Preconditioning strategies for linear dependent generalized component modes in 3D flexible multibody dynamics
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Abstract
The trend away from physical towards virtual prototyping as well as increasing industrial demands require advanced simulation tools for dynamical systems. Virtually all engineering systems are assemblies and are associated with stresses, noise and vibrations; therefore, flexible multibody simulations are inevitable for accurate predictions. However, realworld finite element models contain millions of degrees of freedom that cannot be reasonably handled without model reduction techniques. Generalized component mode synthesis is a promising tool for flexible 3D multibody systems, since the generalized component modes not only represent the deformation modes in any possible orientation, but also rigid body motion, which preserves a linear configuration space, yielding a constant mass matrix, a corotated but constant stiffness matrix, no quadratic velocity vector and a simple structure of the equations of motion. In this novel framework, the displacement is approximated by a linear combination of generalized component modes generated from vibration eigenmodes, undeformed nodal coordinates and the Cartesian base vectors. The emerging system matrices may be illconditioned and may introduce significant numerical errors, because of linearly dependent generalized component modes and due to different orders of magnitude of their Euclidean norm. However, this issue has not received much attention in the open literature despite its importance. Hence, the current contribution sheds light on this problem and derives preprocessing procedures to convert illconditioned into wellconditioned problems, which shall improve the formulation’s applicability. The new findings are illustrated by numerical experiments of simple bodies and a crankshaft.
Keywords
Flexible multibody systems Generalized component mode synthesis Absolute coordinate formulation Finite element method Modal reduction Model order reduction1 Introduction
The trend away from physical towards virtual prototyping as well as increasing industrial demands on reliability and efficiency of modern dynamical engineering devices require advanced modeling techniques during the design process. At present, virtually all such engineering systems are assemblies and, hence, made out of multiple components, which interact with each other during operation. The forces required to execute desired motions are associated with stresses, noise and vibrations of the bodies in the system. Thus, it is insufficient to model multibody systems as rigid bodies and extract boundary forces to perform subsequent standard finite element (FE) analyses. Flexible multibody simulations, where the system is spatially discretized using a finite number of elements, are, therefore, inevitable. However, most FE models of relevant engineering problems contain a huge number of degrees of freedom (DOFs) that cannot be simulated within a reasonable amount of time without employing model reduction techniques.
There are several ways to model flexible multibody systems [19, 24]. Among them the absolute coordinate formulation known as generalized component mode synthesis [16], which is a promising alternative to wellestablished flexible multibody formalisms, such as the floating frame of reference formulation (FFRF) [15, 20, 21], since the socalled generalized component modes not only describe rigid body motion, but can also represent the deformation modes in any possible orientation, which preserves a linear relationship between the global displacement field and the DOFs of the considered domain; this yields a constant mass matrix, a constant but corotated stiffness matrix with one corotational frame for each system body only, a trivial quadratic velocity vector and a simple structure of the governing equations. The mass and stiffness matrix are simply the standard linear FE system matrices pre and postmultiplied with a constant reduction matrix. Hence, the FE system matrices are extracted only once during preprocessing and remain constant during the whole simulation, which is why, the algorithm is fully decoupled from any FE package and easily applicable – it requires just a few lines of code – to any multibody system subjected to large reference motion, but small deformations of the individual components.
Both the FFRF and the generalized component mode synthesis approximate the flexible deformation by a linear combination of component modes. In case of the FFRF, the component modes are, e.g., the eigenmodes of vibration limited to the frequency range of interest, or a combination of eigenmodes and static modes, see, e.g., the pioneering work of [1, 9, 10, 12, 17]. If the deformation is approximated by vibration modes, the reduction matrix containing columnwise the eigenmodes is wellconditioned, since the eigenvectors are linearly independent [2, p. 158] even for repeated eigenvalues – the degeneracy theorem [4, p. 72] allows the generalization of the orthogonality of the eigenvectors in the metrics of the FE mass and stiffness matrices in the case of repeated eigenvalues. Hence, the reduction matrix of the FFRFbased component mode synthesis does not introduce numerical errors; in fact, the condition number is close to one if the eigenvectors are displacement normalized, e.g., smaller than 1.4 for all hereinafter analyzed models, whereas the generalized component mode reduction matrix is in many cases illconditioned, e.g., on the order of \(10^{6}\) to \(10^{17}\) for the hereinafter analyzed beamlike models, because of linear dependencies and Euclidean norms that deviate by orders of magnitude from each other between the generalized component modes, which may arise due to the special structure of the reduction basis \(\boldsymbol{\varPhi }\), see Sect. 2.1. It is well known that linear dependencies and high condition numbers may lead to large errors or even unsolvable problems, since they preclude the factorization of the system Jacobian impossible.
There is past work concerned with a continuummechanicsbased mathematical derivation of the generalized component modes and equations of motion [5, 16], whereas [6] shows how the idea may be employed without model reduction. It is also known from the literature that the inherent properties of the formulation facilitate the construction of energy–momentumconserving time integration schemes [7]. Also, the formulation has been successfully applied to engineering problems, such as fluid–structure interaction [18] and machine parts with large rotations about one axis only [26], and was extended by means of a nullspace projection approach [25] as well as the global modal parametrization [3, 8, 13]. However, the problem of linear dependencies and illconditioned system matrices has not received much attention despite its importance. The issue was marginally reported in [5, 16], but has not been addressed in the available literature. Furthermore, it has been believed that these issues can only arise for symmetric problems, which is, as shown in the present paper, in general not true. Hence, the main objective of this contribution is to analyze this problem associated with the generalized component mode synthesis and to present preprocessing strategies to obtain wellconditioned problems.
The remaining part of the paper is organized as follows: Sect. 2 is devoted to the configuration space – the generalized component modes are derived on a nodalbased level and the linear relationship between the global nodal displacements and DOFs is illustrated. Furthermore, the section contains a traceable and more intuitive derivation of the governing equations of motion with a Lagrangian formulation for a general spatially discretized mechanical system, in contrast to the original continuummechanicsbased derivation reported in the literature. This novel presentation shall also clarify the idea behind the method to some extent. Section 3 presents a short note on the mathematics of linear dependencies and how to handle them, which is required to analyze and eliminate linear dependencies in the arising reduction matrix; followed by an illustrative presentation of the generalized component modes and the problem of linear dependencies in Sect. 4. Sections 5 and 6 investigate the linear dependencies of the flexible part of the reduction matrix and between the flexible and rigid body motion parts, respectively, with the help of simple FE beamlike 3D models. Section 7 applies the presented theory to an FE model of a crankshaft of a twocylinder reciprocal combustion engine, and Sect. 8 gives stepbystep preprocessing strategies for handling the inherent problem of the formulation to convert illconditioned into wellconditioned problems; followed by a conclusion in Sect. 9.
The present paper is a revised and extended version of the conference paper [27] presented at the Fifth Joint International Conference on Multibody System Dynamics (IMSD) 2018.
2 Generalized component mode synthesis
2.1 Reduction basis
The idea behind the generalized component mode formalism is (i) to define a reduction basis that can represent large rigid body translation and rotation, as well as flexible deformation; and (ii) to obtain a linear relationship between the global FE nodal displacements and the DOFs. As already mentioned, the formulation exploits a modal superposition reduction method, where the flexible deformation is approximated by a linear combination of vibration modes, to reduce the system size from a large number of DOFs to a significantly smaller one. This, of course, restricts its applicability to problems where the deformation of the components remains linear within each body frame.
The socalled generalized component modes not only represent the deformation modes in any possible orientation, but also account for large rigid body motion (translation & rotation), yielding a linear configuration space; see Eq. (22). The generalized component modes are generated from the set of Cartesian unit vectors, the undeformed FE nodal coordinates and the original eigenmodes of vibration in the frequency range of interest. However, the linear relationship between the global displacements and the DOFs, i.e., the generalized coordinates \(\boldsymbol{q}\), is obtained at the expense of a ninefold increase in the number of flexible modal coordinates \({}^{m\!}{{\zeta }}\), i.e., nine generalized flexible coordinates per original natural mode of vibration \({}^{m\!}{\boldsymbol{\psi }}\) of the bodies in the system, as shown in the following paragraphs.
The formal rules to generate the generalized translational, rotational and flexible component modes are explicitly stated in Eqs. (8), (13) and (20), respectively, according to [16]. It is shown in [16] that these modes may be obtained by rearranging the equations during the derivation of the FFRFbased component mode synthesis and by introducing new coordinates. However, a more intuitive and concise derivation of the generalized component modes is shown in this contribution.
2.2 Equations of motion
3 A short note on linear dependence
3.1 Definition
As already pointed out in Sect. 1, a straightforward generation of \(\boldsymbol{\varPhi }\) according to Eqs. (8), (13), (20) and Eq. (23) may lead to an illconditioned system due to linearly dependent generalized component modes, which is why this section is devoted to a short note on the mathematics of linear dependence.
3.2 Singularvalue decomposition
Equation (66) shows that the right singular vectors \(\boldsymbol{v}_{p}\) corresponding to vanishing singular values \(\sigma _{p}\) are elements of the nullspace of \(\boldsymbol{\varPhi }\). Hence, the components of \(\boldsymbol{v}_{p}\) are the coefficients \(a_{pi}\) of Eq. (60), i.e., \(\boldsymbol{v} _{p}=\boldsymbol{a}_{p}\), compare Eq. (61) with Eq. (66), and we have identified the nature of the dependencies.
3.3 Matrix condition number
The SVD is not only useful to determine the nullspace of a matrix, but may be also used to determine the condition number of a matrix, \(\mathrm{cond}(\dots )\), which defines how accurately a linear system of equations can be solved. The condition number may be considered as an amplification of error and one has to expect to lose \(\log _{10} [ \mathrm{cond}(\boldsymbol{J})]\), see Eq. (69), digits of accuracy on top of all the other errors, such as finite machine precision [23, p. 95]. If the condition number is small, the matrix is said to be wellconditioned, otherwise illconditioned.
3.4 Cosine similarity
A special case of linear dependence appears whenever two columns of \(\boldsymbol{\varPhi }\) are directly proportional to each other, i.e., if \(\boldsymbol{\varphi }_{i}= b \boldsymbol{\varphi }_{j}\) for some nonzero scalar \(b\).
3.5 Orthogonal reduction basis
The number of nonzero vectors generated by the algorithm is equal to the dimension of the space spanned by the original set of linearly dependent generalized component modes; hence, all zero vectors must be discarded from the reduction basis to obtain the desired independent set of generalized component modes.
3.6 Remark
It should be noted that the condition number of the translational reduction matrix \(\boldsymbol{\varPhi }_{\mathrm{t}}^{\mathrm{}}\) is always equal to one, since the translational modes are always orthogonal to each other, Eq. (8), and that it is neither possible to remove nor alter the rotational part \(\boldsymbol{\varPhi }_{\mathrm{r}}^{\mathrm{}}\) without changing the formulation. Hence, if linear dependencies arise, only generalized flexible modes may be removed or altered without the need to adopt the formulation. However, it is known from the literature [22] that scaling, i.e., adopting the Euclidean norm of modes to similar values, has a positive effect on the condition of a reduction basis, which is possible for both translational and flexible modes, and will be also employed in here, see Sect. 5.2.
Also, as evident from Sect. 2.1, the reduction basis \(\boldsymbol{\varPhi } \in \mathbb{R}^{3 n_{\mathrm{n}} \times (12+9 n_{\mathrm{m}})}\) contains predominately generalized flexible component modes, which is why the main part of the rest of the present paper is devoted to the investigation of the flexible part of the reduction matrix, since \(\boldsymbol{\varPhi }_{\mathrm{f}}^{ \mathrm{}} \in \mathbb{R}^{3 n_{\mathrm{n}} \times 9 n_{\mathrm{m}}}\) accounts for the majority of linear dependencies and the high condition number in the first place.
4 An illustrative example
In this section, a simple example of a 60 mm squaresectioned 900 mm long steel beam should illustrate the generalized component modes, as well as the inherent problem of linear dependencies, to gain a deeper understanding of the formulation and to show the significance of the current contribution, respectively.
These linear dependencies, Eqs. (77) to (81), are attributed to the symmetric crosssection of the analyzed beam and manifest themselves in a high condition number, \(\mathrm{cond}(\boldsymbol{\varPhi })=9.51 \times 10^{17}\). Note that the first two eigenmodes of vibration form a repeated mode pair with equal eigenfrequencies, where the displacements in the \(x\) and \(y\)directions are simply “exchanged” for the first and second bending mode, since any axis through the centroid of a square crosssection is a principal bending axis. Consequently, only nine out of the full set of 18 generalized flexible component modes are, in this case, linearly independent and the system of equations with the initially \(9 n_{ \mathrm{m}}\) generalized flexible coordinates may be further reduced significantly.
 1.
\(l \; \; =1 \to 3\) for every \(k\) and \(m\)
 2.
\(k \; =1 \to 3\) for every \(m\)
 3.
\(m=1 \to n_{\mathrm{m}}\)
Figure 5 proves that the identified dependencies indeed fulfill the condition stated in Eq. (71). Such heatmaps are used for the rest of the analysis to visualize directly proportional columns of \(\boldsymbol{\varPhi }_{\mathrm{f}} ^{\mathrm{}}\).
This example shows the importance of the mode selection process. The systematic investigation of the generalized component modes, which has not been addressed in the available literature, is not only required to obtain an accurate solution or a solvable system of equations, but may also enable a further reduction of the generalized coordinates and, therefore, a gain in efficiency.
5 Dependencies within the flexible reduction matrix
5.1 Detailed analysis of a squaresectioned extruded body
 \(\square \)

Young’s modulus, \(E=1500~\mbox{MPa}\)
 \(\square \)

Poisson’s ratio, \(\mu =0.3\)
 \(\square \)

Density, \(\rho =1000~\mbox{kg}\,\mbox{m}^{3}\)
 \(\square \)

Length, \(l=2~\mbox{m}\)
Figure 8(a) shows the matrix condition number of the flexible reduction matrix \(\mathrm{cond}(\boldsymbol{\varPhi } _{\mathrm{f}}^{\mathrm{}})\) generated from the eight eigenmodes of vibration displayed in Fig. 6 over the threshold value \(c_{\mathrm{th}}\) of the Cosine similarity. Figure 8(b) depicts the corresponding number of removed columns of \(\boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}}\) over \(c_{\mathrm{th}}\). The figures indicate that a threshold value relatively close to one (\(c_{\mathrm{th}}=0.999\)) suffices to reduce the condition number by three orders of magnitude; such a sharp drop can be generally expected if columns are directly proportional, as evident from all analyzed beamlike models discussed in this paper. Further reducing the threshold (\(c_{\mathrm{th}}=0.995\)) leads to a minor improvement by a factor of approximately two. After that, the condition number remains constant within the considered range and removing more columns (from 24 to 27 at \(c_{\mathrm{th}}=0.980\)) does not lower the condition number further. Hence, if the Cosine similarity is employed to eliminate linear dependencies, “convergence” of \(\mathrm{cond}(\boldsymbol{\varPhi } _{\mathrm{f}}^{\mathrm{}})\) should always be checked prior time integration to avoid numerical errors. Similar numerical experiments on the influence of weak “dependencies” on the reduction matrix’s condition number were conducted for all herein analyzed models, see Sect. 5.2, and indicated that a Cosine similarity (absolute) value below 0.950 may be considered as extremely low. In fact, a value of \(c_{\mathrm{th}} = 0.993\) yielded sound results throughout the investigations presented in this paper,^{7} which is why only entries with \(\vert \cos \theta _{ij} \vert \geq 0.993\) in the Cosine similarity matrix are further discussed in here.
The initial threshold value may be chosen to be one, or as the Cosine similarity (absolute) value calculated from a pair of generalized component modes where linear dependence is a priori known, which is the case between the generalized component modes obtained from \({}^{\mathrm{B1} \! }{}{\boldsymbol{\psi }}\) and \({}^{\mathrm{B1^{*}} \! \! \! }{}{\boldsymbol{\psi }}\) (Fig. 6) for the here considered example.
Another and a very elegant possibility to eliminate the linear dependencies within \(\boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}}\) is to apply the “shortened” version of the Gram–Schmidt process, introduced in Sect. 3.5, to the dependent set of modes. This provides an (virtually) orthogonal flexible reduction basis if the arising zero vectors are omitted, as already described in Sect. 3.5. This has the advantage that no threshold value needs to be determined, since the norm of the arising (virtually) zero vectors is by several orders of magnitude smaller than the average norm of the full set of the arising vectors. Another advantage is that the algorithm is only required once, i.e., no iterations, and no user interaction is required at all to obtain a perfectly wellconditioned flexible reduction basis. This algorithm is also faster, not only because no iterations are required, but also since no SVD needs to be performed to estimate the condition number, since the condition of the generated “nonzero set” is a priori sufficiently small. The only drawback is that the arising reduction basis is in general larger than that obtained with a neat Cosine similarity preconditioning. Hence, there is a tradeoff between preprocessing and “online” cost, however, the “shortened” Gram–Schmidt process (Sect. 3.5) should be, in general, preferred, due to dispensable user intervention.
5.2 Extruded bodies with different crosssections
Comparison of the effect of the Cosine similarity (\(c_{ \mathrm{th}}=0.993\)) and the Gram–Schmidt preconditioning on the condition number \(\mathrm{cond} (\boldsymbol{\varPhi }_{ \mathrm{f}}^{\mathrm{}} )\) of the flexible reduction matrix generated from the eigenmode selection, Fig. 6, of the beamlike models, Fig. 9. The table shows the number of excluded modes \(n_{\mathrm{x}}\), the condition number of the reduced reduction matrix \(\mathrm{cond_{x}} ( \boldsymbol{\varPhi } _{\mathrm{f}}^{\mathrm{}} )\) and of the reduced reduction matrix with scaled columns \(\mathrm{cond_{xs}} ( \boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}} )\), for both methods. The modes are scaled after the Cosine similarity or Gram–Schmidt preconditioning
Models  \(\mathrm{cond}( \boldsymbol{\varPhi }_{\mathrm{f}})\)  Cosine similarity precond.  Gram–Schmidt precond.  

\(n_{\mathrm{x}}\)  \(\mathrm{cond_{x}}( \boldsymbol{\varPhi }_{\mathrm{f}} )\)  \(\mathrm{cond_{xs}}( \boldsymbol{\varPhi }_{\mathrm{f}} )\)  \(n_{\mathrm{x}}\)  \(\mathrm{cond_{x}}( \boldsymbol{\varPhi }_{\mathrm{f}})\)  \(\mathrm{cond_{xs}}( \boldsymbol{\varPhi }_{\mathrm{f}} )\)  
Square  2.98 × 10^{6}  24  4.27 × 10^{2}  3.74 × 10^{1}  6  8.12 × 10^{2}  1.00 
Rectangle  9.61 × 10^{5}  18  1.24 × 10^{3}  3.73 × 10^{1}  6  1.04 × 10^{3}  1.00 
Circle  9.99 × 10^{11}  24  2.48 × 10^{9}  3.74 × 10^{1}  12  1.19 × 10^{3}  1.00 
Triangle  2.98 × 10^{6}  18  1.90 × 10^{3}  3.99 × 10^{1}  6  1.35 × 10^{3}  1.00 
Arbitrary  2.39 × 10^{5}  18  1.42 × 10^{3}  2.76 × 10^{2}  3  4.09 × 10^{3}  1.00 
The condition number of the flexible reduction matrices of the analyzed beams ranges from the order of \(10^{5}\) to \(10^{11}\), where the “best” condition number belongs to the arbitrary beam, Fig. 9(e), and the “worst” one to the circular beam, Fig. 9(c). This clearly emphasizes the importance of the present investigations, since one has to expect to lose at least eleven digits of accuracy on top of all the other errors, if the linear dependencies are not handled appropriately.
Removing linearly dependent modes with a Cosine similarity equal to or higher than \(c_{\mathrm{th}}=0.993\) lowers the condition number between two to four orders of magnitude, whereas the “shortened” Gram–Schmidt process (with excluded zero vectors) between two and almost nine orders of magnitude. A subsequent scaling of the columns of \(\boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}}\), i.e., converting all flexible modes to the same Euclidean norm (length), lowers \(\mathrm{cond}(\boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}})\) of all beamlike models down to the order of \(10^{1}\) to \(10^{2}\) after the Cosine similarity preprocessing, or down to virtually one after the “shortened” Gram–Schmidt process. Scaling is a crucial preprocessing step for the formulation, as can be seen especially for the circularsectioned beam, where scaling lowers the condition number by seven orders of magnitude!
The exclusion of linearly dependent modes with \(\vert \cos \theta _{ij} \vert \geq 0.993\), reduces the initially 72 generalized flexible coordinates by 33% in the best case and by at least 25%, even though only eight eigenmodes of vibration are included in the flexible reduction basis here. Hence, the gain in efficiency may be essential especially for larger problems. This shows that a neat preprocessing is not only important for accuracy, but may also for efficiency, due to the significant reduction of the DOFs. However, as already mentioned, the Gram–Schmidt preprocessing yields in general to a smaller reduction in the number of flexible DOFs, which is also reflected here, i.e., only between 4% and 16%, see Table 1.
The FE meshes for the beamlike models were chosen in order to preserve the lines of symmetry of the crosssections, Fig. 9, except for the circle, where the infinite number of symmetry lines is reduced to four, due to the discretization. The arbitraryshaped beam’s crosssection, Fig. 9(e), was generated such that the geometry itself and the FE mesh do not exhibit any symmetries. Nevertheless, the flexible reduction matrices of the circular, the square and the arbitraryshaped beam show the same linear dependencies, except that the correlations between the first torsion mode and the first bending mode pair, see Eq. (87), are slightly stronger for the square beam. Also, the Cosine similarity matrix of the arbitraryshaped beam show additionally some very weak correlations, yet with negligible values. Therefore, it seems that “small” deviations from symmetric geometries and the design of FE meshes have a minor influence on the dependencies of the flexible reduction matrices, and that not the shape but the number of symmetry lines dictates the dependencies. Finally, the rectangular and equilateral triangular beam exhibit the same correlations as the square beam, except for the dependencies within each generalized mode set generated from the bending modes, see Eq. (79).
6 Dependencies between the flexible and the rigid body motion part
6.1 General results
It is possible that the condition number of the reduction matrix \(\mathrm{cond}(\boldsymbol{\varPhi })\) is still unacceptably high after preconditioning its flexible part. This happens whenever linear dependencies between the flexible \(\boldsymbol{\varPhi }_{\mathrm{f}} ^{\mathrm{}}\) and rigid body motion part \([\boldsymbol{\varPhi } _{\mathrm{t}}^{\mathrm{}} \ \boldsymbol{\varPhi }_{\mathrm{r}} ^{\mathrm{}} ]\) of the reduction basis arise. Such dependencies usually appear as linear combinations, i.e., if Eq. (60) is fulfilled with more than two vectors involved. The Cosine similarity fails to identify such linear combinations – also we cannot apply the “shortened” Gram–Schmidt process to the full reduction matrix, since the generalized translational and rotational component modes must remain unchanged (\(\boldsymbol{\varPhi }_{\mathrm{t}}^{\mathrm{}}\) may be scaled). Which is why we have to resort to the SVD, briefly introduced in Sect. 3.2, to determine the nullspace of \(\boldsymbol{\varPhi }\). The nullspace identifies the generalized component modes involved in linear combinations. Therefore, we can iteratively (i) determine \(\mathrm{null}(\boldsymbol{\varPhi })\), (ii) remove one of the generalized flexible component modes involved, and (iii) start with (i) again, as long as the condition number is as low as desired.^{8}
Figure 10 shows the singular values \(\sigma _{i}\) of the unmodified reduction matrix \(\boldsymbol{\varPhi }\) of the circular beamlike model, Fig. 9(c), in descending order; the especially distinctive singular value drop by almost six orders of magnitude occurs for the depicted example between the 72nd and 73rd singular value. The vanishing singular values after the drop indicate that Eq. (60) is fulfilled for nonzero coefficients \(a_{pi}\), i.e., the components of the right singular vectors \(\boldsymbol{v}_{p}\) of \(\boldsymbol{\varPhi }\), see Sect. 3. If \(\boldsymbol{\varPhi }_{ \mathrm{f}}^{\mathrm{}}\) is already wellconditioned (Cosine similarity or Gram–Schmidt preconditioning), linear dependencies between the rigid body motion and flexible part of the reduction matrix exist. These linear dependencies between \(\boldsymbol{\varPhi }_{\mathrm{f}}^{ \mathrm{}}\) and \([\boldsymbol{\varPhi }_{\mathrm{t}}^{\mathrm{}} \ \boldsymbol{\varPhi }_{\mathrm{r}}^{\mathrm{}} ]\) manifest themselves as a sharp drop between two subsequent singular values, which is why no threshold for vanishing singular values needs to be determined, i.e., right singular vectors corresponding to singular values after the drop (here, 73 to 84) compose the nullspace of \(\boldsymbol{\varPhi }\), see Eq. (66). The corresponding right singular vectors indicate the columns of \(\boldsymbol{\varPhi }\) involved in linear dependencies.
6.2 Further results for the beamlike models
For all analyzed beamlike models (Fig. 9), \(\mathrm{cond}(\boldsymbol{\varPhi })\) is one order of magnitude larger than the condition number of the wellconditioned flexible reduction matrix. As already mentioned, it is not possible to scale the rotational part \(\boldsymbol{\varPhi }_{\mathrm{r}}^{\mathrm{}}\) of the reduction matrix, however, it is important to ensure that the generalized component modes have similar norms. This is why the generalized translational and flexible component modes are scaled to the mean value of the norms of the columns of \(\boldsymbol{\varPhi }_{\mathrm{r}} ^{\mathrm{}}\), which lowers the overall condition number of the already preconditioned total reduction matrix down to the order of \(10^{2}\), and no dependencies between the rigid body motion and flexible part emerged for these examples.
7 Analysis of a crankshaft – a relevant engineering example
Comparison of the condition numbers of the reduction matrices generated from the eigenmode selection of the crankshaft, Fig. 11. The table shows the condition of the flexible, \(\mathrm{cond} ( \boldsymbol{\varPhi }_{\mathrm{f}} )\), and full, \(\mathrm{cond} (\boldsymbol{\varPhi } )\), reduction matrices, the number of excluded modes \(n_{\mathrm{x}}\), the condition of the scaled flexible reduction matrix \(\mathrm{cond_{s}} ( \boldsymbol{\varPhi }_{\mathrm{f}} )\), of the full reduction matrix with scaled translational and flexible part (mean value of the norms of the rotational part), of the Gram–Schmidt orthonormalized flexible reduction matrix \(\mathrm{cond_{gs}} ( \boldsymbol{\varPhi }_{\mathrm{f}} )\) and of the full reduction matrix with scaled and orthonormalized flexible and scaled translational part
\(\mathrm{cond} ( \boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}} )\)  cond(Φ)  \(n_{\mathrm{x}}\)  \(\mathrm{cond_{s}} ( \boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}} )\)  \(\mathrm{cond_{s}} ( \boldsymbol{\varPhi } )\)  \(\mathrm{cond_{gs}} ( \boldsymbol{\varPhi }_{\mathrm{f}}^{\mathrm{}} )\)  \(\mathrm{cond_{gs}} ( \boldsymbol{\varPhi } )\) 

8.75 × 10^{2}  6.22 × 10^{4}  0  6.64 × 10^{2}  4.72 × 10^{2}  1.00  4.09 × 10^{1} 
8 Stepbystep strategies to handle illconditioned reduction matrices
The algorithm shown in the present section state strategies how to handle linear dependencies within the reduction matrix \(\boldsymbol{\varPhi }\) and how to improve the overall condition number to convert illconditioned into wellconditioned problems.
The pseudocode^{10} contains the two different strategies discussed in detail in the previous sections, where one employs the Cosine similarity preconditioning (omit red instructions) and the other the “shortened” version of the Gram–Schmidt process (omit blue instructions). As already mentioned, the latter one should be preferred for practical applications, whereas the former allows more freedom during the mode selection process and provides more insight into the nature of the dependencies.
9 Conclusions
The present paper makes a contribution to the understanding and applicability of a promising 3D flexible multibody dynamics formulation known as generalized component mode synthesis. The novel nodalbased derivations of the governing equations of motion and the reduction basis in the already spatially discretized domain, in contrast to the standard continuummechanicsbased derivation available in the literature, are presented in a traceable and intuitive way.
The main part of this contribution identified and resolved the weakness of the formulation, i.e., the possibility of illconditioned reduction matrices, which may introduce large errors or even lead to unsolvable problems – condition numbers up to nearly \(10^{18}\) – if not handled appropriately. The high condition numbers are caused by linear dependencies and Euclidean norms that deviate by orders of magnitude from each other of the generalized component modes – the reduction matrix’s columns – generated from original linearly independent FE eigenmodes of vibration, undeformed FE nodal coordinates and the set of Cartesian base vectors.
It was shown by numerical experiments of simple extruded bodies with different crosssections that such linear dependencies may also arise for fully asymmetric geometries and FE meshes, which is why it is impossible to identify these dependencies a priori. It was shown how the singular value decomposition, the matrix condition number, the Cosine similarity and a “shortened” version of the classical Gram–Schmidt process may be used to identify and eliminate these linear dependencies, yielding not only sufficiently wellconditioned systems, but may also less DOFs and, therefore, a further gain in efficiency. The crankshaft example suggests that complex realworld engineering systems are potentially less likely to result in illconditioned reduction matrices, however, a neat preprocessing is suggested in any case to avoid unexpected errors.
Finally, stepbystep preprocessing procedures were derived to convert ill into wellconditioned problems, which shall improve the formulation’s applicability.
Footnotes
 1.
Note, for a threedimensional FE model discretized with continuum elements, the total number of DOFs and therefore the length of the column matrices in Eq. (3) is equal to \(3 n_{\mathrm{n}}\).
 2.
Note that the size of \(\boldsymbol{A}_{\mathrm{bd}}^{\mathrm{}} \in \mathbb{R}^{3 n_{\mathrm{n}} \times 3 n_{\mathrm{n}}}\) changes to \(\boldsymbol{\widehat{A}}_{\mathrm{bd}}^{\mathrm{}} \in \mathbb{R}^{(12+ 9 n_{\mathrm{m}}) \times (12+ 9 n_{\mathrm{m}})}\) if the order of multiplication is changed, such that a proper matrix multiplication is well defined, i.e., matching dummy indices.
 3.
Note that \(\partial \boldsymbol{q}_{\mathrm{rig}}^{\mathrm{}}/\partial \boldsymbol{q}\) was calculated analytically, see Eq. (35), where \(\boldsymbol{q}_{\mathrm{r}}^{\mathrm{*}}\) was obtained by Gram–Schmidt orthonormalization from \(\boldsymbol{q}_{\mathrm{r}} ^{\mathrm{}}\); \(\boldsymbol{\widehat{A}}_{\mathrm{bd}}^{\mathrm{}}\) is then given by Eq. (38). Equation (57) was then verified for the beamlike models of Sect. 5 at different times during simulation and for arbitrary \(\boldsymbol{q}\)’s.
 4.
Note that the subscript \(p\) should indicate that the vector \(\boldsymbol{a}\) is not unique.
 5.
Note that \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are in general unitary; however, since \(\boldsymbol{\varPhi }\) is real, \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are also real.
 6.
Note that the set \(\bigl\lbrace \boldsymbol{\varphi }_{1}', \dots , \boldsymbol{\varphi}_{(\kappa 1)}' \bigr\rbrace \) has already been orthogonalized by Eq. (73).
 7.
The threshold value of \(c_{\mathrm{th}} = 0.993\) was adequate for the models treated within this paper, however, there is no guarantee that it will yield satisfying results for other problems.
 8.
Note that all preprocessing steps are summarized in the algorithms presented in Sect. 8.
 9.
In this context, vanishing means small compared to large singular values.
 10.
Note that the SVD is expensive for large matrices. However, one may calculate the SVD of the small (compared to \(\boldsymbol{\varPhi }\)) upper triangular matrix \(\boldsymbol{R}\) obtained by the QRdecomposition of \(\boldsymbol{\varPhi }\) to efficiently calculate the desired quantities, since \(\boldsymbol{\varPhi }\) and \(\boldsymbol{R}\) share the same right singular vectors and singular values and \(\boldsymbol{U}\) is not needed here.
Notes
Acknowledgements
Open access funding provided by University of Innsbruck and Medical University of Innsbruck.
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