Quantification and Evaluation of Parameter and Model Uncertainty for Passive and Active Vibration Isolation

Vibration isolation is a common method used for minimizing the vibration of dynamic load-bearing structures in a region past the resonance frequency, when excited by disturbances. The vibration reduction mainly results from the tuning of stiffness and damping during the early design stage. High vibration reduction over a broad bandwidth can be achieved with additional and controlled forces, the active vibration isolation. In this context, “active” does not mean the common understanding that the surroundings are isolated against the machine vibrations. Also in this context, “passive” means that no additional and controlled force is present, other than the common understanding that the machine is isolated against the surroundings. For active vibration isolation, a signal processing chain and an actuator are included in the system. Typically, a controller is designed to enable a force of an actuator that reduces the system’s excitation response. In both passive and active vibration isolation, uncertainty is an issue for adequate tuning of stiffness and damping in early design stage. The two types of uncertainty investigated in this contribution are parametric uncertainty, i.e. the variation of model parameters resulting in the variation of the systems output, and model uncertainty, the uncertainty from discrepancies between model output and experimentally measured output. For this investigation, a simple one mass oscillator under displacement excitation is used to quantify the parameter and model uncertainty in passive and active vibration isolation. A linear mathematical model of the one mass oscillator is used to numerically simulate the transfer behavior for both passive and active vibration isolation, thus predicting the behavior of an experimental test rig of the one mass oscillator under displacement excitation. The models’ parameters that are assumed to be uncertain are mass and stiffness as well as damping for the passive vibration isolation and an additional gain factor for the velocity feedback control in case of active vibration isolation. Stochastic uncertainty is assumed for the parameter uncertainty when conducting a Monte Carlo Simulation to investigate the variation of the numerically simulated transfer functions. The experimental test rig enables purposefully adjustable insertion of parameter uncertainty in the assumed value range of the model parameters in order to validate the model. The discrepancy between model and system output results from model uncertainty and is quantified by the Area Validation Metric and an Bayesian model validation approach. The novelty of this contribution is the application of the Area Validation Metric and Bayes’ approach to evaluate and to compare the two different passive and active approaches for vibration isolation numerically and experimentally. Furthermore, both model validation approaches are compared

The interval uncertain optimization strategy based on Chebyshev meta-model

© Springer International Publishing Switzerland 2015. This paper proposes a new design optimization method for structures subject to uncertainty. Interval model is used to account for uncertainties of uncertain-but-bounded parameters. It only requires the determination of lower and upper bounds of an uncertain parameter, without necessarily knowing its precise probability distribution. The interval uncertain optimization problem containing interval design variables and/or interval parameters will be formulated as a nested double-loop procedure, in which the outer loop optimization updates the midpoint of interval variables while the inner loop optimization calculates the bounds of objective and constraints. However, the nested double-loop optimization strategy will be computationally prohibitive, and it may be trapped into some local optimal solutions. To reduce the computational cost, the interval arithmetic is applied to the inner loop to directly evaluate the bounds of interval functions, so as to eliminate the optimization of the inner loop. The Taylor interval inclusion function is introduced to control the overestimation induced by the intrinsic wrapping effect of interval arithmetic. Since it is hard to evaluate the high-order coefficients in the Taylor inclusion function, a Chebyshev meta-model is proposed to approximate the Taylor inclusion function. Two numerical examples are used to demonstrate the effectiveness of the proposed method in the uncertain design optimization