Identifying parameters of a model using experimental data has been extensively studied in various areas, including for determining mechanical material properties from strain measurements or modal vibration data. In order to find the parameters that make the model agree best with the experiments, the most widely used method is based on minimizing the least-squares error between the experimental data and the model predictions. An alternative way to identify the material properties is through statistical frameworks, based on maximum likelihood or on Bayes' rule. The Bayesian framework is more general since it can include prior knowledge. Isenberg proposed the application of the Bayesian framework for parameter estimation in 1979 and several articles have been published since on the application of this approach to frequency or modal identification in particular, i.e. identifying material properties from vibration test data. Here we compare the least-squares and the Bayesian approach to material properties identification, seeking situations where the properties identified by the two approaches are significantly different. We chose a simple, didactic example of a three bar truss so that the results of the comparison are not clouded by unnecessary complexity or computational issues. For the three bar truss example we seek to identify the Young modulus of the bars from strain measurements on these bars. Due to uncertainty in the loads applied on the truss, which propagate to variability in the strains, the different strain measurements provide conflicting information. The least-squares approach and the Bayesian approach combine this conflicting information differently to come up with the identified Young modulus. We found that the least-squares method can be negatively affected by differences in magnitudes of the experimental data, by differences in uncertainty and by correlation among the experimental data. The least-squares method does indeed implicitly weight strains depending on their sensitivity to Young's modulus. This negatively impacts the results found by the least-squares approach. Different uncertainty in the strains in each bar also negatively affects the least-squares approach, which does not differentiate between a strain with low uncertainty and one with high. Last, the least-squares approach does not handle correlation between the strain measurements. On the other hand the Bayesian correctly handles these three situations, namely different strain magnitude, uncertainty and correlation. We found that in each situation the Bayesian approach systematically outperforms the least-squares approach. In the case when all three effects act combined, we found that the Young modulus identified by the Bayesian approach is on average almost 10 times closer than least-squares to the true value of the Young modulus.

I N an interesting paper, Professor Haftka describes a method for optimum control of static deformations of space structures. However, it must be emphasized that not only are the deformations of the space structures static but, appropriately, the means of control are also static. Having this in mind, it is not apparent to this author why Prof. Haftka had to involve the mass of the structure in his equations. In an approach somewhat different to that given in the paper, it will be shown here that one can develop adequate equations for the static control of a space structure without involving the mass of the structure. It will be assumed that the free space structure is represented by its stiffness matrix K(nxn), usually obtained by using finite element methods. This matrix is needed to calculate the deformations of the structure constrained in a statically definite way. The distortion from the desired shape of the structure is represented by the deformation vector \l/(n x 1). Thermal or force controls are applied on the structure to minimize the distortion. The rigid body displacements of the structure are characterized by the matrix R(nxr). (For space structures, r = 6.) For convenience, the matrix R can be orthonormalized to obtain The total displacement vector caused by the distortion and the controls will be given by

RÉSUMÉ. La méthode d'identification bayesienne présente l'avantage de pouvoir tenir compte de différentes sources d'incertitude présentes dans le problème et de quantifier la confiance avec laquelle les propriétés sont identifiées. Son application sur un problème d’identification de propriétés élastiques orthotropes à partir de mesures des champs des déplacements pose cependant des problèmes de coût numérique. Nous proposons de résoudre ce défi par l'utilisation de méthodes de réduction de dimensionnalité (analyse en composantes principales) et la méthode des surfaces de réponse. Cette approche est testée sur des mesures de champ simulées, résultant dans l’identification de la distribution jointe de probabilité des quatre propriétés élastiques orthotropes.

In probabilistic design, uncertainties (such as material properties or loads on the structure) are considered when calculating the reliability of a structure, and the structure is optimized such that the risk is distributed amongst different modes of failure. The changes in the design that protect against the failure in the different modes often conflict [1]. For example, we can consider a vertical column that experiences both a compressive load and thermal load at its top surface, where failure occurs if the column buckles or the temperature of the bottom face exceeds the allowable. If the column is made thin, the amount of heat transferred to the bottom surface is limited, and the probability of thermal failure is reduced. However, the risk of buckling failure is increased.