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Nonparametric estimation of a quantile $q_{m(X),\alpha }$ of a random variable $m(X)$ is considered, where $m: \mathbb {R}^{d}\rightarrow \mathbb {R}$ is a function, which is costly to compute and $X$ is an $ \mathbb {R}^{d}$ -valued random variable with known distribution. Monte Carlo surrogate quantile estimates are considered, where in a first step, the function $m$ is estimated by some estimate (surrogate) $m_{n}$ , and then, the quantile $q_{m(X),\alpha }$ is estimated by a Monte Carlo estimate of the quantile $q_{m_{n}(X),\alpha }$ . A general error bound on the error of this quantile estimate is derived, which depends on the local error of the function estimate $m_{n}$ , and the rates of convergence of the corresponding Monte Carlo surrogate quantile estimates are analyzed for two different function estimates. The finite sample size behavior of the estimates is investigated in simulations.

Buckling of slender load-bearing beam-columns is a crucial failure scenario in light-weight structures as it may result in the collapse of the entire structure. If axial load and load capacity are unknown, stability becomes uncertain. To compensate this uncertainty, the authors successfully developed and evaluated an approach for active buckling control for a slender beam-column, clamped at the base and pinned at the upper end. Active lateral forces are applied with two piezoelectric stack actuators in opposing directions near the beam-column' clamped base to prevent buckling. A Linear Quadratic Regulator is designed and implemented on the experimental demonstrator and statistical tests are conducted to prove effectivity of the active approach. The load capacity of the beam-column could be increased by 40% and scatter of buckling occurrences for increasing axial loads is reduced. Weibull analysis is used to evaluate the increase of the load capacity and its related uncertainty compensation.

Uncertainty in load carrying systems e.g. may result from geometric and material deviations in production and assembly of its parts. In usage, this uncertainty may lead to not completely known loads and strength which may lead to severe failure of parts or the entire system. Therefore, an analysis of uncertainty is recommended. In this paper, uncertainty is assumed to occur in processes and an approach is presented to describe uncertainty consistently within processes and process chains. This description is then applied to an example which considers uncertainty in the production and assembly processes of a simple tripod system and its effect on the resulting load distribution in its legs. The consistent description allows the detection of uncertainties and, furthermore, to display uncertainty propagation in process chains for load carrying systems.

Uncertainty in usage of load-carrying systems mainly results from not fully knownloads and strength. This article discusses basic approaches to control uncertainty in usage ofload-carrying systems by passive and active means. An active low damped column system critical to buckling is presented in which a slender column can be stabilised actively by piezo stackactuators at one of its ends only. Uncertainty may be controlled in the active column systemby temporarily enhancing the bearable axial load theoretically up to three times compared to the passive column system in case of critical loading. However, in the implementation of theseapproaches, system-speci c uncertainty may also occur. In numerical examinations it is shown, that small deviations in measured axial loading may increase the active force signi cantly to achieve stabilisation. The increase of applied active force might affect lifetime of the piezo stackactuators and thus the stabilising capability of the active column system.

In this contribution, the authors discuss a clear and comprehensive way to deepen the understanding about the comparison of parametric uncertainty for early passive and active vibration isolation design in an adequate probabilistic way. A simple mathematical one degree of freedom linear model of an automobile’s suspension leg, excited by harmonic base point stroke and subject to passive and active vibration isolation purpose is used as an example study for uncertainty comparison. The model’s parameters are chassis mass, suspensions leg’s damping and stiffness for passive vibration isolation, and an additional gain factor for velocity feedback control when active vibration isolation is assumed. Assuming the parameters to be normally distributed, they are non-deterministic input for Monte Carlo-Simulations to investigate the dynamic vibrational response due the deterministic excitation.

Buckling of load-carrying beam-columns is a severe failure scenario in light-weight structures. The authors present an approach to actively stabilize a slender beam-column under static axial load to prevent it from buckling in its first buckling mode. For that, controlled active counteracting forces are applied by two piezoelectric stack actuators near the column’s fixed base, achieving a 40% higher axial critical load and leaving most of the column’s surface free from actuation devices. However, uncertain actuator properties due to tolerances in characteristic maximum free stroke and blocking force capability have an influence on the active stabilization. This uncertainty and its effect on active buckling control is investigated by numerical simulation, based on experimental tests to determine the actual maximum free stroke and blocking force for several piezoelectric stack actuators. The simulation shows that the success of active buckling control depends on the actuator’s variation in its maximum free stroke and blocking force capability.

Buckling of axially loaded beam-columns represents a critical design constraint for light-weight structures. Besides passive solutions to increase the critical buckling load, active buckling control provides a possibility to stabilize slender elements in structures. So far, buckling control by active forces or bending moments has been mostly investigated for beam-columns with rectangular cross-section and with a preferred direction of buckling. The proposed approach investigates active buckling control of a beam-column with circular solid cross-section which is fixed at its base and pinned at its upper end. Three controlled active lateral forces are applied near the fixed base with angles of 120° to each other to stabilize the beam-column and allow higher critical axial loads. The beam-column is subject to supercritical static axial loads and lateral disturbance forces with varying directions and offsets. Two independent modal state space systems are derived for the bending planes in the lateral y- and z-directions of the circular cross-section. These are used to design two linear-quadratic regulators (LQR) that determine the necessary control forces which are transformed into the directions of the active lateral forces. The system behavior is simulated with a finite element model using one-dimensional beam elements with six degrees of freedom at each node. With the implemented control, it is possible to actively stabilize a beam-column with circular cross-section in arbitrary buckling direction for axial loads significantly above the critical axial buckling load.

Uncertainty is an important design constraint when configuring a dynamic mechanical system that is subject to passive or active vibration reduction. Uncertainty can be divided into the categories unknown, estimated and stochastic uncertainty depending on the amount of information, e.g. of the principal mechanical parameter’s deviation in inertia, energy dissipation, compliance and today more and more with active energy feeding to enhance damping. In this paper, these uncertainty categories as well as solutions for uncertainty control in the early design phase will be described and evaluated analytically in a simple but consistent and transparent way on the basis of a mathematical dynamic linear model. The model is a one degree of freedom mass-damper-spring system representing a suspension leg supporting a vehicle’s chassis that is subject to passive and active damping. The amplitude and phase responses in frequency domain are shown analytically in case studies for different assumptions of the effective uncertainty. Amongst others, sample tests are conducted by Monte Carlo Simulations when stochastic uncertainty is considered. The uncertainty examinations on vibration reduction for the selected dynamical model show promising results indicating the predominance of active damping vs. passive damping statistically.

A simulation model of a complex system is considered, for which the outcome is described by $m(p,X)$, where $p$ is a parameter of the system, $X$ is a random input of the system and $m$ is a real-valued function. The maximum (with respect to $p$) of the quantiles of $m(p,X)$ is estimated. The quantiles of $m(p,X)$ of a given level are estimated for various values of $p$ from an order statistic of values $m(p_{i},X_{i})$ where $X,X_{1},X_{2},\dots$ are independent and identically distributed and where $p_{i}$ is close to $p$, and the maximal quantile is estimated by the maximum of these quantile estimates. Under assumptions on the smoothness of the function describing the dependency of the values of the quantiles on the parameter $p$ the rate of convergence of this estimate is analyzed. The finite sample size behavior of the estimate is illustrated by simulated data and by applying it in a simulation model of a real mechanical system.

Buckling is an important design constraint in light-weight structures as it may result in the collapse of an entire structure. When a mechanical beam column is loaded above its critical buckling load, it may buckle. In addition, if the actual loading is not fully known, stability becomes highly uncertain. To control uncertainty in buckling, an approach is presented to actively stabilise a slender flat column sensitive to buckling. For this purpose, actively controlled forces applied by piezoelectric actuators located close to the column's clamped base stabilise the column against buckling at critical loading. In order to design a controller to stabilise the column, a mathematical model of the postcritically loaded system is needed. Simulating postbuckling behaviour is important to study the effect of axial loads above the critical axial buckling load within active buckling control. Within this postbuckling model, different kinds of uncertainty may occur: i) error in estimation of model parameters such as mass, damping and stiffness, ii) non-linearities e. g. in the assumption of curvature of the column's deflection shapes and many more. In this paper, numerical simulations based on the mathematical model for the postcritically axially loaded column are compared to a mathematical model based on experiments of the actively stabilised postcritically loaded real column system using closed loop identification. The motivation to develop an experimentally validated mathematical model is to develop of a model based stabilising control algorithm for a real postcritically axially loaded beam column.