負荷するものであるが、振動荷重負荷装置３は、水や油等の液体を間歇的に噴射するものや、磁力、静電力、電磁誘導による渦電流、音波等によって、振動荷重を負荷するものとすることもできる。また、帯状体１の１点を打撃する装置や、支持ロール２ａ、２ｂのいずれかを加振する装置とすることもできる。 The vibration load device 3 intermittently injects air onto the strip 1 to apply a vibration load, but the vibration load device 3 intermittently injects a liquid such as water or oil. Alternatively, a vibration load can be applied by magnetic force, electrostatic force, eddy current due to electromagnetic induction, sound wave, or the like. Moreover, it can also be set as the apparatus which strikes one point of the strip | belt-shaped body 1, or the apparatus which vibrates either support roll 2a, 2b.

前記変位計４は光反射式のレーザ変位計とされている。帯状体１が導電性を有するものである場合は、帯状体１に生じさせた渦電流の大きさを検出する渦電流式変位計や、帯状体１とセンサヘッド間の静電容量を検出する静電容量式変位計等とすることもできる。また、図１では、便宜上、変位計４を幅方向に等間隔で５台配置するように図示しているが、変位計４の配置台数（測定点１ａの数）は任意に設定することができ、幅方向での配置間隔も、例えば、幅端部を密に、幅中央部を粗くするように不等間隔で配置してもよい。さらに、一部の変位計４を幅方向に移動可能としてもよい。 The displacement meter 4 is a light reflection type laser displacement meter. When the strip 1 is conductive, an eddy current displacement meter that detects the magnitude of the eddy current generated in the strip 1 or the capacitance between the strip 1 and the sensor head is detected. A capacitance displacement meter or the like can also be used. Further, in FIG. 1, for the sake of convenience, five displacement meters 4 are arranged at equal intervals in the width direction, but the number of displacement meters 4 (the number of measurement points 1a) may be arbitrarily set. For example, the arrangement intervals in the width direction may be arranged at unequal intervals so that the width end portions are dense and the width center portion is rough. Furthermore, some displacement meters 4 may be movable in the width direction.

前記演算装置５は、各変位計４で計測された各測定点１ａの振動変位から固有振動数と振動モードを算出する振動特性算出部５ａと、帯状体１を後述する２次元多質点系モデルにモデル化するモデル化部５ｂと、２次元多質点系モデルの固有値を解析する固有値解析部５ｃと、振動特性算出部５ａで算出された固有振動数および振動モードから２次元多質点系モデルの後述する直線ばねおよび回転ばねの各ばね定数を算出するばね定数算出部５ｄと、算出された直線ばねおよび回転ばねの各ばね定数を各測定点１ａにおける張力値および曲げ剛性値に変換するばね定数変換部５ｅと、変換された各測定点１ａの張力値から帯状体１の幅方向の張力分布を算出する張力分布算出部５ｆとで構成されている。 The arithmetic unit 5 includes a vibration characteristic calculation unit 5a that calculates a natural frequency and a vibration mode from the vibration displacement of each measurement point 1a measured by each displacement meter 4, and a two-dimensional multi-mass point system model that describes the band 1 later. The modeling unit 5b for modeling the model, the eigenvalue analysis unit 5c for analyzing the eigenvalue of the two-dimensional multi-mass system model, and the two-dimensional multi-mass system model from the natural frequency and the vibration mode calculated by the vibration characteristic calculation unit 5a. A spring constant calculation unit 5d that calculates spring constants of a linear spring and a rotary spring, which will be described later, and a spring constant that converts the calculated spring constants of the linear spring and the rotary spring into a tension value and a bending rigidity value at each measurement point 1a. The conversion unit 5e and a tension distribution calculation unit 5f that calculates the tension distribution in the width direction of the strip 1 from the converted tension value of each measurement point 1a.

図２は、前記モデル化部５ｂで帯状体１をモデル化する２次元多質点系モデルを示す。この２次元多質点系モデルは、支持部位間の帯状体１について、振動変位の各測定点１ａに対応する各節点１１を、離間する固定面１２に垂直に直線ばね１３で連結し、隣接する節点１１をリンク１４で結合して、隣接するリンク１４同士を回転ばね１５で連結したものであり、各リンク１４はその幅方向部位での質量ｍと慣性モーメントＪを有する。 FIG. 2 shows a two-dimensional multi-mass point system model for modeling the band 1 in the modeling unit 5b. In this two-dimensional multi-mass point system model, the strips 1 between the support parts are connected by connecting the nodes 11 corresponding to the measurement points 1a of the vibration displacement with the linear springs 13 perpendicularly to the fixed surfaces 12 that are spaced apart. The nodes 11 are connected by links 14 and the adjacent links 14 are connected by a rotary spring 15. Each link 14 has a mass m and a moment of inertia J at the width direction portion.

このように、前記２次元多質点系モデルは、帯状体１の張力の大小と固有振動数の大小との間に相関があることに着目し、振動荷重負荷装置３により加振された帯状体１の各計測点１ａにおける振動変位と、各直線ばね１３の各節点１１における振動変位とが等しいものとして、帯状体１の幅方向の張力分布を直線ばね１３のばね定数の変化として把握するとともに、各計測点１ａにおける幅方向の曲げ剛性を、各節点１１における回転ばね１５のばね定数の変化として把握するようにモデル化したものである。 Thus, the two-dimensional multi-mass system model pays attention to the fact that there is a correlation between the magnitude of the tension of the strip 1 and the magnitude of the natural frequency, and the strip that is vibrated by the vibration load device 3. Assuming that the vibration displacement at each measurement point 1a of 1 is equal to the vibration displacement at each node 11 of each linear spring 13, the tension distribution in the width direction of the strip 1 is grasped as a change in the spring constant of the linear spring 13. The bending stiffness in the width direction at each measurement point 1 a is modeled so as to be grasped as a change in the spring constant of the rotary spring 15 at each node 11.

図３は、上述した張力分布測定装置を用いて張力分布を測定する第１の実施形態の手順を示す。まず、帯状体１に振動荷重負荷装置３で振動荷重を負荷し（ステップ１）、変位計４によって帯状体１の振動変位を計測して（ステップ２）、計測された振動変位から、振動特性算出部５ａで帯状体１の固有振動数と振動モードを算出し（ステップ３）、モデル化部５ｂで帯状体１をモデル化する（ステップ４）。こののち、モデルの直線ばね１３と回転ばね１５の各ばね定数の初期値を設定し（ステップ５）、固有値解析部５ｃでモデルの固有振動数と振動モードを算出して（ステップ６）、後述する（１７）式の評価関数を用いて、計測から求められた帯状体１の固有振動数および振動モードとの各差の二乗和を計算する（ステップ７）。つぎに、この二乗和の評価値が最小となるように収束したか否かを判定し（ステップ８）、収束していない場合は、各ばね定数の値を修正してステップ６に戻り、モデルの固有振動数と振動モードを繰り返し計算する。この繰り返し計算のループはばね定数算出部５ｄで行われる。収束した場合は、ばね定数変換部５ｅで、同定された直線ばね１３と回転ばね１５の各ばね定数をそれぞれ張力値と曲げ剛性値に換算し（ステップ９）、張力分布算出部５ｆで張力分布を算出して（ステップ１０）、測定を終了する。 FIG. 3 shows the procedure of the first embodiment for measuring the tension distribution using the tension distribution measuring apparatus described above. First, a vibration load is applied to the band 1 by the vibration load device 3 (step 1), the vibration displacement of the band 1 is measured by the displacement meter 4 (step 2), and vibration characteristics are determined from the measured vibration displacement. The calculation unit 5a calculates the natural frequency and vibration mode of the strip 1 (step 3), and the modeling unit 5b models the strip 1 (step 4). Thereafter, initial values of the spring constants of the linear spring 13 and the rotary spring 15 of the model are set (step 5), and the natural frequency and vibration mode of the model are calculated by the eigenvalue analysis unit 5c (step 6). The sum of squares of the differences between the natural frequency and the vibration mode of the strip 1 obtained from the measurement is calculated using the evaluation function of (17) (Step 7). Next, it is determined whether or not the evaluation value of the sum of squares has converged so as to be minimized (step 8). If not converged, the value of each spring constant is corrected and the process returns to step 6 to return to the model. The natural frequency and vibration mode of are repeatedly calculated. This loop of repeated calculation is performed by the spring constant calculation unit 5d. In the case of convergence, each spring constant of the identified linear spring 13 and rotary spring 15 is converted into a tension value and a bending rigidity value by the spring constant conversion unit 5e (step 9), and the tension distribution calculation unit 5f converts the tension distribution. Is calculated (step 10), and the measurement is terminated.

以下に、上述した第１の実施形態で帯状体１の張力分布を測定する方法を、具体的に説明する。ここでは、測定点１ａおよびモデルの節点１１の数を一般化してｎとする。 Hereinafter, a method for measuring the tension distribution of the band 1 in the first embodiment described above will be described in detail. Here, the number of measurement points 1a and model nodes 11 is generalized to n.

前記変位計４で計測された帯状体１の振動変位ｖは、（１）式のようにモードベクトルｖ_ｉで表される。
ｖ_ｉ＝｛ｖ_i,1 ｖ_i,2 ・・・ｖ_i,n｝^Ｔ（ｉ＝１〜ｍ）（１）
ここに、ｉは幅方向の振動モードのモード次数である。 The vibration displacement v of the displacement gauge 4 strip 1 measured by the is represented by the mode vectors v _i as (1).
v _i = {v _{i, 1} v _{i, 2} ... v _{i, n} } ^T (i = 1 to _m ) (1)
Here, i is the mode order of the vibration mode in the width direction.

一方、前記２次元多質点系モデルの運動方程式は、帯状体１の質量マトリクスをＭ、各節点１１の変位ベクトルをｚ、直線ばね１３のばね定数ｋに相当する張力剛性マトリクスをＫ_Ｔ、回転ばね１５のばね定数τに相当する曲げ剛性マトリクスをＫ_τとすると、（２）式で表される。
ｚ＝｛ｚ₁ ｚ₂ ・・・ｚ_n｝^Ｔ（３） On the other hand, the equation of motion of the two-dimensional multi-mass point system model is that the mass matrix of the strip 1 is M, the displacement vector of each node 11 is z, the tension stiffness matrix corresponding to the spring constant k of the linear spring 13 is K _T , When the bending stiffness matrix corresponding to the spring constant τ of the spring 15 is represented by K _τ , it is expressed by equation (2).
z = {z ₁ z ₂ ... z _n } ^T (3)

質量マトリクスＭは、帯状体１の寸法および物性から算出される既知行列であり、（４）式で表される。
質量マトリクスＭの各要素に含まれる質量ｍと慣性モーメントＪは、長手方向の振動モードを正弦波で近似して縮小した場合の等価質量と等価剛性から算出される。支持ロール２ａ、２ｂ間の距離をＬ、帯状体１の板厚をｔ、密度をρ、リンク１４の長さをｌとすると、支持ロール２ａ、２ｂ間の中心における等価質量ｍは、長手方向の振動モードを１次とすると、（５）式で求められる。
ｍ＝ρｔｌＬ／２（５）
また、慣性モーメントＪは、（５）式で求めた等価質量ｍから（６）式で求められる。
Ｊ＝ｍ（ｔ^２＋ｌ^２）／１２（６） The mass matrix M is a known matrix calculated from the dimensions and physical properties of the strip 1 and is represented by the equation (4).
The mass m and the moment of inertia J included in each element of the mass matrix M are calculated from the equivalent mass and the equivalent stiffness when the longitudinal vibration mode is approximated by a sine wave and reduced. When the distance between the support rolls 2a and 2b is L, the thickness of the strip 1 is t, the density is ρ, and the length of the link 14 is l, the equivalent mass m at the center between the support rolls 2a and 2b is the longitudinal direction. If the vibration mode is primary, it can be obtained by equation (5).
m = ρtlL / 2 (5)
In addition, the moment of inertia J is obtained by the equation (6) from the equivalent mass m obtained by the equation (5).
J = m (t ² + l ² ) / 12 (6)

張力剛性マトリクスＫ_Ｔと曲げ剛性マトリクスＫ_τは未知行列であり、張力剛性マトリクスＫ_Ｔは直線ばね１３のばね定数ｋ_ｊ（ｊ＝１〜ｎ）を用いて（７）式で表される。
また、曲げ剛性マトリクスＫ_τは回転ばね１５のばね定数τ_ｊ（ｊ＝１〜ｎ−２）を用いて（８）式で表される。
曲げ剛性マトリクスＫ_τの各要素に含まれる幅方向の曲げ剛性に相当するばね定数τ_ｉが算出されたときの断面二次モーメントＩ_ｊへの換算式は（９）式で表される。
Ｉ_ｊ＝τ_ｊｌ／Ｅ（９） The tension stiffness matrix K _T and the bending stiffness matrix K _τ are unknown matrices, and the tension stiffness matrix K _T is expressed by equation (7) using the spring constant k _j (j = 1 to n) of the linear spring 13.
Further, the bending stiffness matrix K _τ is expressed by the equation (8) using the spring constant τ _j (j = 1 to n−2) of the rotary spring 15.
A conversion formula to the sectional secondary moment I _j when the spring constant τ _i corresponding to the bending stiffness in the width direction included in each element of the bending stiffness matrix K _τ is calculated is expressed by the following equation (9).
I _j = τ _j 1 / E (9)

つぎに、（１）式の帯状体１の振動変位ｖ_ｉから得られる固有振動数ωおよび幅方向の振動モードから、幅方向の張力分布と曲げ剛性分布を求める方法について述べる。剛性行列Ｋ_Ｔ、Ｋ_τと質量行列Ｍは、一般的に固有振動数と振動モードとの間に（１０）式の関係がある。
ＭΦΩ^２＝（Ｋ_Ｔ＋Ｋ_τ）Φ （１０）
Ω^２は幅方向振動モードの固有振動数ω_ｉ ^２（ｉ＝１〜ｎ）を対角要素とする行列で、（１１）式で表される。
Λはｉ次の固有値λ_ｉ（ｉ＝１〜ｎ）を対角要素とする行列である。また、Φは対応するモードベクトルを列ベクトルφ_ｉ（ｉ＝１〜ｎ）とするモード行列であり、（１２）式で表される。
Φ＝〔φ_１ φ_２・・・ φ_ｎ〕（１２）
φ_ｉ＝｛φ_i,1 φ_i,2 ・・・ φ_i,n｝^Ｔ（１３）
φ_ｉは固有値解析により得られるｉ次の振動モードベクトルで、（１）式で示した計測値のモードベクトルｖ_ｉに相当するが、計測値と解析値を区別するために、便宜上別の記号を用いる。 Next, a method for obtaining the tension distribution and the bending stiffness distribution in the width direction from the natural frequency ω obtained from the vibration displacement v _i of the belt-shaped body 1 of the equation (1) and the vibration mode in the width direction will be described. In general, the stiffness matrices K _T and K _τ and the mass matrix M have the relationship of the equation (10) between the natural frequency and the vibration mode.
MΦΩ ² = (K _T + K _τ ) Φ (10)
Ω ² is a matrix having the natural frequency ω _i ² (i = 1 to n) in the width direction vibration mode as a diagonal element, and is represented by Expression (11).
Λ is a matrix whose diagonal elements are i-th order eigenvalues λ _i (i = 1 to n). Further, Φ is a mode matrix in which the corresponding mode vector is a column vector φ _i (i = 1 to n), and is expressed by Expression (12).
Φ = [φ ₁ φ ₂ ... Φ _n ] (12)
φ _i = {φ _{i, 1} φ _{i, 2} ... φ _{i, n} } ^T (13)
phi _i the i-th order vibration mode vector obtained by eigenvalue analysis, (1) corresponding to the mode vectors v _i of the measurement values shown in the formula, in order to distinguish the analysis values and the measured values, for convenience another symbol Is used.

つぎに、未知の直線ばね１３のばね定数ｋ_ｊ（ｉ＝１〜ｎ）と回転ばね１５のばね定数τ_ｊ（ｉ＝１〜ｎ）を繰り返し計算によって求める方法を説明する。各ばね定数ｋ_ｊ、τ_ｊは未知数であるため、仮の初期値を設定して（１０）〜（１３）式から固有値λ_ｉと振動モードベクトルφ_ｉを計算し、繰り返し計算により計測値と解析値の誤差が最小となる各ばね定数ｋ_ｊ、τ_ｊを算出する。具体的には、（１４）式に示す評価関数を用いて、評価値Ｓが最小となる各ばね定数ｋ_ｊ、τ_ｊを求める。
この評価関数は、計測されたモードベクトルｖ_ｉと振動モードベクトルφ_ｉの各成分の差と、計測から得られた固有振動数ω_ｉの二乗と固有値λ_ｉの差を固有振動数ω_ｉの二乗で除算した値とを計算し、これらを二乗和するものであり、振動モードベクトルφ_ｉと固有値λ_ｉとが直線ばね１３と回転ばね１５の各ばね定数ｋ_ｊ、τ_ｊから計算されることから、パラメータとして各ばね定数ｋ_ｊ、τ_ｊを有している。多質点系モデルの節点１１での各ばね定数ｋ_ｊ、τ_ｊは物理的に正の値をとるので、ｋ_ｊ＞０、τ_ｊ＞０であることを拘束条件として、評価関数の評価値Ｓが最小となるように最適化問題を解く。最適化の方法としては、最急降下法、準ニュートン法等を用いることができ、これにより、各ばね定数ｋ_ｊ、τ_ｊが算出される。 Next, a method for obtaining the spring constant k _j (i = 1 to n) of the unknown linear spring 13 and the spring constant τ _j (i = 1 to n) of the rotary spring 15 by repeated calculation will be described. Since each spring constant k _j , τ _j is an unknown number, a temporary initial value is set and the eigenvalue λ _i and the vibration mode vector φ _i are calculated from the equations (10) to (13). The spring constants k _j and τ _j that minimize the error of the analysis value are calculated. Specifically, the spring constants k _j and τ _j that minimize the evaluation value S are obtained using the evaluation function shown in the equation (14).
This evaluation function calculates the difference between each component of the measured mode vector v _i and the vibration mode vector φ _i and the difference between the square of the natural frequency ω _i obtained from the measurement and the natural value λ _i of the natural frequency ω _i . The value divided by the square is calculated, and these are summed to the square. The vibration mode vector φ _i and the eigenvalue λ _i are calculated from the spring constants k _j and τ _{j of the} linear spring 13 and the rotary spring 15. For this _reason , the spring constants k _j and τ _j are used as parameters. Since the spring constants k _j and τ _j at the node 11 of the multi-mass system model take physically positive values, the evaluation value of the evaluation function is set with k _j > 0 and τ _j > 0 as constraints. Solve the optimization problem so that S is minimized. As an optimization method, a steepest descent method, a quasi-Newton method, or the like can be used, and thereby the spring constants k _j and τ _j are calculated.

つぎに、算出した直線ばね１３のばね定数ｋ_ｊを各節点１１すなわち各測定点１ａでの張力Ｔ_ｊに換算する。距離Ｌの支持ロール２ａ、２ｂ間の帯状体１を幅方向にリンク１４の長さｌで分割した１要素を弦と見なして、各測定点１ａでの張力Ｔ_ｊが（１５）式で与えられる。
また、帯状体１の総張力Ｔ_totalは、各測定点１ａでの張力Ｔ_ｊの合計として（１６）式で与えられる。
Next, the calculated spring constant k _j of the linear spring 13 is converted into the tension T _j at each node 11, that is, each measurement point 1 a. Distance L of the supporting rolls 2a, one element divided by the length l of the link 14 the strip 1 in the width direction between 2b is regarded as the strings, given by the tension T _j at each measurement point 1a is (15) It is done.
Further, the total tension T _total of the strip 1 is given by the equation (16) as the total of the tension T _j at each measurement point 1a.

一方、算出した回転ばね１５のばね定数τ_ｊは、（９）式によって断面二次モーメントＩ_ｊに換算され、幅方向の曲げ剛性分布が算出される。 On the other hand, the calculated spring constant τ _j of the rotary spring 15 is converted into a cross-sectional secondary moment I _j by equation (9), and the bending stiffness distribution in the width direction is calculated.

図４は、上述した張力分布測定装置を用いて張力分布を測定する第２の実施形態の手順を示す。帯状体１をモデル化するステップ４までは、第１の実施形態のものと同じである。第２の実施形態では、こののち、固有値解析部５ｃで、モデルの直線ばね１３と回転ばね１５の各ばね定数に対する係数行列の擬似逆行列を計算し（ステップ５）、特異値分解によって擬似逆行列の誤差成分を分離する（ステップ６）。つぎに、ばね定数算出部５ｄで、誤差成分を除去した擬似逆行列を用いて、最小二乗法によって直線ばね１３と回転ばね１５の各ばね定数を算出する（ステップ７）。各ばね定数を算出した後は第１の実施形態のものと同じであり、ばね定数変換部５ｅで、同定された直線ばね１３と回転ばね１５の各ばね定数をそれぞれ張力値と曲げ剛性値に換算して（ステップ８）、張力分布算出部５ｆで張力分布を算出し（ステップ９）、測定を終了する。 FIG. 4 shows the procedure of the second embodiment for measuring the tension distribution using the above-described tension distribution measuring apparatus. The steps up to step 4 for modeling the band 1 are the same as those in the first embodiment. In the second embodiment, thereafter, the eigenvalue analysis unit 5c calculates a pseudo inverse matrix of a coefficient matrix for each spring constant of the model linear spring 13 and the rotary spring 15 (step 5), and performs pseudo inverse by singular value decomposition. The error component of the matrix is separated (step 6). Next, the spring constant calculation unit 5d calculates the spring constants of the linear spring 13 and the rotary spring 15 by the least square method using the pseudo inverse matrix from which the error component has been removed (step 7). After calculating each spring constant, it is the same as that of the first embodiment, and the spring constant conversion unit 5e converts the spring constants of the identified linear spring 13 and rotary spring 15 to a tension value and a bending rigidity value, respectively. After conversion (step 8), the tension distribution calculation unit 5f calculates the tension distribution (step 9), and the measurement ends.

以下に、上述した第２の実施形態で帯状体１の張力分布を測定する方法を、具体的に説明する。ステップ１〜４およびステップ８〜９は第１の実施形態のものと同じであるので、ここではステップ５〜７についてのみ説明する。また、測定点１ａおよびモデルの節点１１の数は一般化してｎとする。 Hereinafter, a method for measuring the tension distribution of the belt-like body 1 in the above-described second embodiment will be specifically described. Since steps 1 to 4 and steps 8 to 9 are the same as those in the first embodiment, only steps 5 to 7 will be described here. The number of measurement points 1a and model nodes 11 is generalized to n.

この実施形態では、（１０）式の関係を利用して、計測および固有値解析の各々から得られる固有振動数および振動モードが一致する直線ばね１３と回転ばね１５の各ばね定数を計算する。使用する固有振動数と振動モードの次数は、各ばね定数の同定計算を行う上ではいくつでもよく、ここでは、（１２）式におけるモード行列Φに、（１７）式に示すように、１〜ｍ次モードの列ベクトルφ_ｉ（ｉ＝１〜ｍ、ｍ≦ｎ）を使用する場合について述べる。
Φ＝〔φ_１ φ_２・・・ φ_ｍ〕（１７）
（１２）式に（１７）式を代入して書き直すと、モード毎に独立した自由度ｎのベクトル方程式が（１８）式に示すように得られる。
ω_i ^２Ｍφ_ｉ＝Ｋ_Ｔφ_ｉ＋Ｋ_τφ_ｉ（１８）
（１８）式の右辺第１項は、（１９）式に示すように、既知の係数行列Φ_T,iと、未知の直線ばね１３のばね定数ｋ_ｉのベクトルｋとの積にまとめることができる。
（１８）式の右辺第２項も同様に、（２０）式に示すように、既知の係数行列Φ_τ,iと、未知の回転ばね１５のばね定数τ_ｉのベクトルτとの積にまとめることができる。
In this embodiment, the spring constants of the linear spring 13 and the rotary spring 15 that have the same natural frequency and vibration mode obtained from the measurement and eigenvalue analysis are calculated using the relationship of the equation (10). The number of natural frequencies and the order of the vibration modes to be used may be any number for the identification calculation of each spring constant. Here, the mode matrix Φ in the equation (12) is 1 to 1 as shown in the equation (17). The case where the column vector φ _i (i = 1 to m, m ≦ n) in the m- _th mode is used will be described.
Φ = [φ ₁ φ ₂ ... Φ _m ] (17)
When the equation (17) is substituted into the equation (12) and rewritten, a vector equation having n degrees of freedom independent for each mode is obtained as shown in the equation (18).
ω _i ² Mφ _i = K _T φ _i + K _τ φ _i (18)
The first term on the right side of the equation (18) can be summarized into the product of the known coefficient matrix Φ _{T, i} and the vector k of the spring constant k _i of the unknown linear spring 13 as shown in the equation (19). it can.
Similarly, the second term on the right side of the equation (18) is combined into a product of the known coefficient matrix Φ _{τ, i} and the vector τ of the spring constant τ _i of the unknown rotary spring 15 as shown in the equation (20). be able to.

（１９）式と（２０）式を（１８）式に代入してモード毎のベクトル方程式を、未知ベクトルｋとτについてまとめると、（２１）式に示すように、ｎ×ｍ個の独立した方程式が得られる。
（２１）式の左辺の既知ベクトルをｋ_Φ、右辺のベクトルｋの係数行列をΦ_Ｔ、ベクトルτの係数行列をΦ_τとおいて、（２２）式で表す。
Substituting Equation (19) and Equation (20) into Equation (18) to summarize the vector equations for each mode for unknown vectors k and τ, as shown in Equation (21), n × m independent The equation is obtained.
The known vector on the left side of equation (21) is represented by equation (22), with k _Φ , the coefficient matrix of vector k on the right side as Φ _T , and the coefficient matrix of vector τ as Φ _τ .

つぎに、振動モードと固有値の関係、および振動モードの直交性を利用した定式化を行う。（１０）式に左側から転置行列Φ^Ｔを乗じると（２３）式が得られる。
Φ^Ｔ（Ｋ_Ｔ＋Ｋ_τ）Φ＝Φ^ＴＭΦΩ^２（２３）
（２３）式の関係を利用して、計測および固有値解析の各々から得られる固有振動数および振動モードが一致する直線ばね１３と回転ばね１５の各ばね定数ｋ_ｉ、τ_ｉを計算する。（２３）式の両辺は対称行列であるから、上三角成分または下三角成分についてのｍ（ｍ＋１）／２個の独立した等式が（２４）式のように得られる。
また、マトリクスＫ_Ｔは（７）式に示したように対角行列であるから、（２４）式の左辺の第１項について展開し、未知のばね定数ｋ_ｉ（ｉ＝１〜ｎ）についてまとめると、（２５）式のように既知の係数行列Ψ_Ｔと未知のベクトルｋとの積で表される。
同様に、（２４）式の左辺の第２項は、（２６）式のように既知の係数行列Ψ_τと未知のベクトルτとの積で表される。
Next, formulation is performed using the relationship between the vibration mode and the eigenvalue and the orthogonality of the vibration mode. Multiplying equation (10) by the transpose matrix Φ ^T from the left side yields equation (23).
Φ ^T (K _T + K _τ ) Φ = Φ ^T MΦΩ ² (23)
Using the relationship of the equation (23), the spring constants k _i and τ _i of the linear spring 13 and the rotary spring 15 that have the same natural frequency and vibration mode obtained from the measurement and eigenvalue analysis are calculated. Since both sides of equation (23) are symmetric matrices, m (m + 1) / 2 independent equations for the upper or lower triangular component are obtained as in equation (24).
Further, since the matrix _KT is a diagonal matrix as shown in the equation (7), the first term on the left side of the equation (24) is expanded, and the unknown spring constant k _i (i = 1 to n). In summary, it is represented by the product of a known coefficient matrix Ψ _T and an unknown vector k as shown in equation (25).
Similarly, the second term of the left side of (24), is expressed by the product of the known coefficient matrix [psi _tau and unknown vector tau as equation (26).

（２５）式と（２６）式を（２４）式に代入し、右辺の既知ベクトルをｋ_Ψとすると（２７）式が得られる。
最終的に（２２）式と（２７）式を（２８）式のようにまとめることにより、未知数２ｎ−２個に対してｍ（ｍ＋１）／２＋ｎｍ個の等式が得られる。
Substituting Equations (25) and (26) into Equation (24) and assuming the known vector on the right side to be k _ψ , Equation (27) is obtained.
Finally, the equations (22) and (27) are put together as the equation (28), so that m (m + 1) / 2 + nm equations are obtained for the unknown 2n−2.

（２８）式を最小二乗法で解けば直線ばね１３と回転ばね１５の各ばね定数ｋ_ｉ、τ_ｉを求めることができる。ここで、（２８）式の左辺における係数行列をＡ、未知ベクトルをｘ、右辺の既知ベクトルをｂとおいて、（２９）式のように表す。
Ａｘ＝ｂ（２９）
係数行列Ａの擬似逆行列Ａ^†を用いて（２９）式を書き直すと、（３０）式と（３１）式が得られる。
ｘ＝Ａ^†ｂ（３０）
Ａ^†＝（Ａ^ＴＡ）^−１Ａ^Ｔ（３１）
（３１）式で計算されるＡ^†を用いて（３０）式の未知ベクトルｘ、すなわち、各ばね定数ｋ_ｉ、τ_ｉを算出することができる。 If the equation (28) is solved by the method of least squares, the spring constants k _i and τ _i of the linear spring 13 and the rotary spring 15 can be obtained. Here, the coefficient matrix on the left side of equation (28) is A, the unknown vector is x, and the known vector on the right side is b, and is expressed as equation (29).
Ax = b (29)
When the equation (29) is rewritten using the pseudo inverse matrix A ^† of the coefficient matrix A, the equations (30) and (31) are obtained.
x = A ^† b (30)
A ^† = (A ^T A) ⁻¹ A ^T (31)
Using A ^† calculated by the equation (31), the unknown vector x of the equation (30), that is, the spring constants k _i and τ _i can be calculated.

ところで、計算に使用するモード数が少ない場合や、帯状体１の幅方向分割数が大きい場合は、行列Ａのランクが不足し、（３０）式の解から求められる張力が実際の張力分布と一致しなくなることがある。そこで、特異値分解を利用して望ましい解を求める。係数行列Ａは特異ベクトルを列ベクトルとする直行行列Ｐ、Ｑと特異値を対角成分に持つ対角行列Ｄを用いて（３２）式のように分解できる。
Ａ＝ＰＤＱ^Ｔ（３２）
対角行列Ｄを特異値の大きい部分行列Ｄ_ｓと特異値の小さい部分行列Ｄ_ｎに分けると、（３１）式の擬似逆行列Ａ^†は（３３）式のように表される。
Ａ^†＝Ｑ_ｓＤ_ｓ ^−１Ｐ_ｓ ^Ｔ（３３）
この（３３）式で計算されるＡ^†を（３０）式に用いることにより、より良好な結果を得るように、未知ベクトルｘ、すなわち、各ばね定数ｋ_ｉ、τ_ｉを算出することができる。 By the way, when the number of modes used for the calculation is small, or when the number of divisions in the width direction of the strip 1 is large, the rank of the matrix A is insufficient, and the tension obtained from the solution of Equation (30) is the actual tension distribution. May not match. Therefore, a desirable solution is obtained using singular value decomposition. The coefficient matrix A can be decomposed as shown in the equation (32) using an orthogonal matrix P, Q having a singular vector as a column vector and a diagonal matrix D having a singular value as a diagonal component.
A = PDQ ^T (32)
When the diagonal matrix D is divided into a submatrix D _s with a large singular value and a submatrix D _n with a small singular value, the pseudo inverse matrix A ^† of the equation (31) is expressed as the equation (33).
^{_{_{^{A † = Q s D s -1}}}} P s T (33)
By using A ^† calculated by the equation (33) in the equation (30), the unknown vector x, that is, each spring constant k _i , τ _i can be calculated so as to obtain a better result. .

図５は、片伸びの不均一歪みがある帯状体１に対して、２次モードまで用いた（２８）式の係数行列Φ_Ｔを特異値分解した結果の例を示す。この特異値分解結果より、Ｄ_ｓ＝diag(σ₁，・・・σ₁₈)、Ｄ_ｎ＝diag(σ₁₉，σ₂₀，・・・)＝０とした（３３）式の擬似逆行列Ａ^†を用いて算出した張力分布を図６に示す。図６には、特異値分解をせずに（３１）式の擬似逆行列Ａ^†を用いて算出した張力分布と、３次元ＦＥＭ解析によって求めた張力分布も併せて示す。特異値分解なしの張力分布が、片伸びの存在する幅端部で大きく変動しているのに対して、特異値分解ありの張力分布は、正解値とするＦＥＭ解析結果とよく一致している。このことにより、特異値分解によって誤差成分を除去することにより、一般的な擬似逆行列を用いる最小二乗法よりも精度よく張力分布を測定できることが分かる。 5, with respect to the band-like body 1 there is irregular distortion of the strip elongation, illustrating an example of a singular value decomposition as a result of the (28) equation of the coefficient matrix [Phi _T using up to the second order mode. From this singular value decomposition result, pseudo-inverse matrix A of the equation (33) with D _s = diag (σ ₁ ,..., Σ ₁₈ ) and D _n = diag (σ ₁₉ , σ ₂₀ ,...) = 0. ^The tension distribution calculated using ^† is shown in FIG. FIG. 6 also shows the tension distribution calculated using the pseudo inverse matrix A ^† of Equation (31) without performing singular value decomposition and the tension distribution obtained by three-dimensional FEM analysis. While the tension distribution without singular value decomposition fluctuates greatly at the width end where the single elongation exists, the tension distribution with singular value decomposition agrees well with the FEM analysis result as the correct value. . Thus, it can be seen that by removing the error component by singular value decomposition, the tension distribution can be measured with higher accuracy than the least square method using a general pseudo inverse matrix.

実施例として、上述した第２の実施形態の張力測定装置を用いて、帯状体１としての板厚が１．０ｍｍの薄鋼板と板厚が０．５ｍｍのアルミニウム薄板とについて、不均一歪みが存在しないフラットな歪みパターンと、それぞれ中伸びと耳波の不均一歪みが一部顕在する歪みパターンの張力分布を測定した。いずれの場合も板幅は１０００ｍｍ、支持ロール２ａ、２ｂ間の距離Ｌは４５００ｍｍとし、長手方向に付与した総張力は、薄鋼板の場合は２９４００Ｎ、アルミニウム薄板の場合は４９００Ｎとした。また、比較例として、不均一歪みが全て潜在化すると仮定し、幅方向の曲げ剛性を既知とした特願２０１０−２７４３４５で提案した張力測定装置を用いた張力分布の測定も行った。これらの実施例と比較例の張力分布の測定結果を、３次元ＦＥＭ解析で張力分布を求めた解析結果と対比した。 As an example, using the tension measuring apparatus according to the second embodiment described above, non-uniform distortion occurs in a thin steel plate having a thickness of 1.0 mm and an aluminum thin plate having a thickness of 0.5 mm as the band-like body 1. The tension distribution of the flat strain pattern that does not exist and the strain pattern in which the non-uniform strain of the middle stretch and the ear wave are partially revealed was measured. In any case, the plate width was 1000 mm, the distance L between the support rolls 2a and 2b was 4500 mm, and the total tension applied in the longitudinal direction was 29400 N for thin steel plates and 4900 N for aluminum thin plates. As a comparative example, the tension distribution was also measured using the tension measuring device proposed in Japanese Patent Application No. 2010-274345 in which the bending stiffness in the width direction was known, assuming that all non-uniform strains were latent. The measurement results of the tension distributions in these examples and comparative examples were compared with the analysis results obtained for the tension distribution by three-dimensional FEM analysis.

図７（ａ）、（ｂ）、（ｃ）は、薄鋼板に対する張力分布の測定結果を示す。いずれの歪みパターンでも、不均一歪みの一部顕在による幅方向の曲げ剛性の変化を考慮した実施例の各張力分布が正解値とするＦＥＭ解析結果とよく一致しているのに対して、不均一歪みが全て潜在化すると仮定した比較例の各張力分布は、ＦＥＭ解析結果と大きくずれるように張力が変動している。この結果より、本発明に係る張力測定装置は、不均一歪みが一部顕在しても帯状体１の張力分布を精度よく測定できることが分かる。 7A, 7B, and 7C show the measurement results of the tension distribution for the thin steel plate. In any strain pattern, each tension distribution in the example considering the change in the bending stiffness in the width direction due to the partial manifestation of non-uniform strain is in good agreement with the FEM analysis result as the correct value. In each tension distribution of the comparative example that is assumed that all the uniform strains are latent, the tension fluctuates so as to greatly deviate from the FEM analysis result. From this result, it can be seen that the tension measuring device according to the present invention can accurately measure the tension distribution of the belt-like body 1 even if some non-uniform strain is manifested.

図８（ａ）、（ｂ）、（ｃ）は、アルミニウム薄板に対する張力分布の測定結果を示す。この場合も、実施例の各張力分布は、比較例の各張力分布よりも正解値のＦＥＭ解析結果とよく一致している。ただし、比較例の各張力分布のＦＥＭ解析結果とのずれは、薄鋼板の場合よりも小さくなっている。これは、アルミニウムのヤング率が鋼よりも小さいことと、アルミニウム薄板の板厚が薄鋼板よりも薄いことのために、不均一歪みの一部顕在による幅方向曲げ剛性の変化が少ないためと思われる。 FIGS. 8A, 8B, and 8C show the measurement results of the tension distribution for the aluminum thin plate. In this case as well, each tension distribution in the example is in better agreement with the correct FEM analysis result than each tension distribution in the comparative example. However, the deviation from the FEM analysis result of each tension distribution of the comparative example is smaller than that of the thin steel plate. This is because the Young's modulus of aluminum is smaller than that of steel and the thickness of the aluminum thin plate is thinner than that of the thin steel plate. It is.

図９（ａ）、（ｂ）、（ｃ）および図１０（ａ）、（ｂ）、（ｃ）は、それぞれ薄鋼板とアルミニウム薄板の場合について、板幅方向の曲げ剛性分布に対応する板幅方向の断面二次モーメント分布を、本発明に係る張力測定装置で算出した結果を示す。いずれの場合も、中延びの歪みパターンでは、不均一歪みが顕在する幅中央部で断面二次モーメント、すなわち曲げ剛性が増大し、耳波の歪みパターンでは、不均一歪みが顕在する幅両端部で曲げ剛性が増大している。このことより、幅方向曲げ剛性を既知の一定値とする従来の張力測定装置では、不均一歪みが顕在する帯状体の張力分布の測定精度に限界があることが分かる。なお、アルミニウム薄板の場合は、各歪みパターンでの断面二次モーメントの増大量が薄鋼板の場合よりも１オーダー小さくなっており、図８において、比較例の張力分布とＦＥＭ解析結果とのずれが比較的小さかったことを裏付けている。 9 (a), (b), (c) and FIGS. 10 (a), (b), (c) are the plates corresponding to the bending stiffness distribution in the plate width direction for thin steel plates and aluminum thin plates, respectively. The result of having calculated the cross-sectional secondary moment distribution of the width direction with the tension | tensile_strength measuring apparatus which concerns on this invention is shown. In either case, in the middle strain pattern, the cross-sectional secondary moment, that is, the bending rigidity, increases at the center of the width where the non-uniform strain appears, and in the ear wave strain pattern, both ends of the width where the non-uniform strain appears. The bending rigidity has increased. From this, it can be seen that in the conventional tension measuring apparatus in which the bending stiffness in the width direction is a known constant value, there is a limit to the measurement accuracy of the tension distribution of the belt-like body in which the non-uniform strain appears. In the case of an aluminum thin plate, the amount of increase in the secondary moment of section in each strain pattern is one order of magnitude smaller than that in the case of a thin steel plate. In FIG. 8, the difference between the tension distribution of the comparative example and the FEM analysis results. That it was relatively small.

１帯状体
１ａ測定点
２ａ、２ｂ支持ロール
３振動荷重負荷装置
４変位計
５演算装置
５ａ振動特性算出部
５ｂモデル化部
５ｃ固有値解析部
５ｄばね定数算出部
５ｅばね定数換算部
５ｆ張力分布算出部
１１節点
１２固定面
１３直線ばね
１４リンク
１５回転ばね DESCRIPTION OF SYMBOLS 1 Band 1a Measurement point 2a, 2b Support roll 3 Vibration load application apparatus 4 Displacement meter 5 Arithmetic apparatus 5a Vibration characteristic calculation part 5b Modeling part 5c Eigenvalue analysis part 5d Spring constant calculation part 5e Spring constant conversion part 5f Tension distribution calculation part 11 Node 12 Fixed surface 13 Linear spring 14 Link 15 Rotating spring

Claims

In a tension measuring device for measuring a tension distribution in the width direction of a belt-like body to which tension is applied in the longitudinal direction between the support parts supported by two parts in the longitudinal direction, the belt-like shape between the two support parts. Vibration load loading means for applying a vibration load to the body, and displacement measuring means for measuring vibration displacement caused by the vibration load loading means at a plurality of measurement points in the width direction of the band. Modeled into a two-dimensional multi-mass system model that includes a linear spring that simulates the tension acting on each node corresponding to a plurality of measurement points in the direction, and a rotary spring that simulates the bending stiffness in the width direction at each node. And the natural frequency and vibration mode at each node obtained from the eigenvalue analysis of the multi-mass point system model are the natural frequency and vibration frequency obtained from the vibration displacement at each measured measurement point. A spring constant calculation means for calculating the spring constants of the linear spring and the rotary spring that match the mode, and the calculated spring constants of the linear spring and the rotary spring respectively for the tension value and the bending at each measurement point. A tension measuring device having a spring constant converting means for converting into a stiffness value, and obtaining a tension distribution in the width direction of the strip from the converted tension value at each measurement point.

The spring constant calculating means is a sum of squares of differences between the natural frequency and vibration mode at each node obtained from the eigenvalue analysis and the natural frequency and vibration mode obtained from vibration displacement at each measurement point. The tension measuring device according to claim 1, wherein each spring constant of the linear spring and the rotary spring that minimizes is calculated by a least square method using repetitive calculation.

The spring constant calculation means calculates a pseudo inverse matrix of a coefficient matrix of each spring constant of the linear spring and the rotary spring with respect to the vibration mode of the strip, and the natural frequency at each node obtained from the eigenvalue analysis and The spring constants of the linear spring and the rotary spring, in which the vibration mode matches the natural frequency and vibration mode obtained from the vibration displacement at each measurement point, are obtained by the least square method using the pseudo inverse matrix. The tension measuring device according to claim 1, which is to be calculated.

The error component is removed from the calculated pseudo inverse matrix by singular value decomposition, and the pseudo inverse matrix from which the error component is removed is used for calculation of each spring constant by the least square method. Tension measuring device.