Benedict Götz

Lateral vibration attenuation of a beam with circular cross-section by a support with integrated piezoelectric transducers shunted to negative capacitances

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 beams 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.

Active buckling control of a beam-column with circular cross-section using piezo-elastic supports and integral LQR control

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.

Optimally tuned resonant negative capacitance for piezoelectric shunt damping based on measured electromechanical impedance

In this paper, a new tuning method for shunt damping with a series resistance, inductance and negative capacitance is proposed and its validity is investigated. It is based on the measured electromechanical impedance of a piezoelectric system, which is represented through an equivalent electrical circuit that takes into account the characteristics of the piezoelectric transducer and the host structure. Afterwards, an additional circuit representing the shunt is connected and the Norton equivalent impedance is obtained at the terminals that represent the mechanical mode of interest. During the tuning process, the optimal shunt parameters are found by minimizing the maximum absolute value of the Norton equivalent impedance over a defined frequency range through a numerical optimization. Taking benefit from the analogy between electrical impedance and mechanical admittance, the minimization of different mechanical responses (displacement, velocity or acceleration) is also proposed and the different optimum shunt parameters obtained are compared. In view of real technical applications, this method allows the integration of a real negative capacitance circuit, i.e., a negative impedance converter, rather than an ideal component. It is thus possible to use the impedance of this circuit and optimize the individual component values. Since this method is based on one simple measurement, it can be applied to arbitrary structures without the need of complex dynamic tests or expensive finite elements calculations. Finally, an experimental analysis is carried out in order to compare the damping performance of the proposed method and the conventional analytical method that minimizes a mechanical frequency response function.

Model Verification and Validation of a Piezo-Elastic Support for Passive and Active Structural State Control of Beams with Circular Cross-Section

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 beams bearing in any plane perpendicular to the beams 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.

Approach to Evaluate and to Compare Basic Structural Design Concepts of Landing Gears in Early Stage of Development Under Uncertainty

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.

Robust Optimization of Shunted Piezoelectric Transducers for Vibration Attenuation Considering Different Values of Electromechanical Coupling

Structural vibration may occur in mechanical systems leading to fatigue, reduced durability or undesirable noise. In this context, shunting piezoelectric transducers to resonant shunts can be an appropriate measure for attenuating vibrations. The achieved vibration attenuation significantly depends on the tuning of the shunt parameters. Uncertainty in design and application of resonant shunted piezoelectric transducers may result in a detuned attenuation system and loss of attenuation performance. Therefore, we propose an approach based on robust optimization using the Bounded Real Lemma to contain the loss of vibration attenuation due to uncertainty. It is shown for resonant shunts, that for increasing electromechanical coupling coefficients the worst-case maximal vibration amplitudes for non-robust and robust optimization of shunt parameters converge. Furthermore by adding a negative capacitance to the resonant shunt, the worst-case maximal amplitude remains almost constant for all considered coupling coefficients for non-robust and robust optimization of the shunt parameters.

Non-probabilistic Uncertainty Evaluation in the Concept Phase for Airplane Landing Gear Design

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.

Lateral Vibration Attenuation of a Beam with Piezo-Elastic Supports Subject to Varying Axial Tensile and Compressive Loads

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 maximum of quantiles

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 by an order statistic of values m(pi, Xi) where X,X1, X2, . . . are independent and identically distributed and where pi 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. AMS classification: Primary 62G05; secondary 62G30.