A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control

TLDR: A qualitative comparison of these methods is presented, hereby focusing both on theoretical and computational aspects, and the differences are illustrated on a quantitative level by means of application of the model reduction techniques to a common example.

About: This article is published in Journal of Sound and Vibration.The article was published on 2013-09-16 and is currently open access. It has received 217 citations till now. The article focuses on the topics: Reduction (complexity).

Figures

Fig. 5. Comparison of the modal displacement method (MD), balanced truncation (BT) and moment matching (MM) for reduction to K¼10 (k¼20): magnitude of the frequency response function for input u2 and output y2. For balanced truncation and moment matching, the reduced-order model is based on input u1 and output y1.

Fig. 6. Magnitude of the error for reduction to K¼10 (k¼20) for input u2 and output y2. For balanced truncation and moment matching, the reduced-order model is based on input u1 and output y1.

Fig. 1. Actuation frame model.

Fig. 4. Deflection shapes of the first three modes with eigenfrequencies 1:20 103, 2:66 103 and 2:83 103 Hz.

Fig. 3. Magnitude of the error for reduction to K¼10 (k¼20) for input u1 and output y1. Line styles as in Fig. 2.

Fig. 2. Comparison of the modal displacement method (MD), balanced truncation (BT) and moment matching (MM) for reduction to K¼10 (k¼20): magnitude of the frequency response function for input u1 and output y1.

Frequently asked questions

Q1. What is the effect of the mode displacement technique on the reduced-order model?

Since the truss frame system exhibits modal damping, the reduced-order model obtained by the mode displacement technique is guaranteed to be stable. 

Q2. What are the contributions mentioned in the paper "A comparison of model reduction techniques from structural" ?

The model reduction problem has also been studied in the systems and control community, where the analysis of dynamic systems and the design of feedback controllers are of interest. This paper aims at providing a thorough comparison between the model reduction techniques from these three fields, facilitating the choice of a suitable reduction procedure for a given reduction problem. To this end, the most popular methods from the fields of structural dynamics, systems and control and mathematics will be reviewed. In the current paper, popular model reduction techniques from all the three fields mentioned above will be reviewed. The focus of this paper is on this comparison ; it does not aim at presenting a full comprehensive historical review of all method in these three domains. In this paper, the scope will be limited to model-based reduction techniques for linear time-invariant dynamical systems. Also, reduction methods for nonlinear dynamical systems ( see e. g. [ 54,61 ] ) fall outside the scope of this paper. The outline of this paper is as follows. This comparison will be illustrated by means of examples in Section 4, which further clarifies differences and commonalities between methods. 

Q3. What are the common methods used for the reconstruction of a structure?

The most common methods are mode superposition methods [53], in which a limited number of free vibration modes of the structure is used to represent the displacement pattern [10]. 

Q4. What is the importance of preservation of additional properties?

Preservation of additional properties is of importance if the reduced system has to exhibit some physical properties of the model; for instance, when the reduced system has to be a (realizable) circuit consisting out of resistors, inductors and capacitors (a RLC network), just as the original system. 

Q5. What are the modal truncation and moment matching techniques?

The modal truncation and moment matching model reduction techniques from structural dynamics and numerical mathematics have in common that they can be considered as frequency-domain-based (or Laplace-domain-based) techniques.