On the relevance of inertia related terms in the equations of motion of a flexible body in the floating frame of reference formulation
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Abstract
The floating frame of reference formulation is an established method for the description of linear elastic bodies within multibody dynamics. An exact derivation leads to rather complex equations of motion. In order to reduce the computational burden, it is common to neglect certain terms. In the literature this is done by strict application of the small deformation assumption to the kinetic energy. This leads to a remarkably simplified set of equations. In this work, the significance of all terms is investigated at the level of the equations of motion. It is shown that for a large number of applications the previously mentioned set of simple equations is sufficient. Furthermore, scenarios are described in which this simple set is no longer accurate enough. Finally, guidelines are provided, so that engineers can decide which terms should be considered or not. The theoretical conclusions drawn in this work are underlined by qualitative numerical investigations.
Keywords
Flexible multibody dynamics Degrees of freedom Mass matrix Quadratic velocity vector Floating frame of reference formulation1 Introduction
The floating frame of reference formulation (FFRF) is a widely used strategy for the inclusion of a flexible body into multibody system dynamics. The key idea is the separation of the overall motion into nonlinear rigid body motion superimposed by a small elastic deformation. The latter one is described by the superposition of weighted shape functions which are expressed in a body attached coordinate system which translates and rotates with the body. The final equations of motion for each flexible body contain both simple and complex terms. Hence, the numerical evaluation of some terms is cheap, yet expensive for the others. The numerical costly terms are associated with the body’s flexibility, where the numerical effort increases with the number of considered shape functions. For many practical applications it has been observed that the flexibility related terms are of minor significance. Those terms are the inertia of the deformed body, inertia coupling and parts in the centrifugal and the Coriolis forces, which are related to elastic deformations. In the literature two approaches can be found that deal with the question of which terms need to be considered and which need not. The first one is the strict application of a small deformation assumption at the level of the kinetic energy; see, for example, [12] or [21]. With this assumption, a remarkably simple set of equations is obtained. This work demonstrates why the latter description is probably sufficient for many applications, but not for all. The second approach, examples of which can be found in the software package MSC.ADAMS [11], gives the user the possibility to activate or deactivate certain terms in the equations of motion. The final decision is up to the user, but the guidance is not very clear. However, for both approaches there is a final uncertainty which is addressed in this work.
All inertia related terms of a flexible body in the FFRF are investigated on the level of the equations of motion with respect to their relevance. It is shown why the simplest possible formulation, stemming from the strict application of the small deformation assumption at the level of the kinetic energy, is sufficient for many applications. Beyond that, a clear guidance will be given when, in addition, the deformation dependent inertia tensor, inertia coupling, or elastic deformation related parts of the centrifugal and Coriolis forces should be considered. This will be finally summarized in a “set of guidelines”.
The paper is organized as follows: The first section briefly recaps the equations of motion of a free flexible body considering the FFRF. The assumptions, on which this paper is based, together with the resulting implications for the equations of motion, are documented in the next section. The Finite Element structures which are used for the sake of numerical illustration of the theoretical conclusions are introduced in the subsequent section. Following this, the small deformation assumption and its implications for the magnitude of the modal coordinates are discussed. In the next section, the maximum entries of significant matrices are estimated. The latter two estimations will be important for the determination of the significance of certain terms. The subsequent section is the key section of this contribution. All terms will be investigated with respect to their significance. It starts with the inertia of the deformable body followed by the inertia coupling of the rotational and flexible degrees of freedom. Finally, the terms of the quadratic velocity vector will be discussed. All the theoretical considerations in this chapter are accompanied by qualitative numerical investigations. In the subsequent chapter, a set of guidelines is presented in order to simplify the decision when and which term has to be considered or not. The paper will be concluded by summarizing the benefits when certain terms of the equations of motion can be neglected.
2 Brief review of the equations of motion
In order to avoid redundancy, an extensive derivation of the equations of motion is omitted and only relevant equations are reviewed. Details can be found in Shabana [1] and the work of Sherif and Nachbagauer [2]. For better readability with respect to the mentioned publications, the same notation is used in this work. In contrast to [1], a superscript referring to the number of the flexible body will be omitted.
3 Assumptions and simplifications
In this section, some very common assumptions are discussed, since they lead to significant simplifications in the mass and stiffness matrix, as well as in the quadratic velocity vector.
3.1 Euler parameters
3.2 Linearized meanaxis conditions
3.3 Use of (pseudo) free surface modes together with an FF origin fixed to the center of gravity of the undeformed body
Again, just a brief review is given on the implications of the choice of free surface modes and an FF origin which is initially attached to the center of gravity of the undeformed body. More details can be found in [5] and [8].
3.4 Use of mass normalized modes
3.5 Use of central principal axis of inertia
It is assumed that the initial orientation of the axis of the FF matches the principal axis of inertia of the undeformed body.
3.6 Final equations of motion
3.7 Comment on the use of different mode basis
The literature offers several possibilities for the particular choice of shape vectors. The most famous mode base is probably the one of Craig [23]. Comparative studies of other possibilities can be found, among other references, in [24] and [25]. However, at this point of this work it is interesting to note that arbitrary mode shapes can be transformed into a mode base possessing the properties given before, see [8].
3.8 Comments on the complexity of the equations of motion and goal of this work
An obvious disadvantage of the equations of motion (16) are the necessary matrix vector operations related to the invariant matrices \(\boldsymbol{W} _{1}\), \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\) stemming from the body’s flexibility. This increased numerical effort affects the residuum of (16) and the computation of the Jacobian matrix. In order to minimize this computational burden, some suggestions for the negligence of certain terms can be found in the literature and multibody dynamic software packages. As an example, [12] can be cited. This paper refers to the classical paper [13] where the hypothesis of small elastic deformations around the undeformed configuration is applied on the level of kinetic energy. As a consequence, the coupling between the rotational and flexible degrees of freedom in the mass matrix and in the quadratic velocity vector vanishes in the final equations of motion. Another example is the commercially available software package MSC.ADAMS [14], which offers the user a possibility to deactivate certain terms of the mass matrix. The explanation of the terms is quite vague and, hence, the decision is difficult to make for a user without extensive knowledge of the theory of flexible multibody systems.
The goal of this work is a systematic investigation on the relevance of all the terms related to the invariant matrices on the level of the equations of motion. As far as we know, this kind of in depth analysis of the FFRF is a new contribution to the existing literature. Beside the theoretical insights a set of guidelines is provided as practical benefit. Thereby engineers can decide which terms should be considered or not.
4 Illustrative examples used in the paper
In the practical use of multibody simulation, the flexible bodies are mostly modeled via the Finite Element Method. In that context, the displacements are evaluated at certain grid points and instead of shape functions, shape vectors are used to describe the elastic deformation. More comments on the transition from a “continuous formulation” to a formulation based on Finite Element models can be found in [1, Chap. 5.2, Sect. “Lumped mass”]. Although the details for the computation of the invariant matrices \(\boldsymbol{I} _{0}\), \(\boldsymbol{W} _{1}\), \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\) may change, the final equations of motion (16) remain the same. Those matrices are computed based on the result of the Finite Element analysis and are part of the procedure when a flexible body is imported into the multibody simulation software. In MSC.ADAMS [14], for example, those matrices can be found in the socalled *.mtx file.
In this work several theoretical considerations are presented. The drawn conclusions are of fundamental character for all flexible bodies independent of the overall multibody system where the flexible bodies are embedded. Therefore, it is not necessary to present results of time integrations of particular multibody systems. Instead, the presented examples deal with estimations of effects based on the former mentioned invariant matrices, which are flexible body data only.
Details concerning the used FE models
Connection rod  Car body  Jeffcot rotor  Plate  Generic structures  Beam  

In this work referenced as …  Conrod  Car  Jeffcot  Plate  GenStruct1 GenStruct2  Beam 
Number CNM  15  50  10  20  20  38 
Number CIM  12  24  12  24  6  12 
Extension  20 × 210 × 80 [mm]  3.8 × 1.5 × 1.1 [m]  300 × 300 × 750 [mm]  1000 × 1000 × 0.5 [mm]  330 × 200 × 8 [mm]  750 × 20 × 20 [mm] 
Mass in kg  0.9  270  7.4  3.9  0.36/0.4  1.8 
For reasons of confidentially, the picture of the car body can only be shown in lowresolution. The generic structure GenStruct1 has been created in order to underline the effect of a particular structural compliance with respect to centrifugal forces. The second generic structure GenStruct2 is similar as GenStruct1 without having this special sensitivity to centrifugal forces.
5 Assumption of small deformations
All investigated terms are related to effects caused by the flexibility of the body. These effects are increasing with the elastic deformation of the body. Therefore, it is important to specify a realistic upper bound of expectable deformations, and hence an upper bound of the modal coordinates.
5.1 Small elastic deformations with respect to the body’s dimension
Commonly, the mode shapes are obtained by a linear Finite Element analysis. Consequently, the attribute “linear” defines the range of validity. The deformations have to be small enough, so that no geometric nonlinearities and no material nonlinearities take place. In the simple case of a tensile bar, the longitudinal deformation is given as \(\Delta l = \frac{\sigma l_{0}}{E}\), where \(\sigma \) is the stress, \(l _{0}\) the length of the bar and \(E\) the Young modulus. The use of an unrealistic high yield stress limit \(\sigma = 10^{3}~\mbox{N}/\mbox{mm}^{2}\) and \(E = 2\times 10^{5}~\mbox{N}/\mbox{mm}^{2}\) (steel) leads to a maximum displacement of 0.5% of the original length. In cases where slender structures like beams or plates are loaded by bending, geometric nonlinearities are the limiting criterion for the validity of a linear computation. In the case of a twosided clamped beam or a foursided clamped plate, geometric nonlinearities significantly influence the result even if the deflection of the center point is very small. The regime of validity is larger when just a onesided clamped beam or plate is considered. However, in the literature it is often stated that for linear analysis, the maximum deflection should be in the range of the crosssection dimensions. To give an example, for the Finite Element solver in SolidWorks [9], it is suggested to use nonlinear techniques in case of deformations larger than 1/20 times the largest dimension of the body [10]. However, in this paper it is supposed that the elastic deformations are at least one to two magnitudes smaller than the body’s largest extension.
5.2 Magnitude of modes and modal coordinates
Two very frequently used sets of consistent units are kilogram/meter and ton/millimeter, respectively. When those units are used, experience shows that the requirement of small deformations (no material and no geometric nonlinearity) leads to modal scaling factors (socalled “modal coordinates”) typically much less than one. The following considerations show why this assumption mostly holds true.
In case when kilogram and meter are used as units, a mass of 1 kilogram leads to an averaged modal deflection of 1 meter. A structure of 100 kilograms still leads to an averaged deflection of around 0.1 meter. When ton and millimeter are in use, a mass of 0.001 tons (=1 kg) would lead to an averaged modal deflection of around 40 millimeters. A mass of 0.1 tons (=100 kg) implies an averaged deflection of 4 millimeters. Note once again that these are averaged deflections, the maximum deflection is probably at least a factor of 2 higher. In order to fulfill the restriction of small deformations, modal coordinates smaller than one seem to be a valid assumption. However, heavy structures may not fulfill that assumption.
The former conclusion that the modal coordinates are normally smaller than one depends on the units employed; in this case, kilogram and meter and ton and millimeter, respectively. This conclusion may not be valid for all combinations of length and mass units. Nevertheless, the investigations on the relevance of certain terms in the equations of motion are of general nature since these terms describe mechanical effects like the change of inertia due to deformation, coupling between rotation and deformation, centrifugal and Coriolis forces. These phenomena are of a fundamental nature and their relevance does not depend on the units.
5.2.1 Numerical examples
It can be seen that for almost all structures a modal amplitude lower than one can be expected, because otherwise material nonlinearities take place. Only the car and plate require a more detailed investigation.
Car
Plate
The stresses inside the plate caused by the modes scaled with 1 are definitely in the linear range with respect to material nonlinearity. Figure 6 shows the deflection for the first mode when scaled with −1 and 1, together with the undeformed state (black). The maximum displacement of 42 mm is far beyond the plate thickness of 0.5 mm and occurs at the plates corners. It can be assumed that geometric nonlinear effects take place in case of such deformations. Therefore, the assumption that the modal coordinates remain smaller than 1 is probably a good estimate for that structure, too.
6 Magnitude of entries in invariants \(W_{1}\), \(W_{2}\) and \(W_{3}\)
In the key section of this paper, the significance of the terms holding the matrices \(\boldsymbol{W} _{1}\), \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\) is investigated. This chapter is devoted to the magnitude of the entries of the latter matrices in order to estimate whether a certain effect may be negligible with respect to another one.
6.1 Magnitude of entries in matrix \(W_{1}\)
6.1.1 Numerical examples

Maximum absolute entry.

The average of the magnitude of all nonzero entries. This value gives an idea of whether most of the entries are around the maximum entry or much smaller.

Euclidean norms of \(\boldsymbol{W} _{1}\) and \(\boldsymbol{I}_{0}\).
Maximum absolute values in \(\boldsymbol{W} _{1}\) and the Euclidean norms of \(\boldsymbol{W} _{1}\) and \(\boldsymbol{I} _{0}\)
Maximum absolute entry \(\boldsymbol{W}_{1}\)  Average of magnitudes of all nonzero entries in \(\boldsymbol{W}_{1}\)  Euclidean norm of \(\boldsymbol{W}_{1}\)  Euclidean norm of \(\boldsymbol{I}_{0}\)  

GenStruct1  3.2  0.18  2.2  7.5 
GenStruct2  8.5  0.47  2.3  8.4 
Jeffcott rotor  18.6  0.66  10.5  118.2 
Car body  32.6  35.6  444.1  343673 
Beam  18.4  0.13  9.2  85.7 
Plate  42.1  1.58  21.1  65.74 
Conrod  2.5  0.09  1.8  3.3 
It can be seen in Table 2 that the Euclidean norm of \(\boldsymbol{I} _{0}\) is always greater than that of \(\boldsymbol{W} _{1}\). Moreover, it is interesting to observe that the average value of the magnitude of all nonzero entries is always significantly less than the maximum entry. This leads to the conclusion that most of the entries in \(\boldsymbol{W} _{1}\) hold a magnitude which is significantly smaller than the maximum.
6.2 Magnitude of entries in matrices \(\boldsymbol{W}_{2}\) and \(\boldsymbol{W}_{3}\)
6.2.1 Numerical examples

Maximum absolute entry.

The average of the magnitude of all nonzero entries. This value gives an idea whether most of the entries are around the maximum entry or much smaller.

Euclidean norm.
Maximum absolute values in \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W}_{3}\)
Max. abs. entry \(\boldsymbol{W}_{2}\)  Max. abs. entry \(\boldsymbol{W}_{3}\)  Average of magnitudes of all nonzero entries in \(\boldsymbol{W}_{2}\)  Average of magnitudes of all nonzero entries in \(\boldsymbol{W}_{3}\)  Euclidean norm \(\boldsymbol{W}_{2}\)  Euclidean norm \(\boldsymbol{W}_{3}\)  

GenStruct1  1.0  0.8  0.04  0.05  2.0  1.1 
GenStruct2  1.0  0.9  0.05  0.05  1.9  1.2 
Jeffcott rotor  0.02  0.03  6e−4  7e−4  0.07  0.04 
Car body  0.99  0.6  0.05  0.03  1.9  1.1 
Beam  1.0  1.0  0.04  0.02  2.0  1.4 
Plate  1.0  0.6  0.08  0.05  2.0  1.2 
Conrod  0.1  0.05  0.004  0.003  0.15  0.09 
7 Investigations on the relevance of the single terms in the equations of motion
For the following investigations, a free floating flexible body is assumed as described by (14) and (16). If the body is connected to other bodies via constraining or imposed forces, the situation is somehow different and the resulting consequences are discussed as well.
7.1 Rotational inertia of the deformed body
7.1.1 Theoretical considerations

The second and the third terms (\(\boldsymbol{W} _{1} \boldsymbol{Q} _{f}\) and \(\boldsymbol{Q} _{f} ^{\mathrm{T}} \boldsymbol{W} _{2} \boldsymbol{Q} _{f}\)) cannot be neglected if neither the body’s dimension in the direction \(l\) nor \(m\) in (27) is significantly greater than the flexible deformations in the same directions. A long beam with a small crosssection would be such a structure. In this case, the deformation may have a similar magnitude as the body’s dimension in that direction and the two statedependent terms may be important for an accurate result. A close look at (27) gives no reason to assume that the quadratic term is less important than the linear part if such a beamlike structure is considered.

The statedependent terms may also be nonnegligible in case of structures being particularly soft with respect to centrifugal forces, like GenStruct1. For such structures, small displacements may have a considerable impact on the moment of inertia. If such a structure is considered, it can be assumed that the linear term will dominate the quadratic term since the product of the body’s extension and the elastic deformation will dominate the square of the elastic deformation, see Eq. (27).
7.1.2 Numerical examples
7.1.3 Summary
The deformationrelated change of a body’s rotational inertia can be mostly neglected. Exceptions are only possible if the body has either a dominant extension in one direction or a very particular mass distribution in combination with particular (soft) stiffness properties. If one of those two situations takes place, an additional requirement needs to be fulfilled, which is that the body is a free body or at least very softly connected to other bodies, so that its state is mainly determined by its own inertia. Consequently, in nearly all industrial applications the terms \(\boldsymbol{W} _{1} \boldsymbol{Q} _{f}\) and \(\boldsymbol{Q} _{f} ^{\mathrm{T}} \boldsymbol{W} _{2} \boldsymbol{Q} _{f}\) can be neglected, except when the previously mentioned criteria are fulfilled.
7.2 Inertia coupling
7.2.1 Theoretical considerations
Note again that the former considerations hold true for a body which is not stiffly connected to other bodies with remarkable inertia. In that case the coupling effect loses importance and can be neglected for all kinds of flexible body.
7.2.2 Numerical examples
7.2.3 Summary
The inertia coupling is relevant for the same type of structures, which are critical in terms of deformation induced inertia changes. The influence of the inertia coupling decreases with the number of stiff connections to other bodies. Similar to the preceding section, it can be concluded that inertia coupling is an effect of minor practical relevance.
7.3 Quadratic velocity vector
7.3.1 Theoretical considerations
Rotational part of quadratic velocity vector
The Coriolis force couples the modal and the rotational degrees of freedom via a product of the angular and the modal velocities. It stems from the fact that the elastic body is characterized with respect to a corotating reference frame. It can be observed that all three terms of the Coriolis force consist of a matrix–matrix or a matrix–vector product of the angular and modal velocities together with a kind of inertia. One term holds a constant inertia (\(\boldsymbol{W} _{1}\)) while the other terms hold an inertia which is scaled by the modal coordinates (\(\boldsymbol{Q} _{f} ^{\mathrm{T}} \boldsymbol{W} _{2}\) and \(\boldsymbol{Q} _{f} ^{\mathrm{T}} \boldsymbol{W} _{3}\)). From a former section it is known that the entries of \(\boldsymbol{W} _{1}\) are typically significantly higher as those of \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\). Moreover, the latter two matrices are scaled by modal coordinates smaller than 1, which leads to even smaller values. Based on the insights of the former sections, it can be expected that exceptions may occur in case of beamlike bodies and when a body is particularly soft with respect to the centrifugal force.
Flexible part of quadratic velocity vector
It may be interesting to note that the term holding \(\boldsymbol{W} _{2}\) represents a reduction of the structure’s effective stiffness since the centrifugal force leads to an expansion of the body, which leads to a higher centrifugal force. If the rotational speed exceeds a certain limit, the effective stiffness becomes negative and the system is instable. This is a more academic scenario, however, because under such conditions the structure will not fulfill the small deformation assumption.
Once again, it is emphasized that the former considerations do not contain the centrifugal and Coriolis forces due to other bodies which are somehow stiffly connected to the flexible body under consideration. Even if the body is just connected to ground the mounting forces may influence the body’s behavior much more than the latter discussed quadratic velocity vector. Again, a tendency can be assumed: The more the flexible body interacts with the ground or other bodies, the more the previously discussed terms become irrelevant for the overall solution.
7.3.2 Numerical examples
Rotational part of quadratic velocity vector
In the discussion of the Coriolis force, it has been assumed that the term holding \(\boldsymbol{W} _{1}\) is normally dominating. Exceptions are expected in cases of slender structures and structures with a special mass distribution and an extraordinary sensitivity to centrifugal forces.
Ratio of the Euclidean norm of deformation depended terms of the Coriolis force with respect to the nondeformationdependent term
\(\frac{ \boldsymbol{Q}_{f}^{\mathrm{T}}\boldsymbol{W}_{2}\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}} }{ \boldsymbol{W}_{1}\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}} }\)  \(\frac{ \tilde{\bar{\boldsymbol{\omega}}} \boldsymbol{Q}_{f}^{\mathrm{T}}\boldsymbol{W}_{3}\dot{\boldsymbol{q}}_{f} }{ \boldsymbol{W}_{1}\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}} }\)  

GenStruct1  0.2  0.06 
GenStruct2  0.03  0.005 
Jeffcott rotor  0.0005  6e–5 
Car body  0.04  0.003 
Beam  85  0.06 
Conrod  0.003  3e–19 
Flexible part of quadratic velocity vector
Ratio of the Euclidean norm of the deformationdependent and the nondeformationdependent term of the centrifugal force
\(\frac{\left \boldsymbol{W}_{2}\boldsymbol{Q}_{f} \right}{\left \boldsymbol{W}_{1} \right}\)  

GenStruct1  0.2 
GenStruct2  0.02 
Jeffcott rotor  0.0005 
Car body  0.005 
Beam  0.03 
Conrod  0.0009 
Plate  0.07 
7.3.3 Summary
Considering the gyroscopic forces due to the undeformed body is suggested in any case, since they do not couple the equations of motion and are cheap to evaluate. If the body’s widening due to centrifugal force has to be considered, it is necessary to add the term \([ \begin{array}{c@{\ }c@{\ }c} \bar{\omega}_{1}\boldsymbol{I} & \bar{\omega}_{2}\boldsymbol{I} & \bar{\omega}_{3}\boldsymbol{I} \end{array} ][ \frac{1}{2}\boldsymbol{W}_{1}^{\mathrm{T}} ]\bar{\boldsymbol{\omega}} \). For very special rotational soft structures (in the sense of GenStruct1) the term \([ \begin{array}{c@{\ }c@{\ }c} \bar{\omega}_{1}\boldsymbol{I} & \bar{\omega}_{2}\boldsymbol{I} & \bar{\omega}_{3}\boldsymbol{I} \end{array} ][ \boldsymbol{W}_{2}\boldsymbol{Q}_{f} ]\bar{\boldsymbol{\omega}} \) may be relevant as well.
From rotor dynamics it is known that the Coriolis force can be neglected when the ratio of the assembled systems’ first eigenfrequency and the rotational speed are considerable lower than 1. If this is not the case, the Coriolis forces should be taken into account, whereby, for the rotational degrees of freedom, the term \(\boldsymbol{W}_{1}\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}} \) is sufficient. Exceptions are beamlike structures and, once again, rotational soft structures in the sense of GenStruct1, when the inertia forces of those structures are dominated by themselves and not by other bodies which are stiffly attached.
8 “Set of guidelines” for practical use

Moderate angular velocities when the body is neither a beam nor an extraordinary rotational soft structure (in the sense of GenStruct1). Examples are ground vehicles, housings, aircrafts, and so on.

In cases of flexible bodies which are stiffly connected to other bodies, so that the effective inertia or effective stiffness of the assembled system is different to that of the free body. Examples are crankshafts, connecting rods, or a beam with a mounted flywheel.
In industrial operations, the latter cases will probably cover most the applications.

If the body under consideration is very soft with respect to centrifugal forces (in the sense of GenStruct1) and this softness is not influenced by connections to other bodies (or ground), the linear change in inertia should be considered in the form of \(\boldsymbol{I}_{0} + \boldsymbol{W}_{1}\boldsymbol{Q}_{f}\). If, in addition, the widening of such structures due to centrifugal forces is of interest, the full centrifugal force acting on the flexible coordinates should be regarded in the form of \([\bar{\omega}_{1}\boldsymbol{I} \ \bar{\omega}_{2}\boldsymbol{I} \ \bar{\omega}_{3}\boldsymbol{I} ][ \frac{1}{2}\boldsymbol{W}_{1}^{\mathrm{T}} + \boldsymbol{W}_{2}\boldsymbol{Q}_{f} ]\bar{\boldsymbol{\omega}} \).

In cases where a long slender (beamlike) structure with an inertia which is not influenced by connections to other bodies (or ground) the full change of inertia \(\bar{\boldsymbol{G}}^{\mathrm{T}}[ \boldsymbol{I}_{0} + \boldsymbol{W}_{1}\boldsymbol{Q}_{f} + \boldsymbol{Q}_{f}^{\mathrm{T}}\boldsymbol{W}_{2}\boldsymbol{Q}_{f} ]\bar{\boldsymbol{G}}\) should be computed. In addition, the inertia coupling \(\bar{\boldsymbol{G}}^{\mathrm{T}}\boldsymbol{Q}_{f}^{\mathrm{T}}\boldsymbol{W}_{3}\) in mass matrix should be regarded as well.
IV. The Coriolis force should be considered when the ratio of the assembled system’s first eigenfrequency and the rotational speed is not considerably lower than 1 and when the inertia of the rotating body is determined by itself and not by other bodies which are stiffly connected. In such a case, the term \(\bar{\boldsymbol{G}}^{\mathrm{T}}\boldsymbol{W}_{1}\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}} \) is normally sufficient for the rotational degrees of freedom. The full expression (\( \bar{\boldsymbol{G}}^{\mathrm{T}}[ \boldsymbol{W}_{1} + 2\boldsymbol{Q}_{f}^{\mathrm{T}}\boldsymbol{W}_{2} ]\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}}  2\dot{\bar{\boldsymbol{G}}}^{\mathrm{T}}\boldsymbol{Q}_{f}^{\mathrm{T}}\boldsymbol{W}_{3}\dot{\boldsymbol{q}}_{f}\)) needs to be regarded only in cases with beam like structures or rotational soft structures, like GenStruc1. For the flexible coordinates the Coriolis force is then covered by the term \( 2\boldsymbol{W}_{3}^{\mathrm{T}}\dot{\boldsymbol{Q}}_{f}\bar{\boldsymbol{\omega}}\).
9 Benefit
The wellfounded negligence of certain terms in the equations of motion leads to the following benefits:
A second benefit is that the computation of \(\boldsymbol{W} _{1}\), \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\) is not necessary at all when they are not needed. Consequently, the eigenvalues and mode shapes are enough for flexible multibody dynamics. This is standard output of many Finite Element codes, while \(\boldsymbol{W} _{1}\), \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\) are not. This simplifies the numerical implementation of the equations of motion enormously.
The third benefit is that negligence of the terms including \(\boldsymbol{W} _{1}\), \(\boldsymbol{W} _{2}\) and \(\boldsymbol{W} _{3}\) leads to equations of motion where the mass matrix and the quadratic velocity vector are decoupled with respect to the translational, rotational and flexible degrees of freedom. This fact simplifies considerations with respect to separated time integration [19] or model reduction of multibody systems [22] tremendously.
Finally, the set of guidelines given above removes uncertainty concerning the question of which invariant needs to be considered and which not. In available software packages it is common to commit this decision to the user without clear guidance. The former suggestions can be used as such a guidance in order to activate or deactivate the proper invariants.
10 Conclusion
In this work the significance of all inertia related terms of a flexible body in the FFRF were investigated at the level of the equations of motion. It turned out that for a lot of applications a remarkably simple and decoupled set of equations are sufficient. This has already been suggested in the literature when the small deformation assumption is strictly applied at the level of the kinetic energy (see [12] or [21]). However, there are situations which require either the deformationdependent inertia tensor, or the inertia coupling, or the centrifugal and Coriolis forces which are related to the elastic deformation. All of these terms were investigated with respect to their significance and, finally, these results were condensed in a set of guidelines which give simple advice about which term needs to be considered and which not. All theoretical considerations have been underlined by simple numerical investigations which have been applied to a couple of very different Finite Element structures.
Notes
Acknowledgements
Open access funding provided by University of Applied Sciences Upper Austria.
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