JP4429118B2 - Time history response analysis method, apparatus, and program - Google Patents

Time history response analysis method, apparatus, and program Download PDF

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JP4429118B2
JP4429118B2 JP2004253130A JP2004253130A JP4429118B2 JP 4429118 B2 JP4429118 B2 JP 4429118B2 JP 2004253130 A JP2004253130 A JP 2004253130A JP 2004253130 A JP2004253130 A JP 2004253130A JP 4429118 B2 JP4429118 B2 JP 4429118B2
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尚弘 中村
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本発明は時刻歴応答解析方法、装置及びプログラムに係り、特に、物体を振動させる外力と前記物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析方法、該時刻歴応答解析方法を適用可能な時刻歴応答解析装置、及び、コンピュータを前記時刻歴応答解析装置として機能させるための時刻歴応答解析プログラムに関する。   The present invention relates to a time history response analysis method, apparatus, and program, and in particular, performs a time history response analysis of an object in which the relationship between an external force that vibrates the object and the behavior of the object exhibits frequency dependence and nonlinear distortion dependence. The present invention relates to a time history response analysis method, a time history response analysis device to which the time history response analysis method can be applied, and a time history response analysis program for causing a computer to function as the time history response analysis device.

地震時の建物の振動を抑制するために建物に取付けられる制振ダンパーは、粘弾性体が封入された構成が採用されることが多いが、粘弾性体が封入されたダンパー（粘弾性ダンパー）は、外力（反力）と挙動（変位）との関係（動的特性）が周波数依存性と非線形の歪依存性（物体の歪の変化に対して物体の反力が非線形に変化する特性）を各々示すことが知られている。地震時の建物の挙動を解析・評価する地震応答解析等では、解析対象の物体の動的特性が周波数依存性を有している場合は周波数領域で線形の応答解析（周波数応答解析）を行い、解析対象の物体の動的特性が非線形の歪依存性を有している場合は時間領域で非線形の応答解析（時刻歴応答解析）を行うことが一般的であるが、上記の粘弾性ダンパーのように、動的特性が周波数依存性及び非線形の歪依存性を各々有している物体については、単に周波数領域又は時間領域で応答解析を行うのみでは、動的特性の周波数依存性の影響及び非線形の歪依存性の影響を各々評価して物体の挙動を正確に解析することは困難である。   The damping damper attached to the building to suppress the vibration of the building during an earthquake is often configured with a viscoelastic body enclosed, but the damper with the viscoelastic body enclosed (viscoelastic damper) The relationship between the external force (reaction force) and the behavior (displacement) (dynamic characteristics) is frequency-dependent and non-linear strain dependency (characteristic that the reaction force of the object changes nonlinearly with respect to the change of the object strain) Are known to each indicate. In earthquake response analysis that analyzes and evaluates the behavior of buildings during earthquakes, linear response analysis (frequency response analysis) is performed in the frequency domain when the dynamic characteristics of the object to be analyzed have frequency dependence. When the dynamic characteristics of the object to be analyzed have nonlinear strain dependence, it is common to perform nonlinear response analysis (time history response analysis) in the time domain. For an object whose dynamic characteristics have frequency dependence and nonlinear distortion dependence, simply performing response analysis in the frequency domain or time domain will affect the frequency dependence of the dynamic characteristics. In addition, it is difficult to accurately analyze the behavior of an object by evaluating the effects of nonlinear strain dependence and nonlinearity, respectively.

このような問題に対しては、解析対象の物体を単純な物理モデルに置き換えて応答解析を行う方法が知られている（例えば非特許文献１〜３を参照）。物理モデルとしては、単純なばねとダンパーの結合体である一般化マックスウェル(Maxwell)要素が用いられることが多い。マックスウェル要素は、例えば図６に示すＫ２＋Ｃ２のように、ばね(Ｋ２)とダンパー(Ｃ２)が直列に結合されたものであり、一般化マックスウェル要素は、マックスウェル要素にばねやダンパー（図６の例ではばね(Ｋ１)やダンパー(Ｃ２))を並列に結合して構成される。一般化マックスウェル要素には様々なバリエーションがあるが、基本的には、一組のマックスウェル要素に１個以上のばね又はダンパー又は他のマックスウェル要素が並列に結合されて構成される。解析対象の物体のモデルとして、一部のばね又はダンパーに非線形の歪依存性を付加した一般化マックスウェル要素を生成することで、解析対象の物体の動的特性が周波数依存性及び非線形の歪依存性を各々有している場合にも、生成した一般化マックスウェル要素に対して時間領域で非線形の応答解析を行うことで、周波数依存性の影響及び非線形の歪依存性の影響を考慮して物体の挙動を解析することができる。   For such a problem, a method of performing response analysis by replacing an object to be analyzed with a simple physical model is known (see, for example, Non-Patent Documents 1 to 3). As a physical model, a generalized Maxwell element, which is a simple spring / damper combination, is often used. The Maxwell element has a spring (K2) and a damper (C2) coupled in series, for example, K2 + C2 shown in FIG. 6, and the generalized Maxwell element has a spring and a damper (see FIG. In the example of FIG. 6, a spring (K1) and a damper (C2)) are coupled in parallel. There are various variations of generalized Maxwell elements, but basically a set of Maxwell elements is composed of one or more springs or dampers or other Maxwell elements coupled in parallel. By generating a generalized Maxwell element that adds nonlinear strain dependence to some springs or dampers as a model of the object to be analyzed, the dynamic characteristics of the object to be analyzed are frequency dependent and nonlinear distortion. Even in the case of each having a dependency, a nonlinear response analysis is performed on the generated generalized Maxwell element in the time domain to take into account the effects of frequency dependence and nonlinear distortion dependence. The behavior of the object can be analyzed.

なお、上記に関連して本願発明者は、地盤の動的特性（詳しくは、地震動と地盤の挙動との関係を、振動の周波数の変化に応じて実部及び虚部の値が変化する周波数領域の複素関数で表す地盤の動的剛性（地盤インピーダンスともいう))の周波数依存性が強い場合にも、地盤の動的剛性を時間領域で表されるインパルス応答へ精度良く変換できる変換方法として、変位依存と速度依存の両方の時間遅れ成分に加えて加速度依存の同時成分を有する形式を、インパルス応答を用いた反力Ｆ(t)の一般解として設定し、設定した反力Ｆ(t)の一般解と、この一般解からインパルス応答の同時成分及び時間遅れ成分を用いて表される地盤の動的剛性Ｓ(ω)の式に基づき、ω0〜ωnの各周波数におけるＮ（＝ｎ＋１）個の地盤の動的剛性のデータＤ(ωi)を用いて２Ｎ×２Ｎの係数マトリクスを有する連立方程式を立て、この連立方程式を解くことでインパルス応答の各成分を求める変換方法を提案している（非特許文献４を参照）。
東野雅彦，山本雅史，浅野三男，「粘弾性ダンパーの力学特性 その１ 各種粘弾性ダンパーの基本的性状」，日本建築学会大会学術講演梗概集（中国），１９９９年９月，ｐ．９６３−９６４ 東野雅彦，山本雅史，浅野三男，「粘弾性ダンパーの力学特性 その２ 振動数依存性を考慮した粘弾性ダンパーのモデル化」，日本建築学会大会学術講演梗概集（中国），１９９９年９月，ｐ．９６５−９６６ 東野雅彦，山本雅史，浅野三男，「粘弾性ダンパーの力学特性 その３ 劣化型特性および振幅依存性をもつ粘弾性体のモデル化」，日本建築学会大会学術講演梗概集（中国），１９９９年９月，ｐ．９６７−９６８ 中村尚弘，「地盤インピーダンスの時間領域変換による成層地盤に埋込まれた構造物の地震応答解析 その２ 変換法の改良及び離散的地盤モデルに基づく応答解析」，日本建築学会構造系論文集，２００３年１２月，第５７４号，ｐ．９９−１０６ In connection with the above, the inventor of the present application described the dynamic characteristics of the ground (specifically, the relationship between the seismic motion and the ground behavior, the frequency at which the values of the real part and the imaginary part change according to the change in the frequency of vibration. As a conversion method that can convert the dynamic stiffness of the ground to the impulse response expressed in the time domain with high accuracy even when the frequency dependence of the ground dynamic stiffness (also referred to as ground impedance) expressed by a complex function of the region is strong. In addition to the time-delay component that is both displacement-dependent and velocity-dependent, a form having an acceleration-dependent simultaneous component is set as a general solution of the reaction force F (t) using the impulse response, and the set reaction force F (t ) And the equation of the dynamic stiffness S (ω) of the ground expressed by using the simultaneous component and the time delay component of the impulse response from this general solution, N (at each frequency of ω 0 to ω n = N + 1) Dynamic stiffness data D (ω i ) of ground Thus, a conversion method has been proposed in which simultaneous equations having a coefficient matrix of 2N × 2N are established and each component of the impulse response is obtained by solving the simultaneous equations (see Non-Patent Document 4).
Masahiko Higashino, Masafumi Yamamoto, Michio Asano, “Mechanical properties of viscoelastic dampers, Part 1 Basic properties of various viscoelastic dampers”, Annual Meeting of the Architectural Institute of Japan (China), September 1999, p. 963-964 Masahiko Higashino, Masafumi Yamamoto, Mitsuo Asano, “Mechanical properties of viscoelastic dampers, Part 2: Modeling of viscoelastic dampers considering frequency dependence”, Abstracts of Annual Meeting of the Architectural Institute of Japan (China), September 1999, p. 965-966 Masahiko Higashino, Masafumi Yamamoto, Mitsuo Asano, “Mechanical properties of viscoelastic dampers, Part 3: Modeling of viscoelastic bodies with degradation characteristics and amplitude dependence”, Abstracts of Annual Meeting of the Architectural Institute of Japan (China), 1999 September Month, p. 967-968 Naohiro Nakamura, “Earthquake response analysis of structures embedded in stratified soil by time domain transformation of ground impedance, Part 2 Improvement of transformation method and response analysis based on discrete ground model”, Architectural Institute of Japan, 2003 December, 574, p. 99-106

前述のように、解析対象の物体を一般化マックスウェル要素等の物理モデルへ置き換えて応答解析を行う場合には、置き換える物理モデルを構成する各要素の定数を同定する必要があるが、上記の応答解析における解析精度は置き換えた物理モデルの精度に依存するので、物理モデルの各要素の定数は、置き換えた物理モデルが解析対象の物体の特性を精度良く表すように定める必要がある。しかしながら、物理モデルの各要素の定数は単純な計算により一意に定まるものではないので、各要素の定数の同定は試行錯誤を繰り返して行っているのが実情であり、解析対象の物体の特性を精度良く表す適切な物理モデルを得るためには多大な手間と熟練が必要とされるという問題がある。そして、実際の応答解析では、解析対象の物体の特性を精度良く表す適切な物理モデルが得られなかったために、応答解析の解析精度が不足することも多々生じていた。   As described above, when a response analysis is performed by replacing an object to be analyzed with a physical model such as a generalized Maxwell element, it is necessary to identify constants of each element constituting the physical model to be replaced. Since the analysis accuracy in response analysis depends on the accuracy of the replaced physical model, the constants of the elements of the physical model need to be determined so that the replaced physical model accurately represents the characteristics of the object to be analyzed. However, since the constants of each element of the physical model are not uniquely determined by simple calculations, it is a fact that the constants of each element are identified by trial and error, and the characteristics of the object to be analyzed are determined. There is a problem that a great deal of labor and skill are required to obtain an appropriate physical model that can be expressed accurately. In the actual response analysis, since an appropriate physical model that accurately represents the characteristics of the object to be analyzed cannot be obtained, the analysis accuracy of the response analysis is often insufficient.

また、非特許文献４に記載の技術を適用すれば、物体の動的特性（物体を振動させる外力と物体の挙動との関係）が強い周波数依存性を示している場合にも、物体の動的特性を時間領域で表すインパルス応答を演算して応答解析を行うことが可能となるものの、上記技術は物体の動的特性が周波数依存性と非線形の歪依存性を各々示している場合については考慮されていない。   Furthermore, if the technique described in Non-Patent Document 4 is applied, even if the dynamic characteristics of the object (the relationship between the external force that vibrates the object and the behavior of the object) exhibit strong frequency dependence, Although it is possible to calculate the impulse response that expresses the dynamic characteristics in the time domain and perform response analysis, the above technique is applicable to the case where the dynamic characteristics of the object show frequency dependence and nonlinear distortion dependence respectively. Not considered.

本発明は上記事実を考慮して成されたもので、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体について、周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を容易に行うことができる時刻歴応答解析方法、時刻歴応答解析装置及び時刻歴応答解析プログラムを得ることが目的である。   The present invention has been made in consideration of the above facts. For an object in which the relationship between the external force that vibrates the object and the behavior of the object shows frequency dependence and nonlinear distortion dependence, the frequency dependence and nonlinear distortion dependence It is an object to obtain a time history response analysis method, a time history response analysis device, and a time history response analysis program capable of easily performing a highly accurate time history response analysis in consideration of each of the characteristics.

物体の動的特性における非線形の歪依存性は、例えば物体の歪（この歪は物体の変位（挙動）に応じて変化し変位より求まる）の変化に対する反力の変化の傾きが歪の増大に伴って小さくなる現象として現れるが、この非線形の歪依存性は周波数領域では表現できないので、粘弾性ダンパー等のように物体を振動させる外力（反力）と物体の挙動（変位）との関係（動的特性）が周波数依存性及び非線形の歪依存性を示す物体の応答解析を行うにあたり、周波数依存性の影響及び非線形の歪依存性の影響を考慮するためには前記応答解析を時間領域で行う必要があり（時刻歴応答解析）、時刻歴応答解析を行うためには解析対象の物体の動的特性を時間領域で表すインパルス応答を求める必要がある。   Nonlinear strain dependency in the dynamic characteristics of an object is, for example, that the slope of the reaction force changes with respect to the change in the strain of the object (this strain changes according to the displacement (behavior) of the object and is obtained from the displacement). Although it appears as a phenomenon that decreases with this, this nonlinear strain dependence cannot be expressed in the frequency domain, so the relationship between the external force (reaction force) that vibrates the object, such as a viscoelastic damper, and the behavior (displacement) of the object ( When analyzing the response of an object whose dynamic characteristics are frequency-dependent and nonlinear strain-dependent, in order to consider the effects of frequency-dependent and nonlinear strain-dependent, the response analysis is performed in the time domain. It is necessary to perform (time history response analysis), and in order to perform time history response analysis, it is necessary to obtain an impulse response that represents the dynamic characteristics of the object to be analyzed in the time domain.

本願発明者は、前述の非特許文献４に記載の技術を適用すれば、解析対象の物体の動的特性が周波数依存性を示している場合にも、解析対象の物体の動的特性を周波数領域で表す動的剛性を、前記物体の動的特性を時間領域で精度良く表すインパルス応答（周波数依存性が反映されたインパルス応答）を得られることに着目し、非特許文献４で提案されている、インパルス応答を用いて反力Ｆ(t)を規定する数式を、解析対象の物体の動的特性における非線形の歪依存性を表現可能に変形して時刻歴応答解析に用いれば、解析対象の物体の動的特性における周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を簡便に行うことができることに想到して本願発明を成すに至った。   By applying the technique described in Non-Patent Document 4 described above, the inventor of the present application converts the dynamic characteristic of the object to be analyzed into the frequency even when the dynamic characteristic of the object to be analyzed exhibits frequency dependence. Focusing on the fact that an impulse response (impulse response reflecting the frequency dependence) that accurately represents the dynamic stiffness expressed in the region can be obtained in the time domain, the dynamic stiffness of the object is proposed in Non-Patent Document 4. If the mathematical expression that defines the reaction force F (t) using the impulse response is transformed to express the nonlinear strain dependence in the dynamic characteristics of the object to be analyzed and used for time history response analysis, The present invention has been made in view of the fact that it is possible to easily perform a highly accurate time history response analysis in consideration of the frequency dependence and nonlinear distortion dependence in the dynamic characteristics of the object.

上記に基づき請求項１記載の発明に係る時刻歴応答解析方法は、物体を振動させる外力と前記物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析方法であって、前記物体の反力Ｆ(t)を規定する数式として、 Based on the above, the time history response analysis method according to the first aspect of the present invention performs time history response analysis of an object in which the relationship between the external force that vibrates the object and the behavior of the object exhibits frequency dependence and nonlinear distortion dependence. A time history response analysis method to be performed, which is a mathematical formula that defines the reaction force F (t) of the object,

上記(１)式を用い、前記振動の角振動数をωとしたときに、上記(１)式に基づき、前記物体の動的剛性Ｓ(ω)を規定する数式として、When the above equation (1) is used and the angular frequency of the vibration is ω, based on the above equation (1), the mathematical equation defining the dynamic stiffness S (ω) of the object is as follows:

上記(２)式を用い、物体の動的剛性のデータから、前記振動がＮ種の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ωUsing the above equation (2), N complex data S (ω representing the value of the dynamic stiffness when the vibration has N frequencies from the data of the dynamic stiffness of the object. １1 ),…,Ｓ(ω), ..., S (ω ＮN )を抽出し、前記物体の前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(t) And the simultaneous component depending on the displacement of the object among the impulse responses representing the relationship of the objects in the time domain is k (t ０0 )、物体の速度に依存する同時成分をｃ(t), And c (t ０0 )、物体の加速度に依存する同時成分をｍ(t), M (t ０0 )、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(t), The time delay component in increments of Δt depending on the displacement of the object is k (t ｊj )、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(t), The time delay component in increments of Δt depending on the speed of the object is c (t ｊj )（但し、ｊは自然数でt) (Where j is a natural number t ｊj ＝Δt・j）とし、時間領域での物体の変位をｕ(t)、速度をｕ'(t)、加速度をｕ"(t)、物体の変位より求まる物体の歪をγ(t)、物体の歪に依存する物体の剛性低下率をα(γ(t))としたときに、抽出したＮ個の複素データを= Δt · j), the displacement of the object in the time domain is u (t), the velocity is u '(t), the acceleration is u "(t), the distortion of the object obtained from the displacement of the object is γ (t), The extracted N complex data is expressed as α (γ (t)), which is the stiffness reduction rate of the object that depends on the distortion of the object.

前記(１)式及び(２)式から導出される上記(３)式及び(４)式へ代入して演算することで、前記インパルス応答の同時成分ｋ(tThe simultaneous component k (t of the impulse response is calculated by substituting and calculating in the above equations (3) and (4) derived from the equations (1) and (2). ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を各々求め、或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、), And the object displacement u (t), velocity u ′ (t) and acceleration u ″ (t) at a certain time are assumed, and the stiffness reduction rate α (γ (t (t ))

インパルス応答の同時成分ｋ(tSimultaneous component k (t of impulse response ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を代入した上記 (５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、前記物体の時刻歴応答解析を行うことを特徴としている。Substituting the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)) into the above equation (5). When the deviation between the calculated reaction force F ′ (t) and the external force at the certain time is out of the allowable range, the assumed displacement u (t), By correcting the velocity u ′ (t) and the acceleration u ″ (t) and calculating the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), the displacement u at the certain time is repeated. (t), velocity u ′ (t), acceleration u ″ (t), and reaction force F ′ (t) are sequentially obtained for each time in increments of Δt, thereby performing time history response analysis of the object. It is characterized by that.

請求項１記載の発明では、前記物体の反力Ｆ(t)を規定する数式として前記(１)式を用い、前記振動の角振動数をωとしたときに、前記(１)式に基づき、前記物体の動的剛性Ｓ(ω)を規定する数式として前記(２)式を用い、物体の動的剛性のデータから、前記振動がＮ種の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ωIn the first aspect of the present invention, when the equation (1) is used as an equation for defining the reaction force F (t) of the object, and the angular frequency of the vibration is ω, the equation (1) is used. The equation (2) is used as a formula for defining the dynamic stiffness S (ω) of the object, and the value of the dynamic stiffness when the vibration has N kinds of frequencies is expressed from the dynamic stiffness data of the object. N complex data S (ω １1 ),…,Ｓ(ω), ..., S (ω ＮN )を抽出し、抽出したＮ個の複素データを前記(１)式及び(２)式から導出される前記 (３)式及び(４)式へ代入して演算することで、インパルス応答の同時成分ｋ(t) And subtracting the extracted N complex data into the equations (3) and (4) derived from the equations (1) and (2), and calculating Component k (t ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を各々求めている。物体の動的剛性のデータから抽出したＮ個の複素データを上記の(３)式及び(４)式へ代入してインパルス応答の同時成分ｋ(t) Each. N complex data extracted from the dynamic stiffness data of the object is substituted into the above equations (3) and (4), and the simultaneous component k (t ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を各々演算することは、前述の非特許文献４に記載の技術を適用してインパルス応答を求めることに相当し、これにより物体の動的特性における周波数依存性が考慮されたインパルス応答が得られる。) Is equivalent to obtaining an impulse response by applying the technique described in Non-Patent Document 4 described above, thereby obtaining an impulse response in consideration of frequency dependence in the dynamic characteristics of the object. It is done.

また、請求項１記載の発明では、或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、According to the first aspect of the present invention, the object displacement u (t), the velocity u ′ (t) and the acceleration u ″ (t) at a certain time are assumed, and the rigidity reduction rate of the object is calculated from the assumed object displacement. Calculate α (γ (t))
インパルス応答の同時成分ｋ(tSimultaneous component k (t of impulse response ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を代入した前記(５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、物体の時刻歴応答解析を行う。) Is substituted for the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)) into the equation (5). When the deviation between the calculated reaction force F ′ (t) and the external force at the certain time is out of the allowable range, the assumed displacement u (t), By correcting the velocity u ′ (t) and the acceleration u ″ (t) and calculating the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), the displacement u at the certain time is repeated. (t), velocity u ′ (t), acceleration u ″ (t), and reaction force F ′ (t) are sequentially obtained for each time in increments of Δt, thereby performing time history response analysis of the object.

上記の(５)式は、非特許文献４で提案されている、インパルス応答を用いて反力Ｆ(t)を規定する数式の右辺に、物体の動的特性（物体を振動させる外力と物体の挙動との関係）における非線形の歪依存性を表現する係数、すなわち物体の歪に依存する物体の剛性低下率α(γ(t))を乗じた数式であり、この剛性低下率α(γ(t))によって物体の動的特性における非線形の歪依存性を表現することができる。請求項１記載の発明では、物体を振動させる外力と物体の挙動との関係（動的特性）が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行うにあたり、上記の(５)式を用いると共に、 (５)式に代入するインパルス応答の同時成分ｋ(tThe above equation (5) is based on the dynamic characteristics of the object (external force and object that vibrate the object and the object) on the right side of the mathematical expression that defines the reaction force F (t) using the impulse response proposed in Non-Patent Document 4. This is a mathematical expression obtained by multiplying the coefficient representing the nonlinear strain dependency in the relationship of the behavior of the object, that is, the stiffness reduction rate α (γ (t)) of the object depending on the strain of the object, and this stiffness reduction rate α (γ (t)) can represent nonlinear strain dependence in the dynamic properties of an object. In the invention according to claim 1, in performing time history response analysis of an object in which the relationship (dynamic characteristics) between the external force that vibrates the object and the behavior of the object shows frequency dependence and nonlinear distortion dependence, 5) and simultaneous component k (t of impulse response to be substituted into equation (5) ０0 )、ｃ(t), C (t ０0 )、ｍ(t), M (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj )、ｃ(t), C (t ｊj )として、前述の非特許文献４に記載の技術を適用して求めたインパルス応答（物体の動的特性における周波数依存性が考慮されたインパルス応答）の各成分を用いているので、周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を簡便に行うことができる。), Each component of the impulse response (impulse response in which the frequency dependence in the dynamic characteristics of the object is taken into account) obtained by applying the technique described in Non-Patent Document 4 is used. In addition, it is possible to easily perform a highly accurate time history response analysis taking into account nonlinear distortion dependence.

また請求項１記載の発明は、時刻歴応答解析対象の物体の動的特性における周波数依存性及び非線形の歪依存性を考慮した時刻歴応答解析を行うにあたり、時刻歴応答解析対象の物体を一般化マックスウェル要素等の物理モデルへ置き換える（モデル化する）必要もなく、上述の(５)式を用いて前述の演算を行うことで、時刻歴応答解析における解析結果としてのΔt刻みの物体の反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を一意にかつ高精度に求めることができるので、手間がかかりかつ熟練を要する物理モデルの生成作業（物理モデルの各要素の定数を同定する作業）も不要になる。従って、請求項１記載の発明によれば、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体について、周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を容易に行うことができる。According to the first aspect of the present invention, in performing time history response analysis in consideration of frequency dependence and nonlinear distortion dependence in dynamic characteristics of an object to be analyzed for time history response, the object to be analyzed for time history response is generally It is not necessary to replace (model) with a physical model such as a modified Maxwell element, and by performing the above-described calculation using the above-described equation (5), an object in Δt increments as an analysis result in the time history response analysis can be obtained. Since the reaction force F '(t), displacement u (t), velocity u' (t) and acceleration u "(t) can be determined uniquely and with high accuracy, it is time-consuming and requires a skilled physical model. Therefore, the generation work (the work of identifying the constants of each element of the physical model) is also unnecessary.According to the invention described in claim 1, the relationship between the external force that vibrates the object and the behavior of the object is frequency-dependent and nonlinear. For objects that show distortion dependence, The precise time history analysis, each considering strain dependence of the dependence and nonlinearity can be easily performed.

また、請求項１記載の発明において、時刻歴応答解析対象の物体は、動的特性が周波数依存性及び非線形の歪依存性を示す物体であればよいが、前記物体としては、例えば請求項２に記載したように粘弾性体が封入された粘弾性ダンパーが好適である。物体の剛性や減衰は物体の歪に応じて変化するが、粘弾性ダンパーに封入された粘弾性体は、物体の歪の変化に対して物体の減衰率が略一定となる、すなわち減衰が剛性と同様の非線形的変化を示す特性であることが多い。また粘弾性体は、歪の変化に対し、動的特性における周波数依存性の変化が比較的小さい特性であることが多い。前出の(５)式は、特に上記のような特性を示す物体の動的特性における非線形の歪依存性を精度良く表現できるので、本発明を適用することで、上記のような特性を示す物体、例えば粘弾性ダンパーの時刻歴応答解析を特に高精度に行うことができる。 Further, in the invention according to the first SL placing, the object of the time history analysis object may but if the object dynamic characteristic exhibits strain dependence of the frequency-dependent and non-linear, as the object, for example, claims As described in 2 , a viscoelastic damper in which a viscoelastic body is enclosed is suitable. The stiffness and damping of an object change according to the strain of the object, but the damping rate of the viscoelastic body enclosed in the viscoelastic damper is substantially constant with respect to the change of the strain of the object, that is, the damping is rigid. In many cases, the characteristic exhibits a non-linear change. In many cases, viscoelastic bodies have relatively small frequency-dependent changes in dynamic characteristics with respect to changes in strain. The above equation ( 5 ) can express the nonlinear distortion dependency in the dynamic characteristics of the object exhibiting the characteristics as described above with high accuracy. Therefore, by applying the present invention, the characteristics as described above are exhibited. Time history response analysis of an object such as a viscoelastic damper can be performed with particularly high accuracy.

請求項３記載の発明に係る時刻歴応答解析装置は、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析装置であって、前記関係を周波数領域で表す動的剛性のデータから前記振動がＮ種（Ｎ＝ｎ＋１）の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ω１),…,Ｓ(ωＮ)を抽出する抽出手段と、前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(t０)、物体の速度に依存する同時成分をｃ(t０)、物体の加速度に依存する同時成分をｍ(t０)、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(tｊ)、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(tｊ)（但し、ｊは自然数でtｊ＝Δt・j）としたときに、前記抽出手段によって抽出されたＮ個の複素データを、 According to a third aspect of the present invention, there is provided a time history response analyzing apparatus for performing a time history response analysis of an object in which a relationship between an external force that vibrates the object and the behavior of the object exhibits frequency dependence and nonlinear distortion dependence. N analysis data S (ω 1 ) representing dynamic stiffness values when the vibration has N kinds of frequencies (N = n + 1) from dynamic stiffness data representing the relationship in the frequency domain. ),..., S (ω N ), and an impulse response representing the relationship in the time domain, the simultaneous component depending on the displacement of the object is k (t 0 ), and the simultaneous response depending on the velocity of the object. The component is c (t 0 ), the simultaneous component that depends on the acceleration of the object is m (t 0 ), the time delay component of Δt that depends on the displacement of the object is k (t j ), and the Δt that depends on the speed of the object When the time delay component of c (t j ) (where j is a natural number and t j = Δt · j) is extracted, N complex data extracted by the means,

上記(３)式及び(４)式へ代入して演算することで、インパルス応答の同時成分ｋ(t０),ｃ(t０),ｍ(t０)及び時間遅れ成分ｋ(tｊ),ｃ(tｊ)を各々求める第１演算手段と、或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、 By calculating by substituting into the above equations ( 3 ) and ( 4 ), simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay component k (t j ) of the impulse response , c (t j ), respectively, and the object displacement u (t), velocity u ′ (t) and acceleration u ″ (t) at a certain time are assumed, and from the assumed object displacement, Calculate the rigidity reduction rate α (γ (t)) of the object,

インパルス応答の同時成分ｋ(t ０ ),ｃ(t ０ ),ｍ(t ０ )及び時間遅れ成分ｋ(t ｊ ),ｃ(t ｊ )を代入した上記 (５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、前記物体の時刻歴応答解析を行う第２演算手段と、を備えたことを特徴としているので、請求項１記載の発明と同様に、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体について、周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を容易に行うことができる。 The assumed displacement to the above equation (5), which substitutes the simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j ) of the impulse response Substituting u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)), the reaction force F ′ (t) of the object is calculated and calculated. When the deviation between the reaction force F ′ (t) and the external force at a certain time is out of the allowable range, the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) are set as follows. The correction is repeated to calculate the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), and the displacement u (t), velocity u ′ (t), acceleration u ″ at the certain time. (t) and reaction force F ′ (t) are obtained for each time in increments of Δt, thereby providing second calculation means for performing time history response analysis of the object . Therefore, as in the first aspect of the invention, the relationship between the external force that vibrates the object and the behavior of the object. There the object showing the strain dependence of the frequency-dependent and non-linear, the precise time history analysis, each considering strain dependence of the frequency-dependent and non-linear can be easily performed.

請求項４記載の発明に係る時刻歴応答解析プログラムは、コンピュータを、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析装置として機能させる時刻歴応答解析プログラムであって、前記コンピュータを、前記関係を周波数領域で表す動的剛性のデータから前記振動がＮ種（Ｎ＝ｎ＋１）の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ω１),…,Ｓ(ωＮ)を抽出する抽出手段、前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(t０)、物体の速度に依存する同時成分をｃ(t０)、物体の加速度に依存する同時成分をｍ(t０)、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(tｊ)、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(tｊ)（但し、ｊは自然数でtｊ＝Δt・j）としたときに、前記抽出手段によって抽出されたＮ個の複素データを、 According to a fourth aspect of the present invention, there is provided a time history response analysis program that performs a time history response analysis of an object in which the relationship between an external force that vibrates the object and the behavior of the object exhibits frequency dependence and nonlinear distortion dependence. A time history response analysis program for causing a computer to function as a time history response analysis device, wherein the computer is operated when the vibration has N types of frequencies (N = n + 1) from dynamic stiffness data representing the relationship in the frequency domain. Extraction means for extracting N pieces of complex data S (ω 1 ),..., S (ω N ) representing the value of the mechanical stiffness, and the simultaneous component depending on the displacement of the object among the impulse responses representing the relationship in the time domain K (t 0 ), c (t 0 ) the simultaneous component that depends on the velocity of the object, m (t 0 ) the simultaneous component that depends on the acceleration of the object, and the time delay component in increments of Δt that depends on the displacement of the object k (t j ), Δt increment depending on the velocity of the object When the only time delay component is c (t j ) (where j is a natural number, t j = Δt · j), the N complex data extracted by the extracting means are

上記(３)式及び(４)式へ代入して演算することで、インパルス応答の同時成分ｋ(t０),ｃ(t０),ｍ(t０)及び時間遅れ成分ｋ(tｊ),ｃ(tｊ)を各々求める第１演算手段、及び、或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、 By calculating by substituting into the above equations ( 3 ) and ( 4 ), simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay component k (t j ) of the impulse response , c (t j ), respectively, and the object displacement u (t), velocity u ′ (t) and acceleration u ″ (t) at a certain time are assumed and the assumed object displacement is calculated. Calculate the stiffness reduction rate α (γ (t)) of the object from

インパルス応答の同時成分ｋ(t ０ ),ｃ(t ０ ),ｍ(t ０ )及び時間遅れ成分ｋ(t ｊ ),ｃ(t ｊ )を代入した上記 (５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、前記物体の時刻歴応答解析を行う第２演算手段として機能させることを特徴としている。 The assumed displacement to the above equation (5), which substitutes the simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j ) of the impulse response Substituting u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)), the reaction force F ′ (t) of the object is calculated and calculated. When the deviation between the reaction force F ′ (t) and the external force at a certain time is out of the allowable range, the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) are set as follows. The correction is repeated to calculate the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), and the displacement u (t), velocity u ′ (t), acceleration u ″ at the certain time. (t) and reaction force F ′ (t) are obtained sequentially for each time in increments of Δt, thereby functioning as a second computing means for performing time history response analysis of the object .

請求項４記載の発明に係る時刻歴応答解析プログラムは、コンピュータを、上記の抽出手段、第１演算手段及び第２演算手段として機能させるためのプログラムであるので、コンピュータが請求項６記載の発明に係る時刻歴応答解析プログラムを実行することにより、コンピュータが請求項３に記載の時刻歴応答解析装置として機能することになり、請求項１及び請求項３記載の発明と同様に、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体について、周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を容易に行うことができる。 Since the time history response analysis program according to the invention described in claim 4 is a program for causing the computer to function as the extraction means, the first calculation means, and the second calculation means, the computer according to claim 6. by performing the time history response analysis program according to the results in the computer functions as the time history analysis apparatus according to claim 3, similarly to the invention of claim 1 and claim 3, wherein the vibration of the object It is possible to easily perform highly accurate time history response analysis that considers frequency dependence and nonlinear distortion dependence for an object whose relationship between the external force and the behavior of the object shows frequency dependence and nonlinear distortion dependence. it can.

以上説明したように本発明は、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を、物体の前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(t0)、物体の速度に依存する同時成分をｃ(t0)、物体の加速度に依存する同時成分をｍ(t0)、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(tj)、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(tj)（但し、ｊは自然数でtj＝Δt・j）とし、時間領域での物体の変位をｕ(t)、速度をｕ'(t)、加速度をｕ"(t)、物体の変位より求まる物体の歪をγ(t)、物体の歪に依存する物体の剛性低下率をα(γ(t))としたときに、 As described above, the present invention represents the time history response analysis of an object in which the relationship between the external force that vibrates the object and the behavior of the object shows frequency dependence and nonlinear distortion dependence, and the relation of the object in the time domain. Of the impulse response, the simultaneous component that depends on the displacement of the object is k (t 0 ), the simultaneous component that depends on the velocity of the object is c (t 0 ), the simultaneous component that depends on the acceleration of the object is m (t 0 ), The time delay component in increments of Δt depending on the displacement of the object is k (t j ), and the time delay component in increments of Δt depending on the velocity of the object is c (t j ) (where j is a natural number and t j = Δt · j ), The displacement of the object in the time domain is u (t), the velocity is u '(t), the acceleration is u "(t), the distortion of the object obtained from the displacement of the object is γ (t), and the distortion of the object When the rigidity reduction rate of the dependent object is α (γ (t)),

上記(５)式を用い、物体の反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)をΔt刻みで順次演算することにより行うので、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体について、周波数依存性及び非線形の歪依存性を各々考慮した高精度な時刻歴応答解析を容易に行うことができる、という優れた効果を有する。 Since the above equation (5) is used to calculate the reaction force F ′ (t), displacement u (t), velocity u ′ (t) and acceleration u ″ (t) of the object in increments of Δt, the object Easily perform highly accurate time history response analysis considering the frequency dependence and nonlinear distortion dependence of the object whose relationship between the external force that vibrates the object and the behavior of the object shows frequency dependence and nonlinear distortion dependence It has an excellent effect of being able to.

以下、図面を参照して本発明の実施形態の一例を詳細に説明する。図１には本発明を適用可能なパーソナル・コンピュータ（ＰＣ）１０が示されている。ＰＣ１０は、ＣＰＵ１０Ａ、ＲＯＭ１０Ｂ、ＲＡＭ１０Ｃ及び入出力ポート１０Ｄが、データバス、制御バス、アドレスバス等から成るバス１０Ｅを介して互いに接続されて構成されている。また入出力ポート１０Ｄには、各種の入出力機器として、ＣＲＴ又はＬＣＤから成るディスプレイ１２、キーボード１４、マウス１６、プリンタ１８、ハードディスクドライブ（ＨＤＤ）２０、ＣＤ−ＲＯＭ２２からの情報の読み出しを行うＣＤ−ＲＯＭドライブ２４が各々接続されている。   Hereinafter, an example of an embodiment of the present invention will be described in detail with reference to the drawings. FIG. 1 shows a personal computer (PC) 10 to which the present invention can be applied. The PC 10 is configured by connecting a CPU 10A, a ROM 10B, a RAM 10C, and an input / output port 10D to each other via a bus 10E including a data bus, a control bus, an address bus, and the like. In addition, the input / output port 10D is a CD 12 that reads information from a display 12, a keyboard 14, a mouse 16, a printer 18, a hard disk drive (HDD) 20, and a CD-ROM 22, each of which is a CRT or LCD, as various input / output devices. -ROM drives 24 are connected to each other.

ＰＣ１０のＨＤＤ２０には、後述する粘弾性ダンパー時刻歴応答解析処理を行うための粘弾性ダンパー時刻歴応答解析プログラムがインストールされている。この粘弾性ダンパー時刻歴応答解析プログラムは、請求項４記載の発明に係る時刻歴応答解析プログラムに対応している。粘弾性ダンパー時刻歴応答解析プログラムをＰＣ１０にインストール（移入）するには幾つかの方法があるが、例えば粘弾性ダンパー時刻歴応答解析プログラムをセットアッププログラムと共にＣＤ−ＲＯＭ２２に記録しておき、該ＣＤ−ＲＯＭ２２をＣＤ−ＲＯＭドライブ２４にセットし、ＣＰＵ１０Ａに対して前記セットアッププログラムの実行を指示すれば、ＣＤ−ＲＯＭ２２から粘弾性ダンパー時刻歴応答解析プログラムが順に読み出され、読み出された粘弾性ダンパー時刻歴応答解析プログラムがＨＤＤ２０に順に書き込まれることで、粘弾性ダンパー時刻歴応答解析プログラムのインストールが行われる。ＰＣ１０は、ＣＰＵ１０Ａが粘弾性ダンパー時刻歴応答解析プログラムを実行することで、請求項３記載の発明に係る時刻歴応答解析装置として機能する。 A viscoelastic damper time history response analysis program for performing later-described viscoelastic damper time history response analysis processing is installed in the HDD 20 of the PC 10. This viscoelastic damper time history response analysis program corresponds to the time history response analysis program according to the invention of claim 4 . There are several methods for installing (transferring) the viscoelastic damper time history response analysis program to the PC 10. For example, the viscoelastic damper time history response analysis program is recorded in the CD-ROM 22 together with the setup program, and the CD is recorded. When the ROM 22 is set in the CD-ROM drive 24 and the CPU 10A is instructed to execute the setup program, the viscoelastic damper time history response analysis program is sequentially read from the CD-ROM 22, and the read viscoelasticity The damper time history response analysis program is sequentially written in the HDD 20, whereby the viscoelastic damper time history response analysis program is installed. The PC 10 functions as a time history response analysis apparatus according to the invention of claim 3 by the CPU 10A executing the viscoelastic damper time history response analysis program.

なお、請求項４に記載のコンピュータはＰＣ１０に限られるものではなく、例えばワークステーションであってもよいし、汎用の大型コンピュータであってもよい。 Note that the computer according to claim 4 is not limited to the PC 10, and may be a workstation or a general-purpose large computer.

次に本実施形態の作用として、粘弾性ダンパーの時刻歴応答解析の実行を所望しているオペレータによってキーボード１４又はマウス１６を介して粘弾性ダンパー時刻歴応答解析プログラムの実行が指示されることで、ＰＣ１０のＣＰＵ１０Ａで実行される粘弾性ダンパー時刻歴応答解析処理について、図２のフローチャートを参照して説明する。   Next, as an operation of the present embodiment, the execution of the viscoelastic damper time history response analysis program is instructed via the keyboard 14 or the mouse 16 by the operator who desires to execute the time history response analysis of the viscoelastic damper. The viscoelastic damper time history response analysis process executed by the CPU 10A of the PC 10 will be described with reference to the flowchart of FIG.

本実施形態では、解析対象の粘弾性ダンパーについて、該粘弾性ダンパーを振動させる外力（反力）と該粘弾性ダンパーの挙動（変位）との関係を周波数領域で表す動的剛性のデータがＨＤＤ２０に予め記憶されており（粘弾性ダンパーの動的剛性の一例を図３に示す）、ステップ１００では、解析対象の粘弾性ダンパーの動的剛性のデータをＨＤＤ２０から読み込むことで取得し、取得したデータをメモリ（ＲＡＭ１０Ｃ）に記憶させる。なお、粘弾性ダンパーの動的剛性は、例えば、粘弾性ダンパーの変位を一定値に固定した状態で、粘弾性ダンパーを振動させる外力の周波数を変化させて反力の変化を測定することを、粘弾性ダンパーの歪を変化させながら繰り返す実験を行って求めることができるが、これに代えて演算によって求めることも可能である。   In the present embodiment, for the viscoelastic damper to be analyzed, dynamic rigidity data representing the relationship between the external force (reaction force) that vibrates the viscoelastic damper and the behavior (displacement) of the viscoelastic damper in the frequency domain is HDD 20. Is stored in advance (an example of the dynamic stiffness of the viscoelastic damper is shown in FIG. 3). In step 100, the dynamic stiffness data of the viscoelastic damper to be analyzed is acquired by reading from the HDD 20 and acquired. Data is stored in the memory (RAM 10C). The dynamic rigidity of the viscoelastic damper is, for example, measuring the change in the reaction force by changing the frequency of the external force that vibrates the viscoelastic damper in a state where the displacement of the viscoelastic damper is fixed to a constant value. Although it can be obtained by performing an experiment repeatedly while changing the strain of the viscoelastic damper, it can also be obtained by calculation instead.

またステップ１０２では、ステップ１００で取得した解析対象の粘弾性ダンパーの動的剛性のデータをメモリから読み出し、読み出した動的剛性のデータから、予め設定された演算対象の周波数範囲内のＮ種の周波数（Ｎ種の角振動数ω1,…,ωN）における動的剛性の値を表すＮ個の複素データＳ(ω1),…,Ｓ(ωN)を各々抽出し、抽出した複素データをメモリ又はＨＤＤ２０に記憶させる。このステップ１０２は本発明に係る抽出手段に対応している。なお、演算対象の周波数範囲としては、例えば０〜20(Hz)の範囲を適用することができる。また、複素データの抽出を行うＮ種の周波数は、例えば演算対象の周波数範囲の上限及び下限に相当する周波数（例えば演算対象の周波数範囲が０〜20(Hz)であれば、上限周波数である20(Hz)及び下限周波数である０(Hz)）を含むように設定することができる。また、解析対象の粘弾性ダンパーの動的剛性のデータから抽出した複素データはインパルス応答の演算に用いられ、この演算により時刻ｔ＝０及び時刻ｔ＝Δt・j（j=1,2,…)の各時刻における粘弾性ダンパーのインパルス応答を表すインパルス応答データが得られるが、得られるインパルス応答データの個数は演算に用いる複素データの個数に応じて定まり（すなわちjの最大値jmax＝複素データの個数−１(＝ｎ))、得られるインパルス応答データによって表される粘弾性ダンパーのインパルス応答の時刻範囲も演算に用いる複素データの個数に応じて定まる（例えば複素データの個数が21個、Δt＝0.05秒とすると、ｔmax＝Δt・jmax＝0.05×20＝１秒となり、時刻ｔ＝０〜１秒の時刻範囲の粘弾性ダンパーのインパルス応答を表す21個のインパルス応答データが得られる）ことになるので、粘弾性ダンパーの動的剛性から抽出する複素データの個数（複素データの抽出を行う周波数の種類数）は、粘弾性ダンパーのインパルス応答を算出すべき時刻範囲の長さも勘案して予め定めておくことができる。 In step 102, the dynamic stiffness data of the viscoelastic damper to be analyzed acquired in step 100 is read from the memory, and N types of frequency within the preset frequency range of the computation target are read from the read dynamic stiffness data. N complex data S (ω 1 ),..., S (ω N ) representing dynamic stiffness values at frequencies (N kinds of angular frequencies ω 1 ,..., Ω N ) are extracted, and the extracted complex Data is stored in the memory or HDD 20. This step 102 corresponds to the extracting means according to the present invention. In addition, as a frequency range of calculation object, the range of 0-20 (Hz) is applicable, for example. The N types of frequencies for extracting the complex data are, for example, frequencies corresponding to the upper and lower limits of the frequency range to be calculated (for example, if the frequency range to be calculated is 0 to 20 (Hz), the upper limit frequency. 20 (Hz) and the lower limit frequency 0 (Hz)) can be set. Further, the complex data extracted from the dynamic stiffness data of the viscoelastic damper to be analyzed is used for impulse response calculation. By this calculation, time t = 0 and time t = Δt · j (j = 1, 2,... ), Impulse response data representing the impulse response of the viscoelastic damper at each time is obtained. The number of impulse response data obtained is determined according to the number of complex data used in the operation (that is, j maximum value jmax = complex data). -1 (= n)), the time range of the impulse response of the viscoelastic damper represented by the obtained impulse response data is also determined according to the number of complex data used in the calculation (for example, the number of complex data is 21, When Δt = 0.05 seconds, tmax = Δt · jmax = 0.05 × 20 = 1 second, and 21 impulse response data representing the impulse response of the viscoelastic damper in the time range of time t = 0 to 1 second are obtained. The number of complex data extracted from the dynamic stiffness of the viscoelastic damper (the number of types of frequencies at which the complex data is extracted) is the time range in which the impulse response of the viscoelastic damper should be calculated. The length can be determined in advance.

次のステップ１０４では、外力（反力）と粘弾性ダンパーの挙動との関係を周波数領域で表す動的剛性を、外力（反力）と粘弾性ダンパーの挙動との関係を時間領域で表すインパルス応答へ変換するための本発明に係る連立方程式（２Ｎ×２Ｎの係数マトリクスを有する前出の(３)式及び(４)式）をＨＤＤ２０から読み出し、読み出した連立方程式に、ステップ１０２で抽出したＮ個の複素データＳ(ω１),…,Ｓ(ωＮ)を代入し、この連立方程式の解を求めることで、粘弾性ダンパーのインパルス応答を規定するデータとして、同時成分ｋ(t０),ｃ(t０),ｍ(t０)を演算すると共に、時間遅れ成分ｋ(tｊ),ｃ(tｊ)を予め設定されたΔｔ刻みで演算する。このステップ１０６は本発明に係る第１演算手段に対応している。 In the next step 104, the dynamic stiffness representing the relationship between the external force (reaction force) and the behavior of the viscoelastic damper in the frequency domain, and the impulse representing the relationship between the external force (reaction force) and the behavior of the viscoelastic damper in the time domain. The simultaneous equations according to the present invention (the above equations ( 3 ) and (4) having a 2N × 2N coefficient matrix) for conversion into responses are read from the HDD 20 and extracted into the read simultaneous equations in step 102 By substituting N complex data S (ω 1 ),..., S (ω N ) and finding the solution of the simultaneous equations, the simultaneous component k (t 0 is used as data defining the impulse response of the viscoelastic damper. ), c (t 0 ), m (t 0 ), and time delay components k (t j ), c (t j ) are calculated in increments of Δt set in advance. This step 106 corresponds to the first computing means according to the present invention.

なお、(３)式及び(４)式は、粘弾性ダンパーの非線形の歪依存性を考慮せずにインパルス応答Ｆ(t)を規定する(１)式と、この(１)式から導出される、粘弾性ダンパーの動的剛性Ｓ(ω)を規定する(２)式に基づいて導出される演算式であり、(３)式及び(４)式を用いた演算により、粘弾性ダンパーのインパルス応答を規定するデータとして、例として図４(Ａ)〜(Ｃ)にも示すように、粘弾性ダンパーの変位に依存するインパルス応答の（剛性項の）同時成分ｋ(t０)、粘弾性ダンパーの速度に依存するインパルス応答の（減衰項の）同時成分ｃ(t０)、粘弾性ダンパーの加速度に依存するインパルス応答の（質量項の）同時成分ｍ(t０)のデータが得られると共に、粘弾性ダンパーの変位に依存するインパルス応答の（剛性項の）時間遅れ成分ｋ(tｊ)のデータがΔt刻みでｎ個（ｎ＝Ｎ−１）得られ、粘弾性ダンパーの速度に依存するインパルス応答の（減衰項の）時間遅れ成分ｃ(tｊ)のデータがΔt刻みでｎ−１個得られることになる。そして、得られた同時成分ｋ(t０),ｃ(t０),ｍ(t０)及び時間遅れ成分ｋ(tｊ),ｃ(tｊ)はメモリに一時記憶されると共に、次のステップ１０６において、粘弾性ダンパーの非線形の歪依存性を考慮した反力Ｆ'(t)の演算式である前出の(５)式へ代入される。 The expressions ( 3 ) and ( 4 ) are derived from the expression ( 1 ) that defines the impulse response F (t) without considering the nonlinear strain dependence of the viscoelastic damper and the expression ( 1 ). Is an arithmetic expression derived based on the equation ( 2 ) that defines the dynamic stiffness S (ω) of the viscoelastic damper, and is calculated by using the equations ( 3 ) and ( 4 ). As data defining the impulse response, as shown in FIGS. 4A to 4C as an example, the simultaneous component k (t 0 ) of the impulse response (stiffness term) depending on the displacement of the viscoelastic damper, the viscosity Data of the simultaneous component c (t 0 ) of the impulse response (damping term) depending on the velocity of the elastic damper and the simultaneous component m (t 0 ) of the impulse response (mass term) depending on the acceleration of the viscoelastic damper are obtained. And the time delay component of the impulse response that depends on the displacement of the viscoelastic damper (t j) data are n in increments Δt (n = N-1) obtained, the data component delay (the damping term) time of the impulse response that depends on the speed of Viscoelastic Damper c (t j) is Delta] t N-1 pieces are obtained in steps. The obtained simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j ) are temporarily stored in the memory and In step 106, the equation ( 5 ), which is an arithmetic expression of the reaction force F ′ (t) considering the nonlinear strain dependence of the viscoelastic damper, is substituted.

また本実施形態では、解析対象の粘弾性ダンパーにおける変位ｕ−反力Ｆの関係を表す変位−反力特性データがＨＤＤ２０に予め記憶されている（粘弾性ダンパーの変位−反力特性の一例を図５(Ａ)に示す）。図５(Ａ)に示す変位−反力特性Ｆは、変位ｕの増加に対する反力Ｆの増加度合いが徐々に低下する特性を示しており、解析対象の粘弾性ダンパーにおける非線形の歪依存性を表している。なお、解析対象の粘弾性ダンパーの非線形の歪依存性を表す変位−反力特性は、例えば、解析対象の粘弾性ダンパーを振動させる外力の周波数を一定値に固定した状態で、解析対象の粘弾性ダンパーの変位を変化させて反力の変化を測定することを、外力の周波数を変化させながら繰り返す実験を行って求めることができる。ステップ１０８では解析対象の粘弾性ダンパーの変位−反力特性のデータをＨＤＤ２０から取り込む。   In this embodiment, displacement-reaction characteristic data representing the relationship of displacement u-reaction force F in the viscoelastic damper to be analyzed is stored in advance in the HDD 20 (an example of the displacement-reaction force characteristic of the viscoelastic damper). (Shown in FIG. 5 (A)). The displacement-reaction force characteristic F shown in FIG. 5 (A) shows a characteristic in which the increase degree of the reaction force F with respect to the increase of the displacement u gradually decreases, and the nonlinear strain dependence in the viscoelastic damper to be analyzed is shown. Represents. Note that the displacement-reaction force characteristic representing the nonlinear strain dependence of the viscoelastic damper to be analyzed is, for example, a state where the frequency of the external force that vibrates the viscoelastic damper to be analyzed is fixed to a constant value. It is possible to obtain the measurement of the change in the reaction force by changing the displacement of the elastic damper by performing an experiment repeatedly while changing the frequency of the external force. In step 108, the displacement-reaction characteristic data of the viscoelastic damper to be analyzed is fetched from the HDD 20.

またステップ１１０では、ステップ１０８で取り込んだ解析対象の粘弾性ダンパーの変位−反力特性のデータに基づき、解析対象の粘弾性ダンパーの変位ｕが或る値のときの歪γ及び剛性低下率αを演算することを、変位−反力特性のデータにおける変位ｕの全範囲に亘って各々行うことで、解析対象の粘弾性ダンパーにおける歪γと剛性低下率αとの関係(一例を図５(Ｂ)に示す)を求める。解析対象の粘弾性ダンパーの変位ｕ＝ｕ１のときの剛性低下率α１は、例えば解析対象の粘弾性ダンパーの変位ｕ１における解析対象の粘弾性ダンパーの変位−反力特性の変化の傾きｋ１を求め、求めた傾きｋ１を解析対象の粘弾性ダンパーの変位ｕ＝０における変位−反力特性の変化の傾きｋ０で除すことで求めることができる（次の(６)式参照）。
剛性低下率α１＝ｋ１／ｋ０ …(６)
ステップ１１０では、上記の処理を行うことで得られた解析対象の粘弾性ダンパーの歪γ−剛性低下率α特性を表すデータをメモリに一旦記憶させる。 In step 110, the strain γ and the stiffness reduction rate α when the displacement u of the viscoelastic damper to be analyzed is a certain value based on the data of the displacement-reaction characteristics of the viscoelastic damper to be analyzed taken in step 108. Is calculated over the entire range of the displacement u in the displacement-reaction force characteristic data, whereby the relationship between the strain γ and the stiffness reduction rate α in the viscoelastic damper to be analyzed (an example shown in FIG. (Shown in B)) . The stiffness reduction rate α 1 when the displacement u = u 1 of the analysis target viscoelastic damper is, for example, the slope of the change in the displacement-reaction force characteristic of the analysis target viscoelastic damper at the displacement u 1 of the analysis target viscoelastic damper. k 1 is obtained, and the obtained inclination k 1 can be obtained by dividing the obtained inclination k 1 by the inclination k 0 of the change of the displacement-reaction force characteristic at the displacement u = 0 of the analysis target viscoelastic damper (the following equation (6) reference).
Rigidity reduction rate α 1 = k 1 / k 0 (6)
In step 110, data representing the strain γ-stiffness reduction rate α characteristic of the viscoelastic damper to be analyzed obtained by performing the above processing is temporarily stored in the memory.

次のステップ１１２以降では、解析対象の粘弾性ダンパーの時刻歴応答解析を行う。すなわち、ステップ１１２では演算対象時刻ｔを０に初期化する。また、次のステップ１１４では、演算対象時刻ｔにおける解析対象の粘弾性ダンパーの変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を推定する。なお、本実施形態に係る粘弾性ダンパー時刻歴応答解析処理では、解析対象の粘弾性ダンパーの反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を時間Δｔ刻みで順次演算・決定するが、演算対象時刻ｔ＞０における変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)の推定は、演算対象時刻ｔよりも時間Δｔだけ前の時刻について既に演算・決定された反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を用いて行われる。   In step 112 and subsequent steps, a time history response analysis of the viscoelastic damper to be analyzed is performed. That is, in step 112, the calculation target time t is initialized to zero. In the next step 114, the displacement u (t), velocity u ′ (t) and acceleration u ″ (t) of the viscoelastic damper to be analyzed at the calculation target time t are estimated. In the viscoelastic damper time history response analysis processing, the reaction force F ′ (t), displacement u (t), velocity u ′ (t), and acceleration u ″ (t) of the viscoelastic damper to be analyzed are sequentially performed in increments of Δt. Although the calculation / determination is performed, the displacement u (t), the speed u ′ (t) and the acceleration u ″ (t) at the calculation target time t> 0 have already been estimated for a time Δt before the calculation target time t. The calculation is performed using the calculated / determined reaction force F ′ (t), displacement u (t), velocity u ′ (t), and acceleration u ″ (t).

またステップ１１６では、ステップ１１４で推定した演算対象時刻ｔにおける解析対象の粘弾性ダンパーの変位ｕ(t)に基づいて、演算対象時刻ｔにおける解析対象の粘弾性ダンパーの歪γ(t)を演算し、得られた歪γ(t)と先に求めた解析対象の粘弾性ダンパーの歪γ−剛性低下率α特性に基づいて、演算対象時刻ｔにおける解析対象の粘弾性ダンパーの剛性低下率α(γ(t))を求める。そしてステップ１１８では、ステップ１１４で推定した演算対象時刻ｔにおける解析対象の粘弾性ダンパーの変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)及びステップ１１６で演算した演算対象時刻tにおける剛性低下率α(γ(t))を、粘弾性ダンパーの非線形の歪依存性を考慮した反力Ｆ'(t)の演算式である前出の(５)式（先のステップ１０６でインパルス応答の同時成分ｋ(t０),ｃ(t０),ｍ(t０)及び時間遅れ成分ｋ(tｊ),ｃ(tｊ)を代入した後の(５)式）へ代入し、演算対象時刻ｔにおける解析対象の粘弾性ダンパーの反力Ｆ'(t)を演算する。 In step 116, based on the displacement u (t) of the viscoelastic damper to be analyzed at the calculation target time t estimated in step 114, the strain γ (t) of the viscoelastic damper to be analyzed at the calculation target time t is calculated. Then, based on the obtained strain γ (t) and the previously obtained strain γ-stiffness reduction rate α characteristic of the viscoelastic damper to be analyzed, the stiffness reduction rate α of the viscoelastic damper to be analyzed at the calculation target time t Find (γ (t)). In step 118, the displacement u (t), velocity u ′ (t) and acceleration u ″ (t) of the viscoelastic damper to be analyzed at the calculation target time t estimated in step 114 and the calculation target time calculated in step 116. The rigidity reduction rate α (γ (t)) at t is expressed by the above equation ( 5 ) (the previous step 106) which is an arithmetic expression of the reaction force F ′ (t) considering the nonlinear strain dependence of the viscoelastic damper. Then, substitute the simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j ) of the impulse response into ( 5 ))) Then, the reaction force F ′ (t) of the viscoelastic damper to be analyzed at the calculation target time t is calculated.

次のステップ１２０では、演算対象時刻ｔにおける解析対象の粘弾性ダンパーに対する外力を表すデータを取り込む。演算対象時刻ｔにおける外力は、例えば本実施形態に係る時刻歴応答解析が地震動に対する粘弾性ダンパーの時刻歴応答解析であれば、演算対象時刻ｔに粘弾性ダンパーに加わる地震動に相当し、予め想定した地震動のデータから演算対象時刻ｔに粘弾性ダンパーに加わる地震動のデータを抽出することで行うことができる。ステップ１２２では、ステップ１１８で演算した演算対象時刻ｔにおける解析対象の粘弾性ダンパーの反力Ｆ'(t)を、ステップ１２０で取り込んだデータが表す外力と比較し、反力Ｆ'(t)と外力との偏差（反力Ｆ'(t)と外力との釣合の誤差）が許容範囲内か否か判定する。   In the next step 120, data representing an external force applied to the viscoelastic damper to be analyzed at the calculation target time t is captured. For example, if the time history response analysis according to the present embodiment is a time history response analysis of a viscoelastic damper with respect to earthquake motion, the external force at the calculation target time t corresponds to the earthquake motion applied to the viscoelastic damper at the calculation target time t. This can be done by extracting seismic motion data applied to the viscoelastic damper at the calculation target time t from the seismic motion data. In step 122, the reaction force F ′ (t) of the viscoelastic damper to be analyzed at the calculation target time t calculated in step 118 is compared with the external force represented by the data captured in step 120, and the reaction force F ′ (t) It is determined whether or not a deviation between the external force and the external force (an error in balance between the reaction force F ′ (t) and the external force) is within an allowable range.

判定が否定された場合はステップ１１４へ戻り、ステップ１１４において、演算対象時刻ｔでの外力に対する反力Ｆ'(t)の偏差に応じて、先に推定した演算対象時刻ｔにおける解析対象の粘弾性ダンパーの変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正した後に、次のステップ１１６以降の処理を行う。上述したステップ１１４〜１２２の処理は、ステップ１２２の判定が肯定される迄繰り返されるので、演算対象時刻ｔにおける解析対象の粘弾性ダンパーの変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)は、演算対象時刻ｔでの外力に対する反力Ｆ'(t)の偏差を許容範囲内とする値に収束することになる。   If the determination is negative, the process returns to step 114. In step 114, according to the deviation of the reaction force F ′ (t) with respect to the external force at the calculation target time t, the viscosity of the analysis target at the calculation target time t previously estimated is calculated. After correcting the displacement u (t), velocity u ′ (t), and acceleration u ″ (t) of the elastic damper, the following processing from step 116 is performed. The processing of steps 114 to 122 described above is performed in step 122. Since the determination is repeated until the determination is affirmative, the displacement u (t), velocity u ′ (t) and acceleration u ″ (t) of the viscoelastic damper to be analyzed at the calculation target time t are the external forces at the calculation target time t. Therefore, the deviation of the reaction force F ′ (t) with respect to is converged to a value within the allowable range.

ステップ１２２の判定が肯定されるとステップ１２４へ移行し、上記の時刻歴応答解析（各時刻における解析対象の粘弾性ダンパーの反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)の演算）を、時刻歴応答解析対象の時刻範囲の最終時刻迄行ったか（演算対象時刻ｔが最終時刻になったか）否か判定する。判定が否定された場合はステップ１２６へ移行し、演算対象時刻ｔにΔｔを加えることで演算対象時刻ｔを更新してステップ１１４に戻る。これにより、ステップ１２４の判定が肯定される迄ステップ１１４〜１２６が繰り返され、演算対象時刻ｔから時間Δｔ刻みの各時刻について時刻歴応答解析（解析対象の粘弾性ダンパーの反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)の演算）が各々行われることになる。そして、ステップ１２４の判定が肯定されると粘弾性ダンパー時刻歴応答解析処理を終了する。   If the determination in step 122 is affirmed, the routine proceeds to step 124, where the above time history response analysis (reaction force F ′ (t), displacement u (t), displacement u (t), velocity u ′ ( It is determined whether or not (calculation of t) and acceleration u ″ (t)) has been performed until the final time of the time range of the time history response analysis target (the calculation target time t has reached the final time). In this case, the process proceeds to step 126, and Δt is added to the calculation target time t to update the calculation target time t, and the process returns to step 114. Thereby, steps 114 to 126 are repeated until the determination in step 124 is affirmed. , Time history response analysis for each time in increments of Δt from the calculation target time t (reaction force F ′ (t), displacement u (t), velocity u ′ (t) and acceleration u ″ () of the analysis target viscoelastic damper The calculation of t) is performed. If the determination in step 124 is affirmative, the viscoelastic damper time history response analysis process is terminated.

上述した時刻歴応答解析に用いている(５)式は粘弾性ダンパーの歪が変化しても粘弾性ダンパーの周波数依存性が変化しない（動的剛性の実部と虚部の比率や形状が変化しない）と仮定して、粘弾性ダンパーの非線形の歪依存性を考慮した反力Ｆ'(t)を演算する演算式であり、この(５)式を用いることで、粘弾性ダンパーの動的特性における周波数依存性及び非線形の歪依存性を考慮した高精度な時刻歴応答解析を簡便に行うことができる。また、上記の演算では解析対象の粘弾性ダンパーを一般化マックスウェル要素等の物理モデルへモデル化する必要もなく、(５)式等に基づいて各時刻における解析対象の粘弾性ダンパーの反力Ｆ'(t)、変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を一意かつ高精度に求めることができるので、手間がかかりかつ熟練を要する物理モデルの生成作業（物理モデルの各要素の定数を同定する作業）も不要になる。なお、上述したステップ１０６〜１２６は本発明に係る第２演算手段に対応している。 Equation ( 5 ) used for the time history response analysis described above does not change the frequency dependence of the viscoelastic damper even if the strain of the viscoelastic damper changes (the ratio and shape of the real part and the imaginary part of the dynamic stiffness are not changed). It is an arithmetic expression for calculating the reaction force F ′ (t) considering the nonlinear strain dependence of the viscoelastic damper, and using this expression ( 5 ), the motion of the viscoelastic damper is assumed. Highly accurate time history response analysis that takes into account frequency dependence and nonlinear distortion dependence in dynamic characteristics. Further, in the above calculation, it is not necessary to model the viscoelastic damper to be analyzed into a physical model such as a generalized Maxwell element, and the reaction force of the viscoelastic damper to be analyzed at each time based on the equation ( 5 ) or the like. Since F ′ (t), displacement u (t), velocity u ′ (t) and acceleration u ″ (t) can be determined uniquely and with high accuracy, it takes time and labor to generate a physical model that requires skill ( The operation of identifying the constant of each element of the physical model is also unnecessary.Steps 106 to 126 described above correspond to the second computing means according to the present invention.

なお、粘弾性ダンパーの時刻歴応答解析では、ステップ１０４の演算によって得られたインパルス応答のデータ（インパルス応答の同時成分ｋ(t0),ｃ(t0),ｍ(t0)及び時間遅れ成分ｋ(tj),ｃ(tj)）を必ずしも全数用いる必要はなく、解析精度に悪影響を及ぼさない範囲で、時刻歴応答解析に用いる時間遅れ成分のデータの個数をなるべく少なくするようにしてもよい。時刻歴応答解析に用いる時間遅れ成分のデータの個数は、例えば事前に設定しておいてもよいし、粘弾性ダンパーのインパルス応答のデータ（ステップ１０４で演算した同時成分ｋ(t0),ｃ(t0),ｍ(t0)及び時間遅れ成分ｋ(tj),ｃ(tj)のデータ）を周波数領域へ再変換することで粘弾性ダンパーの動的剛性を再現し、先のステップ１００で読み出した動的剛性（元の動的剛性）のデータをメモリ又はＨＤＤ２０から読み出し、再現した動的剛性と元の動的剛性との一致度を演算することを、周波数領域への再変換に用いる時間遅れ成分のデータの個数を変化させながら繰り返すことで、再現した動的剛性が所定値以上の一致度で元の動的剛性と一致する最少の時間遅れ成分のデータの個数を導出するようにしてもよい。これにより、解析対象の粘弾性ダンパーの時刻歴応答解析に際してＰＣ１０（のＣＰＵ１０Ａ）に多大な負荷が加わることも防止することができる。 In the time history response analysis of the viscoelastic damper, impulse response data (simultaneous components k (t 0 ), c (t 0 ), m (t 0 ), and time delay obtained by the calculation in step 104 are used. It is not always necessary to use all of the components k (t j ), c (t j )), and the number of time delay component data used for time history response analysis should be reduced as much as possible within a range that does not adversely affect the analysis accuracy. May be. The number of time delay component data used for the time history response analysis may be set in advance, for example, or the impulse response data of the viscoelastic damper (simultaneous components k (t 0 ), c calculated in step 104) (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j )) are reconverted to the frequency domain to reproduce the dynamic stiffness of the viscoelastic damper, The data of the dynamic stiffness (original dynamic stiffness) read in step 100 is read from the memory or the HDD 20, and the degree of coincidence between the reproduced dynamic stiffness and the original dynamic stiffness is calculated again in the frequency domain. By repeating while changing the number of data of the time delay component used for conversion, the number of data of the minimum time delay component that matches the original dynamic stiffness with the degree of coincidence of the reproduced dynamic stiffness over a predetermined value is derived. You may make it do. Thereby, it is possible to prevent a large load from being applied to the PC 10 (CPU 10A) during the time history response analysis of the viscoelastic damper to be analyzed.

また、上記では予め定められた個数Ｎの複素データを演算対象の動的剛性のデータから抽出する例を説明したが、本発明はこれに限定されるものではない。例えば粘弾性ダンパーの動的剛性が周波数の変化に対して複雑に変化する特性を有している等の場合には、演算対象の動的剛性のデータから抽出する複素データの個数を増加させた後に、インパルス応答（データ）の演算を再度行うようにしてもよい。   In the above description, an example in which a predetermined number N of complex data is extracted from dynamic rigidity data to be calculated has been described. However, the present invention is not limited to this. For example, when the dynamic stiffness of the viscoelastic damper has a characteristic that changes complicatedly with changes in frequency, the number of complex data extracted from the dynamic stiffness data to be calculated is increased. Later, the impulse response (data) may be calculated again.

また、上記では物体のインパルス応答の演算に用いる(１)式においてＮ＝ｎ＋１としていたが、本発明はこれに限定されるものではなく、Ｎ＞ｎ＋１とし、未知数の数より方程式の数が多い連立方程式を立て、最小二乗法等を適用して解くことでインパルス応答を求めることも可能である。 In the above, N = n + 1 in the expression ( 1 ) used for the calculation of the impulse response of the object. However, the present invention is not limited to this, and N> n + 1, and the number of equations is larger than the number of unknowns. It is also possible to obtain an impulse response by setting simultaneous equations and solving by applying the least square method or the like.

更に、上記では物体の反力を規定する数式として、減衰項の時間遅れ成分（第４項）をj=1〜(n-1)の期間に亘って積算すると共に、剛性項の時間遅れ成分（第５項）をj=1〜nの期間に亘って積算する(５),(１)式を用いていたが、これに限定されるものではなく、物体の反力を規定する数式として、以下の(７),(８)式に示すように、減衰項の時間遅れ成分と剛性項の時間遅れ成分の積算期間を逆にした数式を用い、この(７),(８)式と、(８)式から導出される物体の動的剛性を規定する次の(９)式に基づいて、物体のインパルス応答の演算や時刻歴応答解析を行うようにしてもよい。 Further, in the above description, as a mathematical expression for defining the reaction force of the object, the time delay component (fourth term) of the attenuation term is integrated over a period of j = 1 to (n-1), and the time delay component of the stiffness term. ( 5 ) and ( 1 ) are used to integrate (5th term) over a period of j = 1 to n. However, the present invention is not limited to this. As shown in the following equations (7) and (8), this equation (7), (8) and the equation obtained by reversing the integration period of the time delay component of the attenuation term and the time delay component of the stiffness term are Based on the following equation (9) that defines the dynamic stiffness of the object derived from the equation (8), the calculation of the impulse response of the object and the time history response analysis may be performed.

また、上記では物体の反力を規定する数式として、同時成分（第１項）のみから成る質量項を含む(５),(１)式を用いていたが、これに限定されるものではなく、物体の反力を規定する数式として、以下の(10),(11)式に示すように、同時成分（第１項）と時間遅れ成分（第４項）から成る質量項を含む数式を用い（なお、(10),(11)式において、２Ｎ＝ｎ１＋ｎ２＋ｎ３＋３）、この(10),(11)式と、(11)式から導出される物体の動的剛性を規定する次の(12)式に基づいて、物体のインパルス応答の演算や時刻歴応答解析を行うようにしてもよい。 In the above description, the formulas ( 5 ) and ( 1 ) including the mass term consisting only of the simultaneous components (first term) are used as the formulas for defining the reaction force of the object. However, the formula is not limited to this. As an equation that defines the reaction force of an object, as shown in the following equations (10) and (11), an equation including a mass term composed of a simultaneous component (first term) and a time delay component (fourth term) Used (in formulas (10) and (11), 2N = n 1 + n 2 + n 3 +3), and the dynamic stiffness of the object derived from these formulas (10) and (11) and formula (11) The calculation of the impulse response of the object and the time history response analysis may be performed based on the following defined expression (12).

本発明は、上記の(７)〜(９)式又は(10)〜(12)式を用いて物体のインパルス応答の演算や時刻歴応答解析を行うことも権利範囲に含むものである。 The scope of the present invention also includes the calculation of the impulse response of the object and the time history response analysis using the above equations (7) to (9) or (10) to (12).

また、上記では粘弾性ダンパーの動的剛性から本発明を適用して粘弾性ダンパーのインパルス応答を求め、求めたインパルス応答を粘弾性ダンパーの時刻歴応答解析に用いる例を説明したが、これに限定されるものではなく、例えば地盤等のように、動的特性が周波数依存性及び非線形の歪依存性を示す他の物体の動的剛性をインパルス応答へ変換する際に本発明を適用することも可能であることは言うまでもない。   In the above description, the present invention is applied from the dynamic rigidity of the viscoelastic damper to obtain the impulse response of the viscoelastic damper, and the example in which the obtained impulse response is used for the time history response analysis of the viscoelastic damper has been described. The present invention is applied when converting the dynamic stiffness of other objects whose dynamic characteristics are frequency-dependent and nonlinear strain-dependent, such as the ground, into impulse response. It goes without saying that it is possible.

本実施形態に係るＰＣの概略構成を示すブロック図である。It is a block diagram which shows schematic structure of PC concerning this embodiment. 粘弾性ダンパー時刻歴応答解析処理の内容を示すフローチャートである。It is a flowchart which shows the content of a viscoelastic damper time history response analysis process. 粘弾性ダンパーの動的剛性の一例を示す線図である。It is a diagram which shows an example of the dynamic rigidity of a viscoelastic damper. 粘弾性ダンパーのインパルス応答を規定する同時成分ｋ(t0),ｃ(t0),ｍ(t0)及び時間遅れ成分ｋ(tj),ｃ(tj)の演算結果の一例を示す線図である。An example of calculation results of simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j ) that define the impulse response of the viscoelastic damper FIG. (Ａ)は粘弾性ダンパーの変位−反力特性の一例、(Ｂ)は歪−剛性低下率特性の一例を各々示す線図である。(A) is a diagram showing an example of a displacement-reaction force characteristic of a viscoelastic damper, and (B) is a diagram showing an example of a strain-rigidity reduction rate characteristic. 一般化マックスウェル要素の一例を示す概念図である。It is a conceptual diagram which shows an example of a generalized Maxwell element.

符号の説明Explanation of symbols

１０ ＰＣ
１２ ディスプレイ
１４ キーボード
１６ マウス
２０ ＨＤＤ 10 PC
12 Display 14 Keyboard 16 Mouse 20 HDD

Claims (4)

物体を振動させる外力と前記物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析方法であって、
前記物体の反力Ｆ(t)を規定する数式として、

上記(１)式を用い、前記振動の角振動数をωとしたときに、上記(１)式に基づき、前記物体の動的剛性Ｓ(ω)を規定する数式として、

上記(２)式を用い、物体の動的剛性のデータから、前記振動がＮ種の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ω １ ),…,Ｓ(ω Ｎ )を抽出し、
前記物体の前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(t ０ )、物体の速度に依存する同時成分をｃ(t ０ )、物体の加速度に依存する同時成分をｍ(t ０ )、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(t ｊ )、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(t ｊ )（但し、ｊは自然数でt ｊ ＝Δt・j）とし、時間領域での物体の変位をｕ(t)、速度をｕ'(t)、加速度をｕ"(t)、物体の変位より求まる物体の歪をγ(t)、物体の歪に依存する物体の剛性低下率をα(γ(t))としたときに、抽出したＮ個の複素データを

前記(１)式及び(２)式から導出される上記(３)式及び(４)式へ代入して演算することで、前記インパルス応答の同時成分ｋ(t ０ ),ｃ(t ０ ),ｍ(t ０ )及び時間遅れ成分ｋ(t ｊ ),ｃ(t ｊ )を各々求め、
或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、

インパルス応答の同時成分ｋ(t ０ ),ｃ(t ０ ),ｍ(t ０ )及び時間遅れ成分ｋ(t ｊ ),ｃ(t ｊ )を代入した上記 (５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、前記物体の時刻歴応答解析を行うことを特徴とする時刻歴応答解析方法。 A time history response analysis method for performing a time history response analysis of an object in which a relationship between an external force that vibrates an object and the behavior of the object exhibits frequency dependence and nonlinear distortion dependence,
As a formula defining the reaction force F (t) of the object,

When the above equation (1) is used and the angular frequency of the vibration is ω, based on the above equation (1), the mathematical equation defining the dynamic stiffness S (ω) of the object is as follows:

Using the above equation (2), N complex data S (ω 1 ),..., S (ω representing the value of the dynamic stiffness when the vibration has N frequencies from the data of the dynamic stiffness of the object. N )),
Of the impulse response representing the relationship of the object in the time domain, the simultaneous component depending on the displacement of the object is k (t 0 ), the simultaneous component depending on the velocity of the object is c (t 0 ), and depends on the acceleration of the object M (t 0 ), the time delay component in increments of Δt depending on the displacement of the object k (t j ), and the time delay component in increments of Δt depending on the velocity of the object c (t j ) (where j is a natural number, t j = Δt · j), the displacement of the object in the time domain is u (t), the velocity is u ′ (t), the acceleration is u ″ (t), and the object distortion is obtained from the displacement of the object Is the extracted N complex data, where γ (t) is the stiffness reduction rate of the object that depends on the distortion of the object, and α (γ (t)).

The simultaneous components k (t 0 ), c (t 0 ) of the impulse response are calculated by substituting into the above expressions (3) and (4) derived from the expressions (1) and (2 ). , m (t 0 ) and time delay components k (t j ), c (t j )
Assuming the object displacement u (t), velocity u '(t) and acceleration u "(t) at a certain time, the object stiffness reduction rate α (γ (t)) is calculated from the assumed object displacement. ,

The assumed displacement to the above equation (5), which substitutes the simultaneous components k (t 0 ), c (t 0 ), m (t 0 ) and time delay components k (t j ), c (t j ) of the impulse response Substituting u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)), the reaction force F ′ (t) of the object is calculated and calculated. When the deviation between the reaction force F ′ (t) and the external force at a certain time is out of the allowable range, the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) are set as follows. The correction is repeated to calculate the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), and the displacement u (t), velocity u ′ (t), acceleration u ″ at the certain time. A time history response analysis method for performing time history response analysis of the object by sequentially obtaining (t) and reaction force F ′ (t) for each time in increments of Δt . 前記物体は粘弾性体が封入された粘弾性ダンパーであることを特徴とする請求項１記載の時刻歴応答解析方法。The time history response analysis method according to claim 1, wherein the object is a viscoelastic damper in which a viscoelastic body is enclosed. 物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析装置であって、A time history response analysis device for analyzing a time history response of an object in which the relationship between the external force that vibrates the object and the behavior of the object exhibits frequency dependence and nonlinear distortion dependence,
前記関係を周波数領域で表す動的剛性のデータから前記振動がＮ種（Ｎ＝ｎ＋１）の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ωFrom the dynamic stiffness data representing the relationship in the frequency domain, N complex data S (ω representing the value of dynamic stiffness when the vibration has N frequencies (N = n + 1). １1 ),…,Ｓ(ω), ..., S (ω ＮN )を抽出する抽出手段と、Extraction means for extracting)
前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(tOf the impulse response representing the relationship in the time domain, the simultaneous component depending on the displacement of the object is represented by k (t ０0 )、物体の速度に依存する同時成分をｃ(t), And c (t ０0 )、物体の加速度に依存する同時成分をｍ(t), M (t ０0 )、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(t), The time delay component in increments of Δt depending on the displacement of the object is k (t ｊj )、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(t), The time delay component in increments of Δt depending on the speed of the object is c (t ｊj )（但し、ｊは自然数でt) (Where j is a natural number t ｊj ＝Δt・j）としたときに、前記抽出手段によって抽出されたＮ個の複素データを、= Δt · j), the N complex data extracted by the extracting means

上記(３)式及び(４)式へ代入して演算することで、インパルス応答の同時成分ｋ(tThe simultaneous component k (t of impulse response is calculated by substituting into the above equations (3) and (4). ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を各々求める第１演算手段と、) First calculating means for obtaining each)
或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、Assuming the object displacement u (t), velocity u '(t) and acceleration u "(t) at a certain time, the object stiffness reduction rate α (γ (t)) is calculated from the assumed object displacement. ,

インパルス応答の同時成分ｋ(tSimultaneous component k (t of impulse response ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を代入した上記 (５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、前記物体の時刻歴応答解析を行う第２演算手段と、Substituting the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)) into the above equation (5). When the deviation between the calculated reaction force F ′ (t) and the external force at the certain time is out of the allowable range, the assumed displacement u (t), By correcting the velocity u ′ (t) and the acceleration u ″ (t) and calculating the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), the displacement u at the certain time is repeated. (t), velocity u ′ (t), acceleration u ″ (t), and reaction force F ′ (t) are sequentially obtained for each time in increments of Δt, thereby performing time history response analysis of the object. A second computing means;
を備えたことを特徴とする時刻歴応答解析装置。A time history response analyzing apparatus comprising: コンピュータを、物体を振動させる外力と物体の挙動との関係が周波数依存性及び非線形の歪依存性を示す物体の時刻歴応答解析を行う時刻歴応答解析装置として機能させる時刻歴応答解析プログラムであって、A time history response analysis program that causes a computer to function as a time history response analysis device for performing time history response analysis of an object in which the relationship between the external force that vibrates the object and the behavior of the object has frequency dependence and nonlinear distortion dependence. And
前記コンピュータを、The computer,
前記関係を周波数領域で表す動的剛性のデータから前記振動がＮ種（Ｎ＝ｎ＋１）の周波数のときの動的剛性の値を表すＮ個の複素データＳ(ωFrom the dynamic stiffness data representing the relationship in the frequency domain, N complex data S (ω representing the value of dynamic stiffness when the vibration has N frequencies (N = n + 1). １1 ),…,Ｓ(ω), ..., S (ω ＮN )を抽出する抽出手段、Extraction means for extracting)
前記関係を時間領域で表すインパルス応答のうち、物体の変位に依存する同時成分をｋ(tOf the impulse response representing the relationship in the time domain, the simultaneous component depending on the displacement of the object is represented by k (t ０0 )、物体の速度に依存する同時成分をｃ(t), And c (t ０0 )、物体の加速度に依存する同時成分をｍ(t), M (t ０0 )、物体の変位に依存するΔt刻みの時間遅れ成分をｋ(t), The time delay component in increments of Δt depending on the displacement of the object is k (t ｊj )、物体の速度に依存するΔt刻みの時間遅れ成分をｃ(t), The time delay component in increments of Δt depending on the speed of the object is c (t ｊj )（但し、ｊは自然数でt) (Where j is a natural number t ｊj ＝Δt・j）としたときに、前記抽出手段によって抽出されたＮ個の複素データを、= Δt · j), the N complex data extracted by the extracting means

上記(３)式及び(４)式へ代入して演算することで、インパルス応答の同時成分ｋ(tThe simultaneous component k (t of impulse response is calculated by substituting into the above equations (3) and (4). ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を各々求める第１演算手段、) For each of the first calculation means,
及び、或る時刻における物体の変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を仮定し、仮定した物体の変位から物体の剛性低下率α(γ(t))を演算し、Further, assuming the displacement u (t), velocity u ′ (t) and acceleration u ″ (t) of the object at a certain time, the rigidity reduction rate α (γ (t)) of the object is calculated from the assumed displacement of the object. Operate,

インパルス応答の同時成分ｋ(tSimultaneous component k (t of impulse response ０0 ),ｃ(t), c (t ０0 ),ｍ(t), m (t ０0 )及び時間遅れ成分ｋ(t) And time delay component k (t ｊj ),ｃ(t), c (t ｊj )を代入した上記 (５)式へ前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)と演算した剛性低下率α(γ(t))を代入して物体の反力Ｆ'(t)を演算し、演算した反力Ｆ'(t)と前記或る時刻における外力との偏差が許容範囲から外れている場合には前記仮定した変位ｕ(t)、速度ｕ'(t)及び加速度ｕ"(t)を修正して剛性低下率α(γ(t))及び反力Ｆ'(t)を演算することを繰り返して、前記或る時刻における変位ｕ(t)、速度ｕ'(t)、加速度ｕ"(t)及び反力Ｆ'(t)を求めることを、Δt刻みの各時刻について順次行うことで、前記物体の時刻歴応答解析を行う第２演算手段Substituting the assumed displacement u (t), velocity u ′ (t) and acceleration u ″ (t) and the calculated stiffness reduction rate α (γ (t)) into the above equation (5). When the deviation between the calculated reaction force F ′ (t) and the external force at the certain time is out of the allowable range, the assumed displacement u (t), By correcting the velocity u ′ (t) and the acceleration u ″ (t) and calculating the stiffness reduction rate α (γ (t)) and the reaction force F ′ (t), the displacement u at the certain time is repeated. (t), velocity u ′ (t), acceleration u ″ (t), and reaction force F ′ (t) are sequentially obtained for each time in increments of Δt, thereby performing time history response analysis of the object. Second computing means
として機能させることを特徴とする時刻歴応答解析プログラム。Time history response analysis program characterized by functioning as
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