Modal Analysis and Testing

Modal analysis is an increasingly more important engineering tool that was first applied around 1940 in the search for a better understanding of aircraft dynamic behaviour. Till the end of the 60’s developments were slow and experimental techniques were based on the use of expensive and cumbersome narrow band analogue spectrum analysers. The modern era of modal analysis can be taken as starting at the beginning of the 70’s, based upon the commercial availability of Fast Fourier Transform (FFT) spectrum analysers, transfer function analysers (TFA) and discrete acquisition and analysis of data, together with the availability of increasingly smaller, less expensive and more powerful digital computers to process the data.

Signal processing is the science of applying transformations to measurements, to facilitate their use by an observer or a computer and the analysis of data involves the three phases of data acquisition, processing and interpretation. The objective of data analysis is to highlight/extract information contained in a signal that direct observation may not reveal.

This chapter is intended to provide an introduction to some topics in signal processing that are finding wide application. These are optimal (and adaptive) filtering using least squares criteria, deconvolution (including cepstral analysis), non-stationary signals with particular reference to time-frequency methods and, finally, higher-order spectra (for non-Gaussian and nonlinear processes).

Three major structures are involved in laboratory simulations of field vibration environments: the test item, the vehicle, and the vibration exciter. The test item is the structure under study and is attached to the vehicle in the field environment. The word vehicle is used here to refer to any structure used in the field to transport, or simply to support the test item. In laboratory simulations, the test item is attached to one or more vibration exciters. The ultimate goal of a laboratory simulation is to define appropriate test item inputs such that its field dynamic behavior can be reasonably simulated, particularly stress levels, natural frequencies, and mode shapes. This paper describes a general theoretical model for laboratory simulations of field dynamic environments. Frequency domain input-output relationships are used to model the structures involved in the simulation process. Matrix partitioned frequency domain equations of motion are written for each structure. Manipulation of these equations with suitable boundary conditions leads to general expressions for the test item interface forces and motions in both field and laboratory environments. These expressions are used to describe four laboratory test scenarios.

This paper presents a numerical example of the theory presented in Part I [1] where a model is developed to describe the requirements for laboratory simulations of field vibration environments. This example illustrates the application of such a model to simulated data in order to evaluate its accuracy in predicting field interface forces and motions, and in using field data to define appropriate test item inputs in laboratory simulations. It is assumed that no external forces act on the test item in either the field or laboratory environments. All Frequency response functions (FRFs) are calculated for all test structures from a multi degree of freedom (MDOF) discrete linear system. The equations developed in Ref. [1] are employed to estimate field interface forces and test item motions as well as to define test item inputs in the laboratory simulation. The test item motions obtained in all laboratory simulations are compared with the corresponding field motions.

The ultimate goal of a laboratory simulation is to ensure that a given test item will survive when exposed to its field dynamic environment. Two previous papers [1,2] described the process of using field vibration data to define suitable test item inputs in laboratory simulations. The structural interactions that occur when the test item is attached to the vehicle in the field or to the vibration exciters in the laboratory were discussed in Ref. [1]. Numerical examples describing different test scenarios used in laboratory simulations were presented in Ref. [2]. In these test scenarios, the test item was subjected to interface forces only, i.e.; no external forces were applied to the test item while in the field environment. The first part of the present paper discusses laboratory simulations of field data when the test item is subjected to external forces in addition to interface forces in the field environment. Numerical examples of different laboratory test scenarios show that accounting for field external forces in the laboratory is of major importance. The second part of the paper deals with the force identification problem. The pseudo-inverse technique, commonly employed in modal testing is used to predict the test item input forces from field motions. Numerical examples show the feasibility of the pseudo-inverse method in predicting external loads. An experimental analysis is performed on a free beam to discuss the practical implications of this force identification technique when applied to realistic situations. This paper deals only with transient and periodic forces and motions.

This paper discusses motion transmissibility concepts and their application to test environments. When a test item is attached to a vehicle at a single point and field external force effects are negligible, test item accelerations can be predicted by using the acceleration transmissibility frequency response functions (FRF) and the test item’s single point field interface motion is used as the input motion. When the test item has multiple interface points and field external force effects are negligible, the motion transmissibility concept is extended by defining a transformation such that the multiple field interface motions can be used as test item inputs in the laboratory. This transformation is defined by the Q-transmissibility matrix that is obtained from the test item driving and transfer point accelerance FRFs and reduces to the standard single point transmissibility FRF for the case of a single interface point. The Q-transmissibility matrix approach is employed to numerically simulated data to predict the test item external motions and it is shown that the usual laboratory setup employing a single vibration exciter and a rigid test fixture leads to incorrect motion predictions and that multiple vibration exciters must be used to simulate field data. Experimental results indicate that: (i) the Q-transmissibility matrix transformation is feasible when dealing with actual data as long as the solution for the test item motions is carried out in a least squares sense since the test item interface FRF matrix may present rank deficiency problems in the solution process, and (ii) Curve fitted accelerance FRFs can be used to reduce experimental noise effects but the quality of the resulting motions is very sensitive to curve fitting errors, especially in the vicinity of natural frequency peaks and antiresonance valleys.

While in the field environment, the test item is connected to the vehicle at several interface points. The combined structure is then subjected to a variety of field loads that produce motions at the test item interface and external (non-interface) points. A commonly employed laboratory simulation procedure consists in attaching the test item to a single vibration exciter through a test fixture. Choice of appropriate test fixtures and definition of suitable laboratory test item inputs is of major importance in obtaining realistic simulation of a field environment. MIL — STD810D recommends that: (i) The test item be attached to the exciter through a rigid test fixture; (ii) When available, Field data should be used to define the test item inputs in the laboratory environment. In this case, enveloping techniques are frequently used with the field data in order to define the input that will be applied in the laboratory. The objectives of this paper is to show the consequences of using these current laboratory simulation procedures on the test item dynamic response. The test item, the vehicle, and the test fixture are modeled by a multi degree of freedom (MDOF) lumped systems. Random excitation and response signals are used in the simulation process. The test item laboratory inputs are defined in terms of an acceleration auto spectral density (ASD) that is obtained by enveloping field data as recommended in MIL — STD810D. The test item accelerations obtained in the laboratory when the test item is attached to either a rigid or flexible test fixture are compared with the actual motions that occur when the test item is attached to the vehicle in the field. It is shown that both the rigid and the flexible test fixtures seriously affect the resulting test item accelerations and that the current enveloping techniques expose the test item to excessively high dynamic strain levels.

This chapter presents a summary of the analytical tools required to understand the theoretical aspects of modal analysis. From the mathematical point of view, modal analysis results because of the ability to decouple coupled sets of ordinary differential equations. This ability is normally set in the context of linear algebra and the theory of matrices. Hence, much of what follows is based upon manipulations of vectors and matrices. The orientation of this chapter is to provide the elementary background required for the following chapters.

Identification of modal parameters in the time domain offers an attractive alternative to the more classical approaches in the frequency domain. The elimination of the need to perform frequency transformation on the inputs and responses and the associated errors of leakage, truncation, frequency resolution and lengthy time records, among other errors, enhances the accuracy and practicality of the time domain approaches.

As for time domain methods, there are two main categories of identification methods in the frequency domain: indirect methods and direct methods. The former are based on the modal model, i.e, on the modal parameters (natural frequency, damping ratio and modal constants), while the latter are based on the spatial model, i.e, they allow for the evaluation of the mass, stiffness and damping matrix coefficients. There is still a third category, the tuned-sinusoidal methods. This is a very particular case, that will be briefly mentioned in section 4.

A strategy is developed to reduce the poor conditioning of estimation equations used in the parametric identification of finite element models for linear elastodynamic structures. It is based on the simultaneous exploitation of synthesized responses resulting from different boundary condition configurations obtained by introducing fictive constraints. This approach is then applied to an updating procedure employing partial output residuals constructed from the eigensolutions of the model and structure. Two numerical examples illustrate the potential of this method.

In the paper basic procedures for computational updating of analytical model parameters are presented. The procedures have been investigated thoroughly in recent years with respect to

The residuals presented are formed by force and response equation errors, by eigenfrequency and mode shape errors and by frequency response errors. The procedures have been derived to handle incomplete test vectors, where the number of measured degrees of freedom (DOF) is much less than the DOF no. of the computational model. Finally an example of updating a laboratory test structure is reported including some recommendations and experiences.

Before using an initial finite element model for subsequent model updating the user must be aware of the modelling uncertainties arising from three main sources :

Structural damage detection using non-destructive vibration test data has received considerable attention in recent years. The analysis is usually based on the assumption that damage will change the structural (mass, stiffness or damping) properties which urther lead to changes in the dynamic characteristics such as the natural frequencies, damping loss factors and mode shapes. Since the changes on the dynamic characteristics can be measured and studied, it is possible to trace what structural changes have caused the dynamic characteristics to change, thus identifying the damage.

In the most general terms damage can be defined as changes introduced into a system that adversely effect the current or future performance of that system. Implicit in this definition is the concept that damage is not meaningful without a comparison between two different states of the system, one of which is assumed to represent the initial, and often undamaged, state. This discussion is focused on the study of damage identification in structural and mechanical systems. Therefore, the definition of damage will be limited to changes to the material and/or geometric properties of these systems, including changes to the boundary conditions and system connectivity, which adversely effect the current or future performance of that system.

Dynamic characteristics of a structure are usually referred to as its natural frequencies and mode shapes. The ability to alter these characteristics in order to have desired dynamic characteristics for a structure either by design, re-design or by modification has been an enduring quest by structural analysts. The need to change dynamic characteristics of a structure may come from new design requirements, solution for excessive vibration, or the necessity to control the response of the structure. It is unrealistic and unnecessary to attempt to alter all natural frequencies or mode shapes of a structure but selected changes via modification are possible. In many applications, it is often a part of the dynamic characteristics of a structure such as a particular natural frequency that needs to be changed.

This chapter provides an examination of viscoelastic damping normally characterized by hysteresis, a complex modulus or frequency dependent damping. Viscoelastic damping exhibited in polymeric and glassy materials as well as in some enamels. Such materials are often added to structures and devices to increase the amount of damping. Examples are rubber mounts and constrained layer damping treatments. Typically in modal analysis the simplest form of modeling damping is used. This form assumes that the damping is a linear, time invariant phenomena chosen to be viscous, or proportional to velocity, motivated by the ability to solve the equations of motion. With this as a first model, one is lead to conclude that viscoelastic behavior causes frequency dependent damping coefficients resulting in the concept of complex modulus. Thus it is not clear how to perform modal analysis of structures with viscoelastic components. Here an alternative formulation is discussed and presented for multiple degree of freedom systems that allows the treatment of hysteretic damping in dynamic finite element formulations, and hence provides a connection to modal analysis and testing.

Wave propagations and vibrations are associated with the removal of energy by dissipation or radiation. In mechanical systems damping forces causing dissipation are often small compared to restoring and inertia forces. However their influence can be great and is discussed in the present survey paper together with the transmission of energy away from the system by radiation. Viscoelastic constitutive equations with integer and fractional time derivatives for the description of stress relaxation and creep of strain as well as for the description of stress-strain damping hysteresis under cyclic oscillations are compared. Semi-analytical solutions of wave propagation and transient vibration problems are obtained by integral transformation and elasticviscoelastic correspondence principle. The numerical solution of boundary value problems requires discretization methods. Generalized damping descriptions are incorporated in frequency and time domain formulations for the boundary element method and the finite element method.

The eigen characteristics of a dynamical system offer a vector sub-space suitable for performing canonical transformations on the system of equations of the structural dynamics problems being considered. While the concept is mathematically fully developed for both damped and undamped systems, practitioners — at all levels — tend to indiscriminately use the system’s normal modes as a basis for applications containing nonproportional damping. Such a practice in most cases is a reasonable approximation and results in small, if not infinitesimal, errors. However, with the increase of sophistication and accuracy requirements in certain applications of modal analysis, these approximations must be fully analyzed and understood.

In this paper, emphasis is directed to some specific applications for which it is generally a common practice to use normal, or undamped, modes as a vector subspace for use with nonproportionally damped system. These are:

Detailed background on complex modes is presented. Commonly used transformations are examined and error models are derived and quantified.

Modern design and tailored materials have lead to structures with significant reduction of weight going along with an improvement of stiffness properties. But as the properties of such conventional passive systems can not be varied time dependent they have reached their limit of performance in a design state.

The phenomena, related to the existence of acoustic modes, were already known in the ancient world and our ancestors, though instinctively, have even exploited some of the acoustic effects [1]. The first treatments of scientific character of the field date back to the 19thcentury [2,3] while the basics of the modal theory of room acoustics were developed in the first half of this century [4–7]. Nevertheless, a revival of the acoustic modal theory and its experimental aspects seems to be worthwhile for a couple of reasons.

The challenge of model updating is to refine a finite element (FE) description of a structure so that it exhibits the same dynamic behaviour as its experimental counterpart. In this sense, the FE model is optimised with respect to the measurements. The following sections discuss the fundamentals of optimisation and then describes two advanced techniques — genetic algorithms and simulated annealing. These techniques enable minimisation of high order, complex functions of which can be defined in model updating terms by declaring a difference between the measured characteristics and the FE model’s representation. Genetic Algorithms have deservedly received particular attention from the optimisation community in a variety of applications. The technique uses natural selection and genetics to form search algorithms. Simulated Annealing is also based on a physical phenomena, the state of a metal at a particular temperature during cooling. As such these methods differ significantly in form from standard optimisation techniques. So in addition to describing genetic algorithms and simulated annealing along with their applications in modal analysis, this paper will provide a brief introduction to the field of optimisation and the way in which some of the approaches differ. As with many techniques in modal analysis, several may seem appropriate at the outset. The field of optimisation is no exception. However, the introduction of new techniques offers new opportunities which have the potential to exploit the increased use of parallel architecture in computing.

The primary objective of this paper is to lay the foundations for the application of modal analysis and testing technology to rotating machinery. Although this technology is already used to some extent by the manufacturers and users of rotating machines, such use is generally limited to stationary parts of the machine, or to rotating components in a stationary state. Relatively little use of modal testing has been possible until quite recently for a number of reasons: mostly to do with the practical difficulties involved but also because of complications which arise in the he underlying theory, not all of which issues are fully understood by all who need them. In this paper, we shall try to dispel these confusions and show the ways in which modal testing may become more widely used in this important area.

Nonlinearity is a topic which is difficult to avoid as all structures encountered in practice are nonlinear to some degree, the nonlinearity often being a function of factors such as boundary/initial conditions, material properties, previous history, excitation levels etc. Nonlinearity can occur in a global (eg material nonlinearity) or a local sense (eg joints/interfaces).