JP7652056B2 - Measuring device and measuring method - Google Patents
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Description
本発明は、超音波を用いて対象物の厚さと縦波音速を同時計測する計測装置および計測方法に関する。 The present invention relates to a measurement device and a measurement method that simultaneously measures the thickness and longitudinal wave sound velocity of an object using ultrasound.
対象物の厚さと音速を同時に計測する技術として、特許文献1や非特許文献1,2がある。 Technologies for simultaneously measuring the thickness and sound speed of an object include Patent Document 1 and Non-Patent Documents 1 and 2.
特許文献1には、超短パルスコヒーレント光を透明薄膜内で干渉させ、その内部の弾性波によって生じる干渉うねりから音速を算出することが記載されている。また、求めた音速と弾性波の薄膜内での伝搬時間から薄膜の厚さを算出することが記載されている。 Patent document 1 describes how ultrashort pulse coherent light is made to interfere in a transparent thin film, and the sound speed is calculated from the interference undulations caused by the elastic waves inside the film. It also describes how the thickness of the thin film is calculated from the obtained sound speed and the propagation time of the elastic waves in the thin film.
非特許文献1には、対象物に超音波を照射し、対象物の表面及び裏面からの反射波を計測し、その時間差を基に対象物の厚さと音速を算出することが記載されている。 Non-Patent Document 1 describes a method of irradiating an object with ultrasonic waves, measuring the waves reflected from the front and back surfaces of the object, and calculating the thickness and sound speed of the object based on the time difference.
非特許文献2には、基板上の薄膜に超音波を照射し、薄膜の表面と裏面からの反射波が合成した波を計測し、その振幅の周波数スペクトルに基づいて薄膜の厚さと音速を算出することが記載されている。 Non-Patent Document 2 describes a method of irradiating a thin film on a substrate with ultrasonic waves, measuring the combined wave of reflected waves from the front and back surfaces of the thin film, and calculating the thickness and sound speed of the thin film based on the frequency spectrum of the amplitude.
しかし、特許文献1の方法では、対象物内でレーザー光を干渉させる必要があるため、不透明体には適用できないという問題があった。 However, the method in Patent Document 1 requires the laser light to interfere within the object, so there is a problem in that it cannot be applied to opaque objects.
また、非特許文献1では、表面からの反射波と裏面からの反射波とを分離して計測する必要があり、対象物が薄い場合には適用できなかった。 In addition, in Non-Patent Document 1, it was necessary to measure the waves reflected from the front surface and the back surface separately, and this was not applicable when the target object was thin.
非特許文献2では表面からの反射波と裏面からの反射波を分離する必要はないが、基板からの反射波は分離する必要がある。そのため、基板が薄い場合には適用が難しかった。加えて、対象物の厚さと縦波音速の両方とも振幅に依存するため、振幅を基にそれらを分離し、厚さと縦波音速を同時に精度よく算出するためには高周波数・広帯域の超音波センサが必要になる。そのため、非特許文献2の方法は水浸超音波法や接触式の超音波法にしか適用できなかった。 In Non-Patent Document 2, there is no need to separate the waves reflected from the front surface and the back surface, but it is necessary to separate the waves reflected from the substrate. This makes it difficult to apply to thin substrates. In addition, because both the thickness and longitudinal wave sound velocity of the object depend on the amplitude, a high-frequency, wideband ultrasonic sensor is required to separate them based on the amplitude and simultaneously calculate the thickness and longitudinal wave sound velocity with high accuracy. For this reason, the method in Non-Patent Document 2 can only be applied to the water immersion ultrasonic method and the contact ultrasonic method.
そこで本発明の目的は、対象物の厚さと縦波音速を同時に精度よく測定することができる計測装置及び計測方法を提供することである。 The object of the present invention is to provide a measurement device and a measurement method that can simultaneously and accurately measure the thickness and longitudinal wave sound velocity of an object.
本発明は、対象物の厚さと縦波音速を同時に計測する計測装置であって、前記対象物に超音波を照射する送信部と、前記対象物からの透過波または反射波である超音波を時間波形の電気信号として受信する受信部と、前記電気信号を振幅および位相の周波数スペクトルに変換する変換部と、前記対象物の厚さおよび縦波音速を所定の値としたときの振幅および位相の周波数スペクトルを解析式により算出し、前記変換部による振幅および位相の周波数スペクトルと前記解析式による振幅および位相の周波数スペクトルとの適合性を、前記解析式に用いる前記対象物の厚さおよび縦波音速を変えて繰り返し判断し、前記変換部による振幅および位相の周波数スペクトルと解析による振幅および位相の周波数スペクトルとが最も適合するときに解析式に用いた前記対象物の厚さおよび縦波音速を、解として算出する解析部と、を有し、前記解析式は、前記送信部から前記対象物までおよび前記対象物から前記受信部までの間に存在する媒質を考慮し、前記媒質による超音波の位相変化を考慮した式である、ことを特徴とする計測装置である。 The present invention is a measuring device that simultaneously measures the thickness and longitudinal wave sound velocity of an object, comprising: a transmitting unit that irradiates the object with ultrasonic waves; a receiving unit that receives ultrasonic waves, which are transmitted waves or reflected waves from the object, as electrical signals of a time waveform; a converting unit that converts the electrical signals into amplitude and phase frequency spectra; and an analyzing unit that calculates the frequency spectrum of the amplitude and phase when the thickness and longitudinal wave sound velocity of the object are set to predetermined values using an analytical formula, repeatedly determines the compatibility of the frequency spectrum of the amplitude and phase obtained by the converting unit with the frequency spectrum of the amplitude and phase obtained by the analytical formula by changing the thickness of the object and the longitudinal wave sound velocity used in the analytical formula, and calculates the thickness of the object and the longitudinal wave sound velocity used in the analytical formula as a solution when the frequency spectrum of the amplitude and phase obtained by the converting unit and the frequency spectrum of the amplitude and phase obtained by the analysis are most compatible, and the analytical formula is a formula that takes into account the medium existing between the transmitting unit and the object, and the medium, and takes into account the phase change of the ultrasonic waves.
また本発明は、対象物の厚さと縦波音速を同時に計測する計測方法であって、送信部から前記対象物に超音波を照射し、前記対象物からの透過波または反射波である超音波を時間波形の電気信号として受信部で受信し、前記電気信号を振幅および位相の周波数スペクトルに変換し、前記対象物の厚さおよび縦波音速を所定の値としたときの振幅および位相の周波数スペクトルを解析式により算出し、前記電気信号から変換した振幅および位相の周波数スペクトルと前記解析式による振幅および位相の周波数スペクトルとの適合性を、前記解析式に用いる前記対象物の厚さおよび縦波音速を変えて繰り返し判断し、前記電気信号から変換した振幅および位相の周波数スペクトルと解析による振幅および位相の周波数スペクトルとが最も適合するときに前記解析式に用いた前記対象物の厚さおよび縦波音速を、解として算出し、前記解析式は、前記送信部から前記対象物までおよび前記対象物から前記受信部までの間に存在する媒質を考慮し、前記媒質による超音波の位相変化を考慮した式である、ことを特徴とする計測方法である。 The present invention also provides a measurement method for simultaneously measuring the thickness and longitudinal wave sound velocity of an object, comprising: irradiating an ultrasonic wave from a transmitting unit to the object; receiving an ultrasonic wave, which is a transmitted wave or a reflected wave from the object, as an electrical signal of a time waveform at a receiving unit; converting the electrical signal into an amplitude and phase frequency spectrum; calculating the amplitude and phase frequency spectrum when the thickness and longitudinal wave sound velocity of the object are set to predetermined values using an analytical formula; repeatedly determining the compatibility between the amplitude and phase frequency spectrum converted from the electrical signal and the amplitude and phase frequency spectrum by the analytical formula by changing the thickness of the object and the longitudinal wave sound velocity used in the analytical formula; and calculating the thickness and longitudinal wave sound velocity of the object used in the analytical formula as a solution when the amplitude and phase frequency spectrum converted from the electrical signal and the amplitude and phase frequency spectrum by the analysis are most compatible; and the analytical formula is a formula that takes into account the medium existing between the transmitting unit and the object and from the object to the receiving unit, and takes into account the phase change of the ultrasonic wave due to the medium.
本発明において、適合性は、RMSE、最尤推定法、または機械学習により判断することができる。 In the present invention, compatibility can be determined by RMSE, maximum likelihood estimation, or machine learning.
本発明において、媒質は空気であってもよい。 In the present invention, the medium may be air.
本発明において、受信部は、対象物からの透過波を受信するものであってよい。 In the present invention, the receiving unit may receive a transmitted wave from an object.
本発明において、解析式は、分布定数回路モデルにより算出した式であってもよい。 In the present invention, the analytical formula may be a formula calculated using a distributed constant circuit model.
本発明において、対象物は、複数の層で構成された積層体のうちの1層であってもよい。 In the present invention, the object may be one layer of a laminate composed of multiple layers.
本発明によれば、対象物の厚さと縦波音速を同時に精度よく測定することができる。 The present invention makes it possible to measure the thickness and longitudinal wave velocity of an object simultaneously with high accuracy.
(計測装置の構成について)
第1実施形態の計測装置は、対象物10の厚さと縦波音速を同時に計測可能な計測装置である。図1は、第1実施形態の計測装置の構成を示した図である。図1のように、第1実施形態の計測装置は、送受信源1と、送信センサ2と、受信センサ3と、フーリエ変換部4と、解析部5と、を有している。
(About the configuration of the measuring device)
The measurement device of the first embodiment is a measurement device capable of simultaneously measuring the thickness and longitudinal wave sound velocity of an object 10. Fig. 1 is a diagram showing the configuration of the measurement device of the first embodiment. As shown in Fig. 1, the measurement device of the first embodiment has a transmission/reception source 1, a transmission sensor 2, a reception sensor 3, a Fourier transform unit 4, and an analysis unit 5.
対象物10は、単層の材料でもよいが、複数の層からなる材料のうちの一層であってもよい。表面からの反射波と裏面からの反射波を分離できないような薄い材料でも第1実施形態では計測可能である。たとえば、2つの薄い材料を薄い接着剤で接合した構造において、接着剤の厚さおよび縦波音速を同時計測可能である。図1には、第1被着材11、接着剤12、第2被着材13の3層の積層体14を示し、そのうち接着剤12が対象物10である場合を示している。 The object 10 may be a single layer of material, or one layer of a material consisting of multiple layers. In the first embodiment, even a thin material in which the reflected waves from the front and back cannot be separated can be measured. For example, in a structure in which two thin materials are joined with a thin adhesive, the thickness of the adhesive and the longitudinal wave sound velocity can be measured simultaneously. Figure 1 shows a three-layer laminate 14 of a first adherend 11, an adhesive 12, and a second adherend 13, of which the adhesive 12 is the object 10.
送受信源1は、送信センサ2、受信センサ3、およびフーリエ変換部4に接続されている。送受信源1は、超音波の送信、受信を制御する装置である。 The transmission/reception source 1 is connected to a transmission sensor 2, a reception sensor 3, and a Fourier transform unit 4. The transmission/reception source 1 is a device that controls the transmission and reception of ultrasonic waves.
送信センサ2は、送受信源1に接続されている。送信センサ2は、送受信源1からの送信信号(電気信号)を超音波に変換し、超音波を空中に放射して積層体14に照射する装置である。超音波の中心周波数は、たとえば50kHz~100MHzである。また、帯域幅は、たとえば10kHz~20MHzである。また、超音波の波長は、対象物10の厚さよりも長くてもよい。そのため、送信センサ2、受信センサ3として低コストなものを使用できる。 The transmitting sensor 2 is connected to the transmitting/receiving source 1. The transmitting sensor 2 is a device that converts the transmitting signal (electrical signal) from the transmitting/receiving source 1 into ultrasonic waves, and emits the ultrasonic waves into the air to irradiate the laminate 14. The center frequency of the ultrasonic waves is, for example, 50 kHz to 100 MHz. The bandwidth is, for example, 10 kHz to 20 MHz. The wavelength of the ultrasonic waves may be longer than the thickness of the target object 10. Therefore, low-cost sensors can be used for the transmitting sensor 2 and the receiving sensor 3.
受信センサ3は、送受信源1に接続されている。積層体14を透過して空中を伝搬する超音波を受信して電気信号(時間波形)に変換する装置である。変換された電気信号は、送受信源1に入力される。なお、第1実施形態では積層体14の透過波を受信しているが、積層体14からの反射波を受信してもよい。ただし、積層体14からの反射波は透過波に比べて強度が強く、受信センサ3において受信強度が飽和しないように注意する必要がある。 The receiving sensor 3 is connected to the transmission/reception source 1. It is a device that receives ultrasonic waves that pass through the laminate 14 and propagate through the air, and converts them into an electrical signal (time waveform). The converted electrical signal is input to the transmission/reception source 1. Note that in the first embodiment, the transmitted waves of the laminate 14 are received, but it is also possible to receive reflected waves from the laminate 14. However, the reflected waves from the laminate 14 are stronger than the transmitted waves, and care must be taken to ensure that the reception intensity at the receiving sensor 3 does not become saturated.
送信センサ2と受信センサ3は間隔を開けて対向して配置されている。積層体14は、送信センサ2と受信センサ3の間に配置され、送信センサ2からの超音波が垂直に入射するように配置されている。 The transmitting sensor 2 and the receiving sensor 3 are arranged facing each other with a gap between them. The laminate 14 is placed between the transmitting sensor 2 and the receiving sensor 3 so that the ultrasonic waves from the transmitting sensor 2 are incident perpendicularly.
フーリエ変換部4は、送受信源1および解析部5に接続されている。フーリエ変換部4は、送受信源1からの電気信号(時間波形)を振幅および位相の周波数スペクトルにフーリエ変換して出力する装置である。フーリエ変換以外の方法によって時間波形を振幅および位相の周波数スペクトルに変換してもよい。たとえば、振幅、位相は、検波器などで直接計測してもよい。その場合、入力信号は連続波が望ましい。 The Fourier transform unit 4 is connected to the transmission/reception source 1 and the analysis unit 5. The Fourier transform unit 4 is a device that performs a Fourier transform on the electrical signal (time waveform) from the transmission/reception source 1 to convert it into a frequency spectrum of amplitude and phase and outputs it. The time waveform may also be converted into a frequency spectrum of amplitude and phase by a method other than the Fourier transform. For example, the amplitude and phase may be measured directly using a detector or the like. In that case, the input signal is preferably a continuous wave.
解析部5は、フーリエ変換部4に接続されている。解析部5は、フーリエ変換部4からの振幅および位相の周波数スペクトルから対象物10の厚さと縦波音速を算出する装置である。具体的な算出方法については後述する。 The analysis unit 5 is connected to the Fourier transform unit 4. The analysis unit 5 is a device that calculates the thickness and longitudinal wave sound velocity of the object 10 from the frequency spectrum of the amplitude and phase from the Fourier transform unit 4. The specific calculation method will be described later.
なお、第1実施形態では、送信センサ2および受信センサ3と対象物との間は空気であり、空中超音波法で計測するものであるが、媒質は空気に限るものではなく、水など任意の媒質であってよい。 In the first embodiment, the space between the transmitting sensor 2 and the receiving sensor 3 and the object is air, and measurements are taken using an airborne ultrasonic method, but the medium is not limited to air and may be any medium, such as water.
また、送信センサ2と受信センサ3を移動させる機構、あるいは対象物10を移動させる機構を設け、対象物10への超音波の照射位置を変えることにより、対象物10の厚さの分布と縦波音速の分布を計測してもよい。 In addition, a mechanism for moving the transmitting sensor 2 and the receiving sensor 3, or a mechanism for moving the object 10, may be provided, and the thickness distribution and longitudinal wave sound velocity distribution of the object 10 may be measured by changing the position at which the ultrasonic waves are irradiated onto the object 10.
(計測方法について)
次に、第1実施形態の計測装置を用いた対象物10の厚さと縦波音速の計測方法について説明する。
(Measurement method)
Next, a method for measuring the thickness and longitudinal wave velocity of the object 10 using the measurement device of the first embodiment will be described.
まず、送受信源1から送信信号を送信センサ2に送信し、送信センサ2において送信信号を超音波に変換し、送信センサ2から積層体14に対して超音波を照射する。そして、積層体14を透過した超音波を受信センサ3により受信し、超音波を時間波形の電気信号に変換する。その受信した時間波形を送受信源1において電圧値の時間波形に変換し、さらにフーリエ変換部4において振幅と位相の周波数スペクトルに変換する。送受信源1で送信信号が送信された時間を原点とする。たとえば、送信と同時にトリガー信号を出すなどして決定する。 First, a transmission signal is sent from the transmission/reception source 1 to the transmission sensor 2, where it is converted into ultrasonic waves, which are then irradiated onto the laminate 14. The ultrasonic waves that have passed through the laminate 14 are then received by the receiving sensor 3, which converts the ultrasonic waves into an electrical signal with a time waveform. The received time waveform is converted into a time waveform of voltage values by the transmission/reception source 1, and is further converted into a frequency spectrum of amplitude and phase by the Fourier transform unit 4. The time when the transmission signal is sent by the transmission/reception source 1 is set as the origin. For example, this can be determined by issuing a trigger signal simultaneously with the transmission.
次に、解析部5において、振幅と位相の周波数スペクトルから対象物10の厚さと縦波音速を同時に算出する。その詳細について、図2のフローチャートを参照に説明する。 Next, in the analysis unit 5, the thickness of the object 10 and the longitudinal wave sound velocity are simultaneously calculated from the frequency spectrum of the amplitude and phase. The details will be explained with reference to the flowchart in Figure 2.
まず、振幅と位相の周波数スペクトルを、下記式(1)に示す複素数の形にまとめる(図2のステップS1)。式(1)において、Vm(f)は、周波数fにおける複素数表示の計測信号、Am(f)は、周波数fにおける振幅の計測値、θm(f)は、周波数fにおける位相の計測値、jは虚数単位である。 First, the frequency spectrum of amplitude and phase is summarized in the form of a complex number shown in the following formula (1) (step S1 in FIG. 2 ): In formula (1), V m (f) is a measurement signal at frequency f expressed as a complex number, A m (f) is a measurement value of the amplitude at frequency f, θ m (f) is a measurement value of the phase at frequency f, and j is an imaginary unit.
次に、Vm(f)を、事前に求めておいた基準試験片における計測信号Vm0(f)で割り、比複素透過率Tr’m(f)を算出する(図2のステップS2)。基準試験片は、厚さ、縦波音速、密度が既知の材料で、減衰が少ない平板上が好ましい。 Next, V m (f) is divided by the measurement signal V m0 (f) of the reference test piece, which was obtained in advance, to calculate the specific complex transmittance Tr' m (f) (step S2 in FIG. 2). The reference test piece is a material with known thickness, longitudinal wave velocity, and density, and is preferably a flat plate with little attenuation.
次に、ある厚さおよび縦波音速における積層体14の複素透過率Trc(f)を解析的に算出する(図2のステップS3)。その解析モデルの詳細は後述するが、送信センサ2および受信センサ3から対象物10までの間の空気層も考慮したモデルとなっている。Trc(f)は、下記式(2)で表される。式(2)において、Ac(f)は、周波数fにおける振幅の解析値、θc(f)は、周波数fにおける位相の解析値、jは虚数単位である。 Next, the complex transmittance Tr c (f) of the laminate 14 at a certain thickness and longitudinal wave sound velocity is analytically calculated (step S3 in FIG. 2). The details of the analytical model will be described later, but the model also takes into account the air layer between the transmitting sensor 2 and the receiving sensor 3 and the target object 10. Tr c (f) is expressed by the following formula (2). In formula (2), A c (f) is the analytical value of the amplitude at frequency f, θ c (f) is the analytical value of the phase at frequency f, and j is the imaginary unit.
次に、Trc(f)を、事前に解析的に求めておいた基準試験片の複素透過率Trc0(f)で割り、比複素透過率Tr’c(f)を算出する(図2のステップS4)。 Next, Tr c (f) is divided by the complex transmittance Tr c0 (f) of the reference test piece that has been analytically determined in advance to calculate the specific complex transmittance Tr′ c (f) (step S4 in FIG. 2).
なお、Vm(f)、Trc(f)を基準試験片に対する比に変換しているのは、次の理由による。計測信号は送信信号と複素透過率の積に相当し、計測信号と複素透過率とでは単位・スケールが違う。そのため、Vm(f)とTrc(f)のままでは両者を比較できない。そこで、Vm(f)およびTrc(f)を基準試験片に対する比にすることで無次元化とスケール調整を行い、両者を比較できるようにしている。 The reason why V m (f) and Tr c (f) are converted into ratios to the reference test piece is as follows. The measurement signal corresponds to the product of the transmission signal and the complex transmittance, and the measurement signal and the complex transmittance have different units and scales. Therefore, V m (f) and Tr c (f) cannot be compared as they are. Therefore, by making V m (f) and Tr c (f) into ratios to the reference test piece, dimensionlessness and scale adjustment are performed, making it possible to compare the two.
次に、Tr’m(f)およびTr’c(f)のRMSEを、下記式(3)によって算出する(図2のステップS5)。 Next, the RMSE of Tr' m (f) and Tr' c (f) is calculated by the following equation (3) (step S5 in FIG. 2).
式(3)において、nは周波数の総数、fiはi番目の周波数である。nはたとえば10以上であれば十分に精度よく厚さと縦波音速を同時計測できる。nは好ましくは50以上である。n個の周波数の選び方は特に規定するものではないが、たとえば等周波数間隔で選ぶとよい。また、対象物10の共振周波数近傍を含まないように周波数を選ぶとよい。共振周波数近傍では受信センサ3の受信強度が強く、飽和してしまう恐れがあるためである。 In equation (3), n is the total number of frequencies, and fi is the i-th frequency. For example, if n is 10 or more, the thickness and longitudinal wave sound velocity can be measured simultaneously with sufficient accuracy. n is preferably 50 or more. There is no particular restriction on how to select the n frequencies, but it is advisable to select them at equal frequency intervals, for example. It is also advisable to select frequencies so as not to include frequencies near the resonant frequency of the object 10. This is because the reception strength of the receiving sensor 3 is strong near the resonant frequency, and there is a risk of saturation.
ステップS3からS5までを、解析に用いる厚さおよび縦波音速を変更して繰り返し、事前に決めておいたすべての厚さおよび縦波音速の組み合わせでRMSEを算出する(図2のステップS6)。 Steps S3 to S5 are repeated by changing the thickness and longitudinal wave velocity used in the analysis, and the RMSE is calculated for all combinations of thickness and longitudinal wave velocity determined in advance (step S6 in Figure 2).
次に、RMSEが最も小さかったときのTr’c(f)に用いた厚さと縦波音速を、対象物10の厚さおよび縦波音速として出力する(図2のステップS7)。なお、対象物10の密度とポアソン比が既知であれば、縦波音速からヤング率を算出することが可能である。 Next, the thickness and longitudinal wave velocity used in Tr'c (f) when the RMSE was smallest are output as the thickness and longitudinal wave velocity of the object 10 (step S7 in FIG. 2). If the density and Poisson's ratio of the object 10 are known, it is possible to calculate the Young's modulus from the longitudinal wave velocity.
このステップS3~S6は、つまりはTr’m(f)とTr’c(f)がどれだけ適合しているかをRMSEによって評価し、厚さと縦波音速の組み合わせを総当り的に変えて、最も適合するときのTr’c(f)を求め、そのTr’c(f)に用いた厚さと縦波音速の組み合わせを解とするものである。しかし、Tr’m(f)とTr’c(f)の適合度の指標はRMSEに限らず、最も適合するものを選ぶ方法も上記に限らない。たとえば、最尤推定法や機械学習によって最も適合するTr’c(f)を求めてもよい。 In other words, steps S3 to S6 are to evaluate the degree of compatibility between Tr' m (f) and Tr' c (f) by RMSE, to find the most compatible Tr' c (f) by changing the combination of thickness and longitudinal wave speed in a brute force manner, and to use the combination of thickness and longitudinal wave speed used for that Tr' c (f) as the solution. However, the index of compatibility between Tr' m (f) and Tr' c (f) is not limited to RMSE, and the method of selecting the most compatible one is not limited to the above. For example, the most compatible Tr' c (f) may be found by maximum likelihood estimation or machine learning.
(解析式について)
次に、複素透過率Trc(f)およびTrc0(f)の算出に用いる解析式について説明する。
(About the analytical formula)
Next, the analytical formula used to calculate the complex transmittances Tr c (f) and Tr c0 (f) will be described.
複素透過率Trc(f)およびTrc0(f)は、分布定数回路モデルに基づいて導かれた解析式によって算出される。分布定数回路モデルは、ある回路素子での入出力信号の関係を、その回路素子の反射・透過・伝搬特性を表現したSパラメータと呼ばれる式で表したモデルである。Sパラメータである行列Sは、下記式(4)の通りである。 The complex transmittances Tr c (f) and Tr c0 (f) are calculated by an analytical formula derived based on a distributed constant circuit model. The distributed constant circuit model is a model that expresses the relationship between input and output signals in a circuit element using a formula called S-parameters that express the reflection, transmission, and propagation characteristics of the circuit element. The matrix S, which is an S-parameter, is given by the following formula (4).
以下、分布定数回路モデルによる解析式の導出について説明する。第1実施形態における送信センサ2から受信センサ3までの間を分布定数回路モデルで表したのが図3である。図3のように、Sパラメータは、送信センサ2と積層体14との間の空気層がS1、空気層と第1被着材11との界面がS12、第1被着材11がS2、第1被着材11と接着剤12の界面がS23、接着剤12がS3、接着剤12と第2被着材13との界面がS34、第2被着材13がS4、第2被着材13と空気層との界面がS45、第2被着材13と受信センサ3との間の空気層がS5であり、S1、S12、S2、S23、S3、S34、S4、S45、S5の順に縦列接続された回路モデルである。 The derivation of the analytical formula using the distributed constant circuit model will be described below. Fig. 3 shows the section from the transmitting sensor 2 to the receiving sensor 3 in the first embodiment as a distributed constant circuit model. As shown in Figure 3, the S parameters are S1 for the air layer between the transmitting sensor 2 and the laminate 14, S12 for the interface between the air layer and the first adherend 11, S2 for the first adherend 11, S23 for the interface between the first adherend 11 and the adhesive 12, S3 for the adhesive 12, S3 for the interface between the adhesive 12 and the second adherend 13, S4 for the second adherend 13 , S45 for the interface between the second adherend 13 and the air layer, and S5 for the air layer between the second adherend 13 and the receiving sensor 3, and this is a circuit model in which the elements are connected in series in the order of S1 , S12 , S2 , S23 , S3 , S34 , S4 , S45 , and S5 .
各層及び界面のSパラメータは、次の式(5)、(6)で表される。 The S parameters of each layer and interface are expressed by the following equations (5) and (6).
式(5)において、cnは第n層における超音波の縦波音速、tnは第n層の厚さである。ここで、cnは下記式(7)で表される。 In formula (5), cn is the longitudinal wave speed of the ultrasonic wave in the n-th layer, and tn is the thickness of the n-th layer. Here, cn is expressed by the following formula (7).
式(7)において、Enは第n層のヤング率、ρnは第n層の密度、νnは第n層のポアソン比である。式(7)のように、材料の密度とポアソン比が既知であれば、縦波音速からヤング率を算出することができる。 In formula (7), E n is the Young's modulus of the nth layer, ρ n is the density of the nth layer, and ν n is the Poisson's ratio of the nth layer. If the density and Poisson's ratio of the material are known, as in formula (7), the Young's modulus can be calculated from the longitudinal wave sound velocity.
また、式(6)において、Γ(n+1) nは、第n層から第(n+1)層に向かって超音波が伝搬したときの反射率を示し、Tr(n+1) nは、第n層から第(n+1)層に向かって超音波が伝搬したときの反射率を示している。 In addition, in equation (6), Γ (n+1) n indicates the reflectance when ultrasonic waves propagate from the nth layer to the (n+1)th layer, and Tr (n+1) n indicates the reflectance when ultrasonic waves propagate from the nth layer to the (n+1)th layer.
Γ(n+1) n、Tr(n+1) nは、次の式(8)、(9)で表される。 Γ (n+1) n and Tr (n+1) n are expressed by the following equations (8) and (9).
式(8)、(9)においてZnは第n層の音響インピーダンスであり、下記式(10)で表される。 In equations (8) and (9), Z n is the acoustic impedance of the n-th layer and is expressed by the following equation (10).
SパラメータをTパラメータに変換することで、縦列接続は単に行列の掛け算で表すことができる。SパラメータからTパラメータの変換は式(11)で表される。また、送信センサ2から受信センサ3までの全体のTパラメータ(Tall)は式(12)で表される。 By converting the S-parameters to the T-parameters, the cascade connection can be expressed simply by matrix multiplication. The conversion from the S-parameters to the T-parameters is expressed by Equation (11). Moreover, the overall T-parameter (T all ) from the transmitting sensor 2 to the receiving sensor 3 is expressed by Equation (12).
式(12)において、Tnは第n層のTパラメータであり、Tn (n+1)は第n層と第(n+1)層の界面のTパラメータである。 In formula (12), T n is the T parameter of the nth layer, and T n (n+1) is the T parameter of the interface between the nth layer and the (n+1)th layer.
式(12)により求めたTパラメータTallを再びSパラメータに変換して、送信センサ2から受信センサ3までの全体のSパラメータSallが算出される。TallからSallへの変換は式(13)で表される。 The T parameter T all obtained by equation (12) is converted back into an S parameter to calculate the overall S parameter S all from the transmitting sensor 2 to the receiving sensor 3. The conversion from T all to S all is expressed by equation (13).
式(6)と比較するとわかるように、Sallの成分Sall21が、送信センサ2から受信センサ3までの複素透過率であり、Trc(f)に相当する。このようにして、分布定数回路モデルによるTrc(f)の解析式が求まる。なお、Sallの成分Sall11が複素反射率に相当する。 As can be seen by comparing with formula (6), the component S all 21 of S all is the complex transmittance from the transmitting sensor 2 to the receiving sensor 3, and corresponds to Trc (f). In this manner, the analytical formula for Trc (f) using the distributed constant circuit model is obtained. Note that the component S all 11 of S all corresponds to the complex reflectance.
以上のようにして算出されたTrc(f)の解析式は、各界面での反射波・透過波の振幅と位相の関係が考慮されている。つまり、反射波・透過波の重なり方が考慮されている。よって、反射波・透過波が重なり合うような厚さが薄い対象物10であっても、精度よく厚さを求めることができる。 The analytical formula for Tr c (f) calculated as described above takes into consideration the relationship between the amplitude and phase of the reflected wave and the transmitted wave at each interface. In other words, the overlapping manner of the reflected wave and the transmitted wave is taken into consideration. Therefore, even if the object 10 is thin enough that the reflected wave and the transmitted wave overlap, the thickness can be calculated with high accuracy.
また、式(12)のように、送信センサ2および受信センサ3から対象物10までの間の空気層をT1、T5として考慮しており、空気層における超音波の位相変化が考慮されている。そのため、対象物の厚さが複素透過率Trc(f)の位相に強く反映されるようになり、Tr’m(f)とTr’c(f)の適合性もより適切に評価できるようになる。 Furthermore, as in equation (12), the air layers between the transmitting sensor 2 and the receiving sensor 3 and the object 10 are considered as T1 and T5 , respectively, and the phase change of the ultrasonic wave in the air layers is taken into consideration. Therefore, the thickness of the object is strongly reflected in the phase of the complex transmittance Tr c (f), and the compatibility between Tr' m (f) and Tr' c (f) can be more appropriately evaluated.
なお、解析式は必ずしも分布定数回路モデルに基づくものでなくともよく、送信センサ2および受信センサ3から対象物10までの間の空気層が考慮され、その空気層における超音波の位相変化が考慮されていれば任意でよい。たとえば、各層における音圧、粒子速度の伝搬式と、各層間における音圧、粒子速度の関係式から、反射波および透過波の振幅と位相を解析するモデルを用いてもよい。 The analytical formula does not necessarily have to be based on a distributed constant circuit model, and any formula can be used as long as it takes into account the air layer between the transmitting sensor 2 and the receiving sensor 3 and the target object 10, and the phase change of the ultrasonic wave in that air layer. For example, a model that analyzes the amplitude and phase of the reflected wave and transmitted wave from the propagation formula of the sound pressure and particle velocity in each layer, and the relational formula of the sound pressure and particle velocity between each layer, may be used.
以上、第1実施形態では、超音波の振幅だけでなく位相も計測して利用し、さらに送信センサ2、受信センサ3と対象物10との間の媒質(空気)を超音波が伝搬する際の位相変化も考慮して解析を行っている。そのため、対象物10の厚さと縦波音速を同時に精度よく計測することができる。 As described above, in the first embodiment, not only the amplitude of the ultrasonic waves but also the phase is measured and used, and the analysis is performed taking into account the phase change that occurs when the ultrasonic waves propagate through the medium (air) between the transmitting sensor 2, the receiving sensor 3, and the object 10. Therefore, the thickness of the object 10 and the longitudinal wave sound velocity can be measured simultaneously with high accuracy.
また、第1実施形態では、低周波でも精度よく厚さと縦波音速を求めることができるので、非接触の空中超音波法でも利用することができる。また、多層構造体中の薄い層であっても厚さと縦波音速を同時に計測することができる。たとえば、接着接合体における接着剤の厚さと縦波音速も計測することができる。したがって、接触式や水浸式では厚さと縦波音速を同時計測することが困難であった対象物10であっても、第1実施形態によれば計測が可能であり、従来よりも厚さと縦波音速を同時計測可能な対象物の範囲が広くなっている。 In addition, in the first embodiment, the thickness and longitudinal wave sound velocity can be determined with high accuracy even at low frequencies, so it can also be used in non-contact airborne ultrasonic methods. In addition, the thickness and longitudinal wave sound velocity can be measured simultaneously even for thin layers in a multilayer structure. For example, the thickness and longitudinal wave sound velocity of an adhesive in an adhesive joint can also be measured. Therefore, even for an object 10 for which it is difficult to simultaneously measure the thickness and longitudinal wave sound velocity using contact or water immersion methods, measurement is possible according to the first embodiment, and the range of objects for which the thickness and longitudinal wave sound velocity can be measured simultaneously is wider than in the past.
次に、第1実施形態に関する各種実験結果について説明する。 Next, we will explain the results of various experiments related to the first embodiment.
(実験1)
図4、5は、空気層を考慮しないで解析を行い、複素透過率の振幅と位相を算出した結果である。また、図6、7は、空気層を考慮して解析を行い、複素透過率の振幅と位相を算出した結果である。図4(a)、図6(a)は縦波音速と振幅の関係、図4(b)、図6(b)は縦波音速と位相の関係を示している。接着剤の厚さは100μmとし、縦波音速は400~1200m/Sとした。図5(a)、図7(a)は厚さと振幅の関係、図5(b)、図7(b)は厚さと位相の関係を示している。接着剤の縦波音速は1500m/Sとし、厚さは0~800μmとした。また、図4~7において第1被着材11は厚さ1mmのAl板、第2被着材13は厚さ0.8mmの鋼板とし、接着剤12の密度は1264kg/m3、ポアソン比は0.3とした。また、超音波の周波数は330kHzとした。また、図6、7において送信センサ2と受信センサ3間の距離は30mmとし、送信センサ2から対象物10までの距離(第1被着材11までの距離)は14mmとした。また、空気の音速は340m/sとし、音響インピーダンスは408kg/(s・m2)とした。
(Experiment 1)
4 and 5 show the results of the analysis without considering the air layer, and the amplitude and phase of the complex transmittance are calculated. Also, FIG. 6 and FIG. 7 show the results of the analysis with the air layer in mind, and the amplitude and phase of the complex transmittance are calculated. FIG. 4(a) and FIG. 6(a) show the relationship between the longitudinal wave sound velocity and the amplitude, and FIG. 4(b) and FIG. 6(b) show the relationship between the longitudinal wave sound velocity and the phase. The thickness of the adhesive was 100 μm, and the longitudinal wave sound velocity was 400 to 1200 m/S. FIG. 5(a) and FIG. 7(a) show the relationship between the thickness and the amplitude, and FIG. 5(b) and FIG. 7(b) show the relationship between the thickness and the phase. The longitudinal wave sound velocity of the adhesive was 1500 m/S, and the thickness was 0 to 800 μm. 4 to 7, the first adherend 11 is an Al plate having a thickness of 1 mm, the second adherend 13 is a steel plate having a thickness of 0.8 mm, the density of the adhesive 12 is 1264 kg/m 3 , and the Poisson's ratio is 0.3. The frequency of the ultrasonic waves is 330 kHz. In addition, in Figures 6 and 7, the distance between the transmitting sensor 2 and the receiving sensor 3 is 30 mm, and the distance from the transmitting sensor 2 to the object 10 (the distance to the first adherend 11) is 14 mm. The sound speed of air is 340 m/s, and the acoustic impedance is 408 kg/(s·m 2 ).
振幅のみを考慮して厚さと縦波音速を計測していた従来の方法では、空気層を考慮する必要がなかった。これは、図4(a)と図6(a)、および図5(a)と図7(a)を比較するとわかるように、空気層を考慮してもしなくても同じ結果となるためである。しかし、図4(a)と図5(a)を比較するとわかるように、縦波音速が増加したときと厚さが減少したときとで振幅の変化傾向が似ている。つまり、振幅の変化が縦波音速の変化によるものか厚さの変化によるものかの判断が困難である。そのため、振幅のみでは厚さと縦波音速とを分離して正確に算出することが難しい。 In conventional methods that measured thickness and longitudinal wave speed by considering only the amplitude, there was no need to consider the air layer. This is because, as can be seen by comparing Figures 4(a) and 6(a), and Figures 5(a) and 7(a), the results are the same whether the air layer is considered or not. However, as can be seen by comparing Figures 4(a) and 5(a), the tendency of amplitude change is similar when the longitudinal wave speed increases and when the thickness decreases. In other words, it is difficult to determine whether the change in amplitude is due to a change in the longitudinal wave speed or a change in thickness. Therefore, it is difficult to accurately calculate thickness and longitudinal wave speed separately using only the amplitude.
そこで、第1実施形態では、振幅だけでなく位相も含めて解析式との適合性を判断することとした。しかし、位相も考慮することとすると、実際の計測には空気層の位相変化も含まれる。そのため、空気層を考慮しない解析式では適切に適合性を判断することができない。そこで、図6、7のように、解析式に空気層も考慮することとした。 Therefore, in the first embodiment, the suitability with the analytical formula is judged taking into account not only the amplitude but also the phase. However, if the phase is also taken into account, the actual measurement will also include the phase change of the air layer. Therefore, suitability cannot be judged appropriately with an analytical formula that does not take the air layer into account. Therefore, as shown in Figures 6 and 7, the air layer is also taken into account in the analytical formula.
図5(b)と図7(b)を比較するとわかるように、空気層を考慮すると、位相が厚さの変化に対して変化するようになる。送信センサ2と受信センサ3の距離が一定であるため、接着剤12の厚さの変化によって空気層の厚さも変化し、空気層における位相変化が効いてくるためである。このように、空気層を考慮して解析を行うと、位相に厚さの影響が強く表れるようになる。その結果、位相から厚さを正確に算出できるようになる。 As can be seen by comparing Figure 5(b) and Figure 7(b), when the air layer is taken into account, the phase changes with changes in thickness. This is because the distance between the transmitting sensor 2 and the receiving sensor 3 is constant, so that the thickness of the air layer also changes with changes in the thickness of the adhesive 12, and the phase change in the air layer comes into effect. In this way, when an analysis is performed taking the air layer into account, the effect of the thickness on the phase becomes more pronounced. As a result, it becomes possible to accurately calculate the thickness from the phase.
以上のように、解析式に空気層を考慮し空気層における位相変化を考慮したうえで、振幅と位相の両方で計測値と解析式との適合性を判断することで、厚さと縦波音速の同時に算出可能となることが図4~7からわかる。 As described above, it can be seen from Figures 4 to 7 that it is possible to simultaneously calculate thickness and longitudinal wave sound velocity by taking into account the air layer in the analytical formula and taking into account the phase change in the air layer, and then determining the compatibility between the measured values and the analytical formula in terms of both amplitude and phase.
(実験2)
積層体14の透過波を有限要素法でシミュレーションし、そこから接着剤12の厚さと縦波音速を第1実施形態の方法によって算出した(実施例1)。積層体14の各種物性は表1の通りである。
(Experiment 2)
The transmitted waves of the laminate 14 were simulated by the finite element method, and the thickness of the adhesive 12 and the longitudinal wave sound velocity were calculated from the simulation by the method of the first embodiment (Example 1). Various physical properties of the laminate 14 are shown in Table 1.
接着剤12は、厚さや縦波音速の異なる三種類(接着剤A、B、C)を用いた。また、比較のため、空気層を考慮しない解析式を用い、振幅のみを用いて計測値と解析式の適合性をRMSEで判断した場合について接着剤12の厚さと縦波音速を算出した(比較例1)。 Three types of adhesive 12 (adhesives A, B, and C) with different thicknesses and longitudinal wave sound velocities were used. For comparison, an analytical formula that does not take into account the air layer was used, and the thickness and longitudinal wave sound velocity of adhesive 12 were calculated when the compatibility of the measured values and the analytical formula was judged by RMSE using only the amplitude (Comparative Example 1).
有限要素法シミュレーションでは、対象物10に照射する超音波信号として、中心波長330kHzの空中超音波センサから実際にサンプリングした波形を用いた。送信センサ2と受信センサ3間は30mm、送信センサ2から第1被着材11までの距離は14mm、空気の音速は340m/s、音響インピーダンスは408kg/(s・m2)とした。また、基準試験片は厚さ1mmのAl板とした。 In the finite element method simulation, a waveform actually sampled from an airborne ultrasonic sensor with a central wavelength of 330 kHz was used as the ultrasonic signal to be irradiated to the target object 10. The distance between the transmitting sensor 2 and the receiving sensor 3 was 30 mm, the distance from the transmitting sensor 2 to the first adherend 11 was 14 mm, the sound speed of air was 340 m/s, and the acoustic impedance was 408 kg/(s· m2 ). The reference test piece was an Al plate with a thickness of 1 mm.
図8は、受信センサ3により受信した透過波の時間波形を示したグラフである。この時間波形は、有限要素法シミュレーションにより得られたものである。また、図9、10は、この時間波形をフーリエ変換した振幅と位相の周波数スペクトルである。ここで振幅と位相は、基準試験片での振幅、位相を基準とした相対値で示している。なお相対値は、振幅では基準試験片での振幅に対する比、位相では基準試験片での位相との差をとっている。 Figure 8 is a graph showing the time waveform of the transmitted wave received by the receiving sensor 3. This time waveform was obtained by finite element method simulation. Figures 9 and 10 show the frequency spectrum of the amplitude and phase obtained by Fourier transforming this time waveform. Here, the amplitude and phase are shown as relative values based on the amplitude and phase of a reference test piece. Note that the relative value is the ratio to the amplitude of the reference test piece for the amplitude, and the difference from the phase of the reference test piece for the phase.
図9の周波数スペクトルにおいて強い振幅が得られた周波数320~350kHzにおける振幅と位相を用いて、解析式との適合性をRMSEにより評価した。周波数は320~350kHzの間で等間隔に50点とした。解析式の厚さと縦波音速は、厚さ0~500μmの範囲を1μm間隔で走査し、縦波音速はヤング率から式(7)により求めたが、ヤング率を0.5~4.5GPaの範囲で0.01GPaずつ変化させて、730~2308m/sの範囲で走査した。縦波音速を式(7)から求めるのではなく、直接変化させて走査してもよい。 The amplitude and phase at frequencies between 320 and 350 kHz, where strong amplitudes were obtained in the frequency spectrum of Figure 9, were used to evaluate the compatibility with the analytical formula by RMSE. 50 equally spaced frequency points were used between 320 and 350 kHz. The thickness and longitudinal wave velocity of the analytical formula were determined by scanning the thickness range of 0 to 500 μm at 1 μm intervals, and the longitudinal wave velocity was calculated from Young's modulus using formula (7). The Young's modulus was changed in increments of 0.01 GPa in the range of 0.5 to 4.5 GPa, and scanning was performed in the range of 730 to 2308 m/s. The longitudinal wave velocity may be directly changed and scanned instead of being calculated from formula (7).
厚さと縦波音速の算出結果を表2に示す。 The calculated thickness and longitudinal wave velocity are shown in Table 2.
表2のように、比較例1では算出結果が正解と大きく異なっていたのに対し、実施例1では算出結果が正解に近い値となっていた。この結果から、空気層を考慮した解析式を用い、振幅と位相の両方用いて解析式との適合性を判断することで、接着剤12の厚さと縦波音速を同時に精度よく計測できることが分かった。 As shown in Table 2, the calculation result in Comparative Example 1 was significantly different from the correct answer, whereas the calculation result in Example 1 was close to the correct answer. From this result, it was found that by using an analytical formula that takes into account the air layer and determining the compatibility with the analytical formula using both the amplitude and phase, it is possible to measure the thickness of the adhesive 12 and the longitudinal wave sound velocity simultaneously with high accuracy.
本発明は、たとえば接着剤の厚さと縦波音速を測定し、接着剤の未硬化や接着不良を検出するのに利用できる。 The present invention can be used, for example, to measure the thickness and longitudinal wave sound velocity of an adhesive and detect uncured or poor adhesion of the adhesive.
1:送受信源
2:送信センサ
3:受信センサ
4:フーリエ変換部
5:解析部
1: Transmitting/receiving source 2: Transmitting sensor 3: Receiving sensor 4: Fourier transform section 5: Analysis section
Claims (7)
前記対象物に超音波を照射する送信部と、
前記対象物からの透過波または反射波である超音波を時間波形の電気信号として受信する受信部と、
前記電気信号を振幅および位相の周波数スペクトルに変換する変換部と、
前記対象物の厚さおよび縦波音速を所定の値としたときの振幅および位相の周波数スペクトルを解析式により算出し、前記変換部による振幅および位相の周波数スペクトルと前記解析式による振幅および位相の周波数スペクトルとの適合性を、前記解析式に用いる前記対象物の厚さおよび縦波音速を変えて繰り返し判断し、前記変換部による振幅および位相の周波数スペクトルと解析による振幅および位相の周波数スペクトルとが最も適合するときに解析式に用いた前記対象物の厚さおよび縦波音速を、解として算出する解析部と、
を有し、
前記解析式は、前記送信部から前記対象物までおよび前記対象物から前記受信部までの間に存在する媒質を考慮し、前記媒質による超音波の位相変化を考慮した式である、
ことを特徴とする計測装置。 A measuring device for simultaneously measuring the thickness and longitudinal wave velocity of an object, comprising:
A transmitter that irradiates the target with ultrasonic waves;
A receiving unit that receives ultrasonic waves, which are transmitted waves or reflected waves from the object, as electrical signals of a time waveform;
A converter for converting the electrical signal into an amplitude and phase frequency spectrum;
an analysis unit that calculates an amplitude and phase frequency spectrum when the thickness and longitudinal wave speed of the object are set to a predetermined value by an analytical formula, repeatedly determines the compatibility between the amplitude and phase frequency spectrum by the conversion unit and the amplitude and phase frequency spectrum by the analytical formula by changing the thickness and longitudinal wave speed of the object used in the analytical formula, and calculates the thickness and longitudinal wave speed of the object used in the analytical formula as a solution when the amplitude and phase frequency spectrum by the conversion unit and the amplitude and phase frequency spectrum by the analysis are most compatible;
having
The analytical formula is a formula that takes into account a medium existing between the transmitting unit and the target object and between the target object and the receiving unit, and takes into account a phase change of the ultrasonic wave due to the medium.
A measuring device characterized by:
送信部から前記対象物に超音波を照射し、
前記対象物からの透過波または反射波である超音波を時間波形の電気信号として受信部で受信し、
前記電気信号を振幅および位相の周波数スペクトルに変換し、
前記対象物の厚さおよび縦波音速を所定の値としたときの振幅および位相の周波数スペクトルを解析式により算出し、前記電気信号から変換した振幅および位相の周波数スペクトルと前記解析式による振幅および位相の周波数スペクトルとの適合性を、前記解析式に用いる前記対象物の厚さおよび縦波音速を変えて繰り返し判断し、前記電気信号から変換した振幅および位相の周波数スペクトルと解析による振幅および位相の周波数スペクトルとが最も適合するときに前記解析式に用いた前記対象物の厚さおよび縦波音速を、解として算出し、
前記解析式は、前記送信部から前記対象物までおよび前記対象物から前記受信部までの間に存在する媒質を考慮し、前記媒質による超音波の位相変化を考慮した式である、
ことを特徴とする計測方法。 A method for simultaneously measuring a thickness and a longitudinal wave velocity of an object, comprising the steps of:
A transmitting unit irradiates the target with ultrasonic waves;
The ultrasonic wave, which is a transmitted wave or a reflected wave from the object, is received by a receiving unit as an electrical signal having a time waveform;
converting said electrical signal into an amplitude and phase frequency spectrum;
The frequency spectrum of the amplitude and phase when the thickness and longitudinal wave speed of the object are set to predetermined values is calculated by an analytical formula, and the compatibility between the frequency spectrum of the amplitude and phase converted from the electrical signal and the frequency spectrum of the amplitude and phase obtained by the analytical formula is repeatedly determined by changing the thickness of the object and the longitudinal wave speed used in the analytical formula, and when the frequency spectrum of the amplitude and phase converted from the electrical signal and the frequency spectrum of the amplitude and phase obtained by the analysis are most compatible, the thickness of the object and the longitudinal wave speed used in the analytical formula are calculated as a solution,
The analytical formula is a formula that takes into account a medium existing between the transmitting unit and the target object and between the target object and the receiving unit, and takes into account a phase change of the ultrasonic wave due to the medium.
A measuring method comprising:
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