Vibrational characteristics of structures with uncertainty

This thesis is concerned with the prediction of the vibro-acoustic response of structures with uncertain properties in the mid frequency region. The motivation for this research is the growing need of engineers to understand the responses of a group of similar structures ranging from vehicles, aircraft and aerospace structures, to household whitegood appliances. These structures are complex in geometry and may possess variability in their material or geometric properties, as well as variation arising from the assembly and manufacturing processes. Small variations can have a significant effect on a dynamic response of a structure, and the effect of structural uncertainties increases as the frequency increases. Deterministic modelling techniques such as finite element analysis are only suitable to model complex structures at low frequencies. Furthermore, FEA cannot easily account for uncertainty or randomness in structural parameters. High frequency dynamic predictive techniques such as Statistical Energy Analysis can account for structural uncertainty but is limited to structures with high modal density. There exists a frequency range between the two methods in which neither technique can be applied with great confidence. The objective of this thesis is to investigate predictive techniques for mid frequency vibration analysis of dynamic systems with structural uncertainties.The first part of this work is to numerically characterise the effect of a range of uncertainties on the modal statistics of structures. The degree of uncertainty required to achieve universality of the statistical properties is investigated. This is achieved by examining the modal statistics of dynamic systems with a range of uncertainty, corresponding to uncertainty due to mass and stiffness perturbations, uncertainty at the boundaries of a structure, uncertainty in the coupling between structures, uncertainty in the material properties of a structure and uncertainty in the geometry of a structure. Several structures are examined corresponding to a plate with masses and/or linear springs added at random locations, a plate with torsional springs attached at random locations along its boundary edges, two plates coupled by linear springs at random locations, a mass-loaded coupled L-shaped plate, a mass-loaded frame-plate structure, and a plate with varying Young's modulus, density and thickness. The natural frequencies of the aforementioned structures have been derived using either the Lagrange-Rayleigh-Ritz technique, finite element analysis, or the use of interval analysis in conjunction with FEA.The natural frequency statistics of structures with uncertain properties are observed using two statistical measures; the statistical overlap factor and the probability density function of the spacing between successive natural frequencies. The statistical overlap factor is defined by the variation in a natural frequency from its mean value measured across an ensemble of nominally identical structures with uncertainty. For a single ensemble member, the probability density function of the spacing between successive natural frequencies is compared to a Rayleigh distribution of the mean frequency spacing. A Rayleigh distribution of modal spacings is a feature of the universality exhibited by structures with uncertainty. To further investigate the effect of structural uncertainty on the vibrational characteristics of structures, the interval analysis is applied to finite element models of a plate with uncertainty in its material properties and dimensions. Using this method, the Young's modulus, density and thickness of a rectangular plate were set to vary by a small amount within predefined bounds. Using finite element equations, the natural frequencies and modeshapes of the structure were then determined in terms of the Young's modulus, density and plate thickness. For the mass and spring loaded plates, the springs were shown to affect the lower order modes while the masses had a significant effect on the higher order modes. As the frequency increased, only a small amount of perturbation was sufficient to affect the natural frequencies of a structure. Using the interval analysis method, the variation of the natural frequencies from their deterministic value increased as the frequency increased.An ergodic hypothesis was used to examine the responses statistics of structures with uncertainty. Three structures have been computationally studied corresponding to two plates coupled by springs, an L-shaped plate and a frame plate structure. Uncertainty has been generated for the two coupled plates by locating the springs randomly across the surface of the two plates. For the L-shaped plate and a frame plate structure, uncertainty was generated by randomly positioning small masses across the plates. Using the ergodic hypothesis, the frequency averaged response on one member of an ensemble is compare with the ensemble averaged response. It was found that the ensemble averaged response was well predicted by a frequency averaged response of a single ensemble member. The width of the frequency averaging band was shown to have a large influence on the quality of the match between the frequency and ensemble averaged responses. Results were significantly improved using a frequency averaging bandwidth which varies proportionally to frequency.Finally, experiments have been conducted on an L-shaped plate, a frame plate structure and a vehicle to validate the computational results for the natural frequency and response statistics