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フローで考える音振動のモデリング(その1) ~音振動をエネルギー視点で捉える~1Dモデリングの勘所(34)(1/3 ページ)

「1Dモデリング」に関する連載。連載第34回では「フローで考える音振動のモデリング(その1)」と題し、音振動をエネルギー視点で捉える。

» 2024年08月20日 09時00分 公開
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 音振動のモデリングに関しては、さまざまな視点から本連載でも紹介してきた。一方で、音振動はモデリング対象として重要であるにもかかわらず、熱と流れに比べて「1Dモデリング」の方法が確立されていない。一般的には、質点系に置き換える「MCKモデル」が使用されているが、構造物のように質点系への置き換えが適切ではないケースも少なくない。

 そこで、熱と流れをフローで考えることにより、モデリングが容易に、かつ拡張性が向上したのと同じように、音振動に関してもフローで考えることが有効ではないか――。Modelica標準ライブラリ(MSL)では、力をフロー、速度をポテンシャルと定義しているが、これは直感的には理解しづらい。よって、ここではエネルギー(パワー)フローで音振動を考えることにする。今回はその理論的背景と、1Dモデリングへの拡張方法について説明する。

⇒連載バックナンバーはこちら

パワーフローで考える音振動

 冒頭で述べた通り、図1に示すように、力をフロー、速度をポテンシャルと考えて、1Dモデリングを実施する方法は、熱や流れに比べて直感的に理解しにくい。

一般的な一自由度MCKモデルによる振動表現 図1 一般的な一自由度MCKモデルによる振動表現[クリックで拡大]

 そこで、音振動をエネルギー視点で捉え、対象を要素に分割し、要素間のパワーフローを考えることにする。この方法に関しては、本連載でも下記に示すように幾度か紹介している。

 音振動をエネルギー視点で考える方法は、1960年ごろに、SEA(Statistical Energy Analysis:統計的エネルギー解析法)として発案された。それ以来、音振動分野で、実験および解析の両面から専門家を中心に研究が行われ、効果を上げている。

 一方で、一般の音振動技術者にとって、その理論が分かりにくい(難しいというわけではない)ことから、その適用は限定的であった。実は筆者もその一人である。そこで、SEAの理論に深くは立ち入らず、その結果を1Dモデリングに拡張することを考える。

 通常の音振動表現は音圧、力、変位、速度といった音振動固有の状態量を使用するが、SEAでは物理系に共通のエネルギー、パワーを用いる。パワーフローで考える音振動の例として、図2の六面体構造物を考える。

パワーフローで考える音振動 図2 パワーフローで考える音振動[クリックで拡大]

 六面体なので振動のみを考える場合には、6つの長方形板要素に分け、これらの相互のパワーのやりとりを定義することによって、各要素のエネルギーが求まる。詳しくは次節で述べる。

 図1の例では振動部分のみ表記しているが、音を考える場合には、六面体で囲まれた空間を1つの音場と考え、この音場と各長方形板間のパワーのやりとりを考えればよい。連載第8回のランドリーの音振動問題で部屋の音の伝搬を拡散音場で求めたが、これも今回紹介する方法と基本的に同じである。

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