JP4024626B2 - Structure design support system and method, and program for the system - Google Patents

Description

【００２５】
式(1) は図２の(b) に示す音場と構造についての関係を表わしたものである。式(1) に関連して，Ωは図２の(b) でいうところの音場の領域，Γは音場の境界であり，構造である（具体的にはボディ）。Ｓは構造であり，音場の境界でもある（具体的にはシート）。ＰはΩ内の音圧，ｕは構造の変位，ｃは音速，ρ₀は気体の質量密度，Ｄ は構造の剛性率，ρ₁は構造の質量密度，ｍはＺ方向のフーリエ（Fourier）モード数，ω は固有振動数（固有値は１／ω₂ で表わされる）である。音場の境界条件は，Γ上で音圧が０，構造の境界条件は単純支持とする。
【００２６】
式(1-1) は空気中における固有振動数ωと音圧Ｐの関係を表わす音場振動方程式である。
【００２７】
式(1-2) は音場の境界条件であり，境界Γ上で音圧が０（検証を容易にするために０としたが，実際の計算では，式(1-2) に代えて剛壁条件式 (1-2)’を用いてもよい）であることを表わす。
【００２８】
式(1-3) も音場の境界条件である。音圧Ｐによる境界Ｓの法線方向の変化率が，空気質量密度ρ₀と，構造の変位ｕと，固有振動数の２乗ω²との積に等しいことを表わす。構造の変位ｕは，振動によって生ずる構造Ｓのｘ方向への変位を表わすものである。
【００２９】
式(1-4) は図２(b) に示すような境界Ｓが，音圧Ｐを受けた時の構造の固有振動方程式である。
【００３０】
式(1-5) は構造境界条件で，境界Ｓの両端にある支持点は単純支持であるということを表わしている。式(1-1) から式(1-5) をまとめて式(1) とする。
【００３１】
式(1-3) は構造が音場に与える影響を表わす式（音場系の式）であり，式(1-4) は音場が構造に与える影響を表わす式（構造系の式）である。これら２つの式によって音場と構造が結びつけられている。
【００３２】
音場と構造の結びつきの強さを表わすパラメータをεとし，式(1-3) と式(1-4) の右辺にパラメータεを乗じたものをそれぞれ式(2-3) ，(2-4) とする。式(1-1) ，(1-2) ，(1-5) にそれぞれ対応する式(2-1) ，(2-2) ，(2-5) と上記の式(2-3) ，(2-4) の全体を式(2) とする。
【００３３】
ε＝１の場合は完全連成を表わす。ε＝０の場合，音場と構造を関係づける因子はなくなり，完全非連成となり，この場合には，式(2) の音場系の振動方程式，式(2-1) 〜式(2-3) は，次の式(3) で表わされ，構造系の振動方程式，式(2-4) 〜式(2-5) は，次の式(4) で表わされる。
【００３４】
【数２】

【００３５】
【数３】

【００３６】
式(2) を摂動法の原理を用いて解く。まず，一般的な摂動法を説明する。
【００３７】
摂動法とは，ある複雑な系（系とは解析しようとする対象を指す。例えば，自動車の車室の騒音を解析するときの構造であり，音楽ホールの共鳴を解析するときのホールの構造である。）の状態を求める場合に，厳密に求めるのではなく，この系を構成する既知で簡単な系の状態から近似的に求める方法である。このとき，複雑な系は既知の系に小さな変化を与えることによって実現される状態であることを前提としている。摂動法によって求められるある系に微小変動を加えたときの固有値は，既知の系の固有値の級数で表わされる。
【００３８】
式（２）は理論解を求めるものである。実際の解析に応用するために，一般的には，連続系の式は離散化され，有限要素法を用いて解かれる。式（２）を離散化すると，式（５）が得られる。
【００３９】
【数４】

【００４０】
ここで，式（５）について，次の式（６）を用いた置き換えを行うと，
【数５】

式（７）のように書くことができる。式（７）は非対称のままである。上述したように，非対称のマトリックスを解くのは対称マトリックスを解くのに比べ，所要時間もコンピュータ上の一次記憶域も余分に必要となる。
【００４１】
【数６】

【００４２】
最終的な目的は連成系における固有値と固有モードを求めることにあり，連成系における固有値と固有モードは非連成系の（音場と構造の）固有値と固有モードで表現することができる。この表現において摂動法を用い，式(8-1)，(8-2)に示すように，固有値λ_i と固有モードφ_i をそれぞれ結合パラメータεの級数で表わす（摂動法による展開式）。
【００４３】
そこで，式（５）に示す非対称のマトリックスについて，以下に，摂動法を用いて近似解を求める。式（７）のλ_i(ε）とφ_i(ε）について摂動法を用いて近似解を求める式は，上記の通り式(8-1)，(8-2)のようになる。これらの式における高次項の係数は，下記式（９）（式(9-1)〜式(9-4)）で求められる。
【００４４】
【数７】

【００４５】
【数８】

【００４６】
式（８）（式(8-1)〜式(8-2)） において左辺は連成系における固有値λiと固有モードφ_iである。右辺のεの級数の各係数は非連成系における（音場と構造の）固有値と固有モードを用いて算出される。
【００４７】
非連成系における固有モードを求めるための正規化条件としてＫ直交条件を用いることにより（従来はＭ直交条件を用いていた），式（８） が収束すること （項数が増大すればするほど，各項の係数の値が小さくなること）が保証されることが分った。
【００４８】
式（８）が収束することを説明する。下記式（ａ）は，式（７）のεを「０」とした場合，即ち連成前の状態を示すものである。
λ_iＫφ_i ＝ Ｍφ_i …（ａ）
式（ａ）の左からφ_i ^Tを乗ずると下記式（ｂ）のようになる。
λ_iφ_i ^TＫφ_i ＝ φ_i ^TＭφ_i …（ｂ）
【００４９】
ここでＫ直交とは次の式（１０）を満たすことである。ただしＫは剛性マトリクスである。
【００５０】
【数９】

【００５１】
Ｋ直交条件を満たすということは，式（ｂ）の「φ_i ^TＫφ_i 」が「１」に等しいことを意味する。また，高次の固有値「λ_i」は「０」に近づくことが分っている。したがって式（ｂ）は「０」に収束することとなり，構造解析のための計算が収束に向かうことが分かる。
【００５２】
連成系の固有値，固有モードは，下記のとおり既知の非連成系の固有値λ_i(0)，固有モードφ_i(0)を第１項とする級数で表わされる（ε＝１の場合）。
【数１０】

（λ_i(0) は連成前の構造または音場の固有値であり，奇数項は全て０になることが分かっている。φ_i(0)は連成前の構造または音場の固有モードである。）
【００５３】
上記式(c) の第２項以降（補正後）はλ_i ⁽²ⁿ⁾ （但しn＝１，２，３…）と表わすことができ，図３に示す処理フローの第ｎステップの近似計算により順次求られめる。
【００５４】
固有値と同様に，連成系の固有モードも既知の非連成系の固有モードを第１項とする級数で表わされる。φ_i(n) も同様に近似計算によって求められるものである。
【００５５】
固有値の近似計算をする際には，式(c)と式(d)の初項を決定しておく必要がある。初項とは，λ_i(0) やφ_i(0) である。自動車の車室内の騒音について解析するような場合は，人間の耳に不快に感じる周波数帯に着目する。したがって，連成系の固有値をコンピュータにより求める（計算する）に先立ち，音場の固有値と構造の固有値をコンピュータに読み込むが，音場の固有値と構造の固有値を比較して，より不快と感ずる周波数帯をもつ方を上記式の初項とする（これを基準固有値という）。
【００５６】
式(c) はλ_i(0) を除く右辺の項を無限個求めれば等号が成り立つものである。しかし，現実には有限回で計算を打ち切る必要があるので，図３において計算される第ｎステップ終了後の固有値は式(9-2) のように表わすことができる。計算によって求められる固有値は式(9-2) のｎが大きいほど（式(c) の項をより多く求めるほど）真の値に近づく。固有モードについても同様である（式(9-4)）。
【００５７】
図５(a) は，図２(b) に示す音場の領域Ωを１辺がπ／４である32個の正方形に分割し，各正方形の頂点の座標を示すものである。（横軸をｘ，縦軸をｙとし，座標は（ｘ，ｙ）で表わされる。また，π／４を１とする。）
【００５８】
一般的には，図５に小さな黒丸で示す25個の点における音圧の集合が音場の固有モード（固有ベクトル）を表わすものとする。一方，構造については計算結果の精度を確保するために，音場よりも細かい分割をして固有モードを計算することが多い。境界Ｓ上において音場のデータを取得する点は○で囲まれた５点であるのに対して，構造のデータを求める点については構造Ｓ上で三角で表示される９点が設けられている。
【００５９】
図５(b) は，連成系のデータとして採用するデータＤ３と，境界Ｓ上における構造系のデータＤ１および音場系のデータＤ２の関係を図示したものである。構造系のデータはｕ(1) からｕ(9) まで９個のデータがあるが，連成系のデータとして採用するのはｕ(1) ,ｕ(3) ,ｕ(5) ,ｕ(7) ,ｕ(9) の５個であり，音場系のデータは図５(a) で黒丸で示す25個の点のうち，連成系のデータとして採用するのはｐ(1,5) ,ｐ(2,5) ,ｐ(3,5) ,ｐ(4,5) ,ｐ(5,5) の５個である。
【００６０】
上述の説明でわかるように，連成振動現象を解く場合に必要なデータは，音場と構造の接する部分（構造表面）だけでよく，さらに，構造のデータは音場のデータと重なる点のデータだけでよい。
【００６１】
図６(a) は，上記の記述に基づいて，図３のフローチャートの処理10で読み込むデータの並びを示したものである（このデータはコンピュータのメモリに記憶される）。音場の固有値をλs_n で表し，データＤ３のように選択された固有モードを音圧ｐ(i,j)_n で表す。構造の固有値はλp_n で表し，固有モードを変位ｕ(i)_n で表す。ｎは入力するｎ個の固有値の順番を表わす。図６(b) は具体的数値例を示す。
【００６２】
音場の固有値λs_n と音場の固有モードｐ(i,j)_n は式(3) によってあらかじめ求められ，構造の固有値λp_n と構造の固有モードｕ(i)_n は式(4) によってあらかじめ求められる。このとき，上述したように正規化条件としてＫ直交条件が用いられる。
【００６３】
図６(a)により具体的に説明すると，音場の固有値はきわめて多数存在するが，簡単のために５つ（λs_１，λs_２，λs_３，λs_4，λs_５）が示されている。固有値λs_１とペアをなす固有モードがｐ(１,５)_１，ｐ(２,５)_１， ・・・・・，ｐ(５,５)_１であり，これらは図５(b) の縦に配列された５つの点の音圧によって表わされる固有モードである。
【００６４】
図６(b)は図６(a)のそれぞれについて具体的数値例を示すものである。
【００６５】
図３は実施例の処理フロー図である。この処理はコンピュータによって実行される。処理10では予め計算しておいた構造系と音場系それぞれの固有値と図５のＤ３で示すように選択された固有モードをコンピュータに，またはメモリのワークエリアにより読み込む（図６(a)，(b)参照）。図６(a)，(b)において，上段が音場系の固有ペアであり，下段が構造系の固有ペアである。固有ペアの値がＦＤ等に記憶されている場合には，ＦＤからワークエリアに読込めばよいし，ハードディスクに記憶されている場合には，それをメモリのワークエリアに読込めばよい。固有ペアの値を他のコンピュータから伝送してもよい。
【００６６】
処理20では計算の基準とした（構造か音場のいずれか）固有値λ（基準固有値）と，許容誤差ｅおよび繰返し計算打ち切り回数ｍとをコンピュータに入力する。この入力は一般には手操作により行われるが，ＦＤから入力，またはオンラインで入力してもよい。固有値λは周波数ωとの間にλ＝１／ω² の関係がある。許容誤差とは近似計算の結果，ｎ−１回目の計算結果の固有値とｎ回目の計算結果の固有値との差が指定した許容誤差ｅよりも小さければ計算を打ち切らせるための値である。繰返し計算打ち切り回数ｍは，収束計算が期待する許容誤差範囲に無い場合であっても（すなわち，上記のように，ｎ−１回目の値とｎ回目の値との差が許容誤差ｅより小さくなくても）計算を中止させるためのものである。なお，処理における変数ｎは，上記のｎ個の固有値の順番を表わすｎとは意味が異なるものである。
【００６７】
処理30では処理10で入力した複数の固有ペアを固有値をキーにしてメモリ（ワークエリア）上で並べ替えを行う。並べ替えは各固有ペアの固有値と処理20で読み込んだ基準固有値λとの差の絶対値が小さい順番に行う。並べ替えは構造系と音場系それぞれについて行う。その後，音場と構造の共通節点（図５で□を付けた部分）の固有モードを選び出し，メモリに保存する。（これにより，保存量を減らすことができる。図５の場合では，音場の全体節点数25が５になり，構造の全体節点数９が５になる。）
【００６８】
一般に自動車の車室内の騒音について解析するような場合は，人間にとって不快な騒音と感ずる周波数の音を中心に分析を行う。図６の例で，構造固有値λs_4 が不快な音になる周波数の固有値（基準固有値）であるとした場合，処理30ではまず第１番目にλs_4 （これが基準固有値である）を置き，残りの固有値についてλs_4 に近い順番（各固有値とλs_4 との差の絶対値が小さい順）メモリ上でに並べ替える。
【００６９】
処理40ではカウンタを＝０とする。処理50ではカウンタを１プラス（インクレメント）する。処理60では式（９)での近似計算λ_i ⁽²ⁿ⁾ を上記において並べ替えした順序で固有ペアを用いて，行う。式(9) は式(8) における各係数を求めるための計算式である。式(9)において，λ_i ⁽²ⁿ⁾を計算するためには（式(9-2)），λ_iとφ_iを用いる。φ_iは式(9-3)または式(9-4)を用いて計算する。このときλ_iとλ_j（ｉ≠ｊ）を用いる。λ_jのｊは，ｉを固定したときにカウンタによりカウントアップされていく（下記のｎに相当）。このカウンタにより指定される固有値の順番が上記の並べ替えの順番である。
【００７０】
処理70では処理60で求めたλ_i ⁽²ⁿ⁾の絶対値を許容誤差ｅと比較する（λ_i ⁽²ⁿ⁾はｎ回目の計算結果の固有値とｎ−１回目の計算結果の固有値との差である。λ_i ⁽²ⁿ⁾ が許容誤差ｅより大きければ（処理70でNO），処理72に進み，ｎ（カウンタの値）がｍに達したかどうかをチェックする。ｎがｍに達していなければ（処理72でNO），処理50に戻り，カウンタの値をインクレメントして再び近似計算を行う（処理60）。
λ_i ⁽²ⁿ⁾ が許容誤差ｅより小さければ処理80に進む。λ_i ⁽²ⁿ⁾ が許容誤差ｅより小さくならなくても，ｍがｎに一致した場合（ｎ＝ｍになったとき）には（処理72でYES），処理74 に進み，回数制限で計算打切られたことを示すメッセージを出力し，処理80に進む。近似計算は途中で打切られるので計算時間は短くてすむが，下に説明するように収束が確保される。
【００７１】
処理80では連成系の近似固有値λ＝λ_i＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＋…λ_i ⁽²ⁿ⁾を計算して出力する。近似固有モードも同様に求められるのはいうまでもない。
【００７２】
上記の処理10〜80はパラメータεを適当な値に固定して実行される。もし，所望の，または適切な結果が得られない場合には，パラメータεを変更して再度上記の処理を繰返すことが望ましい。これは，たとえば構造の材質を変更するようなことに相当するので，最も適切な材質の選定にも役立つ。
【００７３】
図４(a)，(b)は式(8-1) の第２項以降の項のうちの１つの項について，縦軸にλ，横軸に計算回数を取ったときの収束状況を示す図である。固有値をλに近い順に並べ替えずに計算した場合は，図４(b)の曲線(b)で示すように誤差が大小に変動し，何回で収束したと見なすかの見極めができない。一方，並べ替えを行った場合は，図４(a) の曲線(a1)または(a2)，(a3)のように確実に回を追うごとに収束に向かう形となる。
【００７４】
以下に具体例を示す。
【００７５】
第１の例 （構造からの連成系固有値）
許容誤差ｅ＝10^-6とする。
ｉ）現実的なパラメータを使うケース
Ｄ＝19.230769，ρ₀＝1.2930000，ρ₁＝7850,ｃ＝340，ｍ＝１，ｉ＝１
理論解λ_i(1)＝102.06188，λ_i(0)＝102.05000
ここで，理論解λ_i(1)は下記式（11）のタイプ１で計算されたものである。
理論解λ_i(0)は式(3)で計算されたものである。
１回近似結果＝λ_i(0)＋λ_i ⁽²⁾＝102.06188，
｜λ_i ⁽²⁾｜＝｜0.01188｜＞ｅ
２回近似結果＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＝102.06188，
｜λ_i ⁽⁴⁾｜＝｜0.0000000｜＜ｅ
３回近似結果＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＋λ_i ⁽⁶⁾＝102.06188，
｜λ_i ⁽⁶⁾｜＝｜0.0000000｜＜ｅ
４回近似結果＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＋λ_i ⁽⁶⁾＋λ_i ⁽⁸⁾＝102.06188，
｜λ_i ⁽⁸⁾｜＝｜0.0000000｜＜ｅ
【００７６】
ii）非現実的なパラメータを使うケース
理論解λ_i(1)＝6.6944765756，λ_i(0)＝6.25
Ｄ＝２，ρ₀＝５，ρ₁＝50，ｃ＝2.5，ｍ＝１，ｉ＝１
ここで，理論解λ_i(1)は式（１１）のタイプ１で計算されたものである。
理論解λ_i(0)は式(3)で計算されたものである。
１回近似結果＝λ_i(0)＋λ_i ⁽²⁾＝6.694667334361377，
｜λ_i ⁽²⁾｜＝｜0.44466733｜＞ｅ
２回近似結果＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＝6.694462764982187，
｜λ_i ⁽⁴⁾｜＝｜−0.000204569｜＞ｅ
３回近似結果＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＋λ_i ⁽⁶⁾＝6.694477554000371，
｜λ_i ⁽⁶⁾｜＝｜0.000014789｜＞ｅ
４回近似結果
＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＋λ_i ⁽⁶⁾＋λ_i ⁽⁸⁾＝6.694476478081498，
｜λ_i ⁽⁸⁾｜＝｜0.0000010759｜≒ｅ
【００７７】
第２の例 （音場からの連成系固有値）
許容誤差ｅ＝10^-4とする。
理論解λ_i(1)＝0.069251599，λｉ(0)＝0.07111111
ここで，理論解λ_i(1)は式（11）のタイプ２で計算されたものである。
理論解λ_i(0)は式(2)で計算されたものである。
１回近似結果＝λ_i(0)＋λ_i ⁽²⁾＝6.9075922324487507×10^-2，
｜λ_i ⁽²⁾｜＝｜−2.035188×10^-3｜＞ｅ，
２回近似結果＝λ_i(0)＋λ_i ⁽²⁾＋λ_i ⁽⁴⁾＝6.9265894614179742×10^-2，
｜λ_i ⁽⁴⁾｜＝｜−4.93329×10^-4｜≒ｅ
【００７８】
【数１１】

【００７９】
図７は構造物の設計支援システム（または連成系振動解析装置）の構成側を示すものであり，コンピュータシステムにより実現される。
【００８０】
処理装置（並び替え手段，処理手段）１はコンピュータ本体であり，図３ステップ10〜ステップ80の処理を実行するプログラム１ａを格納するとともに，図６に示すようなデータを格納するメモリ１ｂを備えている。
【００８１】
入力装置（入力手段）２はキーボード，マウス，ＦＤドライブ，通信装置等により実現され，ステップ10，20におけるデータ等の入力のために用いられる。
【００８２】
出力装置（出力手段）３は表示装置，プリンタ，ＦＤドライブ等により実現され，ステップ80の固有値出力に用いられる。
【図面の簡単な説明】
【図１】自動車の断面図であり，この発明に適する現実問題の一例を示すものである。
【図２】図１をモデル化した図であり，(a)は３次元モデルを，(b)は２次元モデルをそれぞれ示す。
【図３】コンピュータにおける処理手順を示す処理フロー図である。
【図４】演算の収束状況を表わすグラフであり，(a)は収束する場合を，(b)は収束しない場合をそれぞれ示す。
【図５】 (a)，(b)は音場系と構造系の共通節点を表わすイメージ図である。
【図６】 (a)，(b)は音場系と構造系の固有ペアのメモリ上の保存状態を表わす図である。
【図７】構造物の設計支援システムまたは連成系振動解析システムの構成例を示すブロック図である。
【符号の説明】
１ 処理装置
１a プログラム
１b メモリ
２ 入力装置
３ 出力装置[0001]
【Technical field】
The present invention relates to a system and method for supporting the design of a structure in consideration of vibration behavior, or to a coupled vibration analysis system and method for a sound field and a structure, and a program for controlling the system.
[0002]
[Technical background]
It is very important to consider vibrations in the design of various structures. For example, in the design of automobiles, it is required to suppress vibrations as much as possible in order to improve comfort when riding. Further, in designing a speaker, characteristics that can obtain the maximum electrical / acoustic conversion efficiency in a necessary frequency band are required in accordance with the required specifications for the speaker.
[0003]
Taking automobile design as an example, the items required as design specifications include improvement of motor performance and reduction of manufacturing costs. In addition, in order to ensure comfort, Measures to reduce vibration (noise) are one of the important requirements that are indispensable. The vibration (noise) felt by a person sitting on a seat in a car cabin is determined by several conditions. The sound field condition, that is, the condition relating to the vehicle interior space, the structure condition, that is, the condition relating to the vehicle body and the seat, and the like. Specific conditions of the sound field include the size (volume) of the passenger compartment and the shape of the space. The structural conditions include the material and rigidity of the members constituting the vehicle body, the method of attaching the members, and the like. These conditions are interrelated, such as the space in the passenger compartment is defined by the car body. The vibration (noise) in the passenger compartment depends on such sound field conditions and structure conditions. Such vibration that occurs depending on a plurality of mutually related conditions is called a coupled vibration phenomenon. Such a method for analyzing a noise or vibration phenomenon is an extremely important method for designing a car or the like, and is positioned as a support for the design.
[0004]
The matrix that expresses the coupled vibration phenomenon in numerical calculations is asymmetric. It takes an enormous amount of time to solve an asymmetric matrix using a computer. There is a method to derive an asymmetric matrix to a symmetric matrix as a method for facilitating calculation, but even if a symmetric matrix can be derived, the symmetric matrix has no band property, so the computation time required for eigenvalue analysis is Long, after all, it takes time to solve the matrix. Here, the band property means the property that non-zero terms in the matrix are concentrated on the diagonal of the matrix.
[0005]
Vibration has an eigenvalue and an eigenmode (also called eigenvector). The eigenvalue and the eigenmode are collectively called an eigenpair. Attempts have been made to express coupled eigenpairs using uncoupled eigenpairs in order to shorten the vibration analysis time, but it is difficult to diverge midway or solve it, and it takes time to solve. It was not right.
[0006]
DISCLOSURE OF THE INVENTION
The present invention provides a structure design support system or a vibration analysis system and method capable of performing vibration (noise) analysis within a practically executable time, and a program for the system. Objective.
[0007]
The structure design support method (or coupled vibration analysis method) according to the present invention uses the K orthogonal condition as a normalization condition from the uncoupled vibration equations of the sound field and structure in the region where the vibration is to be analyzed. The calculated eigenvalues and eigenmodes are input to the computer and stored in the memory. For the eigenvalues and eigenmode eigenpairs stored in the memory, the absolute value of the difference between the eigenvalues in the eigenpairs and the given reference eigenvalue is in ascending order. The coefficients of the expansion equation by the perturbation method expressing the eigenvalues and eigenmodes in the coupled system of the sound field and structure in the above region to be analyzed in the memory and expressed in the series of the coupling parameter ε of the coupled system When the above coefficient is calculated by a computer, the eigenvalue in the coupled vibration equation is replaced with the reciprocal of the frequency based on the discrete coupled vibration equation. Without symmetry of the deposition system vibration equations, while an asymmetrical, is calculated by using the above sorting is intrinsic pair ultimately is to derive a unique pair of said coupled system.
[0008]
The structure design support system (or coupled vibration analysis system) according to the present invention uses the K-orthogonal condition as a normalization condition based on the uncoupled vibration equations of the sound field and structure in the region where vibration is to be analyzed. An input means for accepting the calculated eigenvalue and eigenmode input and storing them in the memory, and for the eigenvalue and eigenmode eigenpair stored in the memory, the absolute value of the difference between the eigenvalue in the eigenpair and the given reference eigenvalue Permutation expressing the eigenvalues and eigenmodes in the coupled system of sound field and structure in the above region to be analyzed by the series of coupling parameter ε of coupled system When calculating the coefficients of the expansion equation by the method using a computer, the above coefficients are multiplied by the eigenvalues in the coupled vibration equation based on the discrete coupled vibration equation. After replacing with the reciprocal of the number, the coupled vibration equation is not symmetrized and is calculated using the rearranged eigenpairs asymmetry, and finally the eigenpairs of the coupled system are obtained. And a means for outputting the derived unique pair.
[0009]
The structure design support program (or coupled system vibration analysis program) according to the present invention uses the K orthogonal condition as a normalization condition from the uncoupled vibration equations of the sound field and structure in the region where the vibration is to be analyzed. The calculated eigenvalues and eigenmodes are input to the computer and stored in the memory. For the eigenvalues and eigenmode eigenpairs stored in the memory, the absolute value of the difference between the eigenvalues in the eigenpairs and the given reference eigenvalue is in ascending order. The coefficients of the expansion equation by the perturbation method expressing the eigenvalues and eigenmodes in the coupled system of the sound field and structure in the above region to be analyzed in the memory and expressed in the series of the coupling parameter ε of the coupled system Is calculated by the computer, the above coefficient is set based on the discrete coupled vibration equation, and the eigenvalue in the coupled vibration equation is set as the reciprocal of the frequency. In addition, without using symmetrization of the coupled system vibration equation, it is calculated using the rearranged eigenpairs while remaining asymmetric, and finally the eigenpairs of the coupled system are derived. is there.
[0010]
In the present invention, it is assumed that the eigen pair of the coupled system is expressed by using the eigen pair of the uncoupled system of the sound field and the structure in the region where the vibration is to be analyzed. In order to obtain an eigenpair of each uncoupled system of sound field and structure, in the present invention, the uncoupled vibration equation is calculated using the K orthogonal condition as a normalization condition. This guarantees convergence in the calculation of the coefficients of the expansion equation by the perturbation method.
[0011]
In addition, according to the present invention, the eigenpairs calculated as described above are rearranged in the order in which the absolute value of the difference from a given reference eigenvalue (the eigenvalue of interest) is small. Is used for approximate calculation of the above coefficients. Since the expansion equation is guaranteed to converge as described above, even if there are many eigenpairs, the approximate calculation can be aborted halfway, and the eigenpairs are used in the above order. Even if the calculation is interrupted, the value obtained is very close to the true value.
[0012]
In this way, according to the present invention, the result of vibration analysis, that is, the eigenpair of the coupled system can be obtained within a time that can be practically executed.
[0013]
Since the eigenvalue in the vibration equation is replaced by the reciprocal of the frequency, it is possible to handle the coupled vibration equation discretized by the finite element method as asymmetric.
[0014]
In order to cancel the approximate calculation, the allowable error e and the number of times m should be input to the computer. In the approximate calculation of the coefficient by the computer, the approximate calculation is stopped when the difference between the previous calculation result and the current calculation result becomes smaller than the allowable error e, or the difference is less than the allowable error e. However, the approximate calculation is stopped when the number of repetitive calculations reaches the number m of truncation.
[0015]
The reference eigenvalue is specified in an eigenvalue within an eigenpair stored in the computer memory.
[0016]
【Example】
FIG. 1 is a sectional view of an automobile, showing a state in which an engine 2 and a seat 3 are installed inside a body 1.
[0017]
Vibration (including sound) in the interior of an automobile includes noise (about 60 to 300 Hz) that makes humans feel uncomfortable, such as a sound generated by an engine as a sound source and a muffled sound when the automobile is stopped. The purpose of the analysis of vibration phenomena is to determine the natural frequency of these noises, the sound pressure and the displacement at the natural frequency. By obtaining these values, if the natural frequency is in the uncomfortable band, measures such as shifting the natural frequency, lowering the sound pressure level, and the displacement level are possible.
[0018]
In order to solve the above coupled vibration phenomenon, the sound field and structure are considered. The sound field is a space defined by the user as a region to be analyzed. Further, the structure is an elastic body, a solid body, or a structural body, and a sheet or the like corresponds to this embodiment.
[0019]
As an example, a space A where the driver is present is analyzed. FIG. 2A is a three-dimensional model of the automobile shown in FIG. The front part of the vehicle body seat 3 is indicated by a solid line, and the rear part is indicated by a broken line. A part of the vehicle body is represented by a cube whose one side is π, and this is the structure (S corresponding to the seat).₀including). Ω₀Indicates the inside of the cube, and in FIG. 1, it is a region of the sound field representing the space A in the passenger compartment.
[0020]
To facilitate the explanation and enable verification, consider a two-dimensional model as shown in Fig. 2 (b).
[0021]
In FIG. 2B, a square of π × π surrounded by Γ and S is a sound field boundary, a square internal Ω is a sound field region, and a sound field boundary (Γ and S) is a structure. Γ is the body and S is the sheet.
[0022]
The sheet S is fixed at the support point P1 and the support point P2, and the sheet S itself is allowed to be bent slightly. Although displacement in the x and y directions is not allowed at both the support points P1 and P2, the support point is allowed to rotate around the support point due to the deflection of the sheet S. (This condition is called simple support.)
[0023]
FIG. 2 (c) shows a state in which the sheet S is slightly changed by the sound pressure P emitted from the sound source in the sound field Ω (a state in which deflection occurs). u represents the amount of displacement in the x direction that occurs when the sheet S represented by the solid line is in the state of S ′ represented by the alternate long and short dash line by the sound pressure P and the point Px1 on the sheet S is displaced to the point Px2 on S ′. Is.
[0024]
[Expression 1]

[0025]
Equation (1) represents the relationship between the sound field and structure shown in FIG. In relation to Equation (1), Ω is the sound field region in FIG. 2 (b), and Γ is the boundary of the sound field and is the structure (specifically, the body). S is the structure and the boundary of the sound field (specifically, a sheet). P is the sound pressure in Ω, u is the displacement of the structure, c is the speed of sound, ρ₀Is the mass density of the gas, D is the stiffness of the structure, ρ₁Is the mass density of the structure, m is the Fourier mode number in the Z direction, ω is the natural frequency (the eigenvalue is 1 / ω₂ It is expressed by The boundary condition of the sound field is 0, and the boundary condition of the structure is simply supported.
[0026]
Equation (1-1) is a sound field vibration equation representing the relationship between the natural frequency ω and the sound pressure P in the air.
[0027]
Equation (1-2) is the boundary condition of the sound field, and the sound pressure is 0 on the boundary Γ (0 for ease of verification, but in actual calculation, instead of Equation (1-2) Rigid wall conditional expression (1-2) 'may be used).
[0028]
Equation (1-3) is also a boundary condition of the sound field. The rate of change in the normal direction of the boundary S due to the sound pressure P is the air mass density ρ₀And the displacement u of the structure and the square of the natural frequency ω²Is equal to the product of The structure displacement u represents the displacement of the structure S caused by vibration in the x direction.
[0029]
Equation (1-4) is the natural vibration equation of the structure when the boundary S as shown in FIG.
[0030]
Equation (1-5) is a structural boundary condition, and indicates that the support points at both ends of the boundary S are simple supports. Equations (1-1) to (1-5) are combined into Equation (1).
[0031]
Equation (1-3) is the equation (sound field equation) that expresses the effect of the structure on the sound field. Equation (1-4) is the equation that expresses the effect of the sound field on the structure (structure equation) It is. The sound field and structure are linked by these two formulas.
[0032]
The parameter representing the strength of the connection between the sound field and the structure is ε, and the right side of Equations (1-3) and (1-4) is multiplied by parameter ε, respectively, to Equations (2-3) and (2- 4). Equations (2-1), (2-2), and (2-5) corresponding to Equations (1-1), (1-2), and (1-5), respectively, and Equations (2-3), Equation (2) is the whole of (2-4).
[0033]
When ε = 1, it indicates complete coupling. When ε = 0, there is no factor that relates the sound field to the structure, and it becomes completely uncoupled. In this case, the vibration equation of the sound field system of Equation (2), Equation (2-1) to Equation (2 -3) is expressed by the following equation (3), and the vibration equations of the structural system, equations (2-4) to (2-5) are expressed by the following equation (4).
[0034]
[Expression 2]

[0035]
[Equation 3]

[0036]
Solve equation (2) using the principle of the perturbation method. First, the general perturbation method is explained.
[0037]
The perturbation method is a complex system (the system refers to the object to be analyzed. For example, it is a structure for analyzing the noise of a car cabin, and a structure for a hall for analyzing the resonance of a music hall. This is a method of approximately obtaining from the state of a known and simple system constituting this system, instead of obtaining it strictly. At this time, it is assumed that a complex system is realized by giving a small change to a known system. The eigenvalue obtained by applying a small variation to a system obtained by the perturbation method is expressed by a series of eigenvalues of a known system.
[0038]
Equation (2) is for obtaining a theoretical solution. In order to apply to actual analysis, generally, the expression of continuous system is discretized and solved using the finite element method. When equation (2) is discretized, equation (5) is obtained.
[0039]
[Expression 4]

[0040]
Here, when the replacement using the following equation (6) is performed on the equation (5),
[Equation 5]

It can be written as equation (7). Equation (7) remains asymmetric. As described above, solving an asymmetric matrix requires more time and primary storage on the computer than solving a symmetric matrix.
[0041]
[Formula 6]

[0042]
The ultimate goal is to find eigenvalues and eigenmodes in a coupled system, and eigenvalues and eigenmodes in a coupled system can be expressed by eigenvalues and eigenmodes in an uncoupled system (sound field and structure) . In this expression, the perturbation method is used, and the eigenvalue λ is expressed as shown in equations (8-1) and (8-2)._i And eigenmode φ_i Is expressed by a series of coupling parameters ε (expansion formula by perturbation method).
[0043]
Therefore, an approximate solution for the asymmetric matrix shown in Equation (5) is obtained below using the perturbation method. Λ in equation (7)_i(ε) and φ_iThe equations for obtaining an approximate solution using the perturbation method for (ε) are as shown in equations (8-1) and (8-2) as described above. The coefficients of higher-order terms in these equations are obtained by the following equation (9) (equation (9-1) to equation (9-4)).
[0044]
[Expression 7]

[0045]
[Equation 8]

[0046]
In equation (8) (equations (8-1) to (8-2)), the left side is the eigenvalue λi and eigenmode φ in the coupled system_iIt is. Each coefficient of the series of ε on the right side is calculated using eigenvalues (sound field and structure) and eigenmodes in an uncoupled system.
[0047]
By using the K-orthogonal condition as a normalization condition for obtaining the eigenmode in the uncoupled system (previously, the M-orthogonal condition was used), Eq. (8) converges (if the number of terms increases) It was found that the coefficient value of each term becomes smaller.
[0048]
It will be explained that the expression (8) converges. The following formula (a) shows the state before the coupling when ε in the formula (7) is “0”.
λ_iKφ_i  = Mφ_i  ... (a)
Φ from the left of equation (a)_i ^TIs multiplied by the following formula (b).
λ_iφ_i ^TKφ_i  = Φ_i ^TMφ_i  ... (b)
[0049]
Here, K orthogonal means that the following equation (10) is satisfied. Where K is a stiffness matrix.
[0050]
[Equation 9]

[0051]
Satisfying the K-orthogonal condition means that “φ in equation (b)_i ^TKφ_i "Is equal to" 1 ". The higher order eigenvalue “λ_i"Is close to" 0 ". Therefore, equation (b) converges to “0”, and it can be seen that the calculation for the structural analysis tends to converge.
[0052]
The eigenvalues and eigenmodes of the coupled system are as follows:_i(0), Eigenmode φ_iIt is expressed as a series with (0) as the first term (when ε = 1).
[Expression 10]

(Λ_i(0) is the eigenvalue of the structure or sound field before coupling, and all odd terms are known to be zero. φ_i(0) is the eigenmode of the structure or sound field before coupling. )
[0053]
The second term and after (after correction) of the above formula (c) is λ_i ⁽²ⁿ⁾ (Where n = 1, 2, 3,...), And can be obtained sequentially by approximate calculation at the nth step of the processing flow shown in FIG.
[0054]
Similar to eigenvalues, coupled eigenmodes are also represented by a series whose first term is a known uncoupled eigenmode. φ_iSimilarly, (n) is obtained by approximate calculation.
[0055]
When performing approximate calculations of eigenvalues, it is necessary to determine the first terms of equations (c) and (d). The first term is λ_i(0) or φ_i(0). When analyzing the noise in the interior of an automobile, focus on the frequency bands that are uncomfortable to the human ear. Therefore, prior to computing (calculating) the eigenvalues of the coupled system, the eigenvalues of the sound field and the eigenvalues of the structure are read into the computer. The one with the band is the first term of the above formula (this is called the reference eigenvalue).
[0056]
Equation (c) is λ_iEqual sign holds if the infinite number of terms on the right side excluding (0) is obtained. However, since it is actually necessary to terminate the calculation at a finite number of times, the eigenvalue after the completion of the n-th step calculated in FIG. 3 can be expressed as in equation (9-2). The eigenvalue obtained by calculation becomes closer to the true value as n in the equation (9-2) is larger (as more terms in the equation (c) are obtained). The same applies to the eigenmode (Formula (9-4)).
[0057]
FIG. 5 (a) shows the coordinates of the vertices of each square by dividing the sound field region Ω shown in FIG. 2 (b) into 32 squares each having a side of π / 4. (The horizontal axis is x, the vertical axis is y, and the coordinates are represented by (x, y). Also, π / 4 is 1.)
[0058]
In general, a set of sound pressures at 25 points indicated by small black circles in FIG. 5 represents an eigenmode (eigenvector) of the sound field. On the other hand, for the structure, in order to ensure the accuracy of the calculation result, the eigenmode is often calculated by dividing it more finely than the sound field. On the boundary S, there are five points for obtaining sound field data surrounded by circles, whereas for points for obtaining structure data, nine points displayed in triangles on the structure S are provided. Yes.
[0059]
FIG. 5B illustrates the relationship between the data D3 employed as the coupled data, the structural data D1 on the boundary S, and the sound field data D2. There are 9 structural data from u (1) to u (9), but u (1), u (3), u (5), u ( 7) and u (9), and the sound field data is p (1,5) which is adopted as the coupled data among the 25 points indicated by black circles in FIG. 5 (a). ), p (2,5), p (3,5), p (4,5), p (5,5).
[0060]
As can be seen from the above explanation, the data necessary for solving the coupled vibration phenomenon is only the part where the sound field and the structure are in contact (structure surface). Furthermore, the structure data overlaps with the sound field data. Just the data.
[0061]
FIG. 6A shows an arrangement of data read in the process 10 of the flowchart of FIG. 3 based on the above description (this data is stored in the memory of the computer). The eigenvalue of the sound field is represented by λs_n, and the eigenmode selected as in the data D3 is represented by the sound pressure p (i, j) _n. The eigenvalue of the structure is represented by λp_n, and the eigenmode is represented by displacement u (i) _n. n represents the order of n eigenvalues to be input. FIG. 6B shows a specific numerical example.
[0062]
The eigenvalue λs_n of the sound field and the eigenmode p (i, j) _n of the sound field are obtained in advance by the equation (3), and the eigenvalue λp_n of the structure and the eigenmode u (i) _n of the structure are obtained in advance by the equation (4). It is done. At this time, as described above, the K orthogonal condition is used as the normalization condition.
[0063]
Specifically, referring to FIG. 6A, there are a large number of eigenvalues of the sound field, but five (λs_1, λs_2, λs_3, λs_4, λs_5) are shown for simplicity. The eigenmodes paired with the eigenvalue λs_1 are p (1,5) _1, p (2,5) _1,..., P (5,5) _1, which are the vertical lines in FIG. It is an eigenmode represented by the sound pressures of five points arranged in the.
[0064]
FIG. 6 (b) shows specific numerical examples for each of FIG. 6 (a).
[0065]
FIG. 3 is a processing flowchart of the embodiment. This process is executed by a computer. In the process 10, the eigenvalues of the structural system and the sound field system calculated in advance and the eigenmode selected as indicated by D3 in FIG. 5 are read into the computer or from the work area of the memory (FIG. 6 (a), (See (b)). 6 (a) and 6 (b), the upper row is a unique pair of sound field systems, and the lower row is a unique pair of structural systems. If the value of the unique pair is stored in the FD or the like, it may be read from the FD into the work area, and if it is stored in the hard disk, it may be read into the work area of the memory. The unique pair value may be transmitted from another computer.
[0066]
In the process 20, the eigenvalue λ (reference eigenvalue) used as a calculation reference (either the structure or the sound field), the allowable error e, and the iterative calculation abort count m are input to the computer. This input is generally performed manually, but may be input from the FD or input online. The eigenvalue λ is between the frequency ω and λ = 1 / ω² There is a relationship. The allowable error is a value for aborting the calculation if the difference between the eigenvalue of the n-1th calculation result and the eigenvalue of the nth calculation result is smaller than the specified allowable error e as a result of the approximate calculation. The iterative calculation abort count m is not within the allowable error range expected by the convergence calculation (that is, as described above, the difference between the (n−1) th value and the nth value is smaller than the allowable error e. (Even if not) to stop the calculation. Note that the variable n in the process has a different meaning from n representing the order of the n eigenvalues.
[0067]
In process 30, the plurality of unique pairs input in process 10 are rearranged on the memory (work area) using the unique value as a key. The rearrangement is performed in ascending order of the absolute value of the difference between the eigenvalue of each eigenpair and the reference eigenvalue λ read in the process 20. Sorting is performed for each of the structural system and the sound field system. After that, the eigenmode of the common node of the sound field and structure (the part marked with □ in FIG. 5) is selected and stored in the memory. (This can reduce the amount of storage. In the case of FIG. 5, the total number of nodes 25 in the sound field is 5, and the total number 9 in the structure is 5.)
[0068]
In general, when analyzing the noise in the passenger compartment of an automobile, the analysis is performed mainly on sounds of frequencies that are perceived as unpleasant to humans. In the example of FIG. 6, if the structure eigenvalue λs_4 is an eigenvalue of a frequency at which an unpleasant sound is generated (reference eigenvalue), the processing 30 first places λs_4 (this is the reference eigenvalue) first, and the remaining eigenvalues Are arranged in the order close to λs_4 (in order of decreasing absolute value of each eigenvalue and λs_4) on the memory.
[0069]
In process 40, the counter is set to = 0. In process 50, the counter is incremented by 1 (incremented). In process 60, the approximate calculation λ in equation (9)_i ⁽²ⁿ⁾ Are performed using the unique pairs in the order rearranged above. Equation (9) is a calculation formula for obtaining each coefficient in Equation (8). In equation (9), λ_i ⁽²ⁿ⁾To calculate (Equation (9-2)), λ_iAnd φ_iIs used. φ_iIs calculated using equation (9-3) or equation (9-4). At this time, λ_iAnd λ_j(I ≠ j) is used. λ_jJ is incremented by a counter when i is fixed (corresponding to n below). The order of the eigenvalues specified by this counter is the above-described order of rearrangement.
[0070]
In process 70, λ obtained in process 60_i ⁽²ⁿ⁾Is compared with the tolerance e (λ_i ⁽²ⁿ⁾Is the difference between the eigenvalue of the nth calculation result and the eigenvalue of the (n-1) th calculation result. λ_i ⁽²ⁿ⁾ Is larger than the allowable error e (NO in process 70), the process proceeds to process 72, and it is checked whether n (counter value) has reached m. If n has not reached m (NO in process 72), the process returns to process 50, the counter value is incremented, and approximate calculation is performed again (process 60).
λ_i ⁽²ⁿ⁾ Is smaller than the allowable error e, the process proceeds to step 80. λ_i ⁽²ⁿ⁾ Even if is not smaller than the allowable error e, if m matches n (when n = m) (YES in process 72), the process proceeds to process 74, indicating that the calculation has been aborted due to the limited number of times. The message shown in FIG. Since the approximate calculation is terminated in the middle, the calculation time is short, but convergence is ensured as described below.
[0071]
In process 80, the approximate eigenvalue λ = λ of the coupled system_i+ Λ_i ⁽²⁾+ Λ_i ^(Four)+ ... λ_i ⁽²ⁿ⁾Is calculated and output. It goes without saying that the approximate eigenmode is obtained in the same manner.
[0072]
The above processes 10 to 80 are executed with the parameter ε fixed to an appropriate value. If a desired or appropriate result cannot be obtained, it is desirable to change the parameter ε and repeat the above process again. This is equivalent to, for example, changing the material of the structure, and is useful for selecting the most appropriate material.
[0073]
Figures 4 (a) and 4 (b) show the convergence status when the vertical axis represents λ and the horizontal axis represents the number of calculations for one of the terms in the second and subsequent terms of equation (8-1). FIG. If the eigenvalues are calculated without rearrangement in the order close to λ, the error varies greatly as shown by the curve (b) in FIG. On the other hand, when rearrangement is performed, the curve tends to converge every time it is reliably followed as shown by the curves (a1), (a2), and (a3) in FIG.
[0074]
Specific examples are shown below.
[0075]
First example (coupled eigenvalues from structure)
Tolerance e = 10^-6And
i) Case of using realistic parameters
D = 19.230769, ρ₀= 1.2930000, ρ₁= 7850, c = 340, m = 1, i = 1
Theoretical solution λ_i(1) = 102.06188, λ_i(0) = 102.05000
Where the theoretical solution λ_i(1) is calculated by the type 1 of the following formula (11).
Theoretical solution λ_i(0) is calculated by equation (3).
1 approximation result = λ_i(0) + λ_i ⁽²⁾= 102.06188,
| Λ_i ⁽²⁾| = | 0.01188 |> e
Approximate result twice = λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)= 102.06188,
| Λ_i ^(Four)| = | 0.0000000 | <e
3 times approximate result = λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)+ Λ_i ⁽⁶⁾= 102.06188,
| Λ_i ⁽⁶⁾| = | 0.0000000 | <e
4th approximation result = λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)+ Λ_i ⁽⁶⁾+ Λ_i ⁽⁸⁾= 102.06188,
| Λ_i ⁽⁸⁾| = | 0.0000000 | <e
[0076]
ii) Case of using unrealistic parameters
Theoretical solution λ_i(1) = 6.6944765756, λ_i(0) = 6.25
D = 2, ρ₀= 5, ρ₁= 50, c = 2.5, m = 1, i = 1
Where the theoretical solution λ_i(1) is calculated by Type 1 of Equation (11).
Theoretical solution λ_i(0) is calculated by equation (3).
1 approximation result = λ_i(0) + λ_i ⁽²⁾= 6.694667334361377,
| Λ_i ⁽²⁾｜ ＝ ｜ 0.44466733 ｜ ＞ e
Approximate result twice = λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)= 6.694462764982187,
| Λ_i ^(Four)| = | −0.000204569 |> e
3 times approximate result = λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)+ Λ_i ⁽⁶⁾= 6.694477554000371,
| Λ_i ⁽⁶⁾| = | 0.000014789 |> e
4 times approximate result
= Λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)+ Λ_i ⁽⁶⁾+ Λ_i ⁽⁸⁾= 6.694476478081498,
| Λ_i ⁽⁸⁾| = | 0.0000010759 | ≒ e
[0077]
Second example (coupled eigenvalue from sound field)
Tolerance e = 10^-FourAnd
Theoretical solution λ_i(1) = 0.069251599, λi (0) = 0.07111111
Where the theoretical solution λ_i(1) is calculated by Type 2 of Equation (11).
Theoretical solution λ_i(0) is calculated by equation (2).
1 approximation result = λ_i(0) + λ_i ⁽²⁾= 6.9075922324487507 × 10^-2,
| Λ_i ⁽²⁾| = | −2.035188 × 10^-3|> E,
Approximate result twice = λ_i(0) + λ_i ⁽²⁾+ Λ_i ^(Four)= 6.9265894614179742 × 10^-2,
| Λ_i ^(Four)｜ ＝ ｜ −4.93329 × 10^-Four| ≒ e
[0078]
## EQU11 ##

[0079]
FIG. 7 shows a configuration side of a structure design support system (or a coupled system vibration analyzer), which is realized by a computer system.
[0080]
A processing device (sorting means, processing means) 1 is a computer main body, and stores a program 1a for executing the processing of steps 10 to 80 in FIG. 3 and a memory 1b for storing data as shown in FIG. ing.
[0081]
The input device (input means) 2 is realized by a keyboard, a mouse, an FD drive, a communication device, etc., and is used for inputting data in steps 10 and 20.
[0082]
The output device (output means) 3 is realized by a display device, a printer, an FD drive or the like, and is used for eigenvalue output in step 80.
[Brief description of the drawings]
FIG. 1 is a sectional view of an automobile and shows an example of a real problem suitable for the present invention.
FIG. 2 is a diagram obtained by modeling FIG. 1, in which (a) shows a three-dimensional model and (b) shows a two-dimensional model.
FIG. 3 is a processing flowchart showing a processing procedure in a computer.
FIGS. 4A and 4B are graphs showing the convergence status of operations, where FIG. 4A shows a case where convergence occurs, and FIG. 4B shows a case where convergence does not occur.
FIGS. 5A and 5B are image diagrams showing common nodes of a sound field system and a structural system.
FIGS. 6A and 6B are diagrams showing a storage state of a unique pair of a sound field system and a structural system on a memory.
FIG. 7 is a block diagram illustrating a configuration example of a structure design support system or a coupled vibration analysis system.
[Explanation of symbols]
1 Processing device
1a program
1b memory
2 input devices
3 Output device

Claims (6)

The input means uses K as the stiffness matrix and φ _i as a normalization condition from the uncoupled vibration equations of the sound field and structure in the region where vibration is to be analyzed. Is a matrix of eigenmodes , and φ _i ^T is a transposed matrix of eigen modes φ _i , and the vibration to be analyzed is calculated using a K orthogonal condition that is a condition in which φ _i ^T Kφ _i is equal to 1. eigenvalues and eigenmodes unique pair in a non-coupled system of the sound field and structure of the region, accepts an input to a computer and stored in a memory,
The sorting means sorts the eigenvalues and eigenmode eigenpairs stored in the memory in the memory in order of decreasing absolute value of the difference between the eigenvalues in the eigenpairs and the given reference eigenvalue. ,
The eigenvalues and eigenmodes in the coupled system of the sound field and structure in the region to be analyzed are obtained from the discretized asymmetric coupled system vibration equation, and the processing means is a series of coupled parameters ε of the coupled system. and the coefficients in the deployable by perturbation method expressed in a coefficient expressed by using the eigenvalues and eigenmodes in Hiren formation system, when calculating by the computer, the eigenvalues in non-coupled systems are arranged on the memory And the eigenpairs of the eigenmodes according to the order of the calculation, the coefficients are calculated in the order of the low-order term coefficients to the high-order term coefficients in the expansion equation , and finally the eigenvalue and eigenmode in the coupled system are calculated. to derive a unique pair of,
Structure design support method. The input means accepts the designation of the reference characteristic value among the eigenvalues of the specific pairs of eigenvalues and eigenmodes in non Coupled System stored in the memory of the computer, the method according to claim 1. The above input means accepts an input of tolerance e to the computer,
Said processing means, in calculation of the coefficient by the computer, stops the calculation when the difference between the results previous calculation result and the current calculated is smaller than the allowable error e, according to claim 1 or 2 The method described in 1. The above input means accepts an input to the computer with the number m of aborts ,
Said processing means, in calculation of the coefficient by the computer, even when the difference between the results previous calculation result and the currently calculated is larger than the allowable error e, the number of times of iterative calculation is to the abort count m when it reaches stops calculation method according to claim 3. As a normalization condition from the uncoupled vibration equations of the sound field and structure in the region where vibration is to be analyzed , K is a stiffness matrix, φ _i Is a matrix of eigenmodes , and φ _i ^T is a transposed matrix of eigen modes φ _i , and the vibration to be analyzed is calculated using a K orthogonal condition that is a condition in which φ _i ^T Kφ _i is equal to 1. An input means for accepting and storing in memory a eigenvalue-eigenmode eigenpair pair in an uncoupled sound field and structure of the region ;
Sorting means for sorting eigenvalues and eigenmode eigenpairs stored in the memory in the memory in order of decreasing absolute values of eigenvalues in the eigenpairs and given reference eigenvalues. ,
Eigenvalues and eigenmodes in the coupled acoustic field and structure in the above region to be analyzed for vibration, obtained from the discretized asymmetric coupled system vibration equation, and the series of coupled parameters ε of coupled system and uncoupled the coefficients in deployable by perturbation method expressed in a coefficient expressed by using the eigenvalues and eigenmodes in the system, when calculating by the computer, the eigenvalues and eigenmodes in non coupled system in a row on the memory By using eigenpairs according to their order of arrangement, the coefficients are calculated in the order of low-order term coefficients to high-order term coefficients in the expansion equation , and finally the eigenvalues and eigenmode eigenpairs in the coupled system are calculated. Processing means for deriving, and output means for outputting eigenvalue-eigenmode eigenpairs in the derived coupled system,
Design support system for structures with The input means uses K as the stiffness matrix and φ _i as a normalization condition from the uncoupled vibration equations of the sound field and structure in the region where vibration is to be analyzed. Is a matrix of eigenmodes , and φ _i ^T is a transposed matrix of eigen modes φ _i , and the vibration should be analyzed using the K orthogonal condition, which is a condition that φ _i ^T Kφ _i is equal to 1. stored in the memory accepts inputs of eigenvalues and eigenmodes unique pair in the sound field and the non-coupled system structure of the region,
The sorting means sorts the eigenvalues and eigenmode eigenpairs stored in the memory in the memory in order of decreasing absolute value of the difference between the eigenvalues in the eigenpairs and the given reference eigenvalue. ,
The eigenvalues and eigenmodes in the coupled system of the sound field and structure in the region to be analyzed are obtained from the discretized asymmetric coupled system vibration equation, and the processing means is a series of coupled parameters ε of the coupled system. and the coefficients in the deployable by perturbation method expressed in a coefficient expressed by using the eigenvalues and eigenmodes in Hiren formation system, when calculating the eigenvalues in non-coupled systems are arranged on the memory-specific By using eigenpairs of modes according to their order of arrangement, the coefficients are calculated in the order of low-order term coefficients to high-order term coefficients in the expansion equation , and finally the eigenvalues and eigenmode eigenvalues in the coupled system are calculated. to derive a pair, said input means, for controlling said rearranging means and the processing means, the program for the coupled system vibration analysis.

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