Rigid Body

A rigid body is defined by a main node and its associated secondary nodes. Mass and inertia may be added to the initial main node location. The main node is then moved to the center of mass, taking into account the main node and all secondary node masses. Figure 1 shows an idealized rigid body.


Figure 1. Idealized Rigid Body

Rigid Body Mass

The mass of the rigid body is calculated by:(1)
m = m M + I m I
The rigid body's center of mass is defined by:(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
Where,
m M
Main node mass
m I
Secondary node masses
x G , y G , z G
Coordinates of the mass center

Rigid Body Inertia

The six components of inertia of a rigid body are computed by:(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@
Where,
I i j
Moment of rotational inertia in the ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaadQ gaaaa@37D3@ direction
J i j M
Main node added inertia

Rigid Body Force And Moment Computation

The forces and moments acting on the rigid body are calculated by:(11)
F = F M + i F i
(12)
M = M M + i M i + i S i G × F i
Where,
F M
Force vector at the main node
F i
Force vector at the secondary nodes
M M
Moment vector at the main node
M i
Moment vector at the secondary nodes
G
Vector from secondary node to the center of mass

Resolving these into orthogonal components, the linear and rotational acceleration may be computed as:

Linear Acceleration(13)
γ i = F i m
Rotational Acceleration(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
Where,
I i
Principal moments of inertia of the rigid body
α 1
Rotational accelerations in the principal inertia frame (reference frame)
ω i
Rotational velocity in the principal inertia frame (reference frame)
M i
Moments in the principal inertia frame (reference frame)

Time Integration

Time integration is performed to find velocities of the rigid body at the main node:(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

Where, v is the linear velocity vector. Rotational velocities are computed in the local reference frame.

The velocities of secondary nodes are computed by:(19)
ν i = ν M + S i G x ω
(20)
ω i = ω M

Boundary Conditions

The boundary conditions given to secondary nodes are ignored. The rigid body has the boundary conditions given to the main node only.

A kinematic condition is applied on each secondary node, for all directions. A secondary node is not allowed to have any other kinematic conditions.

No kinematic condition is applied on the main node. However, the rotational velocities are computed in a local reference frame. This reference frame is not compatible with all options imposing rotation such as imposed velocity, rotational, rigid link.

The only exception concerns the rotational boundary conditions for which a special treatment is carried out. Connecting shell, beam or spring with rotation stiffness to the main node, is not yet allowed either.