Vom 16. bis zum 20. März 2026 fand in Stuttgart das 96. jährliche Treffen der Gesellschaft für Angewandte Mathematik und Mechanik e. V. (GAMM) statt.
Dieses Jahr beteiligte sich das Fachgebiet Elektrische Messtechnik an dieser Tagung gleich mit zwei Beiträgen.
Zunächst stellte Olga Friesen eine Untersuchung zur Messung und simulativen Bestimmung von "Sprungstellen" in piezoelektrischen Systemen bei Leistungsschallanwendungen vor.
Diese können bei hoch-intensiver Anregung von piezokeramischen Bauelementen (zum Beispiel beim Ultraschall-Bonden) auftreten und zeigen ein deutlich nichtlineares Verhalten.
Im zweiten Beitrag präsentierte Jonas Hölscher ebenfalls Ergebnisse aus dem Bereich nichtlinearer piezoelektrischer Systeme, wobei er sich vorwiegend mit dem simulativen Aspekt der Modellierung befasste.
Er stellte eine Methode vor, mit der sich bestimmte nichtlineare Materialmodelle von Piezokeramiken mithilfe der Finiten-Elemente-Methode (FEM) simulieren lassen.
Beide Beiträge sind im Rahmen des Forschungsgruppenprojektes NEPTUN (FOR 5208, DFG-Projekt-Nr. 444955436) entstanden. Die Titel und Kurzfassungen der Beiträge sind nachfolgend aufgeführt.
Experimental and Numerical Investigation of Jump Phenomena in the Frequency Response of Piezoelectric Systems
Authors: Olga Friesen, Jonas Hölscher, Michael B. K. Siegmund, Leander Claes, Bernd Henning
Piezoelectric ceramics are used in a wide range of fields, including high-power ultrasound applications such as ultrasonic bonding.
The behaviour of these components in resonance excitation shows pronounced nonlinear properties.
In practice, these nonlinear effects lead to a number of phenomena, including voltage-dependent resonance shifts and a characteristic tilting of resonance curves.
These phenomena have a significant impact on the performance, stability and control of the components but cannot be described by linear models.
Consequently, the accurate prediction of the behaviour of resonances in piezoelectric components remains a significant challenge when operating at resonance with elevated excitation voltages.
In this contribution, an experimental and numerical study on the nonlinear resonance behaviour of piezoceramic rings is presented.
An experimental setup is developed to measure electrical properties for both increasing and decreasing frequency sweeps under varying excitation voltages.
For this purpose, a network analyser with a suitable power amplifier is used.
The observed outcome includes distortions of the frequency response, a pronounced tilting of resonance curves, and jump phenomenon.
These observations serve as a basis for the development and validation of nonlinear material models.
For the numerical analysis, the piezoelectric constitutive relations are extended by a polynomial nonlinear stress-strain relationship to account for field-dependent material behaviour.
In order to fit the simulated responses to the experimentally measured resonance characteristics a suitable parametrisation of the nonlinear terms is chosen.
Since the excitations are assumed to be harmonic, a harmonic balancing finite element method approach is used assuming axial symmetry.
Due to this simplified nonlinearity model and the usage of the harmonic balancing method, the simulations can be performed efficiently.
A comparison between simulation and experiment demonstrates a reasonable degree of agreement for the resonance under investigation and for a range of excitation voltages.
The results provide insight into the suitability and limitations of polynomial nonlinear material models for resonant piezoelectric systems.
Modeling Nonlinear Acoustic Wave Propagation with Third-Order Elastic Constants Using the Finite Element Method
Authors: Jonas Hölscher, Leander Claes, Bernd Henning
Due to the widespread application of specialized piezoelectric actuators in various applications, accurate simulations have become key components of modern design processes.
Although simulation tools for piezoelectric devices are widely available, nonlinear effects, which occur when exciting near resonance at elevated amplitudes, are often neglected or linearized.
At those resonance peaks the piezoelectric systems exhibit characteristics of highly dynamical nonlinear systems, showing typical nonlinear effects such as frequency shifts, jump phenomena and bifurcations.
To correctly reproduce this behavior, suitable material models must be identified and incorporated into the simulation environment.
Therefore, in this work we present a finite element formulation for nonlinear piezoelectric systems that exhibit a quadratic elastic nonlinearity for the strain-stress relation in the mechanical field. This results in a third-order mechanical material tensor, which can be described using third-order elastic constants. For the electrical field and the piezoelectric coupling linear behavior is assumed. For the model an axisymmetric piezoelectric ring is used, and the simulation is performed in time-domain with harmonic excitation.
Existing methods for incorporating such quadratic nonlinearities simplify or linearize the nonlinear stiffness matrix, since the classical approach of the variational formulation is not suitable. To enable the use of the full set of nonlinear material parameters, the finite element formulation must be derived using tensor notation. This results in tensor-valued element-matrices as well as a tensor-valued nonlinear stiffness matrix, which is unrolled for the simulation process in this work.
Simulation results are presented which show higher order harmonics for the mechanical field in the response spectrum.
This approach can be applied to even higher-order nonlinearities, however, the presented numerical examples focus on quadratic elastic properties.
