88 D. Day et al. Fig. 8.18 Combined beam: models 9 through 12 and range of response ASD for length perturbations Table 8.2 Combined beam: maximum value error summary Perturbation VRMS range (psi/MPa) Maximum VRMS % error Maximum AZRMS % error Error ratio Modulus 1767/12.2 8.2 18.2 0.45 Damping 2304/15.9 7.8 9.9 0.79 Density 13,960/96.3 25.4 25.0 – Length 13,250/91.4 24.4 27.1 – More data is needed to see how results hold for more complex systems, and future work could include multiaxial inputs, input load uncertainty, and coupled parametric effects. It would also be interesting to study results from other modal-based analyses, such as a modal transient solution to see if a bound could be determined even if the modal stress is not directly used. 8.5 Conclusion A relationship between the error in stress and acceleration was calculated for three example problems. For the first case, a linear relation and theoretical bound for the maximum modal von Mises stress error and acceleration error was determined. The results were extended to a random vibration solution that utilized the modal stress and two example cases were considered. An error in acceleration due to changes in modulus of elasticity, damping ratio, density, and length for each example beam was related to the error in VRMS stress. An error ratio was defined to describe and bound the relationship and results were found to be linear if the system mode shapes do not change significantly with perturbation in model parameters. Irregular behavior was observed when new modes were introduced into the input load frequency range, influencing the response between perturbed models. The results of this work support that, given the stability of system modes across uncertainty in model parameters, the error in stress could be related to and bound through the errors in acceleration for a random vibration analysis. These error trends are useful for providing a quantitative measure of accuracy in stress and acceleration predictions.
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