Dynamic stability of a nonlinear multiple-nanobeam system

We use the incremental harmonic balance (IHB) method to analyse the dynamic stability problem of a nonlinear multiple-nanobeam system (MNBS) within the framework of Eringen’s nonlocal elasticity theory. The nonlinear dynamic system under consideration includes MNBS embedded in a viscoelastic medium as clamped chain system, where every nanobeam in the system is subjected to time-dependent axial loads. By assuming the von Karman type of geometric nonlinearity, a system of m nonlinear partial differential equations of motion is derived based on the Euler–Bernoulli beam theory and D’ Alembert’s principle. All nanobeams in MNBS are considered with simply supported boundary conditions. Semi-analytical solutions for time response functions of the nonlinear MNBS are obtained by using the single-mode Galerkin discretization and IHB method, which are then validated by using the numerical integration method. Moreover, Floquet theory is employed to determine the stability of obtained periodic solutions for different configurations of the nonlinear MNBS. Using the IHB method, we obtain an incremental relationship with the frequency and amplitude of time-varying axial load, which defines stability boundaries. Numerical examples show the effects of different physical and material parameters such as the nonlocal parameter, stiffness of viscoelastic medium and number of nanobeams on Floquet multipliers, instability regions and nonlinear amplitude–frequency response curves of MNBS. The presented results can be useful as a first step in the study and design of complex micro/nanoelectromechanical systems

Finite Element Modal Formulation for Panel Flutter at Hypersonic Speeds and Elevated Temperatures

A finite element time domain modal formulation for analyzing flutter behavior of aircraft surface panels in hypersonic airflow has been developed and presented for the first time. Von Karman large deflection plate theory is used for description of the structural nonlinearity and third order piston theory is employed to account for the aerodynamic nonlinearity. The thermal loadings of uniformly distributed temperature and temperature gradients across the panel thickness are incorporated into the finite element formulation. By applying the modal reduction technique, the number of governing equations of motion is reduced dramatically so that the computational time of direct numerical integration is dropped significantly. All possible types of panel behavior, including flat, buckled but dynamically stable, limit cycle oscillation (LCO), periodic motion, and chaotic motion can be observed and analyzed. As examples of the applications of the proposed methodology, flutter responses of isotropic, specially orthotropic and laminated composite panels are investigated. Special emphasis is put on the boundary between LCO and chaos, as well as the routes to chaos. A systematic mode filtering procedure that helps mode selection without specific knowledge of the complex mode shapes is presented and illustrated. Influences of aerodynamic parameters, including aerodynamic damping and Mach number, on the panel flutter responses are studied. The importance of nonlinear aerodynamic terms is examined in detail. The supporting conditions and panel aspect ratio on the onset condition of chaos are also investigated as an illustration of optimization among different design options. Several mathematical tools, including the time history, phase plane plot, Poincaré map, and bifurcation diagram are employed in the chaos study. The largest Lyapunov exponent is also evaluated to assist in detection of chaos. It is found that at low or moderately high nondimensional dynamic pressures, the fluttering panel typically takes a period-doubling route to evolve into chaos, whereas at high nondimensional dynamic pressure, the route to chaos generally involves bursts of chaos and rejuvenations of periodic motions. Various bifurcation behaviors, such as the Hopf bifurcation, pitchfork bifurcation, and transcritical bifurcation, are observed. On the basis of the successful applications presented, the proposed finite element time domain modal formulation and the mode filtering procedure have proven to be an efficient and practical design tool for designers of hypersonic vehicle

Predicting incipent instabilities and bifurcations of nonlinear dynamical systems modelling compliant off-shore structures

For engineers, the two most important aspects of dynamical analysis are high amplitude resonance vibrations and structural stability, i.e. whether a steady state solution is stable under small perturbations. For the former case, a novel and simple method based on Poincare mapping technique has been devised to predict an imminent flip bifurcation. This bifurcation represents the beginning of the second order subharmonic response. For the latter case, we discovered that while classical quantitative analytical techniques work well in establishing the 'local' structural stability of a steady state solution, the global geometric structure of the catchment region can alter dramatically such that even an initial condition close to the steady state can diverge from it rather than being attracted. This phenomenon known as fractal basin boundary occurs when the invariant manifolds of the saddle separating the steady state solution from any remote attractor cross. The critical point in which the invariant manifolds just touch can be accurately predict by the Melinkov's method. Because of the complicated interwoven nature of the invariant manifolds, it is called a tangle. If the invariant manifolds are originated from the same saddle, the crossing is known as a homoclinic tangle, if originated from different saddle, a heteroclinic tangle. The critical point is then known as homoclinic or heteroclinic tangency. Tangles are also intimately related to chaotic behaviour. The creation and destruction of chaotic attractors have been observed through a series of homoclinic and heteroclinic tangency. In fact, after the invariant manifolds of an inverting saddle cross, the unstable manifold becomes the chaotic attractor. This leads us to believe that all chaotic attractors are topologically the same

Electrostatic Micro-Tweezers

This dissertation presents a novel electrostatic micro-tweezers designed to manipulate particles with diameters in the range of 5-14 μm. The tweezers consist of two grip-arms mounted to an electrostatically actuated initially curved micro-beam. The tweezers offer further control, via electrostatic actuation, to increase the pressure on larger objects and to grasp smaller objects. It can be operated in two modes. The first is a traditional quasi-static mode where DC voltage commands the tweezers along a trajectory to approach, hold and release micro-objects. It exploits nonlinear phenomena in electrostatic curved beams, namely snap-through, snap-back and static pull-in and the bifurcations underlying them. The second mode uses a harmonic voltage signal to release, probe and/or interact with the objects held by the tweezers in order to perform function such as cells lysis and characterization. It exploits additional electrostatic MEMS phenomena including dynamic pull-in as well as the orbits and attractors realized under harmonic excitation. Euler-Bernoulli beam theory is utilized to derive the tweezers governing equation of motion taking into account the arm rotary inertia, the electrostatic fringing field and the nonlinear squeeze-film damping. A reduced-order model (ROM) is developed utilizing two, three and five straight beam mode shapes in a Galerkin expansion. The adequacy of the ROM in representing the tweezers response was investigated by comparing its static and modal response to that of a 2D finite element model (FEM). Simulation results show small differences between the ROM and the FEM static models in the vicinity of snap-through and negligible differences elsewhere. The results also show the ability of the tweezers to manipulate micro-particles and to smoothly compress and hold objects over a voltage range extending from the snap-back voltage (89.01 V) to the pull-in voltage (136.44 V). Characterization of the curved micro-beam show the feasibility of using it as a platform for the tweezers. Evidence of the static snap-through, primary resonance and the superharmonic resonances of orders two and three are observed. The results also show the co-existence of three stable orbits around one stable equilibrium under excitation waveforms with a voltage less than the snap-back voltage. Three branches of orbits are identified as a one branch of small orbits within a narrow potential well and two branches of medium-sized and large orbits within a wider potential well. The transition between those branches results in a characteristic of double-peak frequency-response curve. We also report evidence of a bubble structure along the medium sized branch consisting of a cascade of period-doubling bifurcations and a cascade of reverse period-doubling bifurcations. Experimental evidence of a chaotic attractor developing within this structure is reported. Odd-periodic windows also appear within the attractor including period-three (P-3), period- five (P-5) and period-six (P-6) windows. The chaotic attractor terminates in a cascade of reverse period-doubling bifurcations as it approaches a P-1 orbit

Use of Instabilities in Electrostatic Micro-Electro-Mechanical Systems for Actuation and Sensing

This thesis develops methods to exploit static and dynamic instabilities in electrostatic MEMS to develop new MEMS devices, namely dynamically actuated micro switches and binary micro gas sensors. Models are developed for the devices under consideration where the structures are treated as elastic continua. The electrostatic force is treated as a nonlinear function of displacement derived under the assumption of parallel-plate theorem. The Galerkin method is used to discretize the distributed-parameter models, thus reducing the governing partial differential equations into sets of nonlinear ordinary-differential equations. The shooting method is used to numerically solve those equations to obtain the frequency-response curves of those devices and the Floquet theory is used to investigate their stability. To develop the dynamically actuated micro switches, we investigate the response of microswitches to a combination of DC and AC excitations. We find that dynamically actuated micro switches can realize significant energy savings, up to 60 %, over comparable switches traditionally actuated by pure DC voltage. We devise two dynamic actuation methods: a fixed-frequency method and a shifted-frequency method. While the fixed-frequency method is simpler to implement, the shifted-frequency method can minimize the switching time to the same order as that realized using traditional DC actuation. We also introduce a parameter identification technique to estimate the switch geometrical and material properties, namely thickness, modulus of elasticity, and residual stress. We also develop a new detection technique for micro mass sensors that does not require any readout electronics. We use this method to develop static and dynamic binary mass sensors. The sensors are composed of a cantilever beam connected to a rigid plate at its free end and electrostatically coupled to an electrode underneath it. Two versions of micro mass sensors are presented: static binary mass sensor and dynamic binary mass sensor. Sensitivity analysis shows that the sensitivity of our static mass sensor represents an upper bound for the sensitivity of comparable statically detected inertial mass sensors. It also shows that the dynamic binary mass sensors is three orders of magnitude more sensitive than the static binary mass sensor. We equip our mass sensor with a polymer detector, doped Polyaniline, to realize a formaldehyde vapor sensor and demonstrate its functionality experimentally. We find that while the static binary gas sensor is simpler to realize than the dynamic binary gas sensor, it is more susceptible to external disturbances

Dynamical systems : mechatronics and life sciences

Proceedings of the 13th Conference „Dynamical Systems - Theory and Applications" summarize 164 and the Springer Proceedings summarize 60 best papers of university teachers and students, researchers and engineers from whole the world. The papers were chosen by the International Scientific Committee from 315 papers submitted to the conference. The reader thus obtains an overview of the recent developments of dynamical systems and can study the most progressive tendencies in this field of science

Localized buckling of an elastic strut in a visco-elastic medium

Certain types of long, axially compressed structures have the potential to buckle locally in one or more regions rather than uniformly along their length. Here, the potential for localized buckle patterns in an elastic layer embedded in a visco-elastic medium is investigated using a strut-on-foundation model. Applications of this model include the growth of geological folds and other time-dependent instability processes. The model consists of an elastic strut of uniform flexural stiffness supported by a Winkler-type foundation made up of discrete Maxwell elements. Mathematically, this model corresponds to a nonlinear partial differential equation which is fourth­order in space and first-order in time. The nature of the buckling process is charac­terized by an initial period of elastic deformation followed by an evolutionary phase in which both elasticity and viscosity have a role to play. Two different formulations are studied: the first combines linear strut theory with a nonlinear foundation and is valid for small, but finite, deflections; the other incorporates the exact expression for curvature of the strut resulting in geometrical nonlinearities and is capable of modelling large deflections. The evolution of non-periodic buckle patterns in each system is examined under the constraint of controlled end displacement. Two independent methods are used to approximate the solution of the governing equations. Modal solutions, based on the method of weighted residuals, complement accurate numerical solutions obtained with a boundary-value solver. In either case, the results suggest that for the perfect system, localized solutions follow naturally from the inclusion of nonlinear elasticity with softening characteristics. Emphasis throughout is on the qualitative features displayed by the phenomenon of localization rather than specific applications. Nevertheless, the ideas and results are a step towards accounting for the rich variety of deformed shapes exhibited by nature.Open Acces