Flexible Body Generation

The following information is common to all flexible body generation methods and details the initial partitioning step involved.

Dynamic Reduction is used to reduce a finite element model of an elastic body to the interface degrees of freedom and a set of normal modes for inclusion as a flexible body in a multibody dynamics analysis.

For the purpose of deriving the reduced matrices, the displacement vector u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F5@ may be partitioned into displacements of inner, u o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDamaaBa aaleaacaWGHbaabeaaaaa@3807@ and outer, u a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDamaaBa aaleaacaWGHbaabeaaaaa@3807@ interface/attachment degrees of freedom.(1) u = [ u o u a ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2 da9maadmaabaqbaeqabiqaaaqaaiaahwhadaWgaaWcbaGaam4Baaqa baaakeaacaWH1bWaaSbaaSqaaiaadggaaeqaaaaaaOGaay5waiaaw2 faaaaa@3E3B@

Here, the subscript o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamyyaaaa@36DE@ denotes the inner degrees of freedom, and the a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamyyaaaa@36DE@ denotes the interface degrees of freedom (for example, ASET entry).

The interface nodes that are used in the component mode synthesis process for the construction of mode shapes should be coincidental to the set of force-bearing nodes in the subsequent multibody dynamics analysis. In a multibody dynamics model, the flexible body interacts with other components of the model through joints, constraints, or force elements, which are connected or applied on the nodes of the flexible body. Except for body forces due to gravity or acceleration of the flexible body, all nodes that are subject to constraint or applied forces in the multibody dynamics analysis are denoted as force-bearing nodes.

The purpose of specifying the interface nodes for CMS is mainly to account for the static deformation due to constraints or applied forces acting on the interface nodes. A huge number of eigenmodes is required if these static modes are omitted. The flexible deformations due to constraint forces, compared to the deformation due to the body inertia forces, are often dominant in most constrained models; therefore, the inclusion of all force-bearing nodes as interface nodes is an essential step to get accurate results from subsequent flexible multibody dynamics analysis.

The static equilibrium is given as:(2) [ K oo K oa K oa T K aa ][ u o u a ]=[ f o f a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeGacaaabaGaaC4samaaBaaaleaacaWGVbGaam4Baaqabaaakeaa caWHlbWaaSbaaSqaaiaad+gacaWGHbaabeaaaOqaaiaahUeadaqhaa WcbaGaam4BaiaadggaaeaacaWGubaaaaGcbaGaaC4samaaBaaaleaa caWGHbGaamyyaaqabaaaaaGccaGLBbGaayzxaaWaamWaaeaafaqabe GabaaabaGaaCyDamaaBaaaleaacaWGVbaabeaaaOqaaiaahwhadaWg aaWcbaGaamyyaaqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaae aafaqabeGabaaabaGaaCOzamaaBaaaleaacaWGVbaabeaaaOqaaiaa hAgadaWgaaWcbaGaamyyaaqabaaaaaGccaGLBbGaayzxaaaaaa@51CD@
Where,
K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4saaaa@36CB@
Stiffness matrix
f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4saaaa@36CB@
Force vector with the corresponding subscripts depicting the partitioning based on the inner/interface degrees of freedom
The eigenvalue problem for a normal modes analysis of the body using a diagonal mass matrix, M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4saaaa@36CB@ is illustrated as:(3) [ [ K oo K oa K oa T K aa ]λ[ M oo 0 0 M aa ] ][ A o A a ]=[ 0 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada WadaqaauaabeqaciaaaeaacaWHlbWaaSbaaSqaaiaad+gacaWGVbaa beaaaOqaaiaahUeadaWgaaWcbaGaam4BaiaadggaaeqaaaGcbaGaaC 4samaaDaaaleaacaWGVbGaamyyaaqaaiaadsfaaaaakeaacaWHlbWa aSbaaSqaaiaadggacaWGHbaabeaaaaaakiaawUfacaGLDbaacqGHsi slcqaH7oaBdaWadaqaauaabeqaciaaaeaacaWHnbWaaSbaaSqaaiaa d+gacaWGVbaabeaaaOqaaiaaicdaaeaacaaIWaaabaGaaCytamaaBa aaleaacaWGHbGaamyyaaqabaaaaaGccaGLBbGaayzxaaaacaGLBbGa ayzxaaWaamWaaeaafaqabeGabaaabaGaaCyqamaaBaaaleaacaWGVb aabeaaaOqaaiaahgeadaWgaaWcbaGaamyyaaqabaaaaaGccaGLBbGa ayzxaaGaeyypa0ZaamWaaeaafaqabeGabaaabaGaaGimaaqaaiaaic daaaaacaGLBbGaayzxaaaaaa@5C8A@

Where, A are the partitioned eigenvectors of the system. Only eigenvectors pertaining to the non-attachment degrees of freedom, A o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbaabeaaaaa@37E1@ , are used in the subsequent generation of the combined modal matrix.

The goal of the flexbody generation is to find a set of orthogonal modes A o r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ that represent the displacements u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F5@ of the reduced structure such that:(4) u = A o r t q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2 da9iaahgeadaWgaaWcbaGaam4BaiaadkhacaWG0baabeaakiaahgha aaa@3CD9@

Where, q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F5@ is the matrix of modal participation factors or modal coordinates which are to be determined by the analysis.

The Craig-Bampton and Craig-Chang methods of Dynamic Reduction are available.

Craig-Bampton Method

Normal modes analysis of the fixed-interface system yields the diagonal matrix of eigenvalues, D ω and the matrix of eigenmodes, A ω .

In the normal mode analysis, you can select the cut-off frequency or the number of modes to be solved. This determines the column dimension of A ω .

In addition, a static analysis is performed to generate the static modes. With constrained interface degrees of freedom, a unit displacement in each interface degree of freedom is applied while all other interface degrees of freedom are fixed. With unconstrained (freed) interface degrees of freedom, the inertia relief will be performed by applying a unit force in each interface degree of freedom. This yields the static displacement modes/matrix A s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGZbaabeaaaaa@37E5@ and the interface forces, f a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGHbaabeaaaaa@37F8@ .

Reduced modal stiffness, K r e d u c e d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa beaaaaa@3D76@ , and mass, M r e d u c e d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa beaaaaa@3D76@ , matrices are now generated as:

The displacement matrix is written as:(5) S = [ A ω A S ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4uaiabg2 da9maadmaabaqbaeqabeGaaaqaaiaahgeadaWgaaWcbaGaeqyYdCha beaaaOqaaiaahgeadaWgaaWcbaGaam4uaaqabaaaaaGccaGLBbGaay zxaaaaaa@3E7D@
Which yields,(6) K r e d u c e d = S T K S = [ D ω 0 0 f a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa beaakiabg2da9iaahofadaahaaWcbeqaaiaadsfaaaGccaWHlbGaaC 4uaiabg2da9maadmaabaqbaeqabiGaaaqaaiaahseadaWgaaWcbaGa eqyYdChabeaaaOqaaiaaicdaaeaacaaIWaaabaGaaCOzamaaBaaale aacaWGHbaabeaaaaaakiaawUfacaGLDbaaaaa@4B79@ (7) M r e d u c e d = S T M S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCytamaaBa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa beaakiabg2da9iaahofadaahaaWcbeqaaiaadsfaaaGccaWHnbGaaC 4uaaaa@4226@

An orthogonalization step that transforms S into a set of orthogonal modes A o r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ is now performed. See Eigenvector Orthonormalization for more information.

Craig-Chang Method

This method uses a system that is unconstrained (free-free) and therefore has six rigid body modes. Normal modes analysis of the system yields the diagonal matrix of eigenvalue, D ω and the matrix of eigenmodes, A ω .

In the normal mode analysis, you can select the cut-off frequency or the number of modes to be solved. This determines the column dimension of A ω . The eigenmodes A R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGsbaabeaaaaa@37C4@ associated with the rigid body modes will be normalized with respect to the mass matrix such that:(8) A R T M A R =I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaDa aaleaacaWGsbaabaGaamivaaaakiaah2eacaWHbbWaaSbaaSqaaiaa dkfaaeqaaOGaeyypa0JaaCysaaaa@3D2D@
An equilibrated load matrix f e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGLbaabeaaaaa@37FC@ is formed as:(9) f e = [ 1 M A R A R T ] f a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGLbaabeaakiabg2da9maadmaabaGaaGymaiabgkHiTiaa h2eacaWHbbWaaSbaaSqaaiaadkfaaeqaaOGaaCyqamaaDaaaleaaca WGsbaabaGaamivaaaaaOGaay5waiaaw2faaiaahAgadaWgaaWcbaGa amyyaaqabaaaaa@4405@
As seen in previous sections, f a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGLbaabeaaaaa@37FC@ is the force vector at the attachment points. In Craig-Chang method, the attachment point forces is a collection of force vectors which have otherwise all zero entries except a unit force along each degree of freedom of the interface nodes. The equilibrated load matrix f e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWHMbWaaSbaaSqaaiaadwgaaeqaaaaa@3ADC@ is applied in an inertia relief static analysis to remove the rigid body modes. The equilibrated load matrix is applied, as:(10) K A s = f ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWHlbGaaCyqamaaBaaaleaacaWGZbaabeaakiabg2da9iaahAga daWgaaWcbaGaeqyTdugabeaaaaa@3F6B@

The resulting modes A s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGLbaabeaaaaa@37FC@ are called the inertia relief attachment modes.

Reduced modal stiffness and mass matrices are now generated using the displacement matrix (combines static modes and eigenmodes):(11) S = [ A ω A S ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4uaiabg2 da9maadmaabaqbaeqabeGaaaqaaiaahgeadaWgaaWcbaGaeqyYdCha beaaaOqaaiaahgeadaWgaaWcbaGaam4uaaqabaaaaaGccaGLBbGaay zxaaaaaa@3E7D@
The reduced matrices are calculated as,(12) K reduced = S T KS=[ D ω A ω T f e f e T A ω A A T f a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaBa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa beaakiabg2da9iaahofadaahaaWcbeqaaiaadsfaaaGccaWHlbGaaC 4uaiabg2da9maadmaabaqbaeqabiGaaaqaaiaahseadaWgaaWcbaGa eqyYdChabeaaaOqaaiaahgeadaqhaaWcbaGaeqyYdChabaGaamivaa aakiaahAgadaWgaaWcbaGaamyzaaqabaaakeaacaWHMbWaa0baaSqa aiaadwgaaeaacaWGubaaaOGaaCyqamaaBaaaleaacqaHjpWDaeqaaa GcbaGaaCyqamaaDaaaleaacaWGbbaabaGaamivaaaakiaahAgadaWg aaWcbaGaamyyaaqabaaaaaGccaGLBbGaayzxaaaaaa@5811@ (13) M r e d u c e d = S T M S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCytamaaBa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa beaakiabg2da9iaahofadaahaaWcbeqaaiaadsfaaaGccaWHnbGaaC 4uaaaa@4226@

An orthogonalization step that transforms S into a set of orthogonal modes A o r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ is now performed. Refer to Eigenvector Orthonormalization for more information.

Eigenvector Orthonormalization

First, a new eigenvalue problem using the reduced matrices in the previous two sections is solved as:(14) [ K r e d u c e d λ n e w M r e d u c e d ] A n e w = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WHlbWaaSbaaSqaaiaadkhacaWGLbGaamizaiaadwhacaWGJbGaamyz aiaadsgaaeqaaOGaeyOeI0Iaeq4UdW2aaSbaaSqaaiaad6gacaWGLb Gaam4DaaqabaGccaWHnbWaaSbaaSqaaiaadkhacaWGLbGaamizaiaa dwhacaWGJbGaamyzaiaadsgaaeqaaaGccaGLBbGaayzxaaGaaCyqam aaBaaaleaacaWGUbGaamyzaiaadEhaaeqaaOGaeyypa0JaaGimaaaa @5246@
The resulting eigenvectors A n e w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ are used to transform the original shapes/displacement vectors S into the set of orthogonal modes, A o r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ .(15) A ort =S A new MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaOGaeyypa0JaaC4uaiaahgea daWgaaWcbaGaamOBaiaadwgacaWG3baabeaaaaa@3F8C@
It can be shown that the resulting modes are orthogonal with respect to the system/original stiffness matrix K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4saaaa@36CB@ and mass matrix, M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4saaaa@36CB@ .(16) [ KλM ] A ort =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WHlbGaeyOeI0Iaeq4UdWMaaCytaaGaay5waiaaw2faaiaahgeadaWg aaWcbaGaam4BaiaadkhacaWG0baabeaakiabg2da9iaaicdaaaa@41D8@
The Orthogonal modes A o r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ are normalized with respect to the mass matrix M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4saaaa@36CB@ , therefore the final reduced matrices appear as:(17) K r e d u c e d f i n a l = A o r t T K A o r t = λ n e w I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4samaaDa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa baGaamOzaiaadMgacaWGUbGaamyyaiaadYgaaaGccqGH9aqpcaWHbb Waa0baaSqaaiaad+gacaWGYbGaamiDaaqaaiaadsfaaaGccaWHlbGa aCyqamaaBaaaleaacaWGVbGaamOCaiaadshaaeqaaOGaeyypa0Jaeq 4UdW2aaSbaaSqaaiaad6gacaWGLbGaam4DaaqabaGccaWHjbaaaa@533B@ (18) M r e d u c e d f i n a l = A o r t T M A o r t = I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCytamaaDa aaleaacaWGYbGaamyzaiaadsgacaWG1bGaam4yaiaadwgacaWGKbaa baGaamOzaiaadMgacaWGUbGaamyyaiaadYgaaaGccqGH9aqpcaWHbb Waa0baaSqaaiaad+gacaWGYbGaamiDaaqaaiaadsfaaaGccaWHnbGa aCyqamaaBaaaleaacaWGVbGaamOCaiaadshaaeqaaOGaeyypa0JaaC ysaaaa@4E7B@

The orthonormalized modes are input to other multibody dynamics solvers (like MotionSolve), as modal representations of the flexible body in the subsequent analysis.

Load on Flexible Body

If loading is applied to the flexible body in OptiStruct, then it will be reduced out as modal distributed loads and provided as an input to MotionSolve.

The conversion is performed as:

In MotionSolve, any arbitrary displacement field of the flexible body is approximated by:(19) u= A ort q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2 da9iaahgeadaWgaaWcbaGaam4BaiaadkhacaWG0baabeaakiaahgha aaa@3CD9@

Where, q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyCaaaa@36F1@ is the m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGTbaaaa@32AD@ by 1 modal participation vector.

Suppose f i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGPbaabeaaaaa@3800@ is the i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@ th distributed load defined in the FE model ( f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyCaaaa@36F1@ is an n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGTbaaaa@32AD@ by 1 vector, where n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGTbaaaa@32AD@ is the number of nodes on which the load is applied), u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGPbaabeaaaaa@3800@ is the corresponding displacement field, and u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F5@ is any arbitrary displacement field of the flexible body, as mentioned above. The virtual work, due to f i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGPbaabeaaaaa@3800@ is:(20) δ W = δ u T f i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaam 4vaiabg2da9iabes7aKjaahwhadaahaaWcbeqaaiaadsfaaaGccqGH flY1caWHMbWaaSbaaSqaaiaadMgaaeqaaaaa@4184@
Taking variation on both sides of the arbitrary displacement field equation, you have,(21) δu= A ort δq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaaC yDaiabg2da9iaahgeadaWgaaWcbaGaam4BaiaadkhacaWG0baabeaa kiabes7aKjaahghaaaa@4023@
Substituting for u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F5@ :(22) δ W = δ q T ( A o r t T f i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaam 4vaiabg2da9iabes7aKjaahghadaahaaWcbeqaaiaadsfaaaGccqGH flY1daqadaqaaiaahgeadaqhaaWcbaGaam4BaiaadkhacaWG0baaba GaamivaaaakiaahAgadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGL Paaaaaa@47D1@
Now define the modal distributed load, as:(23) f i modal = A ort T f i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaDa aaleaacaWGPbaabaGaamyBaiaad+gacaWGKbGaamyyaiaadYgaaaGc cqGH9aqpcaWHbbWaa0baaSqaaiaad+gacaWGYbGaamiDaaqaaiaads faaaGccaWHMbWaaSbaaSqaaiaadMgaaeqaaaaa@447E@

Each modal distributed load is a m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGTbaaaa@32AD@ by 1 vector.

The above modal distributed loads, along with their load ID in the FE model, should be pre-computed in OptiStruct and written as new modal distributed load blocks in the .h3d file. In addition, the corresponding displacement fields u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOzamaaBa aaleaacaWGPbaabeaaaaa@3800@ should enter the mode shapes S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4uaaaa@36D3@ in the CMS process.

Input and Output

Flexible Body Generation is activated by the presence of the I/O Option CMSMETH.

The I/O Option references a CMSMETH Bulk Data Entry which defines the method, frequency range or number of modes to be calculated. CC and CB methods are used to generate flexible bodies for some Multibody Dynamics solvers, like MotionSolve. Additionally, the CBN method can be used to generate CMS superelements for some third party solvers (refer to Create Output for Third Party Software for more information).

In general, offsets should not be used in flexible body generation. However, PARAM, CMSOFST may be used to allow small offsets for shell elements.

The orthogonal modes A o r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaWGVbGaamOCaiaadshaaeqaaaaa@39D1@ and the corresponding eigenvalues are exported to a flexh3d file by default. The modal stresses and strains can be output optionally using the STRESS and STRAIN output statements. Sets can be applied to reduce the amount of data. The export of the model to this file can be controlled using the MODEL output statement.

Recover MBD Analysis Results

After MotionSolve is run, it is possible to recover displacements, velocities, accelerations, stresses, and strains for the flexbody in OptiStruct in order to create .op2 and .h3d results files for fatigue analysis.

The procedure is explained below.

After running MotionSolve, a residual run can be made to recover displacement, velocity, acceleration, stress, and strain results for interior grids and elements in the Flex Body based on the modal participation results from MotionSolve. After running MotionSolve, a resulting <filename>.mrf file is created that contains MotionSolve results including the modal participation factors at each time step for the Flex Body for transient analysis. In the residual run, the Flex Body <filename>_recov.h3d file and the .mrf results file are specified using the ASSIGN data:
ASSIGN,H3DMBD,30101,’pfbody_1_recov.h3d'
ASSIGN,H3DMBD,30102,’pfbody_2_recov.h3d'
ASSIGN,MBDINP,10,’pfbody.mrf'

Where, the 10 in the ASSIGN,MBDINP data references the SUBCASE for which the MotionSolve results will be used. In SUBCASE 10, instead of performing a transient analysis, OptiStruct will just use the results from MotionSolve.

The 30101 and 30102 in the ASSIGN,H3DMBD data refer to the Flex Body ID's in the .mrf file.

For transient analysis, the number of time steps in the transient analysis run must match the number of time steps used in the MotionSolve analysis. While the transient analysis data is ignored, there must still be some dummy loading data (TLOAD, DAREA, and TABLED data). A sample of input data for a transient analysis run is:
OUTPUT OP2
OUTPUT H3D
ASSIGN,H3DMBD,30103,MBD_pfbody_BODY_2_PROP_6_recov.h3d
ASSIGN,H3DMBD,30102,MBD_pfbody_BODY_1_PROP_9_recov.h3d
ASSIGN,H3DMBD,30104,MBD_pfbody_BODY_3_PROP_10_recov.h3d
ASSIGN,MBDINP,1,MBD_pfbody_mbd.mrf
SUBCASE       1                            
  TSTEP(TIME) =        4
  DLOAD =        3
  DISPLACEMENT = ALL
  STRESS = ALL
  SPC = 1
$
BEGIN BULK
GRID    9999999   $ Dummy GRID since at least one GRID is required
$------+-------+-------+-------+-------+-------+-------+-------+-------
TSTEP          4 300    .0003333   1  
TLOAD1         3       3            DISP       2
TABLED1        2  LINEAR  LINEAR
+            0.0     1.0    10.0     1.0ENDT   
$
$ Dummy load on the dummy grid
SPC            1 9999999       1
SPCD           3 9999999       1  -200.0  
ENDDATA

A dummy grid and a dummy load has been added for the OptiStruct analysis run.