Modal analysis reduction of multi-body systems with generic damping

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Highlights

  • An energy based modeling metric for model reduction is used.

  • A model reduction metric and algorithm are used to decouple modal equations.

  • A single harmonic excitation is used as an input to the linear system.

  • The procedure is developed for multi-body system with non-proportional damping.

  • An illustrative example demonstrates the effectiveness and accuracy of the algorithm.

Abstract

Modal analysis of multi-body systems is broadly used to study the behavior and controller design of dynamic systems. In both cases, model reduction that does not degrade accuracy is necessary for the efficient use of these models. Previous work by the author addressed the reduction of modal representations by eliminating entire modes or individual modal elements (inertial, compliant, resistive). In that work, the bond graph formulation was used to model the system and the modal decomposition was limited to systems with proportional damping. The objective of the current work is to develop a new methodology such that model reduction can be implemented to modal analysis of multi-body systems with non-proportional damping that were not modeled using bond graphs. This extension also makes the methodology applicable to realistic systems where the importance of modal coupling terms is quantified and potentially eliminated. The new methodology is demonstrated through an illustrative example.

Introduction

Current modeling techniques and simulation tools provide engineers with a variety of options when it comes to modeling of new or existing systems. These tools and techniques are powerful and extensively used in everyday engineering, nevertheless further improvements on modeling decisions and model complexity issues would make them more efficient. Specifically, a main disadvantage is that modeling techniques and simulation tools require sophisticated users who are often not domain experts and thus lack the ability to effectively utilize the available tools to uncover the important design trade-offs. Another drawback is that models are often large and complicated with many parameters, making the physical interpretation of the model outputs, even by domain experts, difficult. This is particularly true when “unnecessary” features are included in the model.

A variety of algorithms have been developed and implemented to help automate the production of proper models of dynamic systems. Wilson and Stein [1] developed the model order deduction algorithm (MODA) that deduces the required system model complexity from subsystem models of variable complexity using a frequency-based metric. They also defined proper models as the models with physically meaningful states and parameters that are of necessary but sufficient complexity to meet the engineering and accuracy objectives. Additional work on deduction algorithms for generating proper models in an automated fashion has extended the functionality of MODA [2], [3], [4]. These algorithms have also been implemented and demonstrated in an automated modeling environment [5].

In an attempt to overcome the limitations of the frequency-based metrics, Louca et al. [6] introduced a new model reduction technique that also generates proper models. This approach uses an energy-based metric (element activity) that in general, can be applied to nonlinear systems [7], and considers the importance of all energetic elements (generalized inductance, capacitance and resistance). The contribution of each energy element in the model is ranked according to the activity metric under specific excitation. Elements with small contribution are eliminated in order to produce a reduced model. The activity metric was also used as a basis for even further reduction, through partitioning a model into smaller and decoupled submodels [8].

Beyond the physical-based model reduction, modal decomposition can also be used to model and analyze continuous and discrete systems [9], [10], [11], [12]. One of the advantages of modal decomposition is the ability to directly adjust (i.e., reduce) model complexity since all modes are orthogonal to each other. The reduction of such modal decomposition models is mostly based on frequency, and the user defined frequency range of interest (FROI) determines the frequencies that are important for a specific scenario. In this case, modes with frequencies within the FROI are retained in the reduced model and modes outside this range are eliminated. As expected, mode truncation introduces error in the predictions that can be measured and adjusted based on the accuracy requirements [13], [14].

The element activity metric provides more flexibility than frequency-based metrics, which address the issue of model complexity by only the frequency content of the model. In contrast, the activity metric considers the energy flow in the system, and therefore, the importance of all energy elements in the model can be described. Previous work by the author addressed the development and reduction of modal representations using the bond graph formulation and using the activity metric [15], [16]. This work introduced a methodology that reduces the model complexity by eliminating entire modes or partial modes through modal elements (inertial, compliant, resistive). The identification and elimination of insignificant elements was performed with the use of the activity metric and model order reduction algorithm (MORA). This approach has advantages over frequency-based reduction techniques; however, it has a significant limitation in that it can only be applied to systems with proportional damping and thus is not able to be applied to realistic systems.

The objective of the current work is to develop a methodology that overcomes the limitations of the author's previous work, such that modal analysis and model reduction can be applied to a more general class of systems with non-proportional damping. In addition, the activity metric will be formulated for systems that are modeled with second order ordinary differential equations (ODE), rather than bond graphs and first order ODEs that were used in previous work. Second order ODEs are typically derived from Lagrange's equations or Newton's law. These two additions will make the activity metric a more appealing model reduction methodology that can be applied to realistic systems.

This paper is organized as follows: first, background about the energy-based metric and modal decomposition are provided. Next, the equation formulation and modal decomposition of multi-body systems with non-proportional damping is presented. Then, the activity analysis of all modal elements is introduced, along with the closed-form expressions of the steady state activities. An illustrative example of a linear quarter car model is also presented, in order to demonstrate the development of its modal decomposition and the evaluation of the coupling terms’ importance using the activity metric. Finally, in the last section, discussion and conclusions are given.

Section snippets

Energy-based model reduction

The original work on the energy-based metric for model reduction is briefly described here since it is the foundation of the contributions in this paper. The main idea behind this model reduction technique is to evaluate the “element activity” of the individual energy elements of a complex system model under a stereotypic set of inputs and initial conditions. The activity of each energy element establishes an importance hierarchy for all elements. Those below a user-defined threshold of

Multi-body systems

The activity metric as described in the previous section will now be introduced for multi-body systems. The definition of the activity remains the same as stated in Eq. (1); however, the modeling approaches for multi-body systems typically result in second order ODEs. In addition, the generalized effort and flow variables used for defining the activity now become force and velocity given that we have a mechanical system. Modeling of a multi-body system through either Lagranges's equations or

Illustrative example

A quarter car model, which is extensively used in the automotive industry, is chosen to apply the developed methodology. The model consists of the sprung mass, namely, the major mass supported by the suspension, and the unsprung mass, which includes the wheel and axle masses supported by the tire. The suspension is modeled as a spring and a damper in parallel, which are connected to the unsprung mass. The tire is also modeled as a spring and a damper in parallel, which transfer the road force

Discussion and conclusions

A new methodology for decoupling modal decompositions of multi-body systems is developed in this paper. The methodology uses the activity metric to quantify the importance of coupling terms, which can then be eliminated using MORA if they fall below the user-specified threshold. The activity metric can also be used to identify unimportant elements within a mode and achieve even further reduction in the model size and complexity. The new methodology overcomes the limitation of a previously

Loucas S. Louca received his Diploma in Mechanical Engineering from the National Technical University of Athens, Greece, in 1992. He then moved to the University of Michigan where he received his M.S.E. in 1994 and Ph.D. in 1998, both in Mechanical Engineering. During his graduate studies at the University of Michigan he received a Fulbright scholarship (Cyprus-America Scholarship Program). He continued to work in the Mechanical Engineering department at the University of Michigan as a Research

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  • Cited by (0)

    Loucas S. Louca received his Diploma in Mechanical Engineering from the National Technical University of Athens, Greece, in 1992. He then moved to the University of Michigan where he received his M.S.E. in 1994 and Ph.D. in 1998, both in Mechanical Engineering. During his graduate studies at the University of Michigan he received a Fulbright scholarship (Cyprus-America Scholarship Program). He continued to work in the Mechanical Engineering department at the University of Michigan as a Research Fellow until 2000 when he joined the research faculty of the Mechanical Engineering department as an Assistant Research Scientist. He was contacting research and advising students in the area of intelligent vehicle system dynamics and control within the Automotive Research Center at the University of Michigan. He joined the faculty of the University of Cyprus in January 2004 and he is currently and Assistant Professor.

    His research interests lie in the areas of system dynamics and control, bond graph theory, physical system modeling and model reduction of large scale systems, modeling of automotive systems, multi-body dynamics, computer aided modeling and simulation, and haptic interfaces and rehabilitation. He is the author of CAMBAS (Computer Aided Model Building Automation System), an automated modeling software that enables the rapid development of efficient models for linear systems and its used for the teaching of courses in modeling of dynamic systems.

    He is an active member of the bond graph research community and organizes focused sessions at modeling related conferences. He is also a member of the American Society of Mechanical Engineers (ASME) and the vice-chairman of the Modeling and Identification technical panel of the Dynamic Systems and Control Division. He is also a member of the Institute of Electrical and Electronics Engineers (IEEE), Society for Modeling & Simulation International (SCS), and Society of Automotive Engineers (SAE).

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