剛性定式化

スプリングの剛性は、それぞれ異なって定義された自由度で複数の方法で定義されます。

力とモーメント

線形剛性と減衰の挙動

最もシンプルな定式化は線形弾性スプリング剛性で、ここでは内力が相対変位に比例します。この場合、定数の剛性パラメータ K i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadMgaaeqaaaaa@3848@ およびオプションの減衰パラメータ C i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadMgaaeqaaaaa@3848@ のみが入力されます。

線形剛性の場合、力とモーメントは:


  • F ( δ ) = K i δ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiTdq2aaWbaaSqabeaacaWGPbaaaaaa@4012@
  • M ( θ ) = K i θ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGnbWaae WaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiUde3aaWbaaSqabeaacaWGPbaaaaaa@403B@
線形ダッシュポットの場合、力とモーメントは:


  • F ( δ ) = C i δ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaWGdbWaaSbaaSqa aiaadMgaaeqaaOGafqiTdqMbaiaadaahaaWcbeqaaiaadMgaaaaaaa@4013@
  • M ( θ ) = C i θ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGnbWaae WaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpcaWGdbWaaSbaaSqa aiaadMgaaeqaaOGafqiUdeNbaiaadaahaaWcbeqaaiaadMgaaaaaaa@403C@
線形ダッシュポットおよびダッシュポットの場合、力とモーメントは:


  • F ( δ ) = K i δ i + C i δ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiTdq2aaWbaaSqabeaacaWGPbaaaOGaey4kaS Iaam4qamaaBaaaleaacaWGPbaabeaakiqbes7aKzaacaWaaWbaaSqa beaacaWGPbaaaaaa@45B3@
  • M ( θ ) = K i θ i + C i θ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGnbWaae WaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiUde3aaWbaaSqabeaacaWGPbaaaOGaey4kaS Iaam4qamaaBaaaleaacaWGPbaabeaakiqbeI7aXzaacaWaaWbaaSqa beaacaWGPbaaaaaa@45ED@

非線形挙動

スプリングの力とモーメントは次のように計算されます:(1)
F i ( δ i ) = f ( δ i A s c a l e i ) [ A i + B i ln ( max ( 1 , | δ ˙ i D i | ) ) + E i g ( δ ˙ i F i ) ] + C i δ ˙ i + H s c a l e i h ( δ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiGacA eadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabes7aKnaaCaaaleqa baGaamyAaaaaaOGaayjkaiaawMcaaiabg2da9iGacAgadaqadaqaam aalaaabaGaeqiTdq2aaWbaaSqabeaacaWGPbaaaaGcbaGaamyqaiaa dohacaWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaadMgaaeqaaa aaaOGaayjkaiaawMcaamaadmaabaGaamyqamaaBaaaleaacaWGPbaa beaakiabgUcaRiaadkeadaWgaaWcbaGaamyAaaqabaGcciGGSbGaai OBamaabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiil amaaemaabaWaaSaaaeaacuaH0oazgaGaamaaCaaaleqabaGaamyAaa aaaOqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaaGccaGLhWUaayjc SdaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaey4kaSIaamyramaaBa aaleaacaWGPbaabeaakiGacEgadaqadaqaamaalaaabaGafqiTdqMb aiaadaahaaWcbeqaaiaadMgaaaaakeaacaWGgbWaaSbaaSqaaiaadM gaaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaiaaywW7 cqGHRaWkcaaMe8Uaam4qamaaBaaaleaacaWGPbaabeaakiqbes7aKz aacaWaaWbaaSqabeaacaWGPbaaaOGaey4kaSIaamisaiaadohacaWG JbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaadMgaaeqaaOGaciiAam aabmaabaWaaSaaaeaacuaH0oazgaGaamaaCaaaleqabaGaamyAaaaa aOqaaiaadAeadaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaa aaaaa@82E9@

ここで、 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@374C@ は並進自由度: 1、2、3

(2)
M i ( θ i ) = f ( θ i A s c a l e i ) [ A i + B i ln ( max ( 1 , | θ ˙ i D i | ) ) + E i g ( θ ˙ i F i ) ] + C i θ ˙ i + H s c a l e i h ( θ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiGac2 eadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabeI7aXnaaCaaaleqa baGaamyAaaaaaOGaayjkaiaawMcaaiabg2da9iGacAgadaqadaqaam aalaaabaGaeqiUde3aaWbaaSqabeaacaWGPbaaaaGcbaGaamyqaiaa dohacaWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaadMgaaeqaaa aaaOGaayjkaiaawMcaamaadmaabaGaamyqamaaBaaaleaacaWGPbaa beaakiabgUcaRiaadkeadaWgaaWcbaGaamyAaaqabaGcciGGSbGaai OBamaabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiil amaaemaabaWaaSaaaeaacuaH4oqCgaGaamaaCaaaleqabaGaamyAaa aaaOqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaaGccaGLhWUaayjc SdaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaey4kaSIaamyramaaBa aaleaacaWGPbaabeaakiGacEgadaqadaqaamaalaaabaGafqiUdeNb aiaadaahaaWcbeqaaiaadMgaaaaakeaacaWGgbWaaSbaaSqaaiaadM gaaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaiaaywW7 cqGHRaWkcaaMb8UaaGjbVlaadoeadaWgaaWcbaGaamyAaaqabaGccu aH4oqCgaGaamaaCaaaleqabaGaamyAaaaakiabgUcaRiaadIeacaWG ZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaWGPbaabeaaki GacIgadaqadaqaamaalaaabaGafqiUdeNbaiaadaahaaWcbeqaaiaa dMgaaaaakeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkai aawMcaaaaaaa@84E0@

ここで、 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@374C@ は回転自由度: 4、5、6

力とモーメントの式内の変数は以下を表します:

f ( δ i A s c a l e i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaae WaaeaadaWcaaqaaiabes7aKnaaCaaaleqabaGaamyAaaaaaOqaaiaa dgeacaWGZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaWGPb aabeaaaaaakiaawIcacaGLPaaaaaa@4238@ スプリングの力 vs fct_IDの変位関数入力1i

f ( θ i A s c a l e i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaae WaaeaadaWcaaqaaiabeI7aXnaaCaaaleqabaGaamyAaaaaaOqaaiaa dgeacaWGZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaWGPb aabeaaaaaakiaawIcacaGLPaaaaaa@4249@ スプリングの力 vs fct_ID1iの回転関数入力

A i B i D i E i  and  F i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadgeadaWgaaWcbaGaamyAaaqabaGccaqGSaGaaeiiaiaadkea daWgaaWcbaGaamyAaaqabaGccaqGSaGaaeiiaiaadseadaWgaaWcba GaamyAaaqabaGccaqGSaGaaeiiaiaadweadaWgaaWcbaGaamyAaaqa baGccaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaadAeadaWgaaWcba GaamyAaaqabaaaaa@4ACC@ スケーリング係数

ln ( max ( 1 , | δ ˙ i D i | ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGSbGaai OBamaabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiil amaaemaabaWaaSaaaeaacuaH0oazgaGaamaaCaaaleqabaGaamyAaa aaaOqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaaGccaGLhWUaayjc SdaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@4785@ 速度の増大に伴いスプリング剛性をスケーリングする対数関数


図 1. パラメータ入力

g ( δ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbWaae WaaeaadaWcaaqaaiqbes7aKzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DA6@ fct_IDの線形入力の関数として剛性をスケーリング2i

g ( θ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbWaae WaaeaadaWcaaqaaiqbeI7aXzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DB7@ fct_IDの回転速度入力の関数として剛性をスケーリング2i

この入力は、スプリング剛性の非線形ひずみ速度効果をモデル化するために使用できます。


図 2. g関数入力

C i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgaaeqaaaaa@3840@ 速度の関数としてスプリング剛性を増大させるために使用される線形減衰係数

h ( δ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGObWaae WaaeaadaWcaaqaaiqbes7aKzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DA7@ または h ( θ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGObWaae WaaeaadaWcaaqaaiqbeI7aXzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DB8@ fct_IDとしての線形減衰関数4i

速度の関数としての線形または非線形減衰は、線形減衰係数またはユーザー定義の関数のいずれかを用いてスプリングの力に付与することも可能です。


図 3. h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ 関数入力

関数 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ は共に、スプリングの減衰挙動を表します。ただし、 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ 関数はスプリング剛性関数 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ をスケーリングしますが、 h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ 関数は、スプリング剛性関数 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ に付与します。

時間ステップ

スプリングの時間ステップ:(3)
Δ t = M K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WG0bGaeyypa0ZaaOaaaeaadaWcaaqaaiaad2eaaeaacaWGlbaaaaWc beaaaaa@3C9A@
ダッシュポットの時間ステップ:(4)
Δ t = M 2 C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WG0bGaeyypa0ZaaSaaaeaacaWGnbaabaGaaGOmaiaadoeaaaaaaa@3C29@
スプリングとダッシュポットの時間ステップ:(5)
Δ t = C 2 + K M C K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WG0bGaeyypa0ZaaSaaaeaadaGcaaqaaiaadoeadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaWGlbGaeyyXICTaamytaaWcbeaakiabgkHiTi aadoeaaeaacaWGlbaaaaaa@4306@
ここで、
M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
粒子の質量
K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
線形剛性または max [ d F d δ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGTbGaai yyaiaacIhadaWadaqaamaalaaabaGaamizaiaadAeaaeaacaWGKbGa eqiTdqgaaaGaay5waiaaw2faaaaa@3F76@
C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
線形減衰または max [ g ( d δ d t ) ( d δ d t ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGTbGaai yyaiaacIhadaWadaqaamaalaaabaGaeyOaIyRaci4zamaabmaabaWa aSaaaeaacaWGKbGaeqiTdqgabaGaamizaiaadshaaaaacaGLOaGaay zkaaaabaGaeyOaIy7aaeWaaeaadaWcaaqaaiaadsgacqaH0oazaeaa caWGKbGaamiDaaaaaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaaaaa@4AFF@ (非線形減衰の場合)

非線形スプリングの場合、 K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ が時間ステップ計算および一定剛性に用いられます。 K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ がスプリングのプロパティで定義されていない場合、fct_ID1iの最大傾き( max [ d F d δ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGTbGaai yyaiaacIhadaWadaqaamaalaaabaGaamizaiaadAeaaeaacaWGKbGa eqiTdqgaaaGaay5waiaaw2faaaaa@3F76@ )が時間ステップの計算に自動的に用いられます。挙動は、減衰剛性 C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@ についても同じです。