Rolf Lammering

Multi-field variational formulations and related finite elements for piezoelectric shells

Smart structures technology characterized by structurally integrated sensors and actuators has recently expanded significantly especially as regards lightweight constructions in aeronautics and robotics, e.g. to allow vibration suppression and noise attenuation. In order to be capable of solving these complex issues the finite element method as a well established design tool has to be extended. This paper focuses on shallow sandwich composite shell structures with thin piezoelectric patches bonded to the surfaces. For the proper design of plate and shell structures with integrated piezoelectric materials, various variational formulations and corresponding finite elements are presented. The starting point is the well known two-field variational formulation where the linear piezoelectric effect is taken into account so that the displacements and the electric potential serve as independent variables. Here, the mostly assumed linear variation of the electric potential through the thickness is assumed. Next, it is shown that a quadratic variation of the electric potential through the thickness can be deduced directly from the charge conservation condition. This quadratic variation of the electric potential in the thickness direction is compared with the linear gradient of the first two-field variational formulation. Moreover, in order to allow the implementation of alternative formulations of the constitutive equations by switching of the independent variables and nonlinear material behaviour, a three-field variational formulation is presented in analogy to the Hu–Washizu principle. Adopting this variational principle a hybrid finite element is derived where the dielectric displacement is formulated as an additional degree of freedom. This independent variable can be condensed on the element level and does not enter the system of equations. For the first time all these different variational formulations are developed for a Reissner–Mindlin shallow shell element formulation and compared with each other.

Investigation on piezoelectrically induced Lamb wave generation and propagation

An analytical solution for transient Lamb wave generation with piezoelectric wafers is presented. The time harmonic problem is solved in the wavenumber domain after Fourier transformation and subsequent inverse transformation by the residue theorem. To evaluate the solution for transient loading the Fourier transform is used again, this time with respect to the time variable. A standard numerical algorithm is utilized to evaluate the necessary integrals as well as the discrete Fourier transform. The solution is valid for the whole waveguide continuum and satisfies the underlying partial differential equations as well as the transient boundary conditions. A numerical example shows that the analytical solution coincides with the finite element result of the problem. The results of the analytical model are also confirmed by experimental investigations.

Micro and macromechanical investigations of CuAlNi single crystal and CuAlMnZn polycrystalline shape memory alloys

The micro and macromechanical transformation behavior of single and polycrystalline shape memory alloys was investigated using in situ optical microscopy. An interference filter was used on the microscope to enhance observation of grain boundaries and martensitic plate formation and growth. Experiments on CuAlMnZn polycrystals as well as CuAlNi single crystal shape memory alloys were performed and results are reported in this paper. Cyclic behavior, temperature effects and strain rate effects were studied in detail. Excellent correlation between stress relaxation and temperature decay at several strain rates was achieved. At high strain rates, a dramatic redistribution of martensitic variants was seen in the single crystal specimens after loading to a fixed strain. Tests were performed at different temperatures to help identify the mechanism for variant redistribution.

Experimental investigations on the damping capacity of NiTi components

In the study presented, the damping capacity of shape memory alloys in the austenitic range was considered. The investigations on the dynamic behaviour of pseudoelastic shape memory alloys start with tensile tests on NiTi specimens at frequencies of up to 4 Hz. These experiments show that the beginning of the transformation process is more difficult to observe and that the critical stress value for the transition to martensite becomes obviously smaller at higher strain rates. Furthermore, the area of the hysteresis loop is reduced with increasing strain rate. The damping behaviours of NiTi and steel are directly compared in experiments on a pre-strained string. The system is excited by hammer impact as well as by an electrodynamic shaker. It is pointed out that the damping capacity is highly dependent on the vibration amplitude. Further experiments are performed with coil springs made from NiTi and steel. A stiff beam on which additional masses are mounted is supported by two coil springs so that the whole assembly can be considered as a spring-mass system. The results which were obtained in the string experiments are verified by these investigations.

Lamb wave excitation and propagation in elastic plates with surface obstacles: proper choice of central frequencies

Experimental and theoretical investigations of Lamb wave excitation and sensing using piezo patch transducers and the laser vibrometer technique have been performed, aiming at the development of adequate mathematical and computer models for the interpretation of sensing data and for the choice of optimal parameters for structural health monitoring. The proposed models are validated by experimental results. Furthermore, a methodology is presented which allows for the determination of central frequencies at which maximal values of the structural response spectrum can be expected in the case of wave propagation monitoring with laser vibrometry.

Selective Lamb mode excitation by piezoelectric coaxial ring actuators

This paper describes an omnidirectional multi-element transducer for selective Lamb wave mode excitation. It is composed of several coaxial ring-shaped piezoelectric elements actuated by n-cycled sinusoidal tone bursts. Mode selection is achieved through a special choice of the amplitudes and time delays of the input driving signals. The method for the determination of these parameters is based on strict analytical consideration. In the limiting case of an infinite number of cycles and with a sufficient number of actuators one can generate a single required mode and completely eliminate all undesired ones. It is shown that within the range of applicability of a simplified model for the piezoelectric elements, i.e. when the actuating contact traction is replaced by a pair of concentrated opposite ring tangential forces, no time delays are needed, and the mode selection is achieved through an appropriate choice of the driving input-signal amplitudes applied in phase or out of phase to different transducer elements.

Modeling and Simulation of Lamb Wave Generation with Piezoelectric Plates

An analytical solution for Lamb wave generation with piezoelectric wafers is presented for both plane strain and axisymmetric cases utilizing integral transform methods. In the case of the plane strain problem a solution for transient loading is presented. Here, the Fourier transform is used with respect to the time variable and a standard numeric algorithm is utilized to evaluate the necessary integrals. Numerical examples show that the analytical solutions coincide with the Finite Element results of the respective problems. The solutions are valid for the whole waveguide continuum and satisfy the underlying partial differential equations and the transient boundary conditions.

Impact Damage Detection in Composite Structures Considering Nonlinear Lamb Wave Propagation

To detect micro-structural damages like fiber/matrix cracks and delaminations in composite materials accurately new methods are developed. Previous works have investigated and shown that the acoustical nonlinearity parameter is a good tool to detect micro-structural damages like plasticity in metal and fatigue damages in metal and composites. In this work, the second harmonic generation is used to analyze composite specimens for impact damages. Therefore, Lamb waves are launched and detected by a piezoelectric actuator and sensor, respectively, at a certain frequency to generate cumulative second harmonic modes. The excitation frequency has to meet special conditions. The signal processing is done by using the wavelet transform to avoid misinterpretations that may occur using the short-time Fourier transform. The Morlet wavelet is used as the mother wavelet. The results of the relative acoustical nonlinearity parameter are compared to the development of the group velocity due to impact damages. It is shown that the relative acoustical nonlinearity parameter is more sensitive to impact damages than the development of the group velocity.

Group velocity of cylindrical guided waves in anisotropic laminate composites

An explicit expression for the group velocity of wave packets, propagating in a laminate anisotropic composite plate in prescribed directions, is proposed. It is based on the cylindrical guided wave asymptotics derived from the path integral representation for wave fields generated in the composites by given localized sources. The expression derived is theoretically confirmed by the comparison with a known representation for the group velocity vector of a plane guided wave. Then it is experimentally validated against laser vibrometer measurements of guided wave packets generated by a piezoelectric wafer active sensor in a composite plate.

A novel methodology for the development of compliant mechanisms with application to feed units

This paper highlights the process of developing compliant mechanisms. Because of their high precision and the absence of friction compliant mechanisms are especially suitable for the use in micro manufacturing. In the following a systematic methodology of the development process is presented, including the design process, the realization of a mechanism and an application for feed units. At the first stage of design the topology of a compliant mechanism and the geometry of the flexure hinges are generated applying optimization methods. Afterwards efficient modeling strategies are applied to ensure dynamic models with high accuracy and manageable size for further investigations. Using these models an open-loop control is designed by means of mathematical trajectory optimization. At the final stage models and open-loop control are applied to design a closed-loop control completing the feed unit. This holistic approach reduces time and effort of the development process significantly. It supports the developer in creating reliable and innovative compliant mechanisms for a wide range of applications.