The vibration and acoustical behavior of a mechanical structure is determined by its dynamic characteristics. This dynamic behavior is typically described with a linear system model. The inputs to the system are forces (loads), and the outputs are the resulting displacements or accelerations. System poles usually occur in complex conjugate pairs, corresponding to structural vibration 'modes.' The pole's imaginary part relates to the resonance frequency and the real part to the damping. Structural damping is typically very low (a few percent of critical damping). The system's eigenvectors, expressed on the basis of the structural coordinates, correspond to characteristic vibration patterns or "mode shapes." System identification from input-output measurements yields the modal model parameters.1 This approach is now a standard part of the mechanical product engineering process.
However, several constraints make the system identification process for structural dynamics more complex than in electrical engineering or process control. A key issue is the difficulty of selecting the correct model order and the corresponding validation of the obtained system poles. First of all, a continuous structure has an infinite number of modes. In practice, the analyst is interested only in a limited number of these, up to a certain frequency or only in a certain frequency band. Still, model orders of more than 100 are no exception. Furthermore, while some of the modes are separated in resonance frequency, others may be very close, leading to highly overlapping responses. The standard approach of selecting a model order and then deriving the corresponding poles is in general not applicable, and over-specification of the model order is needed. Finally, the size of the problem often requires more than 1000 responses to be processed (e.g. a car body is discretized by more than 500 nodes and measured in three directions) and using large data segments to reduce the measurement noise. The consequence of these constraints is that classical system identification approaches, extracting the parameters of a discrete-time state-space model or of an ARMA model directly from the sampled input-output data, are often neither practical nor feasible. Specific procedures are then needed for modal analysis.
Modal analysis users face the following challenges:
* The ever increasing complexity of the tested structures: e.g. fully assembled vehicles instead of components, in-situ instead of laboratory measurements.
* The changing role of testing in the product development cycle,2 implying a reduction of time available for testing and analysis and a demand for increased accuracy adequate for use with hybrid or FE applications,
* Specific to the modal parameter estimation process itself: inconsistency between estimates of different operators, the tedious task of selecting obvious poles in a stabilization diagram and the time-consuming iterations required to validate a modal model.
A key requirement for experimental modal analysis (EMA) is that a reliable analysis of complex datasets should be possible with minimal, or even excluding, user interaction. This is the context of the methodology developed here. The following sections discuss:
* Using a stabilization diagram to solve the order determination problem.
* An automatic procedure is expanded to heavily rely on the stabilization diagram concept.
* The methodology is validated using industrial examples.
Stabilization Diagram
So in EMA, the problem of determining model order boils down to deciding how many modes n to use for fitting the FRFs. Note that in classical system identification literature, many formal procedures exist to solve the problem of determining model order. Models of different order are identified and compared according to quality criteria such as Akaike's final prediction error or Rissanen's minimum description length criterion. Most of these techniques were developed in the context of control theory, where it is the aim to identify optimal low-order models. But in structural dynamics, the order of the models is typically chosen much higher to reduce the bias on the estimates and to capture all relevant characteristics of the structure, even in the presence of large amounts of measurement noise. As a consequence of order over-specification, the physically meaningful poles are completed with a set of 'mathematical' poles, modelling model, data and process noise but without having a relation to the structural problem.
