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Load redistribution in smart load bearing mechanical structures can be used to reduce negative effects of damage or to prevent further damage if predefined load paths become unsuitable. Using controlled friction brakes in joints of kinematic links can be a suitable way to add dynamic functionality for desired load path redistribution. Therefore, adequate friction models are needed to predict the friction behavior. Possible models that can be used to model friction vary from simple static to complex dynamic models with increasing sophistication in the representation of friction phenomena. The LuGre-model is a widely used dynamic friction model for friction compensation in high precision control systems. It needs six parameters for describing the friction behavior. These parameters are coupled to an unmeasurable internal state variable, therefore, parameter identification is challenging. Conventionally, optimization algorithms are used to identify the LuGre-parameters deterministically. In this paper, the parameter identification and calibration is formulated to achieve model prediction that is statistically consistent with the experimental data. By use of the R2 sensitivity analysis, the most influential parameters are selected for calibration. Subsequently, the Bayesian inference based calibration procedure using experimental data is performed. Uncertainty represented in former wide parameter ranges can be reduced and, thus, model prediction accuracy can be increased.

For slender beam-columns loaded by axial compressive forces, active buckling control provides a possibility to increase the maximum bearable axial load above that of a purely passive structure. In this paper, the potential of active buckling control of an imperfect beam-column with circular cross-section using piezo-elastic supports is investigated numerically. Imperfections are given by an initial deformation of the beam-column caused by a constant imperfection force. With the piezo-elastic supports, active bending moments in arbitrary directions orthogonal to the beam-column's longitudinal axis can be applied at both beam- column's ends. The imperfect beam-column is loaded by a gradually increasing axial compressive force resulting in a lateral deformation of the beam-column. First, a finite element model of the imperfect structure for numerical simulation of the active buckling control is presented. Second, an integral linear-quadratic regulator (LQR) that compensates the deformation via the piezo-elastic supports is derived for a reduced modal model of the ideal beam-column. With the proposed active buckling control it is possible to stabilize the imperfect beam-column in arbitrary lateral direction for axial loads above the theoretical critical buckling load and the maximum bearable load of the passive structure.

In this paper, a two mass oscillator, a translatoric moving mass connected to a rigid beam by a spring-damper system, is used to numerically and experimentally investigate the capability of load path redistribution due to controlled semi-active guidance elements with friction brakes. The mathematical friction model will be derived by the L u G re approach. The rigid beam is embedded on two supports and is initially aligned with evenly distributed loads in beam and supports by the same stiffness condition. With the semi-active auxiliary guidance elements it is possible to provide additional forces to relieve one of the beam’s supports. Two control strategies are designed and tested to induce additional forces in the auxiliary guidance elements to bypass a proportion of loading away from the spring-damper system towards the now kinetic auxiliary guidance elements. The control strategies I and II depend on the different control inputs: I beam misalignment and II desired reaction force ratio in the supports. The beam’s misalignment and the supports’ reaction forces are calculated numerically and measured experimentally for varying stiffness parameters of the supports and are compared with and without semi-active auxiliary kinematic guidance elements. The structure’s moving mass is loaded with a force according to a step-function. Thus, undesired misalignment caused by varying stiffness as well as undesired load distribution in the structure’s supports can be reduced by redistributing load between the supports during operation.

Calibration of computer models and the use of those design models are two activities traditionally carried out separately. This paper generalizes existing Bayesian inverse analysis approaches for computer model calibration to present a methodology combining calibration and design in a unified Bayesian framework. This provides a computationally efficient means to undertake both tasks while quantifying all relevant sources of uncertainty. Specifically, compared with the traditional approach of design using parameter estimates from previously completed model calibration, this generalized framework inherently includes uncertainty from the calibration process in the design procedure. We demonstrate our approach to the design of a vibration isolation system. We also demonstrate how, when adaptive sampling of the phenomenon of interest is possible, the proposed framework may select new sampling locations using both available real observations and the computer model. This is especially useful when a misspecified model fails to reflect that the calibration parameter is functionally dependent upon the design inputs to be optimized.

In this paper, active buckling control of an imperfect slender beam-column with circular cross-section by piezo-elastic supports and Linear Parameter-Varying (LPV) control is investigated experimentally. The beam-column is loaded by a time-varying axial compressive load resulting in a lateral deflection of the beam-column due to imperfections. A finite element model of the beam-column under axial load is designed as an LPV system. A reduced and augmented modal model is used to design a quadratically stable gain scheduled LPV control. The control is implemented in an experimental test setup and the maximum bearable loads of the beam-column are obtained. Two cases are tested: with and without LPV control or, respectively, active and passive configuration. With the proposed active LPV buckling control it is possible to compensate the influence of beam-column imperfections and to compensate uncertainty in mounting and loading that in passive configuration without LPV control may lead to early buckling. Eventually, the maximum bearable axial compressive load is increased above the theoretical critical buckling load.

This contribution continues ongoing own research on uncertainty quantification in structural vibration isolation in early design stage by various deterministic and non-deterministic approaches. It takes into account one simple structural dynamic system example throughout the investigation: a one mass oscillator subject to passive and active vibration isolation. In this context, passive means that the vibration isolation only depends on preset inertia, damping, and stiffness properties. Active means that additional controlled forces enhance vibration isolation. The simple system allows a holistic, consistent and transparent look into mathematical modeling, numerical simulation, experimental test and uncertainty quantification for verification and validation. The oscillator represents fundamental structural dynamic behavior of machines, trusses, suspension legs etc. under variable mechanical loading. This contribution assesses basic experimental data and mathematical model form uncertainty in predicting the passive and enhanced vibration isolation after model calibration as the basis for further deterministic and non-deterministic uncertainty quantification measures. The prediction covers six different damping cases, three for passive and three for active configuration. A least squares minimization (LSM) enables calibrating multiple model parameters using different outcomes in time and in frequency domain from experimental observations. Its adequacy strongly depends on varied damping properties, especially in passive configuration.

Buckling of slender beam-columns subject to axial compressive loads represents a critical design constraint for light-weight structures. Active buckling control provides a possibility to stabilize slender beam-columns by active lateral forces or bending moments. In this paper, the potential of active buckling control of an axially loaded beam-column with circular solid cross-section by piezo-elastic supports is investigated experimentally. In the piezo-elastic supports, lateral forces of piezoelectric stack actuators are transformed into bending moments acting in arbitrary directions at the beam-column ends. A mathematical model of the axially loaded beam-column is derived to design an integral linear quadratic regulator (LQR) that stabilizes the system. The effectiveness of the stabilization concept is investigated in an experimental test setup and compared with the uncontrolled system. With the proposed active buckling control it is possible to stabilize the beam-column in arbitrary lateral direction for axial loads up to the theoretical critical buckling load of the system.

Uncertainty affects all phases of the product life cycle of technical systems, from design and production to their usage, even beyond the phase boundaries. Its identification, analysis and representation are discussed in the previous chapter. Based on the gained knowledge, our specific approach on mastering uncertainty can be applied. These approaches follow common strategies that are described in the subsequent chapter, but require individual methods and technologies. In this chapter, first legal and technical aspects for mastering uncertainty are discussed. Then, techniques for product design of technical systems under uncertainty are presented. The propagation of uncertainty is analysed for particular examples of process chains. Finally, semi-active and active technical systems and their relation to uncertainty are discussed.

This chapter describes the various approaches to analyse, quantify and evaluate uncertainty along the phases of the product life cycle. It is based on the previous chapters that introduce a consistent classification of uncertainty and a holistic approach to master the uncertainty of technical systems in mechanical engineering. Here, the following topics are presented: the identification of uncertainty by modelling technical processes, the detection and handling of data-induced conflicts, the analysis, quantification and evaluation of model uncertainty as well as the representation and visualisation of uncertainty. The different approaches are discussed and demonstrated on exemplary technical systems.

The goal of this chapter is to define different types of uncertainty in technical systems and to provide a unified terminology for this book. Indeed, uncertainty comes in different disguises. The first distinction is made with respect to the knowledge on the source of uncertainty: stochastic uncertainty, incertitude or ignorance. Then three main occurrences of uncertainty are discussed: data, model and structural uncertainty.

Model form uncertainty often arises in structural engineering problems when simplifications and assumptions in the mathematical modelling process admit multiple possible models. It is well known that all models incorporate a model error that is captured by a discrepancy due to missing or incomplete physics in the mathematical model. As an example, this discrepancy can be modelled as a function based upon Gaussian processes and its confidence bounds can be seen as a measure of adequacy for the respective model. Assessment of model form uncertainty can be conducted by comparing the confidence bounds of competing discrepancy functions. In this paper, a modular active spring-damper system is considered that was designed to resemble a suspension strut as part of an aircraft landing gear and is excited by dynamic drop tests. In previous research about the suspension strut, different mathematical system models with respect to different linear and non-linear assumptions for damping and stiffness properties to describe the dynamic system behaviour of the suspension strut were compared by means of the confidence intervals of their discrepancy functions. The results indicated that the initial conditions used for exciting the system model were inadequate. The initial conditions themselves constitute a mathematical model, so that model form uncertainty inherent to the initial condition model can effect the system model. The propagation of model form uncertainty within the model will be analysed in this paper by considering two cases: In the first case, the system model is excited with an inadequate initial condition model, while in the second case, experimentally measured initial conditions will be employed that represent the true value except for measurement errors. The comparison of both shows how model form uncertainty propagates through the model chain from the initial condition model to the system model.

Load-bearing structures with kinematic functions like a suspension of a vehicle and an aircraft landing gear enable and disable degrees of freedom and are part of many mechanical engineering applications. In most cases, the load path going through the load-bearing structure is predetermined in the design phase. However, if parts of the load-bearing structure become weak or suffer damage, e.g. due to deterioration or overload, the load capacity may become lower than designed. In that case, load redistribution can be an option to adjust the load path and, thus, reduce the effects of damage or prevent further damage. For an adequate numerical prediction of the load redistribution capability, an adequate mathematical model with calibrated model parameters is needed. Therefore, the adequacy of an exemplary load-bearing structure’s mathematical model is evaluated and its predictability is increased by model parameter uncertainty quantification and reduction. The mathematical model consists of a mechanical part, a friction model and the electromagnetic actuator to achieve load redistribution, whereby the mechanical part is chosen for calibration in this paper. Conventionally, optimization algorithms are used to calibrate the model parameters deterministically. In this paper, the model parameter calibration is formulated to achieve a model prediction that is statistically consistent with the data gained from an experimental test setup of the exemplary load-bearing structure. Using the R2 sensitivity analysis, the most influential parameters for the model prediction of interest, i.e. the load path going through the load-bearing structure represented by the support reaction forces, are identified for calibration. Subsequently, BAYESIAN inference based calibration procedure using the experimental data and the selected model parameters is performed. Thus, the mathematical model is adjusted to the actual operating conditions of the experimental load-bearing structure via the model parameters and the model prediction accuracy is increased. Uncertainty represented by originally large model parameter ranges can be reduced and quantified.

Axial and lateral loads of lightweight beam truss structures e.g. used in automotive engineering may lead to undesired structural vibration that can be reduced near a structural resonance frequency via resonant piezoelectric shunt-damping. In order to tune the electrical circuits to the desired structural resonance frequency within a model-based approach, an adequate mathematical model of the beam truss structure is required. Piezo-elastic truss supports with integrated piezoelectric stack transducers can transfer the axial and lateral forces and may be used for vibration attenuation of single beams or whole beam truss structures. For usage in a single beam test setup, the piezo-elastic support’s casing is clamped rigidly and is connected to the beam via a membrane-like spring element that allows for rotation as well as axial and lateral displacements of the beam. In this contribution, the piezo-elastic support is integrated into a two-dimensional beam truss structure comprising seven beams, where its casing is no longer clamped rigidly but is subject to axial, lateral and rotational displacements. Based on the previously verified and validated model of the single beam test setup, two different complex mathematical models of the piezo-elastic support integrated in the two-dimensional beam truss structure are derived in this contribution. The two mathematical models differ in their number of degrees of freedom for the piezo-elastic support as well as in the assumption of rigid or compliant casing. By comparing numerically and experimentally determined structural resonance frequencies and vibration amplitudes, the model that more adequately predicts the truss structure’s vibration behavior is selected on basis of the normalized root mean squared error. For future works, the more adequate model will be used to tune electrical circuits for resonant piezoelectric shunt-damping in a three-dimensional truss structure.

Active buckling control of compressively loaded beam-columns provides a possibility to increase the maximum bearable axial load compared to passive beam-columns. Reliable mathematical beam-column models that adequately describe the lateral dynamic behavior are required for the model-based controller synthesis in order to avoid controller instability for real testing and application. This paper presents an adequate mathematical beam-column model for the active buckling control in a tetrahedron truss structure. Furthermore, it discusses model form uncertainty arising from model simplification of the global tetrahedron model to three local beam-column models. An experimental tetrahedron truss structure that comprises three passive beams and three active beam-columns with piezo-elastic supports for active buckling control is investigated. The tetrahedron is clamped at the three base nodes and free at the top node. In the two piezo-elastic supports of each active beam-column, integrated piezoelectric stack actuators compensate lateral deflections due to increasing axial compressive loads and may, thus, prevent buckling. In previous works, active buckling control was investigated for a single beam-column that was clamped rigidly in an experimental test setup. A verified and validated single beam-column model with compliant boundary conditions was used to represent the piezo-elastic supports for active buckling control. The mathematical model of the active beam-columns is calibrated with experimental data from all three nominally identical active beam-columns to account for uncertainty in manufacturing, assembly or mounting. Subsequently, they are compared with respect to the transfer functions and the first eigenfrequencies. It is shown that the boundary conditions of the single beam-column model may be calibrated to adequately describe the boundary conditions within the tetrahedron truss structure. Thus, it will be used for the model-based controller synthesis in future investigations on the active buckling control of the tetrahedron truss structure.

Mathematical models of a suspension strut such as an aircraft landing gear are utilized by engineers in order to predict its dynamic response under different boundary conditions. The prediction of the dynamic response, for example the external loads, the stress and the strength as well as the maximum compression in the spring-damper component aids engineers in early decision making to ensure its structural reliability under various operational conditions. However, the prediction of the dynamic response is influenced by model uncertainty. As far as the model uncertainty is concerned, the prediction of the dynamic behavior via different mathematical models depends upon various factors such as the model's complexity in terms of the degrees of freedom, material and geometrical assumptions, their boundary conditions and the governing functional relations between the model input and output parameters. The latter can be linear or nonlinear, axiomatic or empiric, time variant or time-invariant. Hence, the uncertainty that arises in the prediction of the dynamic response of the resulting different mathematical models needs to be quantified with suitable validation metrics, especially when the system is under structural risk and failure assessment. In this contribution, the authors utilize the Bayesian interval hypothesis-based method to quantify the uncertainty in the mathematical models of the suspension strut.

The starting point in uncertainty quantification is a stochastic model, which is fitted to a technical system in a suitable way, and prediction of uncertainty is carried out within this stochastic model. In any application, such a model will not be perfect, so any uncertainty quantification from such a model has to take into account the inadequacy of the model. In this paper, we rigorously show how the observed data of the technical system can be used to build a conservative non‐asymptotic confidence interval on quantiles related to experiments with the technical system. The construction of this confidence interval is based on concentration inequalities and order statistics. An asymptotic bound on the length of this confidence interval is presented. Here we assume that engineers use more and more of their knowledge to build models with order of errors bounded by log(n)/n. The results are illustrated by applying the newly proposed approach to real and simulated data.

Different mathematical models can be developed to represent the dynamic behavior of structural systems and assess properties, such as risk of failure and reliability. Selecting an adequate model requires choosing a model of sufficient complexity to accurately capture the output responses under various operational conditions. However, as model complexity increases, the functional relationship between input parameters varies and the number of parameters required to represent the physical system increases, reducing computational efficiency and increasing modeling difficulty. The process of model selection is further exacerbated by uncertainty introduced from input parameters, noise in experimental measurements, numerical solutions, and model form. The purpose of this research is to evaluate the acceptable level of uncertainty that can be present within numerical models, while reliably capturing the fundamental physics of a subject system. However, before uncertainty quantification can be performed, a sensitivity analysis study is required to prevent numerical ill-conditioning from parameters that contribute insignificant variability to the output response features of interest. The main focus of this paper, therefore, is to employ sensitivity analysis tools on models to remove low sensitivity parameters from the calibration space. The subject system in this study is a modular spring-damper system integrated into a space truss structure. Six different cases of increasing complexity are derived from a mathematical model designed from a two-degree of freedom (2DOF) mass spring-damper that neglects single truss properties, such as geometry and truss member material properties. Model sensitivity analysis is performed using the Analysis of Variation (ANOVA) and the Coefficient of Determination R2. The global sensitivity results for the parameters in each 2DOF case are determined from the R2 calculation and compared in performance to evaluate levels of parameter contribution. Parameters with a weighted R2 value less than .02 account for less than 2% of the variation in the output responses and are removed from the calibration space. This paper concludes with an outlook on implementing Bayesian inference methodologies, delayed-acceptance single-component adaptive Metropolis (DA-SCAM) algorithm and Gaussian Process Models for Simulation Analysis (GPM/SA), to select the most representative mathematical model and set of input parameters that best characterize the system’s dynamic behavior.

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.

In the field of structural engineering, mathematical models are utilized to predict the dynamic response of systems such as a suspension strut under different boundary and loading conditions. However, different mathematical models exist based on their governing functional relations between the model input and state output parameters. For example, the spring-damper component of a suspension strut is considered. Its mathematical model can be represented by linear, nonlinear, axiomatic or empiric relations resulting in different vibrational behaviour. The uncertainty that arises in the prediction of the dynamic response from the resulting different approaches in mathematical modelling may be quantified with Bayesian inference approach especially when the system is under structural risk and failure assessment. As the dynamic output of the suspension strut, the spring-damper compression and the spring-damper forces as well as the ground impact force are considered in this contribution that are taken as the criteria for uncertainty evaluation due to different functional relations of models. The system is excited by initial velocities that depend on a drop height of the suspension strut during drop tests. The suspension strut is a multi-variable system with the payload and the drop height as its varied input variables in this investigation. As a new approach, the authors present a way to adequately compare different models based on axiomatic or empiric assumptions of functional relations using the posterior probabilities of competing mathematical models. The posterior probabilities of different mathematical models are used as a metric to evaluate the model uncertainty of a suspension strut system with similar specifications as actual suspension struts in automotive or aerospace applications for decision making in early design stage. The posterior probabilities are estimated from the likelihood function, which is estimated from the cartesian vector distances between the predicted output and the experimental output.

Beams in lightweight truss structures are subject to axial and lateral loads that may lead to undesired structural vibration or failure by buckling. The axial and lateral forces may be transferred via the truss supports that offer possibilities for state control of single beams and larger structures. In earlier own studies, the concept of a piezo-elastic support for active buckling control and resonant shunt damping has been investigated. An elastic spring element is used to allow a rotation in the beam's bearing in any plane perpendicular to the beam's longitudinal axis. The rotation is laterally transferred to an axial displacement of piezoelectric stack transducers that are either used to generate active lateral forces for active buckling control or to attenuate vibrations with a resonant shunt. In this paper, the model verification and validation of the elastic properties of the piezo-elastic support for passive and active structural control of beams with circular cross-section is presented. The rotational and lateral spring element stiffness is investigated numerically and experimentally and the existing models are updated in the verification process. The model is validated by comparing the numerical results and experimental ability for vibration attenuation.

Load-bearing structures with kinematic functions enable and disable degrees of freedom and are part of many mechanical engineering applications. The relative movements between a wheel and the body of a car or a landing gear and an aircraft fuselage are examples for load-bearing systems with defined kinematics. In most cases, the load is transmitted through a predetermined load path of a host structure to the structural support interfaces. However, uncertainty due to unexpected load peaks or varying health condition, e.g. changes in stiffness or damping parameters over time of the structure’s components may require an adjustment of the load path for safety reasons. Load paths transmitted through damaged or weakened components can be the reason for reduced comfort or even failure. For example, reduced support stiffness can lead to uncertain and undesirable misalignment in the structure. In this paper, a two mass oscillator, a translatoric moving mass connected to a rigid beam by a spring-damper system, is used to numerically investigate the capability of load path adaption due to controlled semi-active guidance elements with friction brakes. The mathematical friction model will be derived by the LuGre approach. The rigid beam is embedded on two supports and is initially aligned with evenly distributed loads in beam and supports by the same stiffness condition. However, if uneven support stiffness occurs, e.g. by damage or fatigue, the beam becomes misaligned. One sided lowering of the beam may follow. Two auxiliary kinematic guidance elements are used to redirect the load path depending on the beam’s alignment condition. With the semi-active auxiliary kinematic guidance elements it is possible to provide additional forces to relieve one of the beam’s support if it changes its stiffness. The beams’s misalignment is calculated numerically for varying stiffness parameters of the supports and is compared with and without semi-active auxiliary kinematic guidance elements. The structure is loaded with a force according to a step-function and a simple signal-based feedback PID-controller is designed to induce additional forces in the auxiliary guidance elements to bypass portions of loading away from supports with decreasing stiffness. Thus, uncertainty due to unacceptable misalignment caused by varying stiffness conditions of the structure’s supports can be reduced by shifting load between the supports during operation.

Predicting the kinematic and dynamic behavior of complex load bearing structures with high safety requirements such as landing gears is time consuming. For that, mathematical analytic, finite element or multi body surrogate models are needed for numeric simulation purposes. Today, these models take into account both deterministic and non-deterministic approaches. However, before adequate and verified simulation begins, the modeling of the mathematical surrogates requires most of the time for adequate prediction, including model verification, before even more costly experimental testing phase begins. This contribution investigates an approach based on Info-Gap analysis to predict critical performance requirements of major landing gear design alternatives in an early design stage. This analysis uses only simple analytical but comparable and sufficient adequate models for four major design concept alternatives according to basic design rules found in relevant literature. The concepts comprise one telescopic and three different trailing link designs. It is the aim to make decisions in selecting the most suitable design as early as possible in the design stage with taking into account uncertainty—before time consuming efforts in modeling finite element and multi body models for detailed prediction are conducted. Particularly, the authors evaluate the robustness to uncertainty or how much of an uncertainty horizon by means of uncertain compression stroke ability due to varied stiffness properties can be tolerated with the four different concepts, until the absolute maximum allowable compression stroke limit is reached. This contribution continues the authors’ prior work presented at IMAC 2016. In there, the authors evaluated and compared the performance requirements like compression stroke ability and ride quality, elastic force retention, structure strength, and weight of mechanisms for main and nose landing gears resulting from the four significant structural design concepts in mathematical physical models in an analytic deterministic way.

Load-bearing mechanical structures like trusses face uncertainty in loading along with uncertainty in stress and strength, which are due to uncertainty in their development, production, and usage. According to the working hypothesis of the German Collaborative Research Center SFB 805, uncertainty occurs in processes that are not or only partial deterministic and can only be controlled in processes. The authors classify, compare, and evaluate four different direct methods to describe and evaluate the uncertainty of normal stress distribution in simple truss structures with one column, two columns, and three columns. The four methods are the direct Monte Carlo (DMC) simulation, the direct quasi-Monte Carlo (DQMC) simulation, the direct interval, and the direct fuzzy analysis with α-cuts, which are common methods for data uncertainty analysis. The DMC simulation and the DQMC simulation are categorized as probabilistic methods to evaluate the stochastic uncertainty. On the contrary, the direct interval and the direct fuzzy analysis with α-cuts are categorized as possibilistic methods to evaluate the nonstochastic uncertainty. Three different truss structures with increasing model complexity, a single-column, a two-column, and a three-column systems are chosen as reference systems in this study. Each truss structure is excited with a vertical external point load. The input parameters of the truss structures are the internal system properties such as geometry and material parameters, and the external properties such as magnitude and direction of load. The probabilistic and the possibilistic methods are applied to each truss structure to describe and evaluate its uncertainty in the developing phase. The DMC simulation and DQMC simulation are carried out with full or “direct” sample sets of model parameters such as geometry parameters and state parameters such as forces, and a sensitivity analysis is conducted to identify the influence of every model and state input parameter on the normal stress, which is the output variable of the truss structures. In parallel, the direct interval and the direct fuzzy analysis with α-cuts are carried out without altering and, therefore, they are direct approaches as well. The four direct methods are then compared based on the simulation results. The criteria of the comparison are the uncertainty in the deviation of the normal stress in one column of each truss structure due to varied model and state input parameters, the computational costs, as well as the implementation complexity of the applied methods.

Mathematical models of a suspension strut such as an aircraft landing gear are utilized by engineers in order to predict its dynamic response under different boundary conditions. The prediction of the dynamic response, for example the external loads, the stress and the strength as well as the maximum compression in the spring-damper component aids engineers in early decision making to ensure its structural reliability under various operational conditions. However, the prediction of the dynamic response is influenced by model uncertainty. As far as the model uncertainty is concerned, the prediction of the dynamic behavior via different mathematical models depends upon various factors such as the model’s complexity in terms of the degrees of freedom, material and geometrical assumptions, their boundary conditions and the governing functional relations between the model input and output parameters. The latter can be linear or nonlinear, axiomatic or empiric, time variant or time-invariant. Hence, the uncertainty that arises in the prediction of the dynamic response of the resulting different mathematical models needs to be quantified with suitable validation metrics, especially when the system is under structural risk and failure assessment. In this contribution, the authors utilize the Bayes factor as a validation metric to quantify the model uncertainty of a suspension strut system with similar specifications as actual suspension struts in automotive or aerospace applications for decision making in early design stage.

In this paper, vibration attenuation of a beam with circular cross-section by resonantly shunted piezo-elastic supports is experimentally investigated for varying axial tensile and compressive beam loads. Varying axial beam loads manipulate the effective lateral bending stiffness and, thus, lead to a detuning of the beams resonance frequencies. Furthermore, varying axial loads affect the general electromechanical coupling coefficient of transducer and beam, an important modal quantity for shunt-damping. The beam’s first mode resonance frequency and coupling coefficient are analyzed for varying axial loads. The values of the resonance frequency and the coupling coefficient are obtained from a transducer impedance measurement. Finally, frequency transfer functions of the beam with one piezo-elastic support either shunted to a RL-shunt or to a RL-shunt with negative capacitance, the RLC-shunt, are compared for varying axial loads. It is shown that the beam vibration attenuation with the RLC-shunt is less influenced by varying axial beam loads.

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.

Structural design concepts for load bearing mechanical systems vary due to individual usage requirements. Particularly strut-configurations for landing gears in airplanes push the envelope according to tight requirements in shock absorption, normal, lateral and torsional load capacity, rolling stability, storage dimensions, low drag, low weight, and maintenance as well as reliability, safety and availability. Since the first controlled and powered flight of the Wright-Brothers in 1903, design evolution generated different structural design concepts. Today’s structures may have, seemingly, reached mature conformity with distinct load path architectures that have been prevailed. In the proposed contribution, the authors evaluate and compare distinctive performance requirements like stroke ability and ride quality, elastic force retention, structure strength, and weight of mechanisms resulting from significant structural design concepts for main and nose landing gears. Loads in landing gears have always been distributed in struts with high and low amounts of strut members such as rods, beams, torque links, and joints as well as different types of absorbers. This paper’s goal is to clarify pros and cons of the four different concepts with respect to their vulnerability due to uncertainty. Here, uncertainty mainly occurs due to variations in elastic force retention and their effect on the performance requirements. For that, simple mathematical models are derived to evaluate and compare the most significant characteristics of the four concepts in the earliest stage of development in order to make early decisions for or against a concept before time and cost consuming detailed development work including manufacturing and test takes over.

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 many applications, kinematic structures are used to enable and disable degrees of freedom. The relative movement between a wheel and the body of a car or a landing gear and an aircraft fuselage are examples for a defined movement. In most cases, a spring-damper system determines the kinetic properties of the movement. However, unexpected high load peaks may lead to maximum displacements and maybe to locking. Thus, a hard clash between two rigid components may occur, causing acceleration peaks. This may have harmful effects for the whole system. For example a hard landing of an aircraft can result in locking the landing gear and thus damage the entire aircraft. In this paper, the potential of adaptive auxiliary kinematic guidance elements in a spring-damper system to prevent locking is investigated numerically. The aim is to provide additional forces in the auxiliary kinematic guidance elements in case of overloading the spring-damper system and thus to absorb some of the impact energy. To estimate the potential of the load redistribution in the spring-damper system, a numerical model of a two-mass oscillator is used, similar to a quarter-car-model. In numerical calculations, the reduction of the acceleration peaks of the masses with the adaptive approach is compared to the Acceleration peaks without the approach, or, respectively, when locking is not prevented. In addition, the required force of the adaptive auxiliary kinematic guidance elements is calculated as a function of the masses of the system and the drop height, or, respectively, the impact energy.

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.

Undesired vibration may occur in lightweight structures due to low damping and excitation. For the purpose of vibration attenuation, tuned mass dampers (TMD) can be an appropriate measure. A similar approach uses resonantly shunted piezoelectric transducers. However, uncertainty in design and application of resonantly shunted piezoelectric transducers and TMD can be caused by insufficient mathematical modeling, geometric and material deviations or deviations in the electrical and mechanical quantities. During operation, uncertainty may result in detuned attenuation systems and loss of attenuation performance. A consistent and general approach to display uncertainty in load carrying systems developed by the authors is applied to describe parametric uncertainty in vibration attenuation with resonantly shunted piezoelectric transducers and TMD. Mathematical models using Hamilton>'s principle and Ritz formulation are set up for a beam, clamped at both ends with resonantly shunted transducers and TMD to demonstrate the effectiveness of both attenuation systems and investigate the effects of parametric uncertainty. Furthermore, both approaches lead to additional masses, piezoelectric material for shunt damping and compensator mass of TMD, in the systems. It is shown that vibration attenuation with TMD is less sensitive to parametric uncertainty and achieves a higher performance using the same additional mass.

Load-bearing structures with kinematic functions enable and disable degrees of freedom and are part of many mechanical engineering applications. The relative movement between a wheel and the body of a car or a landing gear and an aircraft fuselage are examples for load-bearing systems with defined kinematics. In most cases, the load is transmitted through a predetermined load path to the structural support interfaces. However, unexpected load peaks or varying health condition of the system's supports, which means for example varying damping and stiffness characteristics, may require an active adjustment of the load path. However, load paths transmitted through damaged or weakened supports can be the reason for reduced comfort or even failure. In this paper a simplified 2D two mass oscillator with two supports is used to numerically investigate the potential of controlled adaptive auxiliary kinematic guidance elements in a load-bearing structure to adapt the load path depending on the stiffness change, representing damage of the supports. The aim is to provide additional forces in the auxiliary kinematic guidance elements for two reasons. On the one hand, one of the two supports that may become weaker through stiffness change will be relieved from higher loading. On the other hand, tilting due to different compliance in the supports will be minimized. Therefore, shifting load between the supports during operation could be an effective option.

Undesired vibration may occur in lightweight structures due to excitation and low damping. For the purpose of lateral vibration attenuation in beam structures, piezoelectric transducers shunted to negative capacitances can be an appropriate measure. In this paper, a new concept for lateral vibration attenuation by integrated piezoelectric stack transducers in the elastic support of a beam with circular cross-section is presented. In the piezoelastic support, bending of the beam in an arbitrary direction is transformed into a significant axial deformation of three stack transducers and, thus, the beam's surface may remain free from transducers. For multimodal vibration attenuation, each piezoelectric transducer is shunted to a negative capacitance. It is shown by numerical simulation and experiment that the concept of an elastic beam support with integrated shunted piezoelectric stack transducers is capable of reducing the lateral vibration of the beam in an arbitrary direction.

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.

Fatigue cracks in light-weight shell or panel structures may lead to major failures when used for sealing or load-carrying purposes. This paper describes investigations into the potential of piezoelectric actuator patches that are applied to the surface of an already cracked thin aluminum panel to actively reduce the propagation of fatigue cracks. With active reduction of fatigue crack propagation, uncertainties in the cracked structure's strength, which always remain present even when the structure is used under damage tolerance conditions, e.g. airplane fuselages, could be lowered. The main idea is to lower the cyclic stress intensity factor near the crack tip with actively induced mechanical compression forces using thin low voltage piezoelectric actuator patches applied to the panel's surface. With lowering of the cyclic stress intensity, the rate of crack propagation in an already cracked thin aluminum panel will be reduced significantly. First, this paper discusses the proper placement and alignment of thin piezoelectric actuator patches near the crack tip to induce the mechanical compression forces necessary for reduction of crack propagation by numerical simulations. Second, the potential for crack propagation reduction will be investigated statistically by an experimental sample test examining three cases: a cracked aluminum host structure (i) without, (ii) with but passive, and (iii) with activated piezoelectric actuator patches. It will be seen that activated piezoelectric actuator patches lead to a significant reduction in crack propagation.