This thesis examines the relative importance of the inertial and geometric nonlinearities in modelling the dynamics of a flexible aircraft. Inertial nonlinearities are derived by employing an exact definition of the velocity distribution and lead to coupling between the rigid body and elastic motions. The geometric nonlinearities are obtained by applying nonlinear theory of elasticity to the deformations. Peters' finite state unsteady aerodynamic model is used to evaluate the aerodynamic forces. Three approximate models obtained by excluding certain combinations of nonlinear terms are compared with that of the complete dynamics equations to obtain an indication of which terms are required for an accurate representation of the flexible aircraft behavior. A generic business jet model is used for the analysis. The results indicate that the nonlinear terms have a significant effect for more flexible aircraft, especially the geometric nonlinearities which leads to increased damping in the dynamics.

This thesis examines the relative importance of the inertial and geometric nonlinearities in modelling the dynamics of a flexible aircraft. Inertial nonlinearities are derived by employing an exact definition of the velocity distribution and lead to coupling between the rigid body and elastic motions. The geometric nonlinearities are obtained by applying nonlinear theory of elasticity to the deformations. Peters' finite state unsteady aerodynamic model is used to evaluate the aerodynamic forces. Three approximate models obtained by excluding certain combinations of nonlinear terms are compared with that of the complete dynamics equations to obtain an indication of which terms are required for an accurate representation of the flexible aircraft behavior. A generic business jet model is used for the analysis. The results indicate that the nonlinear terms have a significant effect for more flexible aircraft, especially the geometric nonlinearities which leads to increased damping in the dynamics.

The main objective of this thesis is to understand the mechanics of a grand piano soundboard and to investigate the influence of several parameters on its modal and vibrational behaviors. Various analysis techniques are made possible by a development of a fully parameterized Finite Element model of the soundboard which allows the user to modify most geometric and material parameters. In addition, two crucial features, namely crowning and downbearing, are included in the model and their individual and combined effects are observed. This study also accounts for the influence of geometric nonlinearities due to downbearing. The piano soundboard includes a large number of unknown factors stemming from its construction process and the material properties of the wood. This thesis uses sensitivity techniques to investigate the influence of various factors on the variability of the modal outputs (i.e., natural frequencies and mode shapes). The different aspects of the model are described in details and various sensitivity methods are tested in this context.

The Alpha Magnetic Spectrometer is an International Space Station Experiment that features a unique nonlinear support system with no previous flight heritage. The experiment consists of multiple straps with piecewise-linear stiffness curves that support a cryogenic magnet in three-dimensional space inside of a vacuum chamber. The stiffness curves for each strap are essentially bilinear and switch between two distinct slopes at a specified displacement. This highly nonlinear support system poses many questions in regards to feasible computational methods of analysis and possible response behavior. This thesis develops a numerical model for a multi-bilinear-spring support system motivated by the Alpha Magnetic Spectrometer design. Methods of analysis applied to the single bilinear oscillator served as the foundation of the model developed in this thesis. The model is developed using MATLAB and proves to be more computationally efficient than ANSYS finite element software. Numerical simulations contained herein demonstrate the variety of response behaviors possible in a multi-bilinear-spring support system, thus aiding future endeavors which may use a support system similar to the Alpha Magnetic Spectrometer. Classic nonlinear responses, such as subharmonic and chaotic, were found to exist.

In the past few years a new class of layered piezoceramic and piezoceramic-composite actuators, known as RAINBOW and GRAPHBOW, respectively, that are capable of achieving 100 times greater out-of-plane displacements than previously available has been developed. Prior to the development of RAINBOW and GRAPHBOW, large stacks of piezoelectric actuators, requiring complicated electronic drive circuits, were necessary to achieve the displacement now possible through the use of a single RAINBOW actuator. The major issues with both RAINBOW and GRAPHBOW are the prediction of their room-temperature shapes after processing, and their deformation response under application of electric field. In this research, a methodology for predicting the manufactured shapes of rectangular and disk-style RAINBOW and GRAPHBOW is developed. All of the predictive analyses developed are based on finding approximate displacement responses that minimize the total potential energy of the devices through the use of variational methods and the Rayleigh-Ritz technique. These analyses are based on classical layered plate theory and assumed the various layers exhibited linear elastic, temperature-independent behavior. Geometric nonlinearities are important and are included in the strain-displacement relations. Stability of the predicted shapes is determined by examining the second variation of the total potential energy. These models are easily modified to account for the deformations induced by actuation of the piezoceramic. The results indicate that for a given set of material properties, rectangular RAINBOW can have critical values of sidelength-to-thickness ratio (Lx/H or Ly/H) below which RAINBOW exhibits unique, or single-valued, spherical or domed shapes when cooled from the processing temperature to room temperature. For values of sidelength-to-thickness ratio greater than the critical value, RAINBOW exhibits multiple room-temperature shapes. Two of the shapes are stable and are, in general, near-cylindrical. The third shape is spherical and is unstable. Similarly, disk-style RAINBOW can have critical values of radius-to-thickness ratios (R/H) below which RAINBOW exhibits axisymmetric room-temperature shapes. For values of R/H greater than the critical value, disk-style RAINBOW exhibits two stable near-cylindrical shapes and one unstable axisymmetric shape. Moreover, it is found that for the set of material properties used in this study, the optimal reduced layer thickness would be at 55%, since the maximum change in curvature is achieved under the application of an electric field, while the relationship between the change in curvatures and the electric field is kept very close to being linear. In general, good agreement is found for comparisons between the predicted and manufactured shapes of RAINBOW. A multi-step thermoelastic analysis is developed to model the addition of the fiber-reinforced composite layer to RAINBOW to make GRAPHBOW. Results obtained for rectangular RAINBOW indicate that if the bifurcation temperature in the temperature-curvature relation is lower than the composite cure temperature, then a unique stable GRAPHBOW shape can be obtained. If the RAINBOW bifurcation temperature is above the composite cure temperature, multiple room-temperature GRAPHBOW shapes are obtained and saddle-node bifurcations can be encountered during the cooling to room temperature of [0°/RAINBOW], [RAINBOW/0o], and [0o2/RAINBOW]. Rectangular [RAINBOW/0o/90o] seems to be less likely to encounter saddle-node bifurcations. Furthermore, the unstable spherical RAINBOW configuration is converted to a stable near-cylindrical configuration. For the case considered of disk-style GRAPHBOW, three stable room-temperature shapes are obtained and the unstable axisymmetric RAINBOW configuration is also converted to a stable near-cylindrical configuration. For both rectangular and disk-style GRAPHBOW, the relationship between the major curvature and the electric field is shown to be very close to being linear. This characteristic would aid any dynamic analysis of RAINBOW or GRAPHBOW. / Ph. D.

This research discusses the mechanical characteristics of laminated piezoelectric actuators that are manufactured at an elevated temperature, to cure the adhesive bonding the layers together, or to cure the layers made of polymeric composite material, and then cooled to a service temperature. Mainly discussed are actuators that are composed of layers of passive materials and a layer of piezoelectric material. THUNDER (THin layer UNimorph ferroelectric DrivER and sensor) and LIPCA (LIghtweight Piezo-composite Curved Actuator) actuators, which consist of layers of metal, adhesive and piezoelectric material, and carbon-epoxy, glass-epoxy and piezoelectric material, respectively, are studied and investigated in detail to understand the thermal effects due to the elevated manufacturing temperature. Owing to the large out-of-plane deformations of the THUNDER actuators as a result of cooling to the service temperature, inclusion of geometric nonlinearities in the kinematic relations is taken into consideration for prediction of the thermally-induced deformations and residual stresses. The deformations and residual stresses are predicted by using a 23-term Rayleigh-Ritz approach and more rigorous, time-consuming, finite-element analyses performed with ABAQUS. The thermally-induced deformations of THUNDER actuators can result in multiple room-temperature manufactured shapes, whereas those of LIPCA actuators (LIPCA-C1 and LIPCA-C2) exhibit single room-temperature manufactured shape. Actuation responses of these actuators caused by a quasi-static electric field applied to the piezoelectric layer are also studied with the Rayleigh-Ritz approach. It is shown that geometrical nonlinearities play an important role in the actuation responses, and these nonlinearities can be controlled by the choice of actuator geometry and the materials in the passive layers. In addition, blocking forces representing load-carrying capability of THUNDER and LIPCA actuators are determined. Support conditions and again geometrical nonlinearities are vital factor in load-resisting performances. Amongst the actuators considered, the actuated deflection and blocking forces are compared. Finally, based on the outcome of this study, new criteria for designing a new type of laminated piezoelectric actuators with improvement of performance characteristics are proposed. / Ph. D.

Trial and error and the use of highly time-consuming methods are often necessary for modeling, simulating and characterizing nonlinear dynamical systems. However, for the rather common special case when a nonlinear system has linear relations between many of its degrees of freedom there are particularly interesting opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient methods for the theoretical and experimental study of mechanical systems that include significant zero-memory or hysteretic nonlinearities related to only small parts of the whole system. The basic idea is to take advantage of the fact that most of the system is linear and to use much of the linear theories behind forced response simulations. This is made possible by modeling the nonlinearities as external forces acting on the underlying linear system. The result is very fast simulation routines where the model is based on the residues and poles of the underlying linear system. These residues and poles can be obtained analytically, from finite element models or from experimental measurements, making these forced response routines very versatile. Using this approach, a complete nonlinear model contains both linear and nonlinear parts. Thus, it is also important to have robust and accurate methods for estimating both the linear and nonlinear system parameters from experimental data. The results of this work include robust and user-friendly routines based on sinusoidal and random noise excitation signals for characterization and description of nonlinearities from experimental measurements. These routines are used to create models of the studied systems. When combined with efficient simulation routines, complete tools are created which are both versatile and computationally inexpensive. The developed methods have been tested both by simulations and with experimental test rigs with promising results. This indicates that they are useful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.

For repeated transient solutions of geometrically nonlinear structures the numerical effort often poses a major obstacle. Thus, the introduction of a reduced order model, which takes the nonlinear effects into account and accelerates the calculations considerably, is often necessary.This work yields a method that allows for rapid, accurate and parameterisable solutions by means of a reduced model of the original structure. The structure is discretised and its dynamic equilibrium described by a matrix equation. The projection on a reduced basis is introduced to obtain the reduced model. A comprehensive numerical study on several common reduced bases shows that the simple introduction of a constant basis is not sufficient to account for the nonlinear behaviour. Three requirements for an rapid, accurate and parameterisable solution are derived. The solution algorithm has to take into account the nonlinear evolution of the solution, the solution has to be independent of the nonlinear finite element terms and the basis has to be adapted to external parameters.Three approaches are provided, each responding to one requirement. These approaches are assembled to the integrated method. The approaches are the update and augmentation of the basis, the polynomial formulation of the nonlinear terms and the interpolation of the basis. A Newmark-type time-marching algorithm provides the frame of the integrated method. The application of the integrated method on test-cases with geometrically nonlinear finite elements confirms that this method leads to the initial aim of a rapid, accurate and parameterisable transient solution.

Pour les solutions transitoires répétées des structures géométriquement nonlinéaires l’effort numérique présente souvent une contrainte importante. Ainsi, l’introduction d’un modèle d’ordre réduit, qui prend en compte les effets nonlinéaires et qui accélère considérablement les calculs, s’avère souvent nécessaire.Ce travail aboutit à une méthode qui permet des solutions transitoires accélérées, fidèles et paramétrables, à travers d’un modèle réduit de la structure initiale. La structure est discrétisée et son équilibre dynamique décrit par une équation matricielle. La projection sur une base réduite est introduite afin d’obtenir un modèle réduit. Une étude numérique complète sur plusieurs bases communes démontre que la simple introduction d’une base constante ne suffit pas pour prendre en compte le comportement nonlinéaire. Trois exigences sont déduites pour une solution transitoire accélérée, fidèle et paramétrable. L’algorithme de solution doit permettre un suivi de l’évolution nonlinéaire de la solution transitoire, la solution doit être autonome des termes nonlinéaires en éléments finis et la base doit être adaptée à des paramètres externes.Trois approches sont mises en place, chacune répondant à une exigence. Ces approches sont assemblées dans la méthode intégrée. Les approches sont la mise-à-jour et augmentation de la base , la formulation polynomiale des termes nonlinéaires et l’interpolation de la base. Un algorithme de type Newmark forme le cadre de la méthode intégrée. L’application de la méthode intégrée sur des cas test en élément finis géométriquement nonlinéaires confirme qu’elle répond au but initial d’obtenir des solutions transitoires accélérées, fidèles et paramétrables. / For repeated transient solutions of geometrically nonlinear structures the numerical effort often poses a major obstacle. Thus, the introduction of a reduced order model, which takes the nonlinear effects into account and accelerates the calculations considerably, is often necessary.This work yields a method that allows for rapid, accurate and parameterisable solutions by means of a reduced model of the original structure. The structure is discretised and its dynamic equilibrium described by a matrix equation. The projection on a reduced basis is introduced to obtain the reduced model. A comprehensive numerical study on several common reduced bases shows that the simple introduction of a constant basis is not sufficient to account for the nonlinear behaviour. Three requirements for an rapid, accurate and parameterisable solution are derived. The solution algorithm has to take into account the nonlinear evolution of the solution, the solution has to be independent of the nonlinear finite element terms and the basis has to be adapted to external parameters.Three approaches are provided, each responding to one requirement. These approaches are assembled to the integrated method. The approaches are the update and augmentation of the basis, the polynomial formulation of the nonlinear terms and the interpolation of the basis. A Newmark-type time-marching algorithm provides the frame of the integrated method. The application of the integrated method on test-cases with geometrically nonlinear finite elements confirms that this method leads to the initial aim of a rapid, accurate and parameterisable transient solution.

The aim of the current dissertation is the development of finite element models capable of predicting the damping and the damped structural dynamic response of laminated composite blades and beams. The present thesis is divided into two main parts, of which the first one studies the material coupling effect on the static and modal characteristics of composite structures. New damping coupling terms are formulated and incorporated into a linear beam finite element to better capture the composite material and structural coupling effects. The second part describes the theoretical framework for predicting the nonlinear damping and damped vibration of laminated composite structures due to large in-plane tensile and compressive forces. A nonlinear beam finite element for composite strips is developed capable of capturing the effects of geometric nonlinearity on the damping of composite laminates. The damping mechanics consider a strain based Kelvin viscoelastic model and Green-Lagrange nonlinear strain expressions, which introduce geometric nonlinearity into the formulation. Incorporation of first-order shear deformation theory into the equations of motion provides the linear and new nonlinear cross-section stiffness and damping terms. Within each element, the stain field is approximated by linear interpolation shape functions. An incremental-iterative methodology is formulated into the finite element solver, based on the Newton-Raphson technique in order to obtain the system solution at each iteration, till the final convergence is achieved. For the sake of completeness, a series of experimental measurements were carried out for the composite strip, subject to tensile and buckling loads. Correlations with theoretical predictions gave credence to the ability of the nonlinear finite element to predict damping of composite structures undergoing large displacements and rotations in the nonlinear regime. The finite element was further extended to include the nonlinear analysis of large-scale hollow composite structures. New first- and second-order stiffness and damping terms were developed and incorporated into an updated nonlinear beam finite element, capable of capturing the effect of rotational stresses on the static and modal characteristics of composite beams and blades. / Σκοπός της παρούσας διδακτορικής διατριβής με τίτλο: "Ανάπτυξη Μη-Γραμμικού Προτύπου Πεπερασμένου Στοιχείου με Απόσβεση για τη Στατική και Δυναμική Ανάλυση Πτερυγίων Ανεμογεννητριών" είναι η ανάπτυξη προτύπων πεπερασμένων στοιχείων με απόσβεση ικανών να προβλέπουν τη στατική και δυναμική απόκριση δοκών και πτερυγίων από σύνθετα υλικά. Η εργασία επικεντρώνεται σε δύο κύριες κατευθύνσεις, που αφορούν τόσο την εισαγωγή νέων όρων στο μητρώο απόσβεσης ενός πεπερασμένου στοιχείου δοκού, όσο και την ανάπτυξη ενός μη-γραμμικού κώδικα πεπερασμένου στοιχείου για τη μελέτη της μη-γραμμικής συμπεριφοράς δοκών και πτερυγίων από σύνθετα υλικά που υπόκεινται σε μεγάλες μετατοπίσεις και περιστροφές. Στο πρώτο μέρος της διατριβής μελετάται η επίδραση της σύζευξης, λόγω της ανισοτροπίας του σύνθετου υλικού, τόσο στη στατική απόκριση όσο και στα μορφικά χαρακτηριστικά κατασκευών από σύνθετα υλικά, διαφόρων διατομών και γεωμετριών. Διατυπώνονται νέοι όροι απόσβεσης που εκφράζουν την εν λόγω σύζευξη και οι οποίοι καθιστούν το γραμμικό πεπερασμένο στοιχείο δοκού πιο πλήρες στην επίλυση προβλημάτων όπου η σύζευξη υλικού επηρεάζει τη συμπεριφορά της κατασκευής. Στο δεύτερο και πλέον σημαντικό μέρος της παρούσας διατριβής αρχικά περιγράφεται το θεωρητικό υπόβαθρο για την πρόβλεψη της μη-γραμμικής δυναμικής απόσβεσης λεπτών δοκών κατασκευασμένα από σύνθετα υλικά οι οποίες υπόκεινται σε μεγάλα συν-επίπεδα εφελκυστικά φορτία ή φορτία λυγισμού. Αναπτύσσεται νέο πεπερασμένο στοιχείο ικανό να περιγράψει την επίδραση της γεωμετρικής μη-γραμμικότητας στην απόσβεση και τη δυσκαμψία της δοκού. Εφαλτήριο για την ανάπτυξη αυτής της μεθοδολογίας ήταν η ανάγκη της πρόβλεψης της δυναμικής απόσβεσης σε κατασκευές από σύνθετα υλικά με πιο πολύπλοκη και εύκαμπτη γεωμετρία, όπως αυτή των πτερυγίων ανεμογεννητριών. Η ανάπτυξη του μη-γραμμικού πεπερασμένου στοιχείου ξεκινά από το επίπεδο της στρώσης του υλικού, όπου διατυπώνονται οι καταστατικές εξισώσεις θεωρώντας το ιξωδοελαστικό πρότυπο του Kelvin για το υλικό της κατασκευής. Στη συνέχεια εισάγονται οι Green-Lagrange εξισώσεις συμβιβαστού οι οποίες εκφράζουν τη γεωμετρική μη-γραμμικότητα καθώς και οι εξισώσεις κίνησης. Σε επίπεδο διατομής, οι κινηματικές υποθέσεις βασίζονται στις παραδοχές της διατμητικής θεωρίας δοκού πρώτης τάξης. Η πρόβλεψη της μη-γραμμικής απόκρισης μιας κατασκευής από σύνθετα υλικά επιτυγχάνεται μέσω της προσομοίωσης της με έναν επαρκή αριθμό πεπερασμένων στοιχείων. Στο εσωτερικό κάθε στοιχείου οι παραμορφώσεις προσεγγίζονται από γραμμικές συναρτήσεις μορφής, οι οποίες οδηγούν στη μητρωική μορφή των μη-γραμμικών εξισώσεων του συστήματος. Λόγω του γεγονότος ότι οι εξισώσεις αυτές εξαρτώνται από τη λύση, δεν μπορούν να λυθούν απευθείας κάτι που καθιστά αναγκαία τη χρήση μιας σταδιακής-επαναληπτικής τεχνικής. Στην παρούσα διατριβή εισάγεται στο λύτη του μη-γραμμικού κώδικα η Newton-Raphson τεχνική. Το επόμενο βήμα αφορά τη σύνθεση των ολικών δομικών μητρών του συστήματος και την εφαρμογή των συνοριακών συνθηκών. Σε κάθε επανάληψη λαμβάνει χώρα η επίλυση των γραμμικοποιημένων εξισώσεων και ο υπολογισμός των πραγματικών και εφαπτομενικών μη-γραμμικών μητρώων δυσκαμψίας και απόσβεσης της κατασκευής, τα οποία τελικώς επιλύονται με τη μέθοδο της αριθμητικής ολοκλήρωσης κατά Gauss. Το πεπερασμένο στοιχείο δοκού εξελίχθηκε περαιτέρω ώστε να συμπεριλάβει τη μη-γραμμική ανάλυση μεγάλων λεπτότοιχων κατασκευών από σύνθετα υλικά, όπως αυτά των πτερυγίων ελικοπτέρων και ανεμογεννητριών. Η εισαγωγή της πλήρους έκφρασης της αξονικής μη-γραμμικής Green-Lagrange παραμόρφωσης στη διατύπωση των κινηματικών υποθέσεων οδηγεί στην πλήρη έκφραση των πραγματικών και εφαπτομενικών δομικών μητρών της κατασκευής. Οι νέοι μη-γραμμικοί όροι δυσκαμψίας και απόσβεσης πρώτης και δεύτερης τάξης μπορούν να περιγράψουν την επίδραση των εσωτερικών εφελκυστικών τάσεων στα μορφικά χαρακτηριστικά δοκών και πτερυγίων. Το μη-γραμμικό πεπερασμένο στοιχείο είναι ικανό να χαρακτηρίσει τη στατική συμπεριφορά και την αποσβενυμένη ταλάντωση δοκών από σύνθετα υλικά. Η επαλήθευση του μη-γραμμικού κώδικα πραγματοποιήθηκε μέσω μιας σειράς πειραματικών μετρήσεων που αφορούσαν τη μέτρηση της φυσικής συχνότητας και της μορφικής απόσβεσης σε λεπτές δοκούς από σύνθετα υλικά τόσο σε εφελκυσμό όσο και σε συνθήκες λυγισμού. Τα πειραματικά αποτελέσματα έρχονται σε πολύ καλή συμφωνία με τις θεωρητικές προβλέψεις του κώδικα κάτι που εξασφαλίζει την αξιοπιστία του μη-γραμμικού πεπερασμένου στοιχείου.