Lothar Gaul

A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations

Abstract A minimal two degree of freedom model is used to clarify from an intuitive perspective the physical mechanisms underlying the mode-coupling instability of self-excited friction induced oscillations. It is shown that simultaneous out-of-phase oscillations of friction force and displacement tangential to the friction force may lead to energy transfer from the frictional system to vibrational energy. Also it is shown that the friction force acts like a cross-coupling force linking motion normal to the contact surface to motion parallel to it and that a necessary condition for the onset of instability is that these friction-induced cross-coupling forces balance the corresponding structural cross-coupling forces of the system. Finally the origin and the role of phase shifts between oscillations normal and parallel to the contact surface is clarified with respect to the mode-coupling instability. It may be expected that the intuitive picture gained will be of considerable help for practical design purposes.

Finite Element Formulation of Viscoelastic Constitutive Equations Using Fractional Time Derivatives

Fractional time derivatives are used to deduce a generalization ofviscoelastic constitutive equations of differential operator type. Theseso-called fractional constitutive equations result in improvedcurve-fitting properties, especially when experimental data from longtime intervals or spanning several frequency decades need to be fitted.Compared to integer-order time derivative concepts less parameters arerequired. In addition, fractional constitutive equations lead to causalbehavior and the concept of fractional derivatives can be physicallyjustified providing a foundation of fractional constitutive equations.First, three-dimensional fractional constitutive equations based onthe Grünwaldian formulation are derived and their implementationinto an elastic FE code is demonstrated. Then, parameter identificationsfor the fractional 3-parameter model in the time domain as well as inthe frequency domain are carried out and compared to integer-orderderivative constitutive equations. As a result the improved performanceof fractional constitutive equations becomes obvious. Finally, theidentified material model is used to perform an FE time steppinganalysis of a viscoelastic structure.

A boundary element method for transient piezoelectric analysis

In this paper, a boundary element (BE) formulation is developed originally which treats three-dimensional problems of transient piezoelectricity. The approach at hand uses the fundamental solution of the static piezoelectric operator instead of the transient one. This results in a domain integral appearing in the representation formula, which contains the inertia term. This domain integral can be transformed to the boundary using the dual reciprocity method (DRM), which leads to a system of ordinary differential equations in time domain, similar to the systems obtained in standard finite element methods (FEM). The DRM has been chosen because of the difficulties and big computational effort involved in a BE implementation, which makes use of the anisotropic transient piezoelectric fundamental solution. It is an approach that appears to be much too time-consuming for use in a commercial BE code, in which computational costs is an important issue. The method presented in this paper is validated by a numerical example for transient piezoelectricy, which demonstrates excellent agreement with FE computations for the generalized displacements, and an improved accuracy for the flux quantities such as electric field and elastic stresses.

On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems

The numerical evaluation of fractional derivatives requires a high number of computations due to their nonlocal character. Particularly in the case of a multi-degree-of-freedom (mdof) system that is described by a fractional differential equation in time and is solved numerically by time integration, the numerical effort and the storage requirements explode. Therefore, a new method for the numerical evaluation of fractional derivatives is presented which reduces the numerical effort of mdof systems drastically. The algorithm is based on the Grunwald definition of fractional derivatives and, in contrast to some other concepts, maintains its benefits. The algorithm is applied to the equation of motion of a viscoelastic member, the properties of which are described by a fractional constitutive equation. Comparative calculations demonstrate the accuracy of the algorithm and the reduction in computation time and storage requirements.

A coupled symmetric BE-FE method for acoustic fluid-structure interaction

Abstract A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamiltons principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C 1 -continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.

Identification of a bolted-joint model with fuzzy parameters loaded normal to the contact interface

Abstract Modeling and identification of a joint with load normal to the contact interface of two connected rods is discussed in this paper. An experimental setup for the analysis of the joint is proposed and measurement results are presented. The perception that both the damping behavior and the stiffness of the joint are influenced by a large number of effects that can hardly be modeled motivates the use of a rather simple model, but with fuzzy-valued model parameters, instead of crisp ones. In this concept, the uncertainty and variability of the model parameters can be taken into account by representing the parameters as fuzzy numbers that can be identified on the basis of the measured data. The identification of the fuzzy parameters proves to be a non-trivial problem which can be solved by applying the transformation method as a special implementation of fuzzy arithmetic.

Vibration and sound radiation controls of beams using layered modal sensors and actuators

A new smart structural configuration for beams with symmetrically embedded modal sensors and actuators is proposed. Pairs of symmetrically installed modal sensors and actuators having the same shape of electrode profile are individually connected by a control unit. They are used to control each corresponding vibration mode of the beam independently. Dynamic response of the structure is investigated by the modal analysis method. Numerical simulations for vibration and sound radiation controls presented for illustrations are limited to simply supported beams. Results indicate that modal sensors and actuators, properly selected on the basis of the location and frequency of external forces, can effectively suppress the beam vibration and reduce radiated sound pressure in the far-field.

Shape Control of Composite Plates by Bonded Actuators with High Performance Configuration

The electrical fields and configuration parameters such as location, size and thickness of piezoelectric actuators are very important for shape control of structures. By using a performance index, the configuration parameters of actuators and adhesive can be chosen by some rules for improving performance. In this study a procedure combining a finite element model and the gradient projection algorithm is proposed for optimum design of shape control. This finite element model contains the actuator element, the adhesive interface element, and an eight-node isoparametric plate element which is employed for the calculation of plate shape. The gradient projection algorithm is used to choose the optimal values of design variables. Some numerical examples are illustrated and results reveal that the present method is effective and can be applied to the problem of shape control.

Numerical treatment of acoustic problems with the hybrid boundary element method

Abstract The symmetric hybrid boundary element method in the frequency and time domain is introduced for the computation of acoustic radiation and scattering in closed and infinite domains. The hybrid stress boundary element method in a frequency domain formulation is based on the dynamical Hellinger–Reissner potential and leads to a Hermitian, frequency-dependent stiffness equation. As compared to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix are communicated. On the other hand, the hybrid displacement boundary element method for time domain starts out from Hamiltons principle formulated with the velocity potential. The field variables in both formulations are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions, generated by Dirac distributions, and generalized loads, that are time dependent in the transient case. The domain is modified such that small spheres centered at the nodes are subtracted. Then the property of the Dirac distribution, now acting outside the domain, cancels the remaining domain integral in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. In the time domain formulation, an analytical transformation is employed to transform the remaining domain integral into a boundary one. This approach results in a linear system of equations with a symmetric stiffness and mass matrix. Earlier 2D results are generalized in the present paper by a 3D implementation. Numerical results of transient pressure wave propagation in a closed domain are presented.

Variability and Repeatability of Jointed Structures with Frictional Interfaces

Bolted joints are found in almost every assembled system. Damping due to the friction in the interface of the bolted joints dominates the overall damping in these systems. Therefore, in order to accurately model assembled systems, the correct amount of damping as well as the nonlinear characteristics of the bolted joint must be appropriately accounted for. The level of damping, however, is sensitive to many factors, such as the interface condition and the residual stresses. The formulation of the equations of motion hereby has to involve the local properties of the interfacial damping. In this contribution, two different approaches are applied to a two-beam structure coupled by three bolted joint connections: the discontinuous basis function method and a frequency based substructuring formulation. Measurements of three related systems are used to assess the two different modeling approaches: a monolithic beam, a monolithic beam with three interfaces, and a jointed beam with three bolted joints. The FRFs of the three systems are measured in order to quantify the effect of the bolted interfaces, and future work will investigate the ability of the models to predict the FRFs.