# A generalized constraint reduction method for reduced order MBS models

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## Abstract

In this paper we deal with the problem of ill-conditioned reduced order models in the context of redundant formulated nonlinear multibody system dynamics. Proper Orthogonal Decomposition is applied to reduce the physical coordinates, resulting in an overdetermined system. As the original set of algebraic constraint equations becomes, at least partially, redundant, we propose a generalized constraint reduction method, based on the ideas of Principal Component Analysis, to identify a unique and well-conditioned set of reduced constraint equations. Finally, a combination of reduced physical coordinates and reduced constraint coordinates are applied to one purely rigid and one partly flexible large-scale model, pointing out method strengths but also applicability limitations.

## Keywords

Proper Orthogonal Decomposition Constraint reduction Galerkin projection Redundant coordinates Model order reduction## 1 Introduction

Multibody system simulation (MBS) nowadays is an important part in various industrial fields like the automotive industry, aerospace, training simulators, and several more. Within the automotive industry, the main areas of application are load data acquisition for further investigations like finite element computations, component design or the identification of production-related uncertainties like inertia parameters, the body mass, the exact location of the center of mass, etc. As many of those applications are of a repetitive nature, it is desirable to optimize such recurring simulation tasks.

In this sense, model order reduction of the underlying system of differential algebraic equations (DAEs), created by the MBS pre-processor is one potential technique. We point out that the present work covers multibody system models, created by automated modeling strategies using a redundant set of coordinates. Further, such models are processed by solvers using the Hilber–Hughes–Taylor [9] (HHT) algorithm, as it is the case for commercial software like MSC Adams [18] or the freeware tool FreeDyn [7]. Suitable model order reduction techniques, like Proper Orthogonal Decomposition (POD) [2, 14, 28] or Smooth Orthogonal Decomposition [4] cannot be applied a priori, and need at least one full system simulation. Hence, model order reduction is only meaningful in terms of repetitive simulations. In the past, both methods were applied to dynamical systems by several authors, and the interested reader is referred to [3, 5, 6, 10, 11, 13, 17] and references therein.

In [17], POD model order reduction was recently applied to a suspension system. Therein, the kinematics and dynamics are treated with special attention and hence, the governed equations of motion are very distinct from those treated in this work. We recently compared POD and SOD [26] and further proposed a physical and constraint coordinates reduction method [27] for MBS models. Therein, we focused on rather good-natured models, subject to idealistic boundary conditions. The present contribution demonstrates the applicability and difficulties concerning redundant constraint equations, when dealing with practical, rigid, and flexible models, subject to superimposed vibrations or resonance effects. To overcome the issue of redundant constraint equations in terms of reduced order models, we propose a generalization of the constraint reduction method introduced in [27], based on a singular value decomposition of the reduced constraint Jacobian of the system under consideration. The use of the singular value decomposition method in terms of the elimination of Lagrange multipliers or to handle redundant constraint equations was discussed in the past by various authors; see, for instance, [8, 12, 15, 22, 30, 31].

In the first part of this work, the system of DAEs under consideration is presented. Following [27], the second part of this contribution introduces the model order reduction technique, and presents a generalization of the constraint reduction method, which improves the conditioning of the reduced order model. In the third part, the reduction process is first applied to an oscillatory supported, rigid crank drive, which introduces superimposed vibrations into the model. The second example shows a flexible offroad vehicle suspension, which is loaded by a frequency sweep at its wheel hub. Finally, the model order reduction method is discussed critically, pointing out the strengths, but also known limitations to the method’s applicability.

## 2 Redundant multibody systems

The MBS model is generated using our in-house MBS software package FreeDyn [7] and both time integration and model order reduction are carried out in Scilab 5.5.1 [23], utilizing a Hilber–Hughes–Taylor (HHT) solver algorithm, which is able to directly handle the present system of DAEs. For further insight into the topic of multibody system modeling, the interested reader is referred to [24].

## 3 The reduced order model

A redundant MBS model, as introduced in Sect. 2, often consists of too many coordinates and/or constraint equations. A simple example would be a 2D mechanism modeled in 3D space. Hence, the MBS solver has to deal with a larger set of equations than actually necessary. To reduce this set of equations, a POD model order reduction method, in combination with a modified Galerkin projection and a constraint reduction method following [27], is applied.

### 3.1 A brief review on the Singular Value Decomposition

The model order reduction methods applied to both the physical coordinate and the constraint coordinate reduction are based on the Singular Value Decomposition (SVD) of data, from which we derive basis vectors characterizing the reduction subspaces to project onto. Hence, we briefly summarize the SVDs main features as they are needed for the model order reduction method.

^{ i }) and dependent (superscript

^{ d }) left singular vectors. The matrix of right singular vectors \(\mathbf{R}^{\mathsf{T}}\in \mathbb{R}^{n \times n}\) spans the row-space and can be split accordingly. With no loss of precision, see, e.g., [1, 28], Eq. (2) can be rewritten in terms of the independent components as

Beside their usefulness to calculate the matrix rank, the left singular vectors, together with the related singular values, give information about the unique content of the entry matrix. As the singular values are collected in descending order, the largest (first) singular value relates to the first left singular vector, which gives the highest representation of the entry data information. The second singular value relates to the second left singular vector, which gives the highest representation of the set of remaining left singular vectors, and so on. These features can be used to generate a low-rank approximation of the information stored in the entry matrix by a set of left singular vectors.

### 3.2 Physical coordinate reduction based on POD

The physical coordinate reduction method was presented in [27], and it is used in this work with no changes.

The subspace of the reduced order model is found by investigating the POVs decay. Unfortunately, there is no general algorithm or criteria available, which allows to automatically identify the essential number of POVs/POMs. Actually, one has to get a feeling of POVs and their descending behavior. We experienced good results by first normalizing the POVs vector and then searching for drops in POV magnitude by several powers of ten. However, we can give no clear statement on the exact drop-size, which characterizes a sufficient drop. In other cases, if there is no clear drop detectable, it can be helpful to define a maximum error in signal energy and sum up the POVs “from back to front” until the maximum error is reached. The remaining first POVs then characterize the reduced order model’s dimension. Again, the maximum error in signal energy is no hard fact but model-dependent. It might also be the case that one has to try out various sets of POMs to achieve the best reduced order model. Further, as the POVs are not only model dependent but depend also on the simulation inputs, the set of POVs and POMs is not necessarily unique. This fact must be kept in mind, if changes to the model or the inputs are made without evaluating the reduction subspace. Therefore, we will deal with the topic of subspace-robustness in the context of parameter identification in the future.

At this point, we briefly summarize our reasons, originally presented in [27], to apply POD for each coordinate-type separately and on the velocity level:

Zero-valued POVs describe possible simplifications in the orientation of a body-fixed coordinate frame, as motion along the corresponding POM is zero. In other words, the body coordinate frame orientation is marginal in terms of the acting motion if the corresponding POV is zero. Hence, a more significant frame orientation can be indicated by POVs, as, for instance, a 3D motion along an arbitrary vector in space may be reduced to a one-dimensional motion along a new directional vector, which is described by the corresponding POM. For this reason, we decide to handle translational, rotational, and flexible DOFs separately.

Further, while POD is frequently applied to position data, we use velocity data as input to the POD process. Velocity coordinate data contains more “dynamic information” of the system under consideration. Especially when investigating models which combine small (fast) fluctuations with large (slow) body motions, position data may overvalue large movements, which was, at least to us, unascertainable using velocity data. For a more detailed discussion on our choice of the POD input-data, and the reduction method in general, the interested reader is referred to [27].

### 3.3 Constraint reduction

To overcome this issue, we proposed a constraint reduction procedure in [27], which uses the projection of the constraint Jacobian onto the subspace spanned by \(\mathbf{U}\). Therein, if \(\mathbf{U}^{\mathsf{T}}\cdot \mathbf{C}_{\mathbf{q}}^{\mathsf{T}}\) equals zero for all instances of time, a smaller set of constraint equations \(\mathbf{C}_{l} ( \mathbf{q} ) \in \mathbb{R}^{l}\) with \(l< m\) can be defined. However, this method tends to fail if the constraint equations are defined in any direction other than the global axis directions. So, in terms of a more complex model, which consists of boundary conditions in various directions in space, hardly any constraint equations might be identified as reducible by the algorithm proposed in [27].

In order to overcome this issue, the constraint reduction method is generalized in the following section, ensuring that the reduced MBS system is determined and well-conditioned.

#### 3.3.1 A generalized constraint reduction method

### 3.4 Comments with respect to the constraint forces in the reduced model

The model order reduction to physical and constraint coordinates, as presented in this section, allows us to find a reduced order model, which reproduces the original systems coordinate movement with high consistency. Nevertheless, the advantage of finding a smaller set of coordinates by a linear combination of the original coordinates, goes hand in hand with a loss of system information, as we truncate the POM matrix.

To ensure correct constraint forces is still an open field of research and several authors (see [8, 12, 30, 31]) deal with the problem of redundant constraint equations, whether or not they result from singular configuration, overconstrained modeling (as in the case of a door being modeled using both hinges), etc. Also in the present case of the reduced order multibody system, the constraint forces turn out to differ from the constraint forces of the original system. This is a major limitation, as the reduced order model cannot be used to generate, for instance, load data in joints, as they might be needed for special FEM simulations. Nevertheless, we can show that the change in the constraint forces is already caused by the physical coordinate reduction (the reduction of \(\mathbf{q}\)) and not introduced due to the constraint reduction (the reduction of \(\mathbf{C}\), etc.).

As a matter of fact, if any original coordinate is weighted with factor zero in the reduced order model, which is a zero entry in all related entries of the projection matrix \(\mathbf{U}\), forces acting on this coordinate no longer enter the reduced order model. Therefore, the corresponding constraint force \(\mathbf{U}\mathbf{C}_{\mathbf{q}}^{\mathsf{T}}\boldsymbol{\lambda }\) equals to zero.

With these limitations in mind, we point out that this model reduction method is not usable for simulations concerning constraint forces as important results.

## 4 Numerical examples

### 4.1 Example: rigid V8 crank drive under superimposed vibrations

### 4.2 Example: flexible front axis suspension

The flexible suspension model, see Fig. 3(a), consists of the flexible lower wishbone, the flexible pushrod and the rigid components tie-rod, top wishbone, steering knuckle, and suspension linkage. The flexible lower wishbone consists of four interaction nodes, representing the connection to the frame, to the push-rod and to the steering knuckle. The flexible pushrod consists of two interaction nodes, representing the connection to the linkage and the lower wishbone. The modal basis of each flex body is derived by MSC. Nastran [19] and consists of 30 modes for the wishbone and 26 modes for the pushrod. Note that flexible MBS simulation in FreeDyn needs the so-called zero-inertia bodies. These bodies, with mass and inertia equal to zero, are needed to establish constraints between a flex bodies’ interface node and any other body or ground. Hence, any boundary condition between a flex body and a rigid body brings up a set of seven coordinates and at least seven constraint equations, including the inner Euler-constraint [21]. Summing up, the model consists of four rigid, two flexible, and six zero-inertia bodies. It should be noted that flexible bodies, modeled in FreeDyn, consist not only of the flexible coordinates but also of seven physical coordinates which describe the rigid body motion. Hence, together with the 56 flexible coordinates, a total of 140 coordinates arise. As the vehicle’s frame is not modeled, the suspension is fixed to the ground, bringing up a total of 84 constraint equations. The high number of constraint equations originates from the zero-inertia bodies, which are coupled to the flexible bodies using fix joints, with six constraint equations each. As the flexible lower wishbone has four coupling points, and the flexible pushrod two, 36 constraint equations arise between these points and the zero-inertia bodies. Further, due to the use of Euler parameters, each body brings up the need of one internal constraint equation, which causes another six constraint equations. The remaining 42 constraint equations result from the actual constraints between the single physical/zero-inertia bodies and the environment.

The suspension strut is replaced by its characteristic spring and damping force curves, and the frictionless model is actuated at the wheel hub by a frequency sweep from 1 to 15 Hz. Due to the chosen mass and damping parameters, the simulation passes through a resonant frequency of the model, see Fig. 3(b).

## 5 Discussion and conclusion

This paper extends the model order reduction method, presented in [27], to models in arbitrary position in space. As the method originally proposed focuses mainly on rigid models, with constraints modeled in global axis directions, its application to models other than these is not as straightforward as expected. Especially when dealing with vibrations or resonance phenomena, the resulting tumbling behavior critically influences the constraint reduction potential. Therefore, we herein propose a generalization to the constraint reduction method in [27], which is based on the ideas of principal component analysis, showing that the original constraint reduction idea is a special case of the present one. The proposed constraint reduction method ensures that the resulting reduced order models are determined and well-conditioned. One no longer has to distinguish between internal and external constraint equations, as the constraint reduction method also allows the Euler-constraints to be reduced.

However, the reduced order models are no longer close to the minimal set of coordinates representation. Finally, it must be stated that, although the reduced order models show high consistency in the system states, the constraint forces of the reduced order model are no longer comparable to the original system. Hence, this method is not suitable to deliver the same joint reaction forces, which are then used for upcoming considerations like FEM simulations, as the original model.

Further, the accuracy of the reduced order models is closely related to the processed snapshots. In cases with critically changing excitation directions or model parameters, omitted constraints may become necessary. In such a case, it might be necessary to collect a new set of snapshots in order to renew the reduction subspace and the set of unique constraints.

The numerical examples, with which we demonstrated the method, are modeled arbitrarily in space. The proper orthogonal value plots suggest that for practical examples, the reduced order models are not necessarily indicated by a set of high-valued POVs or huge drops in magnitude, but could also be indicated by, e.g., flat spots. We point out that in such a case it may be necessary to identify the reduced order model by repetitive simulation runs, using a varying number of reduced coordinates. The numerical examples show that although vibrating systems and resonance phenomena complicate the identification of a reduced order model, the reduction method still performs well.

## Notes

### Acknowledgement

This work was supported by the Austrian funding agency (FFG) under Grant number 839074.

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