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The aim of the present paper is to associate a new symmetric boundary integral method to well-known symmetric domain methods with the purpose of improving solutions. Multifield problems arise from acoustic and hydroacoustic radiation of mechanical systems with vibrating surfaces. Instead of discretising the mechanical system with finite elements, the proposed boundary integral method is effective because the boundary data are of primary interest. Thus, the problem dimension is reduced by one and symmetry is preserved. As a method is based on test functions, which analytically fulfil the homogeneous field equations, high accuracy is gained. From a single-field variational principle for a 3-D linear, elastodynamic state, a three-field hybrid principle is developed by Hamilton's principle and by decoupling displacements in the domain from those on the boundary. Compatibility is enforced in a weak sense. For investigating steady-state vibrations, the functional is transformed in the frequency domain. Superimposed singular fundamental solutions of the Lame-Navier field equations generated by Dirac functions and weighted by generalised loads are used as test functions in the domain. In the absence of body forces, they cancel the remaining domain integral in the hybrid principle and lead to a boundary integral formulation. The boundary variables are discretised by boundary elements. A symmetric dynamic stiffness matrix equation is gained which relates nodal displacements and tractions on the boundary. An application of the hybrid boundary integral method is derived. Because acoustic and hydroacoustic radiation in 2-D is predominantly generated by bending waves, the 3-D hybrid method is adopted for 1-D beams. As the boundary of a finite beam degenerates to two nodes, no shape functions are needed. This is why the theory is shown to give analytical results of dynamical beam analysis.

The nonlinear transfer behaviour of an assembled structure such as a large lightweight space structure is caused by the nonlinear influence of structural connections. Bolted or riveted joints are the primary source of damping compared to material damping, if no special damping treatment is added to the structure. Simulation of this damping amount is very important in the design phase of a structure. Several well known lumped parameter joint models used in the past to describe the dynamic transfer behaviour of isolated joints by Coulomb friction elements are capable of describing global states of slip and stick only. The present paper investigates the influence of joints by a mixed experimental and numerical strategy. A detailed Finite Element model is established to provide understanding of different slip-stick mechanisms in the contact area. An advanced lumped parameter model is developed and identified by experimental investigations for an isolated bolted joint. This model is implemented in a Finite Element program for calculating the dynamic response of assembled structures incorporating the influence of micro- and macroslip of several bolted joints.

Mixed boundary value problems of solid mechanics are treated by numericalsolutions of Boundary Integral Equations (BIE) in time domain with theBoundary Element Method (BEM) thus reducing the spatial problem dimensionby one. Viscoelastic constitutive behaviour is implemented by means of aLaplace transform technique based on an elastic--viscoelasticcorrespondence principle. The concept of fractional differintegrationgeneralizes conventional constitutive equations and provides improvedcurve fitting of measured material response with fewer parameters. As theimplementation of viscoelasticity is provided in each time step in theLaplace domain, efficient algorithms for the inverse transformation intime domain are needed. This is why the performance of adapted algorithmsby Talbot, Durbin and Crump are compared. The impact responseof a base plate bonded on a viscoelastic soil halfspace is discussed as anumerical example. Viscous forces increase the velocities of surface wavepropagation and cause attenuation in addition to the so called geometricaldamping by radiation.

The vibration behavior of ships is noticeably influenced by the surrounding water, which represents a fluid of high density. In this case, the feedback of the fluid pressure onto the structure cannot be neglected and a strong coupling scheme between the fluid domain and the structural domain is necessary. In this work, fast boundary element methods are used to model the semi-infinite fluid domain with the free water surface. Two approaches are compared: A symmetric mixed formulation is applied where a part of the water surface is discretized. The second approach is a formulation with a special half-space fundamental solution, which allows the exact representation of the Dirichlet boundary condition on the free water surface without its discretization. Furthermore, the influence of the compressibility of the water is investigated by comparing the solutions of the Helmholtz and the Laplace equation. The ship itself is modeled with the finite element method. A binary interface to the commercial finite element package ANSYS is used to import the mass matrix and the stiffness matrix. The coupled problems are formulated using Schur complements. To solve the resulting system of equations, a combination of a direct solver for the finite element matrix and a preconditioned GMRES for the overall Schur complement is chosen. The applicability of the approach is demonstrated using a realistic model problem.Copyright © 2013 by ASME

Assembled structures often include local nonlinear behavior depending on the interface properties which can strongly change the overall dynamics. A typical example is a high local damping due to friction in bolted joints. A further example are bushing elements, like those which are widely-used in the automotive industry and which often show complicated characteristics like nonlinear stiffness and frequency dependency. In this contribution, measured and simulated FRFs of different systems are considered in order to identify nonlinear joint characteristics. The computation of the FRFs is established by the Harmonic Balance Method using a substructure formulation. A detection of the influence of the local damping properties is done by the Hilbert Transformation. For the computation of FRFs of systems with progressive stiffness, the Harmonic Balance Method is extended by a Continuation Method in order to be able to capture multiple solutions and is applied to a numerical example system. As an application an assembly consisting of two beam-like substructures coupled by a bolted joint is used in order to identify the friction characteristic of a real system. The results are used for further investigations of coupled axle parts of a wheel brake.

Bei der Konzeption und Auslegung technischer Bauteile spielt die numerische Simulation heute oft eine wesentliche Rolle, da einerseits Experimente oftmals zu kostspielig oder aber technisch nicht durchfuhrbar sind, andererseits in der Computerwelt rasende Fortschritte erzielt werden. Um Experimente zu erganzen oder sogar zu ersetzen, mussen die Simulationsverfahren einer Reihe hoher Anforderungen genugen. Die wesentliche Forderung besteht darin, das die Berechnung auf effizientem Weg moglichst genaue Ergebnisse liefert, d. h. das reale System, seine Belastung und seine Reaktion auf diese Belastung moglichst genau wiederspiegelt.

Eine Einteilung der Fertigungsverfahren ist in der DIN 8580 allgemeingultig beschrieben. Fertigungsverfahren werden in sechs Hauptgruppen eingeteilt. Jede Hauptgruppe basiert auf spezifischen Merkmalen.

The sound radiation from a vibrating structure in water is investigated. The radiation depends on fluid boundaries, the geometry of structure, and associated mode shapes generated by driving forces on the structure. In the lower frequency range, the structure can be divided into regions having high and low radiation. These regions represent a distribution of sources and sinks of noise. The sound pressure level in the fluid domain and the velocity field is determined by using a boundary element method with input data of surface velocities obtained from the modal analysis of from a finite element calculation. A consideration of fluid structure interaction is possible. Alternatively, noise sources can be identified by intensity vector measurements in the acoustic near field. For a comparison of these approaches, the sound radiation from a vibrating plate calculated by the boundary element method (BEM) was compared with the results obtained by a superposition of pulsating spheres. The second approach is based on analytical results and the acoustic near field can be described in terms of energy flow.

In this paper damping induced by extensive friction occurring in the interface between bolted structures is considered by simulations and experiments. A friction damper is attached to a beam-like flexible structure by screws such that the normal force in the interface can be varied by the clamping force of the screws. Contact and friction force parameters are identified by the comparison of simulated and experimentally determined FRFs for a particular normal force. Afterward a prediction of damping for different configurations is established. For simulations a finite element model is used where suitable contact and friction models are implemented. A time simulation of the system is expensive due to the large number of DoFs of the discretized substructures and the required small step size due to the high contact stiffness. Therefore model reduction methods are used. A further reduction of the computation time can be achieved by using the Harmonic Balance Method (HBM) for a direct frequency domain computation of FRFs. This enables an efficient procedure to approximate the reachable damping as well as to search the optimal damper position and the optimal normal force. The dependency of the friction to the vibration amplitude is therefore taken into account. A more detailed investigation of the nonlinear effects, e.g. higher harmonic response, is then accomplished by transient simulations for the optimal configured system in the time domain and the results are compared to experimental results.Copyright © 2012 by ASME

Dynamic substructuring offers the possibility to simulate assembled systems efficiently. The coupling of the substructures can be established either by Component Mode Synthesis (CMS), or Frequency Response Functions (FRF) can be used to couple the substructures by Frequency Based Substructuring (FBS). In real systems, coupling is done by joints which can influence the dynamics of the assembled system significantly due to local damping and nonlinearities caused by friction. In this contribution the coupling of two beam-like substructures, which are assembled by a bolted joint, is considered using both coupling methods. While the substructures are linear, the implementation of the nonlinear friction forces requires special attendance in the equations of motion. The Harmonic Balance Method is therefore used to efficiently compute FRFs. Using FBS, the coupling is established directly in the frequency domain. The method provides the possibility to replace the dynamics of individual substructures by measured FRFs of the uncoupled system and combining numerical and experimental models. Alternatively, Component-Mode-Synthesis is used.

Acoustic fluid-structure interaction is a common issue in automotive applications. An example is the pressure-induced structure-borne sound of piping and exhaust systems. Efficient model order reduction and substructuring techniques accelerate the finite element analysis and enable the vibro-acoustic optimization of such complex systems with acoustic fluid-structure interaction. This tutorial reviews the application of the Craig-Bampton and Rubin method to fluid-structure coupled systems and presents two automotive applications. First, a fluid-filled brakepipe system is assembled by substructures according to the Craig-Bampton method. Fluid and structural partitions are fully coupled in order to capture the interaction between the pipe shell and the heavy fluid inside the pipe. Second, a rear muffler with an air-borne excitation is analyzed. Here, the Rubin and the Craig-Bampton method are used to separately compute the uncoupled component modes of both the acoustic and structural domain. These modes are then used to compute a reduced model which incorporates full acoustic-structure coupling. For both applications, transfer functions are computed and compared to the results of dynamic measurements. The vibro-acoustic behavior of ship-like structures is noticeably influenced by the surrounding water and thus represents a multi-field problem. In this tutorial, fast boundary element methods are applied for the semi-infinite fluid domain. As an advantage, the Sommerfeld radiation condition is satisfied in an exact way and only the boundary, i.e. the ship hull, has to be discretized. To overcome the draw-back of fully populated matrices, fast boundary element methods are applied. The focus is on the comparison of the multipole method with hierarchical matrices, which are set up by adaptive cross approximation. In both cases, a half-space fundamental solution is used to incorporate the water surface, which is treated as pressure-free. The structural domain is discretized with the finite element method. A binary interface to the commercial finite element package ANSYS is used to import the mass and stiffness matrices. The coupled problems are formulated as Schur complements, which are solved by a combination of iterative and direct solvers. Depending on the applied fast boundary element method, different strategies arise for the preconditioning and the overall solution. The applicability of these approaches is demonstrated using a realistic model problem.

It was Nov 10, 1982 when Prof. Federn, Prof. Gaul, Prof. Mahrenholtz, and Dr. Pieper VDI decided to work out a guideline on damping in the VDI/FANAK C13 Committee “Material Damping”. They were joined by Prof. Ottl, Prof. Kraemer, Prof. Pfeiffer, Prof. Markert, Prof. Wallaschek, and Mr. Hilpert VDI lateron in their names order. The idea was to comprise distributed theoretical and experimental knowledge and to homogenize the nomenclature of this subject.

Precise information of the current state of motion of a vibrating system is crucial for every effective active vibration control. Due to the limitation of the number of sensors in the final operational setup, a state estimator is needed to reconstruct the current state of vibration at important degrees of freedom from a small set of sensors. If the excitation of the structure (system input) cannot be measured directly, a sensor based estimator has to be used, which operates on response data only (system output). Core piece of this estimator is a reduced model of the vibrating system. If additionally some sensors are only capable to measure spatially distributed deflection instead of the deflection at single degrees of freedom (like piezo-electric foils), the reduced model must incorporate both those sensors as well as the important degrees of freedom as outputs. For structures which cannot be excited by a measurable excitation even in the development phase, the reduced model can still be identified by an output-only modal analysis. In this work, the identification technique is demonstrated for a curved plate submerged in water and exited by a surrounding sound field.

Reduction of structural vibrations is of major interest in mechanical engineering for lowering sound emission of vibrating structures, improving accuracy of machines and increasing structure durability. Besides optimization of the mechanical design or various types of passive damping treatments, active structural vibration control concepts are efficient means to reduce unwanted vibrations. In this contribution, two different semi-active control concepts for vibration reduction are proposed that adapt the normal force of attached friction dampers. Thereby, semi-active control concepts generally possess the advantage over active control that the closed loop is intrinsically stable and that less energy is required for the actuation than in active control. In the chosen experimental implementation, a piezoelectric stack actuators is used to apply adjustable normal forces between a structure and an attached friction damper. Simulation and experimental results of a benchmark structure with passive and semi-actively controlled friction dampers are compared for stationary narrow-band excitation.

The influence of the acoustic field on the structural dynamics is a common issue in automotive applications. An example is the pressure-induced structure-borne sound of piping and exhaust systems. Efficient model order reduction and substructuring techniques accelerate the finite element analysis and enable the vibro-acoustic optimization of such complex systems with acoustic fluid-structure interaction. This research reviews the application of the Craig-Bampton and Rubin method to fluid-structure coupled systems and presents two automotive applications. First, a fluid-filled brake-pipe system is assembled by substructures or superelements according to the Craig-Bampton method. Fluid and structural partitions are fully coupled in order to capture the interaction between the pipe shell and the heavy fluid inside the pipe. Second, a rear muffler with an air-borne excitation is analyzed. Here, the Rubin and the Craig-Bampton method are used to separately compute the uncoupled component modes of both the acoustic and structural domain. These modes are then used to compute a reduced model which incorporates full acoustic-structure coupling. For both applications, transfer functions are computed and compared to the results of dynamic measurements.

The boundary element method (BEM) provides a powerful tool for the calculation of elastodynamic response in frequency and time domain. Field equations of motion and boundary conditions are cast into integral equations, which are discretized only at the boundary. The boundary data often are of primary interest because they govern the transfer dynamics of members and the energy radiation into a surrounding medium. Formulations of BEM currently include conventional viscoelastic constitutive equations in the frequency domain. In the present paper viscoelastic behaviour is implemented in a time domain approach as well. The constitutive equations are generalized by taking fractional order time derivatives into account.

Formulations of the boundary element method (BEM) currently include conventional viscoelastic constitutive equations in the frequency domain. The aim of the present paper is to implement viscoelastic behaviour in a time domain approach as well. The elastic Stokes fundamental solution is converted to a viscoelastic one by adopting a correspondence principle. A novel viscoelastic fundamental solution is obtained analytically by inverse Laplace transformation. A frequency domain BE approach is generalized by taking viscoelastic constitutive equations with fractional order time derivatives into account. It is shown that the boundary matrix ϵij for non smooth boundaries in a dynamic formulation equals the elastostatic matrix. The transfer behaviour of a mounting system has been calculated by adopting the developed BE formulation to a viscoelastic resilient support mount.

In this paper scattering problems with elastic obstacles that are hit by an incident acoustic wave are discussed. Underwater acoustics mainly differs from air acoustics in the fact that a strong coupling scheme between the structural part and the acoustic domain is necessary. Such a scheme is discussed, using the fast multilevel multipole boundary element method (FMBEM) to model the exterior acoustic fluid. The structural part is modeled with the finite element method (FEM). To obtain a high flexibility, an interface to a commercial FE package is established. For a high efficiency, an iterative solver with preconditioning is applied. The numerical results are compared with an analytical solution for a model problem.

Reduction of structural vibrations is of major interest in mechanical engineering for lowering sound emission of vibrating structures, improving accuracy of machines and increasing structure durability. Besides design optimization and passive damping treatments, active structural vibration control can be applied to reduce unwanted vibrations. In this contribution, two semi-active control laws for control of friction dampers are derived and investigated in simulations and experiments. Thereby, semi-active control concepts have the advantage over active control to yield intrinsically stable closed-loop systems and low energy consumption. In the experimental implementation, the control makes use of piezoelectric stack actuators to apply adjustable normal forces between structure and attached friction dampers. The control uses an observer based on reduced finite-element models to estimate the unknown relative displacement beneath the normal force actuator from acceleration measurements. Experimental results of the control algorithms for a structure with attached friction damper show the effectiveness of the proposed control algorithms.Copyright © 2009 by ASME

Fluid–structure interaction plays a crucial role for simulating vibro–acoustic, multi–field problems. The paper starts with the fundamentals of the boundary element method for acoustics, including the Burton–Miller formulation for the solution of exterior problems. The fast multilevel–multipole algorithm is applied together with a preconditioned iterative solver yielding a quasi–linear numerical complexity. The structural part is modelled by the finite element method. Various coupling schemes are discussed depending on the influence of the feedback of the fluid onto the structure. In a first example, an expansion chamber, which is exposed to structural vibrations caused by interior pressure fluctuations, is presented. A weak coupling scheme is used for simulating the surface radiated sound. In a second example, the vibro–acoustic behavior of a totally submerged submarine–like structure is investigated. Two strong coupling formulations are compared for their numerical efficiency. The examples show, that the presented coupling algorithms are well–suited for simulating large–scale industrial applications.

Piezoelectric materials are known to exhibit nonlinear effects if they are operated outside their linear small-signal regime, which significantly degrades the performance of structural control concepts. In this contribution, these nonlinear effects are experimentally investigated in the context of a feedforward control application, where a control command is designed to steer the tip of a cantilever beam by means of a piezoelectric patch actuator from initial to a prescribed desired final stationary deflection. As expected, the encountered nonlinear effects degrade the control performance with increasing applied electrical field. Hence, a modified feedforward control design procedure is proposed. The overall nonlinear system model is recast in a series connection of an input nonlinearity and the linear dynamics of the mechanical structure, which the feedforward control to be designed in two steps: First, a feedforward control for the linear model part is derived based on an approach exploting the notion of flatness in combination with modal analysis of the linear dynamics. It uses the finite-element method to derive the linear dynamics of the piezoelectric structure. Secondly, an inverse filter is designed to compensate for the nonlinear piezoelectric hysteresis and creep effects. By insertion of this inverse filter at the system input, i.e. by filtering the feedforward control, very good tracking control performance is recovered in both small and large-signal operation of the piezoelectric actuator. This filter itself is derived by inversion of a model of the nonlinearities in the discrete-time domain. The chosen model for the hysteresis is based on polynomial approximations of the hysteresis loops, appropriate scaling of these loops to the actual point of operation by keeping track of the input signal reversals and on implementation of the physically motivated Madelung rules that the piezoelectric hysteresis obeys. The creep is found to behave according to a Kelvin-Voigt viscoelastic model. Various experiments for the piezoelectrically actuated beam show that the modified feedforward control design yields very good tracking performance also outside of the small-signal regime by application of the compensation filter. The excellent feedforward tracking control performance predicted by simulations of the full (linear) finite-element model is verified. The designed feedforward control realizes very fast rest-to-rest transition in less than half of the period of the first structural mode. As an interesting application of such a feedforward control, a two-degree-of-freedom control concept combining the presented feedforward control and additional feedback control is investigated. By use of the feedforward control, the feedback can be relatively simple because it is only responsible for disturbance rejection and adding robustness.Copyright © 2007 by ASME

In the fields of high-resolution metrology and manufacturing, effective anti-vibration measures are required to obtain precise and repeatable results. This is particularly true when the amplitudes of ambient vibration and the dimensions of the investigated or manufactured structure are comparable, e.g. in sub-micron semiconductor chip production, holographic interferometry, confocal optical imaging, and scanning probe microscopy. In the active antivibration system examined, signals are acquired by extremely sensitive vibration detectors, and the vibration is reduced using a feedback controller to drive electrodynamic actuators. This paper deals with the modeling and control of this anti-vibration system. First, a six-degree-of-freedom rigid body model of the system is developed. The unknown parameters of the unloaded system, including actuator transduction constants, spring stiffness, damping, moments of inertia, and the vertical position of the center of mass, are determined by comparing measured transfer functions to those calculated using the updated model. Finally, two different strategies for actively controlling the vibration isolation system are considered.

The spoilers on an aircraft are responsible for several tasks, including execution of roll maneuvers, lift dumping (aerodynamic "spoiling"), and braking. The examined spoiler is manufactured from carbon fiber reinforced composite material and is attached to the wing by four bearing hinges and one actuator hinge. Correct spoiler design involves knowledge of the loads acting on the spoiler, calculation of stresses and strains, and examination of possible failures. Additionally, the fatigue and damage tolerance evaluation of such a spoiler has to follow established certification protocols. This study defines a load cycle based on in-service loadings, including aerodynamic loading, wing bending, inertial loading, and actuator loading. A finite element model of the spoiler is used to calculate the reaction forces in the hinges and the strain in the carbon fiber components occurring during the load cycle. The Miner Rule is used to calculate the fatigue life of the hinges based on the computed stress. A damage tolerance evaluation is then performed assuming that different hinges have failed. Finally, a certification test for fatigue and damage tolerance evaluation of a spoiler is discussed.

In the fields of high-resolution metrology and manufacturing, effective anti-vibration measures are required to obtain precise and repeatable results. This is particularly true when the amplitudes of ambient vibration and the dimensions of the investigated or manufactured structure are comparable, e.g. in sub-micron semiconductor chip production, holographic interferometry, confocal optical imaging, and scanning probe microscopy. In the active anti-vibration system examined, signals are acquired by extremely sensitive vibration detectors, and the vibration is reduced using a feedback controller to drive electrodynamic actuators. This paper deals with the modeling of this anti-vibration system. First, a six-degree-of-freedom rigid body model of the system is developed. The unknown parameters of the unloaded system, including actuator transduction constants, spring stiffness, damping, moments of inertia, and the location of the center of mass, are determined by comparing measured transfer functions to those calculated using the updated model. The model is then re-updated for the case of an arbitrarily loaded system. The responses predicted by the final updated model agree well with the experimental measurements, thereby giving confidence in the model and the updating procedure.

Every physical system is described by parameters, and one goal of the present contribution is to study the movements of eigenvalues in the complex plane depending on the system inherent parameters. A main focus lies in tracing the boundary curve which separates unstable from marginally stable domains in the parameter space. Hence, there is no need to study the whole parameter space but a certain subset which can be characterized by a zero eigenvalue. The method is illustrated by means of a static stability problem arising in the study of rotating blades.Copyright © 2005 by ASME

Valve action and pump fluctuation in piping systems can lead to undesired excitation of structural components by propagating sound waves in the fluid path. This vibro-acoustic problem is addressed by studying the dynamics as well as excitation mechanism. Fluid-structure interaction has a significant influence on both hydroacoustics and on structural deformation. Therefore, pipe models are generated in three dimensions by using Finite Elements to include higher-order deflection modes and fluid modes. The acoustic wave equation in the fluid is hereby fully coupled to the structural domain at the fluid-structure interface. These models are used for simulating transient response and for performing numerical modal analysis. Unfortunately, such 3D models are large and simulation runs turn out to be very time consuming. To overcome this limitation, reduced pipe models are needed for efficient simulations. The proposed model reduction is hereby based on a series of modal transformations and modal truncations, where focus is placed on the treatment of the nonsymmetric system matrices due to the coupling. Afterwards, dominant modes are selected based on controllability and observability considerations. Furthermore, modal controllabilities are used to quantify the excitation of vibration modes by white noise at the pipe inlet representing acoustic sources. The excitation of structural elements connected to the piping system can therefore be predicted without performing transient simulations. Numerical results are presented for spatially arranged complex piping systems including elbow pipes and joints connected to target structures to demonstrate the usefulness of the presented method for vibro-acoustic investigations. The method is to support the design and the analysis of fluid-filled elastic piping systems and its environment in the presence of acoustic sources such as in hydraulic systems.

This work presents a semi-passive concept to reduce structural vibrations over a wide frequency regime. Therefore, a purely resistive passive electrical network is designed and connected to a piezoelectric element. This concept allows an enhancement in structural damping without using sensitive sensor electronics and amplifiers. Furthermore, it yields constant vibration reduction over a wide temperature range; limitations arise for temperatures above the Curie temperature. Firstly, the damping capabilities of a resistively shunted piezoelectric element are discussed and the optimal resistance for the passive electrical network is outlined. Next, a design concept for optimal placement of the piezoelectric elements is presented. This concept is designed for placement on two dimensional structures, such as plates. It is based on an energy ratio, which is defined for each structural mode. In this context, the effective strain energy of a two dimensional piezoelectric element is introduced. It allows calculation of the electrical energy, generated by the piezoelectric element as it is mechanically loaded. Adaption of the shunt resistor, and thus, maximal reduction of structural vibrations, is obtained by a new concept, which uses digital potentiometers in combination with a feedforward control concept. Depending on the excitation signal, two different resistances can be realized by the adaptive passive electrical network. In this manner two structural modes can be optimally damped. Finally, experiments are conducted, in which fixed resistors as well as adaptive resistors are implemented. Results show the advantage of using adaptive resistors for the passive electrical network in terms of enhanced vibration reduction capabilities.

The present approach for vibration suppression of flexible structures is based on friction damping in semi-active joints. At optimal locations conventional rigid connections of a large truss structure are replaced by semi-active friction joints. Two different concepts for the control of the normal forces in the friction interfaces are implemented. In the first approach each semi-active joint has its own local feedback controller, whereas the second concept uses a global, clipped-optimal controller. Simulation results of a 10-bay truss structure show the potential of the proposed semi-active concept.

Ultrasonic waves travel in rope structures over long distances as guided waves, allowing for effective health monitoring. In order to localize and characterize defects, an exact knowledge of the propagation, reflection, and transmission properties of the ultrasonic waves is required. These properties can be obtained using the Finite Element Method by modeling a segment of the periodic waveguide with a periodicity condition. The solution of the corresponding eigenvalue problem leads to all propagating modes of the waveguide as well as locally generated evanescent modes. The Boundary Element Method (BEM) is used in combination with the Finite Element Method for characterizing the wave propagation. The mode conversion at discontinuities, such as cracks or notches, can be subsequently described by reflection and transmission coefficients. The simulation results are the corresponding coefficients as a function of frequency and enable the selection of adequate modes for an effective defect detection. Additionally, it is demonstrated that along with the localization of cracks, conclusions about the crack geometry can be made with the help of reflection and transmission coefficients. The reliability and numerical accuracy of the simulation results are verfied by comparison with experimental findings. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this chapter, we describe the Hybrid Displacement Boundary Element Method for the time domain and derive formulations for elastodynamics and acoustics. These formulations lead to symmetric systems of equations with time-invariant mass and stiffness matrices. The symmetry is exploited to formulate a symmetric coupled system of equations for fluid-structure interaction in time domain.

The governing differential equations of continuum mechanics can be derived by evaluating balance equations of vector fields, such as the momentum. A second approach for obtaining the governing differential equations, so-called analytical mechanics, is based on axioms defined in scalar quantities such as work and energy (Lanczos (1970), Riemer et al (1993), Szabo (1987)). We can obtain the governing vector equations by using variational calculus, outlined in Section 2.5. The two approaches are mechanically equivalent but when numerical solutions for complex systems are required, the two approaches provide significant differences in suitability and efficiency.

Having described the basic principles of the Boundary Element Method through the 2-D versions of Laplace’s and Poisson’s equations, we now extend the formulation to the consideration of 3-D problems in elastomechanics. In this context, we will also deal with anisotropic materials and give an introduction to the analysis of coupled field problems by means of static piezoelectricity.

In this chapter we use the Hybrid Displacement Boundary Element Method (HDBEM) to investigate several static and dynamic field problems. The method is based on variational principles, rather than on integral equations as the BEM from Part I and II, which is based on weighted residual statements. To avoid confusion, the weighted residual method will be referred to as Direct BEM for the remainder of the book.

A mortar coupling algorithm is proposed for acoustic-structure-interaction. Finite plate elements are coupled to a boundary element discretization of the acoustic domain. The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non-matching grids. The method is demonstrated on the example of a cavity backed elastic panel.

In Chapter 1, we introduced the Boundary Element Method by means of a simple one-dimensional example for which we sketched the basic steps from the differential equation to the boundary integral equation. However, since the boundary in a 1-D continuum degenerates to two points, we have neither encountered the special problems arising from the singularity of the fundamental solutions, nor could we describe the discretisation process with boundary elements. With the mathematical and physical background given in Chapters 2 and 3, we can now direct our attention to the more interesting and practically relevant two- and three-dimensional problems. In the present chapter, we will deal with simple potential problems in 2-D, so that we can concentrate on the basic features of the Boundary Element formulation. In Chapter 5, we then apply the method to 3-D problems of elastomechanics and static piezoelectricity.

One of the cornerstones of the Boundary Element Method is its use of fundamental solutions to eliminate the domain integral over the adjoint operator. In the absence of source terms, the resulting formulation contains only boundary variables, so that only the boundary of the body has to be discre-tised and not its domain. This reduces the number of nodal unknowns and greatly simplifies the meshing and remeshing process as compared to domain discretisation methods such as Finite Elements or Finite Differences.