![(a) Fundamental and (b) third harmonic amplitude response curves for the undamped Duffing oscillator, with variations in the DNF detuning, using ω n = 1, α = 0.5, and γ ∈ [0, 1]. This figure has been reproduced from [1].](https://www.researchgate.net/profile/Alexander-Elliott-2/publication/337717413/figure/fig2/AS:832054904225792@1575388756079/a-Fundamental-and-b-third-harmonic-amplitude-response-curves-for-the-undamped_Q320.jpg)
![The first five symmetric mode shapes for the clamped-clamped beam, as found using the numerical values in [2].](https://www.researchgate.net/profile/Alexander-Elliott-2/publication/337717413/figure/fig4/AS:832054904242178@1575388756159/The-first-five-symmetric-mode-shapes-for-the-clamped-clamped-beam-as-found-using-the_Q320.jpg)
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Accurate Approximations for Nonlinear Vibrations
Abstract and Figures
As global issues such as climate change and overpopulation continue to grow, the role of the
engineer is forced to adapt. The general population now places an emphasis not only on the
performance of a mechanical system, but also the efficiency with which this can be achieved.
In pushing these structures to their maximum efficiency, a number of design and engineering
challenges can arise. In particular, the occurrence of geometric nonlinearities can lead to
failures in the linear modelling techniques that have traditionally been used.
The aim of this thesis is to increase the understanding of a number of widely-used, nonlinear
methods, so that they may eventually be used with the same ease and confidence as
traditional linear techniques.
A key theme throughout this work is the notion that nonlinear behaviour is typically
approximated in some way, rather than finding exact solutions. This is not to say that exact
solutions cannot be found, but rather that the process of doing so, or the solutions themselves,
can be prohibitively complicated. Across the techniques considered, there is a desire to
accurately predict the frequency-amplitude relationship, whether this be for the free or forced
response of the system.
Analytical techniques can be used to produce insight that may be inaccessible through
the use of numerical methods, though they require assumptions to be made about the structure.
In this thesis, a number of these methods are compared in terms of their accuracy and
their usability, so that the influence of the aforementioned assumptions can be understood.
Frequency tuning is then used to bring the solutions from three prominent methods in line
with one another.
The Galerkin method is used to project a continuous beam model into a discrete set
of modal equations, as is the traditional method for treating such a system. Motivated by
microscale beam structures, an updated approach for incorporating nonlinear boundary conditions
is developed. This methodology is then applied to two example structures to demonstrate
the importance of this procedure in developing accurate solutions.
The discussion is expanded to consider non-intrusive reduced-order modelling techniques,
which are typically applied to systems developed with commercial finite element software.
By instead applying these methods to an analytical nonlinear system, it is possible to compare
the approximated results with exact analytical solutions. This allows a number of observations
to be made regarding their application to real structures, noting a number of situations
in which the static cases applied or the software itself may influence the solution accuracy.
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Approximate analytical methods, such as the multiple scales (MS) and direct normal form (DNF) techniques, have been used extensively for investigating nonlinear mechanical structures, due to their ability to offer insight into the system dynamics. A comparison of their accuracy has not previously been undertaken, so is addressed in this paper. This is achieved by computing the backbone curves of two systems: the single-degree-of-freedom Duffing oscillator and a non-symmetric, two-degree-of-freedom oscillator. The DNF method includes an inherent detuning, which can be physically interpreted as a series expansion about the natural frequencies of the underlying linear system and has previously been shown to increase its accuracy. In contrast, there is no such inbuilt detuning for MS, although one may be, and usually is, included. This paper investigates the use of the DNF detuning as the chosen detuning in the MS method as a way of equating the two techniques, demonstrating that the two can be made to give identical results up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^2$$\end{document}ε2 order. For the examples considered here, the resulting predictions are more accurate than those provided by the standard MS technique. Wolfram Mathematica scripts implementing these methods have been provided to be used in conjunction with this paper to illustrate their practicality.
Electronic supplementary material
The online version of this article (10.1007/s11071-018-4534-1) contains supplementary material, which is available to authorized users.
In this work, the nonlinear behaviour of a parametrically excited system with electromagnetic excitation is accurately modelled, predicted and experimentally investigated. The equations of motion include both the electromechanical coupling factor and the electromechanical damping. Unlike previous studies where only linear time-varying stiffness due to electromagnetic forces was presented, in this paper the effect of the induced current is studied. As a consequence, nonlinear parameters such as electromechanical damping, cubic stiffness and cubic parametric stiffness have been included in the model. These parameters are also observed experimentally by controlling the direct current (DC) and alternating current (AC) passed through the electromagnets. In fact, the proposed apparatus allows to control both linear and nonlinear stiffnesses and the independent effect of each parameter on the response is presented. In particular the effect of the cubic parametric stiffness on the parametric resonance amplitudes and the influence of cubic stiffness on the frequency bandwidth of the parametric resonance are shown. This model improves the prediction of parametric resonance, frequency bandwidth, and the response amplitude of parametrically excited systems and it may lead to refined design of electromagnetic actuators, filters, amplifiers, vibration energy harvesters, and magnetic bearings.
The objective of the present paper is to provide experimental evidence of isolated resonances in the frequency response of nonlinear mechanical systems. More specifically, this work explores the presence of isolas, which are periodic solutions detached from the main frequency response, in the case of a nonlinear set-up consisting of two masses sliding on a horizontal guide. A careful experimental investigation of isolas is carried out using responses to swept-sine and stepped-sine excitations. The experimental findings are validated with advanced numerical simulations combining nonlinear modal analysis and bifurcation monitoring. In particular, the interactions between two nonlinear normal modes are shown to be responsible for the creation of the isolas.
Procedure for designing the wind tunnel model of a high aspect ratio (HAR) wing containing geometric nonlinearities is described in this paper. The design process begins with identification of basic features of the HAR wing as well as its design constraints. This enables the design space to be narrowed down and consequently, brings ease of convergence towards the design solution. Parametric studies in terms of the spar thickness, the span length and the store diameter are performed using finite element analysis for both undeformed and deformed cases, which respectively demonstrate the linear and nonlinear conditions. Two main criteria are accounted for in the selection of the wing design: the static deflections due to gravitational loading should be within the allowable margin of the size of the wind tunnel test section and the flutter speed of the wing should be much below the maximum speed of the wind tunnel. The findings show that the wing experiences a stiffness hardening effect under the nonlinear static solution and the presence of the store enables significant reduction in linear flutter speed.
The superharmonic resonance of second order of microelectro-mechanical system (MEMS) circular plate resonator under electrostatic actuation is investigated. The MEMS resonator consists of a clamped circular plate suspended over a parallel ground plate under an applied Alternating Current (AC) voltage. The AC voltage is characterized as hard excitation, i.e. the magnitude is large enough, and the operating frequency is near one-fourth of the natural frequency of the resonator. Reduced Order Model (ROM), based on the Galerkin procedure, transforms the partial differential equation of motion into a system of ordinary differential equations in time using mode shapes of vibration of the circular plate resonator. Three numerical methods are used to predict the voltage-amplitude response of the MEMS plate resonator. First, the Method of Multiple Scales (MMS) is directly applied to the partial differential equation of motion which is this way transformed into zero-order and first-order problems. Second, ROM using two modes of vibration is numerical integrated using MATLAB to predict time responses, and third, the AUTO 07P software for continuation and bifurcation to predict the voltage-amplitude response. The nonlinear behavior (i.e. bifurcation and pull-in instability) of the system is attributed to the inclusion of viscous air damping and electrostatic force in the model. The influences of various parameters (i.e. detuning frequency and damping) are also investigated in this work.
This paper investigates the voltage-amplitude response of superharmonic resonance of second order (order two) of alternating current (AC) electrostatically actuated microelectromechanical system (MEMS) cantilever resonators. The resonators consist of a cantilever parallel to a ground plate and under voltage that produces hard excitations. AC frequency is near one-fourth of the natural frequency of the cantilever. The electrostatic force includes fringe effect. Two kinds of models, namely reduced-order models (ROMs), and boundary value problem (BVP) model, are developed. Methods used to solve these models are (1) method of multiple scales (MMS) for ROM using one mode of vibration, (2) continuation and bifurcation analysis for ROMs with several modes of vibration, (3) numerical integration for ROM with several modes of vibration, and (4) numerical integration for BVP model. The voltage-amplitude response shows a softening effect and three saddle-node bifurcation points. The first two bifurcation points occur at low voltage and amplitudes of 0.2 and 0.56 of the gap. The third bifurcation point occurs at higher voltage, called pull-in voltage, and amplitude of 0.44 of the gap. Pull-in occurs, (1) for voltage larger than the pull-in voltage regardless of the initial amplitude and (2) for voltage values lower than the pull-in voltage and large initial amplitudes. Pull-in does not occur at relatively small voltages and small initial amplitudes. First two bifurcation points vanish as damping increases. All bifurcation points are shifted to lower voltages as fringe increases. Pull-in voltage is not affected by the damping or detuning frequency.
In this paper, we summarize research activities and technological progress in the field of current CMOS-MEMS resonators and oscillators for portable sensor node and timing applications. By employing CMOS-based fabrication technologies, we can monolithically integrate MEMS devices and their associated application-specific integrated circuits (ASICs), thereby enhancing their overall performance as compared with their stand-alone counterparts. Owing to the diversity of post-CMOS processing techniques, in this review, we mainly focus on the devices achieved by so-called foundry-orientated CMOS-MEMS platforms. On the basis of how the mechanical structure is achieved, in the paper we focus on two major strategies: (i) additive back-end-of-line (BEOL) compatible layers and (ii) machining of standard CMOS layers for device fabrication. Given their superior CMOS integration capability, the circuitry design concepts, testing results, and potential merits of state-of-the-art CMOS-MEMS oscillators are also presented.
In order to ensure integrity of thermal protection system (TPS) subjected to a combination of thermal and acoustic loadings, a thin composite plate resting on a two-parameter elastic foundation is used to characterize the behavior of the thin top facesheet of TPS. The nonlinear dynamic response of a thermal loaded, acoustic excited plate is investigated. A theoretical model is developed based on Kirchhoff thin plate assumptions and von Kármán-type equation. General static condensation and Galerkin's method are used to derive a set of ordinary differential equations with cubic nonlinearity related to nonlinear coupling between mid-plane stretching and transverse deflection. The reduced-order model has been validated by comparison of postbuckled displacements with those obtained from full-order FEM analysis. Variations of transverse displacement and in-plane strain statistics with acoustic loading level and temperature rising are presented. It is proposed that the in-plane strain located on the plate surface is dominated by the competition of the linear and quadratic nonlinear modal amplitude terms, thus the characteristic of the strain histogram can be used to identify oscillation transition from no snap-through to persistent dynamic snap-through for the thermally buckled plate. The skewness of the strain histogram can be used to evaluate the degree of dynamic geometrical nonlinearity quantitatively for the postbuckled plate with symmetric snap-through motion.
The natural frequencies and mode shapes of the flapwise and chordwise vibrations of a rotating cracked Euler-Bernoulli beam are investigated using a simplified method. This approach is based on obtaining the lateral deflection of the cracked rotating beam by subtracting the potential energy of a rotating massless spring, which represents the crack, from the total potential energy of the intact rotating beam. With this new method, it is assumed that the admissible function which satisfies the geometric boundary conditions of an intact beam is valid even in the presence of a crack. Furthermore, the centrifugal stiffness due to rotation is considered as an additional stiffness, which is obtained from the rotational speed and the geometry of the beam. Finally, the Rayleigh-Ritz method is utilised to solve the eigenvalue problem. The validity of the results is confirmed at different rotational speeds, crack depth and location by comparison with solid and beam finite element model simulations. Furthermore, the mode shapes are compared with those obtained from finite element models using a Modal Assurance Criterion (MAC).
Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that the authors have presented an extensive revision of the first edition of the Averaging Methods in Nonlinear Dynamical Systems book. There are many changes, corrections and updates in chapters on Basic Material and Asymptotics, Averaging, and Attraction. Chapters on Periodic Averaging and Hyperbolicity, Classical (first level) Normal Form Theory, Nilpotent (classical) Normal Form, and Higher Level Normal Form Theory are entirely new and represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are surveys on invariant manifolds in Appendix C and averaging for PDEs in Appendix E. Since the first edition, the book has expanded in length and the third author, James Murdock has been added.
Review of First Edition
"One of the most striking features of the book is the nice collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with profuse, illuminating diagrams." - Mathematical Reviews