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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, 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.

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.

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.

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.

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.

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.

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.

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.

This paper gives a general view on some aspects of the influence of uncertainty in model-based monitoring of loadcarrying structures. The advantages and relevance of monitoring for the prediction of reliability will be clarified and the difference between uncertainty and reliability is discussed. Solving inverse Problems is a particular challenge in monitoring systems. Therefore, different categories of inverse problems are discussed. A generally valid extended difference equation, which describes the transfer behavior of the structure, will be derived as the basis for digital signal processing of model-based monitoring. This equation also considers changes in the structures dynamic properties, e.g due to damage or temperature. With this equation, the influence of uncertainty due to measurement noise to the functionability of monitoring is discussed and some possibilities are shown to control this uncertainty when determining ideal sensor-positions for monitoring.