剛体

剛体は、1つのメイン節点とそれに付随する複数のセカンダリ節点で定義します。当初のメイン節点位置に慣性と質量が付加されます。つづいて、メイン節点の質量とすべてのセカンダリ節点の質量を考慮した質量中心へメイン節点が移動します。 図 1 に理想化した剛体を示します。


図 1. 理想化した剛体

剛体の質量

剛体の質量は次の式で求められます。(1)
m = m M + I m I
剛体の質量中心は次の式で定義できます。(2)
x G = m M x M + m I x I m
(3)
y G = m M y M + m I y I m
(4)
z G = m M z M + m I z I m
ここで、
m M
メイン節点の質量
m I
セカンダリ節点の質量
x G y G z G
質量中心の座標

剛体の慣性

剛体の慣性を構成する6つの成分は次の式で求められます。(5)
I xx = J xx M + m M ( ( y M y G ) 2 + ( z M z G ) 2 )+ i ( I xx i + m i ( ( y i y G ) 2 + ( z i z G ) 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG4bGaamiEaaqabaGccqGH9aqpcaWGkbWaa0baaSqaaiaa dIhacaWG4baabaGaamytaaaakiabgUcaRiaad2gadaahaaWcbeqaai aad2eaaaGcdaqadaqaamaabmaabaGaamyEamaaBaaaleaacaWGnbaa beaakiabgkHiTiaadMhadaWgaaWcbaGaam4raaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaaiaadQha daWgaaWcbaGaamytaaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaadE eaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGccaGL OaGaayzkaaGaey4kaSYaaabuaeaadaqadaqaaiaadMeadaqhaaWcba GaamiEaiaadIhaaeaacaWGPbaaaOGaey4kaSIaamyBamaaCaaaleqa baGaamyAaaaakmaabmaabaWaaeWaaeaacaWG5bWaaSbaaSqaaiaadM gaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGhbaabeaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaam OEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadQhadaWgaaWcbaGa am4raaqabaaakiaawIcacaGLPaaadaahaaWcbeqaamaaCaaameqaba GaaGOmaaaaaaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqaaiaa dMgaaeqaniabggHiLdaaaa@7007@
(6)
I yy = J yy M + m M ( ( x M x G ) 2 + ( z M z G ) 2 )+ i ( I yy i + m i ( ( x i x G ) 2 + ( z i z G ) 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG5bGaamyEaaqabaGccqGH9aqpcaWGkbWaa0baaSqaaiaa dMhacaWG5baabaGaamytaaaakiabgUcaRiaad2gadaahaaWcbeqaai aad2eaaaGcdaqadaqaamaabmaabaGaamiEamaaBaaaleaacaWGnbaa beaakiabgkHiTiaadIhadaWgaaWcbaGaam4raaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaaiaadQha daWgaaWcbaGaamytaaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaadE eaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGccaGL OaGaayzkaaGaey4kaSYaaabuaeaadaqadaqaaiaadMeadaqhaaWcba GaamyEaiaadMhaaeaacaWGPbaaaOGaey4kaSIaamyBamaaCaaaleqa baGaamyAaaaakmaabmaabaWaaeWaaeaacaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGhbaabeaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaam OEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadQhadaWgaaWcbaGa am4raaqabaaakiaawIcacaGLPaaadaahaaWcbeqaamaaCaaameqaba GaaGOmaaaaaaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqaaiaa dMgaaeqaniabggHiLdaaaa@7009@
(7)
I zz = J zz M + m M ( ( x M x G ) 2 + ( y M y G ) 2 )+ i ( I zz i + m i ( ( x i x G ) 2 + ( y i y G ) 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG6bGaamOEaaqabaGccqGH9aqpcaWGkbWaa0baaSqaaiaa dQhacaWG6baabaGaamytaaaakiabgUcaRiaad2gadaahaaWcbeqaai aad2eaaaGcdaqadaqaamaabmaabaGaamiEamaaBaaaleaacaWGnbaa beaakiabgkHiTiaadIhadaWgaaWcbaGaam4raaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaaiaadMha daWgaaWcbaGaamytaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaadE eaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGccaGL OaGaayzkaaGaey4kaSYaaabuaeaadaqadaqaaiaadMeadaqhaaWcba GaamOEaiaadQhaaeaacaWGPbaaaOGaey4kaSIaamyBamaaCaaaleqa baGaamyAaaaakmaabmaabaWaaeWaaeaacaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGhbaabeaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaam yEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadMhadaWgaaWcbaGa am4raaqabaaakiaawIcacaGLPaaadaahaaWcbeqaamaaCaaameqaba GaaGOmaaaaaaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqaaiaa dMgaaeqaniabggHiLdaaaa@700C@
(8)
I xy = J xy M + m M ( ( x M x G ) + ( y M y G ) )+ i ( I xy i m i ( ( x i x G ) + ( y i y G ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG4bGaamyEaaqabaGccqGH9aqpcaWGkbWaa0baaSqaaiaa dIhacaWG5baabaGaamytaaaakiabgUcaRiaad2gadaahaaWcbeqaai aad2eaaaGcdaqadaqaamaabmaabaGaamiEamaaBaaaleaacaWGnbaa beaakiabgkHiTiaadIhadaWgaaWcbaGaam4raaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaaaakiabgUcaRmaabmaabaGaamyEamaaBaaa leaacaWGnbaabeaakiabgkHiTiaadMhadaWgaaWcbaGaam4raaqaba aakiaawIcacaGLPaaadaahaaWcbeqaaaaaaOGaayjkaiaawMcaaiab gUcaRmaaqafabaWaaeWaaeaacaWGjbWaa0baaSqaaiaadIhacaWG5b aabaGaamyAaaaakiabgkHiTiaad2gadaahaaWcbeqaaiaadMgaaaGc daqadaqaamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaakiabgk HiTiaadIhadaWgaaWcbaGaam4raaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaaaakiabgUcaRmaabmaabaGaamyEamaaBaaaleaacaWGPb aabeaakiabgkHiTiaadMhadaWgaaWcbaGaam4raaqabaaakiaawIca caGLPaaadaahaaWcbeqaamaaCaaameqabaaaaaaaaOGaayjkaiaawM caaaGaayjkaiaawMcaaaWcbaGaamyAaaqab0GaeyyeIuoaaaa@6D1E@
(9)
I yz = J yz M + m M ( ( y M y G ) + ( z M z G ) )+ i ( I yz i m i ( ( y i y G ) + ( z i z G ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG5bGaamOEaaqabaGccqGH9aqpcaWGkbWaa0baaSqaaiaa dMhacaWG6baabaGaamytaaaakiabgUcaRiaad2gadaahaaWcbeqaai aad2eaaaGcdaqadaqaamaabmaabaGaamyEamaaBaaaleaacaWGnbaa beaakiabgkHiTiaadMhadaWgaaWcbaGaam4raaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaaaakiabgUcaRmaabmaabaGaamOEamaaBaaa leaacaWGnbaabeaakiabgkHiTiaadQhadaWgaaWcbaGaam4raaqaba aakiaawIcacaGLPaaadaahaaWcbeqaaaaaaOGaayjkaiaawMcaaiab gUcaRmaaqafabaWaaeWaaeaacaWGjbWaa0baaSqaaiaadMhacaWG6b aabaGaamyAaaaakiabgkHiTiaad2gadaahaaWcbeqaaiaadMgaaaGc daqadaqaamaabmaabaGaamyEamaaBaaaleaacaWGPbaabeaakiabgk HiTiaadMhadaWgaaWcbaGaam4raaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaaaakiabgUcaRmaabmaabaGaamOEamaaBaaaleaacaWGPb aabeaakiabgkHiTiaadQhadaWgaaWcbaGaam4raaqabaaakiaawIca caGLPaaadaahaaWcbeqaamaaCaaameqabaaaaaaaaOGaayjkaiaawM caaaGaayjkaiaawMcaaaWcbaGaamyAaaqab0GaeyyeIuoaaaa@6D2C@
(10)
I xz = J xz M + m M ( ( x M x G ) + ( z M z G ) )+ i ( I xz i m i ( ( x i x G ) + ( z i z G ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG4bGaamOEaaqabaGccqGH9aqpcaWGkbWaa0baaSqaaiaa dIhacaWG6baabaGaamytaaaakiabgUcaRiaad2gadaahaaWcbeqaai aad2eaaaGcdaqadaqaamaabmaabaGaamiEamaaBaaaleaacaWGnbaa beaakiabgkHiTiaadIhadaWgaaWcbaGaam4raaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaaaakiabgUcaRmaabmaabaGaamOEamaaBaaa leaacaWGnbaabeaakiabgkHiTiaadQhadaWgaaWcbaGaam4raaqaba aakiaawIcacaGLPaaadaahaaWcbeqaaaaaaOGaayjkaiaawMcaaiab gUcaRmaaqafabaWaaeWaaeaacaWGjbWaa0baaSqaaiaadIhacaWG6b aabaGaamyAaaaakiabgkHiTiaad2gadaahaaWcbeqaaiaadMgaaaGc daqadaqaamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaakiabgk HiTiaadIhadaWgaaWcbaGaam4raaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaaaakiabgUcaRmaabmaabaGaamOEamaaBaaaleaacaWGPb aabeaakiabgkHiTiaadQhadaWgaaWcbaGaam4raaqabaaakiaawIca caGLPaaadaahaaWcbeqaamaaCaaameqabaaaaaaaaOGaayjkaiaawM caaaGaayjkaiaawMcaaaWcbaGaamyAaaqab0GaeyyeIuoaaaa@6D25@
ここで、
I i j
ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaadQ gaaaa@37D3@ 方向の回転慣性のモーメント
J i j M
メイン節点に付加された慣性

剛体の荷重とモーメントの計算

剛体に作用する荷重とモーメントは次の式で求められます。(11)
F = F M + i F i
(12)
M = M M + i M i + i S i G × F i
ここで、
F M
メイン節点における荷重ベクトル
F i
セカンダリ節点における荷重ベクトル
M M
メイン節点におけるモーメントベクトル
M i
セカンダリ節点におけるモーメントベクトル
G
セカンダリ節点から質量中心へ向かうベクトル

これらの式を直交成分に解くと、次のように線加速度と回転加速度を計算できます。

線加速度(13)
γ i = F i m
回転加速度(14)
I 1 α 1 = M 1 ( I 3 I 2 ) ω 2 ω 3
(15)
I 2 α 2 = M 2 ( I 1 I 3 ) ω 1 ω 3
(16)
I 3 α 3 = M 3 ( I 2 I 1 ) ω 1 ω 2
ここで、
I i
剛体の主慣性モーメント
α 1
主慣性フレーム(参照フレーム)での回転加速度
ω i
主慣性フレーム(参照フレーム)での回転速度
M i
主慣性フレーム(参照フレーム)でのモーメント

時間積分

時間積分を実行して、次のようにメイン節点における剛体の速度を求めます。(17)
ν ( t + Δ t 2 ) = ν ( t Δ t 2 ) + γ ( t ) Δ t
(18)
ω ( t + Δ t 2 ) = ω ( t Δ t 2 ) + α ( t ) Δ t

v は線速度ベクトルです。回転速度は局所参照フレームで計算します。

セカンダリ節点の速度は次の式で求められます。(19)
ν i = ν M + S i G x ω
(20)
ω i = ω M

境界条件

セカンダリ節点に指定した境界条件は無視されます。剛体の境界条件は、メイン節点に指定した境界条件のみとなります。

各セカンダリ節点には、すべての方向に運動条件が適用されます。これ以外の運動条件をセカンダリ節点に指定することはできません。

メイン節点には運動条件が適用されません。ただし、回転速度は局所参照フレームで計算します。この参照フレームでは、回転を適用するどのオプション(適用する速度、回転、剛結など)も使用できません。

唯一の例外は、特殊な処理の実行対象とする回転境界条件に関するオプションです。回転剛性を設定したシェル、ビーム、スプリングをメイン節点に結合することもできません。