Nonlinear superharmonic resonance analysis of a nonlocal beam on a fractional visco-Pasternak foundation

This paper investigates the dynamic behavior of a geometrically nonlinear nanobeam resting on the fractional visco-Pasternak foundation and subjected to dynamic axial and transverse loads. The fractional-order governing equation of the system is derived and then discretized by using the single-mode Galerkin discretization. Corresponding forced Mathieu-Duffing equation is solved by using the perturbation multiple time scales method for the weak nonlinearity and by the semi-numerical incremental harmonic balance method for the strongly nonlinear case. A comparison of the results from two methods is performed in the validation study for the weakly nonlinear case and a fine agreement is achieved. A parametric study is performed and the advantages and deficiencies of each method are discussed for order two and three superharmonic resonance conditions. The results demonstrate a significant influence of the fractional-order damping of the visco-Pasternak foundation as well as the nonlocal parameter and external excitation load on the frequency response of the system. The proposed methodology can be used in pre-design procedures of novel energy harvesting and sensor devices at small scales exhibiting nonlinear dynamic behavior

Analysis and design of large space structures with nonlinear joints

Issued as Final report, Project no. E-25-62

An Investigation into Dynamic Stability of Waterborne Aircraft on Take-off and Landing

This research contributes to the knowledge of dynamic stability of waterborne aircraft and ground effect phenomenon. Hereto an analytical and computational study has been performed during which the motion of waterborne aircraft in take-off and landing is predicted. An analytical tool that can be used to predict the nonlinear heaving and pitching motions of seaplanes is presented. First, the heaving and pitching equations of motion are presented in their general Lagrangian form. Then, the equations are simplified to a form of nonlinear equations known as the forced Duffing equations with cubic nonlinearity. The system of motion is assumed to be driven by a sinusoidal head sea wave. The equations are then solved using the Poincare-Lindstedt perturbation method. The analytical solution is verified with CFD simulations performed on Ansys Fluent and AQWA. The solution is used to extend Savitsky’s method to predict porpoising which is a form of dynamic instability found in high-speed boats and seaplanes. The results of the analytical tool are in very good agreement with the results obtained from Fluent and AQWA. However, as the motion is assumed to be 2D in Fluent, heaving amplitude is slightly over predicted. Moreover, the frequency of oscillations of the 2D simulations is found to be unsteady. The unsteadiness in frequency increases with the increase of the length of the hull. Nevertheless, the amplitude of the pitch motion is slightly less than the amplitude predicted analytically. The discrepancy in the results is due to the characteristics of the 2D simulations that assumes that sea water will only pass underneath the hull which will make the buoyancy force greater as less damping is experienced. This is also a consequence of the fact that parameters within the analytical model of heave and pitch are calculated using a strip theory which considers only hydrodynamic effects, while Fluent also incorporate aerodynamic contributions. Similarly, AQWA is a 3D platform that only takes in consideration hydrodynamic effects. Hence, the results of AQWA are slightly less in amplitude than that predicted analytically. In addition, it was found that the frequency of oscillations obtained using AQWA increases with time while in the analytical approach, the frequency of oscillations can only be assumed to be constant for the whole period of motion. The increment in the oscillations indicates that porpoising is taking place. Nevertheless, it was found that heaving terms control the amplitude of motion and pitching terms control frequency of oscillations. The pitching nonlinear term has an effect on the amplitude of motion but not significant. Finally, the analytical method of Savitsky that is used to predict the porpoising stability limit is extended to find the porpoising limit for a wider range of pitch angles. In addition, the porpoising limit is predicted for a planing hull that is moving under the effect of head sea waves. When the seaplane is moving through head sea waves at a fixed pitch angle, porpoising takes place at a lower speed than what Savitsky has predicted

Tuning of the Active Hair Bundle

The organs of the inner ear rely upon a population of several thousand sensory hair cells to amplify and transduce acoustic, seismic, and kinesthetic signals. Each hair cell detects mechanical disturbances by means of its hair bundle, a motile organelle consisting of actin-filled, villous projections (called stereocilia) endowed with assemblies (called adaptation motors) of mechano-sensitive ion channels and myosin molecules that power both spontaneous and evoked movements. Active hair-bundle motility serves two functions: it mechanically amplifies sensory stimuli; and it regulates their transduction into electrical signals that drive the hair-cell synapse. To characterize these two functions, we consider here a model of the mechanical and electrical dynamics of the hair bundle of the bullfrog sacculus. Under simplifying assumptions, we reduce this model of a muscle fiber, and outline a procedure for estimating its parameters from experiment. We delineate the bifurcation structure of this simplified model, and analyse by perturbation methods its behavior in various dynamical regimes, notably in the relaxation-oscillation regime that displays prominently the hair bundle\u27s active process; and in the near-Hopf-bifurcation regime at which auditory hair cells are thought to operate in vivo. We find close similarities between the dynamics of the active hair bundle and those of simplified models of a spiking neuron. In light of this analysis, we offer an account of the biophysical mechanisms underlying the spontaneous oscillations, frequency specificity, nonlinear gain, and self-tuning predicted for auditory hair bundles poised near a Hopf bifurcation

Spline based controller for nonlinear systems

The objective of this thesis is to apply spline theory to implement controllers for nonlinear systems. Two different systems, forced Duffing oscillators and power systems, are investigated. The spline method is used to mimic the controller which drives a state of the Duffing system toward a desired path. The spline-based nonlinear controller has piecewise polynomial segments with different order of polynomials on each segment. Controller efforts for different order of polynomial interpolants and power spectral densities of the controller signals are compared with the exact feedback linearizaton method.;The first objective for power systems is to design nonlinear excitation controllers for a multi-machine power system using Direct Feedback Linearization. The designed controllers, whose parameters are obtained, require the internal variables of the machines. These variables are verified by using a proposed internal variable identifying algorithm. The objective is to design nonlinear excitation controllers for power system stability enhancement. Spline techniques are used to approximate the nonlinear controllers obtained through feedback linearization by piecewise polynomials while enhancing the stability of the system

Dynamic Characterization of Inertant Meta-Structures

In establishing the traditional analogies between mechanical and electrical networks, lack of preservation of topology while transitioning from a mechanical to its equivalent electrical network was rectified by so-called mobility or force-current analogy. One drawback was that the mass, which is the equivalent of the grounded capacitor, cannot represent a two-terminal device when described in an inertial frame. This was remedied by a two-terminal mechanical device termed the 'inerter' postulated at the turn of this century. The inerter is a mechanical element in which applied force is proportional to the relative acceleration across its terminals. The proportionality constant is termed 'inertance'. Practically, the inerter is realized by storing energy using a flywheel and can deliver a dynamic mass presence a few orders of magnitude greater than the device mass leading to considerable interest in inerters in recent years. In this study, inspired by acoustic metamaterials (AM), the dynamic characteristics of 'meta-structures' employing inerters are explored. Firstly, improved analytical models incorporating component inertias and sizing and parametric studies for two prominent embodiments of the inerter viz. the rack-and-pinion and the ball-screw inerter are considered. The dependence of specific inertance (ratio of inertance to static mass) on key parameters are brought out through simulations. A prototype rack-and-pinion inerter with specific inertance of 90 was designed, fabricated and tested under low-rate excitations. Measured specific inertance was found to display an exponential decline with increase in excitation frequency. Using a phase-matching procedure, estimation of internal stiffness and damping in the prototype reveals influences of the excitation frequency and ultra-low frequency meandering effects. Further, motivated by challenges in miniaturizing rotary components for microscale inerters, a potential kinematically-simpler structure based on the von Mises truss is investigated. Its nonlinear equation of motion is derived using Hamilton's principle. Potential inertant mechanisms for this structure under harmonic inputs are probed using simulations. Finally, while studies on inherent nonlinearities in inerters are more widely available, those concerning the use of intentionally nonlinear inerters are scarce. In this context, the mechanical wave manipulation characteristics of Nonlinear Inertant Acoustic Metamaterials (NLIAM) are studied. Based on notional concepts for inertant devices, frequency and acceleration-dependent nonlinear inertant models are advanced. Dispersion characteristics of NLIAM with frequency-dependent inverse square law (ISL) and power law (PL) inertance are examined. While a tuned ISL model ensures existence of band gap over almost the entire bandwidth of interest, its limiting inertances are challenging to realize in practice. A potentially more practical PL approximation is proposed and shown to have a widening of the band gap by more than 100% towards frequencies below the lower bound of the band gap for the AM with frequency invariant inertance. Further, drawing inspiration from the Duffing-type stiffness, first order dispersion corrections are obtained for an NLIAM with acceleration-dependent inertance using a perturbation approach. Dispersion curves shifts are found to enable this NLIAM to act as a passive adaptive filter for mechanical waves based solely on excitation amplitude. Practical manifestations of NLIAM could help realize extraordinary wave manipulation capabilities especially suitable for low frequency structural dynamic applications.Mechanical and Aerospace Engineerin

3D-printing technology applied to the development of bio-inspired functional acoustic systems

Examples of bio-inspired technology can be found almost everywhere in society: robots with specific capabilities, materials with unique physical and chemical properties, aerodynamic systems, and architectonic structures are a few examples of taking profit of evolution-driven processes to solve common engineering problems. One field of research taking advantage of bio-inspiration is that of acoustical engineering, aiming to find solutions to problems arising from the miniaturisation of microphones and loudspeakers. Studying the auditory organs of insects to seek inspiration for new design structures is one of the best ways to solve such an important problem. Another discipline of science that has experienced a research boom is that of materials science, as development of new materials has attracted the attention of researchers. In addition, three-dimensional (3D) printers have contributed to further development in materials science making the production process more efficient. The aim of this research is to bring these fields of science together to develop novel bioinspired, polymer-based sensors presenting functional specific acoustic properties after 3D-printing. While the study of complex biological hearing systems provides inspiration to develop sensors featuring specific properties, the use of polymer-based materials allows the customization of the manufacturing process, as the produced parts adapt to the desired needs. In this thesis one such insect auditory system that has been thoroughly studied is that of the desert locust Schistocerca gregaria as it presents a simple structure that allows for acoustic frequency selectivity and displays nonlinear acoustic phenomena. Prior to the development of a bio-inspired system, a mathematical description of the mechanical response of such a structure is presented. Furthermore, the physical behaviours measured on the locust tympanal membrane have been studied using finite element analysis. The 3D-printed functional sensors have been used to determine the degree of accuracy between experimental and theoretical results.Examples of bio-inspired technology can be found almost everywhere in society: robots with specific capabilities, materials with unique physical and chemical properties, aerodynamic systems, and architectonic structures are a few examples of taking profit of evolution-driven processes to solve common engineering problems. One field of research taking advantage of bio-inspiration is that of acoustical engineering, aiming to find solutions to problems arising from the miniaturisation of microphones and loudspeakers. Studying the auditory organs of insects to seek inspiration for new design structures is one of the best ways to solve such an important problem. Another discipline of science that has experienced a research boom is that of materials science, as development of new materials has attracted the attention of researchers. In addition, three-dimensional (3D) printers have contributed to further development in materials science making the production process more efficient. The aim of this research is to bring these fields of science together to develop novel bioinspired, polymer-based sensors presenting functional specific acoustic properties after 3D-printing. While the study of complex biological hearing systems provides inspiration to develop sensors featuring specific properties, the use of polymer-based materials allows the customization of the manufacturing process, as the produced parts adapt to the desired needs. In this thesis one such insect auditory system that has been thoroughly studied is that of the desert locust Schistocerca gregaria as it presents a simple structure that allows for acoustic frequency selectivity and displays nonlinear acoustic phenomena. Prior to the development of a bio-inspired system, a mathematical description of the mechanical response of such a structure is presented. Furthermore, the physical behaviours measured on the locust tympanal membrane have been studied using finite element analysis. The 3D-printed functional sensors have been used to determine the degree of accuracy between experimental and theoretical results

Multiphysics modelling and experimental validation of microelectromechanical resonator dynamics

The modelling of microelectromechanical systems provides a very challenging task in microsystems engineering. This field of research is inherently multiphysics of nature, since different physical phenomena are tightly intertwined at microscale. Typically, up to four different physical domains are usually considered in the analysis of microsystems: mechanical, electrical, thermal and fluidic. For each of these separate domains, well-established modelling and analysis techniques are available. However, one of the main challenges in the field of microsystems engineering is to connect models for the behavior of the device in each of these domains to equivalent lumped or reduced-order models without making unacceptably inaccurate assumptions and simplifications and to couple these domains correctly and efficiently. Such a so-called multiphysics modelling framework is very important for simulation of microdevices, since fast and accurate computational prototyping may greatly shorten the design cycle and thus the time-to-market of new products. This research will focus on a specific class of microsystems: microelectromechanical resonators. MEMS resonators provide a promising alternative for quartz crystals in time reference oscillators, due to their small size and on-chip integrability. However, because of their small size, they have to be driven into nonlinear regimes in order to store enough energy for obtaining an acceptable signal-to-noise ratio in the oscillator. Since these resonators are to be used as a frequency reference in the oscillator circuits, their steady-state (nonlinear) dynamic vibration behaviour is of special interest. A heuristic modelling approach is investigated for two different MEMS resonators, a clamped-clamped beam resonator and a dog-bone resonator. For the clamped-clamped beam resonator, the simulations with the proposed model shows a good agreement with experimental results, but the model is limited in its predictive capabilities. For the dogbone resonator, the proposed heuristic modelling approach does not lead to a match between simulations and experiments. Shortcomings of the heuristic modelling approach serve as a motivation for a first-principles based approach. The main objective of this research is to derive a multiphysics modelling framework for MEMS resonators that is based on first-principles formulations. The framework is intended for fast and accurate simulation of the steady-state nonlinear dynamic behaviour of MEMS resonators. Moreover, the proposed approach is validated by means of experiments. Although the multiphysics modelling framework is proposed for MEMS resonators, it is not restricted to this application field within microsystems engineering. Other fields, such as (resonant) sensors, switches and variable capacitors, allow for a similar modelling approach. In the proposed framework, themechanical, electrical and thermal domains are included. Since the resonators considered are operated in vacuum, the fluidic domain (squeeze film damping) is not included. Starting from a first-principles description, founded on partial differential equations (PDEs), characteristic nonlinear effects from each of the included domains are incorporated. Both flexural and bulk resonators can be considered. Next, Galerkin discretization of the coupled PDEs takes place, to construct reduced-order models while retaining the nonlinear effects. The multiphysics model consists of the combined reduced-order models from the different domains. Designated numerical tools are used to solve for the steady-state nonlinear dynamic behaviour of the combined model. The proposed semi-analytical (i.e. analytical-numerical) multiphysics modeling framework is illustrated for a full case study of an electrostatically actuated single-crystal silicon clamped-clamped beam MEMS resonator. By means of the modelling framework, multiphysics models of varying complexity have been derived for this resonator, including effects like electrostatic actuation, fringing fields, shear deformation, rotary inertia, thermoelastic damping and nonlinear material behaviour. The first-principles based approach allows for addressing the relevance of individual effects in a straightforward way, such that the models can be used as a (pre-)design tool for dynamic response analysis. The method can be considered complementary to conventional finite element simulations. The multiphysics model for the clamped-clamped beam resonator is validated by means of experiments. A good match between the simulations and experiments is obtained, thereby giving confidence in the proposed modelling framework. Finally, next to themodelling approach for MEMS resonators, a technique for using these nonlinear resonators in an oscillator circuit setting is presented. This approach, called phase feedback, allows for operation of the resonator in its nonlinear regime. The closedloop technique enables control of both the frequency of oscillation and the output power of the signal. Additionally, optimal operation points for oscillator circuits incorporating a nonlinear resonator can be defined

Experimental and Numerical Studies on the Structural Dynamics of Flapping Beams

The nonlinear structural dynamics of slender cantilever beams in flapping motion is studied through experiments, numerical simulations, and perturbation analyses. A flapping mechanism which imparts a periodic flapping motion of certain amplitude and frequency on the clamped boundary of the appended cantilever beam is constructed. Centimeter-size thin aluminum beams are tested at two amplitudes and frequencies up to, and slightly above, the first bending mode to collect beam tip displacement and surface bending strain data. Experimental data analyzed in time and frequency domains reveal a planar, single stable (for a given flapping amplitude-frequency combination) periodic beam response with superharmonic resonance peaks. Numerical simulations performed with a nonlinear beam finite element corroborate the experiments in general with the exception of the resonance regions where they overpredict the experiments. The discrepancy is mainly attributed to the use of a linear viscous damping model in the simulations. Nonlinear response dynamics predicted by the simulations include symmetric periodic, asymmetric periodic, quasi-periodic, and aperiodic motions. To investigate the above-mentioned discrepancy between experiment and simulation, linear and nonlinear damping force models of different functional forms are incorporated into a nonlinear inextensible beam theory. The mathematical model is solved for periodic response by using a combination of Galerkin and a time-spectral numerical scheme; two reduced order methods which, along with the choice of the inextensible beam model, facilitate parametric study and analytical analysis. Additional experiments are conducted in reduced air pressure to isolate the air damping from the material damping. The frequency response curves obtained with different damping models reveal that, when compared to the linear viscous damping, the nonlinear external damping models better represent the experimental damping forces in the regions of superharmonic and primary resonances. The effect of different damping models on the stability of the periodic solutions are investigated using the Floquet theory. The mathematical models with nonlinear damping yield stable periodic solutions which is in accord with the experimental observation. The effect of excitation and damping parameters on the steady-state superharmonic and primary resonance responses of the flapping beam is further investigated through perturbation analyses. The resonance solutions of the spatially-discretized equation of motion (via 1-mode Galerkin approximation of the inextensible beam model), which involves both quadratic and cubic nonlinear terms, are constructed as first-order uniform asymptotic expansions via the method of multiple time scales. The critical excitation amplitudes leading to bistable solutions are identified and are found to be consistent with the experimental and numerical results. The approximate analytical results indicate that a second harmonic is required in the boundary actuation spectra in order for a second order superharmonic response to exist. The perturbation solutions are compared with numerical time-spectral solutions for different flapping amplitudes. The first-order perturbation solution is determined to be in very good agreement with the numerical solution up to 5° while above this angle differences in the two solutions develop, which are attributed to phase estimation accuracy