Abstract
Design of mechanical systems often necessitates the use of dynamic simulations to calculate the displacements (and their derivatives) of the bodies in a system as a function of time in response to dynamic inputs. These types of simulations are especially prevalent in the shock and vibration community where simulations associated with models having complex inputs are routine. If the forcing functions as well as the parameters used in these simulations are subject to uncertainties, then these uncertainties will propagate through the models resulting in uncertainties in the outputs of interest. The uncertainty analysis procedure for these kinds of time-varying problems can be challenging, and in many instances, explicit data reduction equations (DRE's), i.e., analytical formulas, are not available because the outputs of interest are obtained from complex simulation software, e.g. FEA programs. Moreover, uncertainty propagation in systems modeled using nonlinear differential equations can prove to be difficult to analyze. However, if (1) the uncertainties propagate through the models in a linear manner, obeying the principle of superposition, then the complexity of the problem can be significantly simplified. If in addition, (2) the uncertainty in the model parameters do not change during the simulation and the manner in which the outputs of interest respond to small perturbations in the external input forces is not dependent on when the perturbations are applied, then the number of calculations required can be greatly reduced. Conditions (1) and (2) characterize a Linear Time Invariant (LTI) uncertainty model. This paper seeks to explain one possible approach to obtain the uncertainty results based on these assumptions.
Highlights
The uncertainty analysis of time-varying simulation results presents challenges not typically faced when conducting routine detailed uncertainty analyses of steady-state results
Time-varying uncertainty methodology using the impulse response and convolution integral was introduced for Linear Time Invariant (LTI) uncertainty models
Two examples were given to derive and show the advantages of the methodology with respect to drastically reducing the number of simulation runs needed to resolve the uncertainty in a set of time-varying simulation results
Summary
The uncertainty analysis of time-varying simulation results presents challenges not typically faced when conducting routine detailed uncertainty analyses of steady-state results. One way to handle time-varying uncertainties associated with LTI engineering applications is by taking advantage of the well-accepted mathematics related to the response of a linear system to an arbitrary excitation. This theory is commonly used in the shock and vibration community while dealing with response spectrum analysis or shock response spectrums (SRS). Assuming a LTI uncertainty model using the principle of superposition, one can derive the equation for the LTI system response to any arbitrary excitation with the help of the impulse response through a convolution process [4] This approach can be used to derive an efficient way to calculate time-varying uncertainties associated with simulation results. The second example illustrates the use of the transient uncertainty analysis methodology in a more realistic and complex time-varying problem involving the finite element analysis (FEA) of a double isolated air compressor
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