マルチボディシステムの運動方程式

このセクションでは、運動方程式について説明します。

次の方法で定義した機構システムを検討します。
  • 設計変数のセット b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@
  • 状態のセット x ( b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamiEaiaacIcacaWGIbGaaiykaaaa@403A@
  • 制約条件のセット ϕ ( x ( b ) ,   b ,   t ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqy1dyMaaiikaiaadIhacaGGOaGaamOyaiaa cMcacaGGSaGaaeiiaiaadkgacaGGSaGaaeiiaiaadshacaGGPaGaey ypa0JaaGimaaaa@49A2@
状態のセットxは、以下の値で構成されます:
  • 変位
  • 速度
  • Lagrange乗数:
  • ユーザー微分方程式に使用する、ユーザー変数などの補助変数と状態
この表記に基づいて、機構システムの運動方程式は次のように表現できます:
  • x ˙ ( t ) = f ( x ( b ) , b , t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGabmiEayaacaGaaiikaiaadshacaGGPaGaeyyp a0JaamOzamaabmaabaGaamiEaiaacIcacaWGIbGaaiykaiaacYcaca WGIbGaaiilaiaadshaaiaawIcacaGLPaaaaaa@4A4C@
  • ϕ ( x ( b ) , b , t ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqy1dyMaaiikaiaadIhacaGGOaGaamOyaiaa cMcacaGGSaGaamOyaiaacYcacaWG0bGaaiykaiabg2da9iaaicdaaa a@485B@
この機構システムの初期状態は次のように指定されます:(1)
x ( t 0 ) = x 0 ( b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamiEaiaacIcacaWG0bWaaSbaaSqaaiaaicda aeqaaOGaaiykaiabg2da9iaadIhadaWgaaWcbaGaaGimaaqabaGcca GGOaGaamOyaiaacMcaaaa@4670@

表記を簡素化するために、以降は x ( b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamiEaiaacIcacaWGIbGaaiykaaaa@403B@ を単に x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamiEaaaa@3DFB@ と表現します。状態 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamiEaaaa@3DFB@ は、上記の方程式をすべての時点で満たす必要があります。