Uncertainty propagation using a component modal method

Typically a basic mid-frequency analysis will use the physical properties of a structure to model the modal response, commonly using Finite Element Analysis (FEA). The Frequency Response Function (FRF) for the structure is then found using modal summation. If some uncertainty exists in the physical properties of the structure then a common route would be to use Monte Carlo simulations to allow the properties to vary randomly and the analysis repeated many times. This process is represented by the flowchart in Figure 6-1.

Figure 6-1 Typical Monte Carlo analysis of variability.

However, this process is expensive and time consuming as a full eigensolution must be found for each cycle of the MC analysis. Potential areas for approximation include the use of

Physical Properties

Global Modal Properties

FRF Variability FEA Modal Summation FRF Statistics

eigensolution approximations, such as the perturbational approach used in chapter 5, and the use of a reduced or truncated set of global modes in the modal summation to obtain the FRF.

An alternative method is to use CMS to divide the structure into substructures or components which can then be analysed separately and these results assembled into a global model. A representation of this process is shown in Figure 6-2. The theory behind the CMS process and two common types of CMS analysis, fixed interface analysis and free interface analysis, are discussed in detail in section 6.2. Briefly, an eigensolution is performed on each substructure to obtain a set of component modes, which depending on the type of CMS analysis may include normal modes of free vibration, rigid-body modes, constraint modes or attachment modes. These are combined in some form (dependent on the chosen method) to create a set of assumed modes or trial vectors for the full structure.

Figure 6-2 Basic CMS analysis of a structure.

If all the component modes are used then the CMS process does not offer any reduction in computational cost over a standard global FEA outlined in Figure 6-1. However, one of the main benefits of the CMS analysis is the possibility to use a reduced set of component modes, referred to as ‘kept’ modes, to describe the subsystem behaviour. In addition there is some benefit when modelling extremely large or complex structures in that the components may be

Physical Properties

Local Modal Properties

FRF Variability FEA/CMS Global Eigensolution FRF Statistics

Global Modal Properties

Modal Summation Kept Modes

supplied from different manufacturers, which offers the possibility of assembling modal models from different suppliers. Typically the component modes are obtained from an FE analysis, however they may also be obtained experimentally.

Possible approximations may also be made by simplification of the physical uncertainties. For example, the joints between components will exhibit some uncertainty, which may be highly variable according to the manufacturing process. This source of uncertainty, if ignored, will approximate the behaviour of the built-up structure.

A further extension of this method is the Local Mode Perturbational method (LMP) first proposed by Mace and Shorter [6.3]. This method introduces uncertainty directly into the component eigenvalues, thereby eliminating the consideration of physical property variations. These component eigenvalue variations are assumed to be small and an eigensolution approximation is used to estimate the global eigenvalues and eigenvectors. In this case a perturbational approximation is used. A representation of this process is shown in Figure 6-3.

Figure 6-3 Local Mode Perturbational Method.

The method as proposed by Mace and Shorter uses a fixed interface CMS method. Three main areas of approximation are introduced. The first two of these are involved with the CMS method

Physical Properties

Local Modal Properties

FRF Variability FEA/CMS Perturbation FRF Statistics

Global Modal Properties

Modal Summation Kept Modes

(explained in detail in section 6.2) which uses local component fixed interface modeshapes and constraint modes. As the LMP method only considers variability in the component eigenvalues, the variability in the component modeshapes and constraint modes is not included. Thirdly, an eigensolution approximation is used to obtain the global modal response. The LMP method will be applied in this thesis to generate expressions for the statistics of the global modal response, in terms of the statistics of the local modal behaviour.

Two further potential areas for approximations are proposed here; the use of free interface component eigenvalues statistics in place of the fixed interface values and the disregard of covariance between components. Both of these proposals will be examined in further detail in chapter 8.

In summary the potential areas for data reduction and approximation are as follows:

1) Simplification or approximation of the physical uncertainties such as ignoring joint uncertainties

2) Truncation of the set of local modes to a reduced set of kept modes

3) Assuming constant local eigenvectors e.g. assuming fixed interface local modeshapes remain unchanging

4) Assuming constant local constraint modeshapes

5) Simplification or approximation of the component uncertainties such as ignoring covariance between components

6) Use of free interface component statistics to approximate fixed interface component statistics

7) Truncation of the set of global modes to a reduced set of kept modes 8) Use of eigen solution approximations such as perturbational expansions

These areas for approximation are summarised in Figure 6-4. The LMP method will be used in this thesis as a basis for further examining the propagation of uncertainty in complex structures. In the following two chapters the transmission of uncertainty from the physical properties to the global FRF will be divided into two stages. Chapter 7 will examine the relationship between the variations in the physical properties of a component to the resultant variations in the modal properties. Chapter 8 will examine the relationship between variations in the local modal properties and the resultant variations in the global modal properties and the FRF response.

Figure 6-4 Potential areas for data reduction and approximation.

In both chapters 7 and 8, the propagation of statistics and the PDF of the response will be considered. The effect on both eigenvalues and eigenvectors will be discussed along with examples considering both correlated and uncorrelated sources of uncertainty.