Energy harvesting of ambient and aeroelastic vibrations is important for reducing the dependence of wireless sensing and networks on batteries. We develop a configuration for a piezoelectric energy harvester with the capability to wirelessly communicate vibration measurements while using those vibrations to power the sensing and communication devices. Particularly, we perform experiments that aim at identifying challenges to overcome in the development of such a configuration. Towards that objective, we successfully tested a self-powered real-time point-to-point wireless communication system between a vibration sensor and transmission and receiving modules. The sensing device and transmission module are powered by the vibrating object using a piezoelectric energy harvester. The communication is established by using two XBee modules. In the second part of this dissertation, we address the optimization of the output power of piezoelectric energy harvesters of aeroelastic vibrations. Given the complexity of high-fidelity simulations of the coupling between the fluid flow, structural response and piezoelectric transduction, we develop and experimentally validate a phenomelogical reduced-order model for energy harvesting from wake galloping. We also develop a high-fidelity simulation for the same phenomena. The modeling and high-fidelity simulations can be a part of a multi-disciplinary optimization framework to be used in the design and operation of galloping-based energy harvesters. / Doctor of Philosophy / Energy harvesting of ambient or flow-induced vibrations is important for reducing the dependence on batteries in wireless sensing and networks to monitor deterioration conditions, environmental pollution or wildlife conservation. Balancing the benefits and shortcomings of a specific approach, namely piezoelctric transduction, for energy harvesting from vibrations, we address a specific challenge related to the development of a configuration that allows for communicating measured vibrations using their power. Furthermore, given the low levels of output power from piezoelectric transduction, we address the need to optimize power output levels through the development of predictive models that depend on geometry and speed of the fluid flow.

We model the nonlinear dynamics of a cantilever beam with tip mass system subjected to different excitation and exploit the nonlinear behavior to perform sensitivity analysis and propose a parameter identification scheme for nonlinear piezoelectric coefficients. First, the distributed parameter governing equations taking into consideration the nonlinear boundary conditions of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. We solve the distributed parameter and discretized equations separately using the method of multiple scales. We determine that the cantilever beam tip mass system subjected to parametric excitation is highly sensitive to the detuning. Finally, we show that assuming linearized boundary conditions yields the wrong type of bifurcation. Noting the highly sensitive nature of a cantilever beam with tip mass system subjected to parametric excitation to detuning, we perform sensitivity of the response to small variations in elasticity (stiffness), and the tip mass. The governing equation of the first mode is derived, and the method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications. Finally, we consider a cantilever beam with tip mass and piezoelectric layer and propose a parameter identification scheme that exploits the vibration response to estimate the nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and piezoelectric layer by considering an enthalpy that accounts for quadratic and cubic material nonlinearities. We then use the method of multiple scales to determine the approximate solution of the response to direct excitation. We show that approximate solution and amplitude and phase modulation equations obtained from the method of multiple scales analysis can be matched with numerical simulation of the response to estimate the nonlinear piezoelectric coefficients. / Master of Science

There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.

Dans un contexte d’amélioration des dispositifs pour la réduction de bruit, l’étude sur le transfert d’énergie irréversible en utilisant des résonateurs purement acoustiques à comportement non linéaire a été réalisée. Les résonateurs acoustiques classiques en régime linéaire agissent comme un Amortisseur de Masse Accordée (TMD, en anglais) et ils sont efficaces pour une gamme de fréquence très étroite. Cependant, lorsqu’ils sont soumis à des excitations très fortes (régime non-linéaire) ils peuvent devenir efficaces pour une plus large gamme de fréquences si des termes non linéaires peuvent être activés. Dans un premier temps, une étude sur ce comportement non-linéaire d’un résonateur d’Helmholtz modifié a été réalisée expérimentalement. Ensuite, l’équation dynamique gouvernante de tels résonateurs ont été développées en prenant en compte les non-linéarités de la force de rappel et d’amortissement. Une approximation de la solution analytique de l’équation gouvernante du résonateur acoustique a été déterminée en utilisant les méthodes des échelles multiples du temps et de transformation du temps non régulière. Dans un deuxième temps, une étude du couplage entre un mode acoustique en basses fréquences et un résonateur (celui étudié précédemment) à comportement non-linéaire a été réalisée. Pour ce faire, des mesures expérimentales avec un montage du système couplé ont permis de vérifier l’atténuation acoustique produite par le résonateur en régime forcé et libre. Une modélisation analytique du couplage a permis d’identifier l’expression de la variété invariante lente, ce qui a permis d’étudier les possibles points d’équilibre et points singuliers du système. Les modèles analytiques développés ont également été vérifiés par des simulations numériques. / Nowadays, there is a need of new types of technologies for sound reduction because of the growing of different industries. In this context, we have studied the targeted energy transfer using a purely acoustic resonator. These acoustic resonators act, in the linear regime, as a Tuned Masse Damper (TMD) and they are efficient for a narrow frequency band. But, when they are excited with high forces, in the nonlinear regime, they are efficient for a wider frequency band if the nonlinear terms are activated. First, an experimental study about the nonlinear behavior of a modified Helmholtz Resonator was done. Then, the governing equation of such resonators were developed considering the nonlinearities in the restitution force and damping. An approximation of the analytical solution of the governing equation of the acoustical resonator is derived using the multiples scales of time method and the non-smooth time transformation method. In a second part, a study about the coupling between an acoustic mode in low frequencies and a resonator (the one studied in the previous part) with a nonlinear behavior is done. In order to do this, experimental measurements of the coupled system to confirm acoustic attenuation by the resonator in forced and free regime were done. Then, an analytical modelling of the coupled system allowed to derive the expression of the Slow Invariant Manifold (SIM), in order to identify the possible equilibrium points and singular points of the system. Derived analytical models were verified by numerical simulations.

This thesis involves the study of four problems in the area of nonlinear vibrations, using the asymptotic method of multiple scales(MMS). Accordingly, it consists of four sequentially arranged parts. In the first part of this thesis we study some nonlinear dynamics related to the amplitude control of a lightly damped, resonantly forced, harmonic oscillator. The slow flow equations governing the evolution of amplitude and phase of the controlled system are derived using the MMS. Upon choice of a suitable control law, the dynamics is represented by three coupled ,nonlinear ordinary differential equations involving a scalar free parameter. Preliminary study of this system using the bifurcation analysis package MATCONT reveals the presence of Hopf bifurcations, pitchfork bifurcations, and limit cycles which seem to approach a homoclinic orbit. However, close approach to homoclinic orbit is not attained using MATCONT due to an inherent limitation of time domain-based continuation algorithms. To continue the limit cycles closer to the homoclinic point, a new algorithm is proposed. The proposed algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. The algorithm incorporates variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented showing favorable comparisons with MATCONT near saddle homoclinic points. The algorithm is also formulated with infinitesimal parameter increments resulting in ordinary differential equations, which gives some advantages like the ability to handle fold points of periodic solution branches upon suitable re-parametrization. Extensions to higher dimensions are outlined as well. With the new algorithm, we revisit the amplitude control system and continue the limit cycles much closer to the homoclinic point. We also provide some independent semi-analytical estimates of the homoclinic point, and mention an a typical property of the homoclinic orbit. In the second part of this thesis we analytically study the classical van der Pol oscillator, but with an added fractional damping term. We use the MMS near the Hopf bifurcation point. Systems with (1)fractional terms, such as the one studied here, have hitherto been largely treated numerically after suitable approximations of the fractional order operator in the frequency domain. Analytical progress has been restricted to systems with small fractional terms. Here, the fractional term is approximated by a recently pro-posed Galerkin-based discretization scheme resulting in a set of ODEs. These ODEs are then treated by the MMS, at parameter values close to the Hopf bifurcation. The resulting slow flow provides good approximations to the full numerical solutions. The system is also studied under weak resonant forcing. Quasiperiodicity, weak phase locking, and entrainment are observed. An interesting observation in this work is that although the Galerkin approximation nominally leaves several long time scales in the dynamics, useful MMS approximations of the fractional damping term are nevertheless obtained for relatively large deviations from the nominal bifurcation point. In the third part of this thesis, we study a well known tool vibration model in the large delay regime using the MMS. Systems with small delayed terms have been studied extensively as perturbations of harmonic oscillators. Systems with (1) delayed terms, but near Hopf points, have also been studied by the method of multiple scales. However, studies on systems with large delays are few in number. By “large” we mean here that the delay is much larger than the time scale of typical cutting tool oscillations. The MMS up to second order, recently developed for such large-delay systems, is applied. The second order analysis is shown to be more accurate than first order. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy. A key point is that although certain parameters are treated as small(or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation. In the present analysis, infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. Lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS. The strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly. In the last part of this thesis, we study the weakly nonlinear whirl of an asymmetric, overhung rotor near its gravity critical speed using a well known two-degree of freedom model. Gravity critical speeds of rotors have hitherto been studied using linear analysis, and ascribed to rotor stiffness asymmetry. Here we present a weakly nonlinear study of this phenomenon. Nonlinearities arise from finite displacements, and the rotor’s static lateral deflection under gravity is taken as small. Assuming small asymmetry and damping, slow flow equations for modulations of whirl amplitudes are developed using the MMS. Inertia asymmetry appears only at second order. More interestingly, even without stiffness asymmetry, the gravity-induced resonance survives through geometric nonlinearities. The gravity resonant forcing does not influence the resonant mode at leading order, unlike typical resonant oscillations. Nevertheless, the usual phenomena of resonances, namely saddle-node bifurcations, jump phenomena and hysteresis, are all observed. An unanticipated periodic solution branch is found. In the three dimensional space of two modal coefficients and a detuning parameter, the full set of periodic solutions is found to be an imperfect version of three mutually intersecting curves: a straight line, a parabola, and an ellipse. To summarize, the first and fourth problems, while involving routine MMS involve new applications with rich dynamics. The second problem demonstrated a semi-analytical approach via the MMS to study a fractional order system. Finally, the third problem studied a known application in a hitherto less-explored parameter regime through an atypical MMS procedure. In this way, a variety of problems that showcase the utility of the MMS have been studied in this thesis.

L'utilisation de systèmes légers non linéaires permet de réaliser le contrôle vibratoire de structures subissant des oscillations non acceptables en termes de confort pour l'usager ou de sécurité de l'ouvrage. L'étude des puits d'énergie non linéaires, ou « Nonlinear Energy Sinks » (NES), a notamment fait l'objet de nombreuses recherches depuis le début des années 2000. Sa non-linéarité lui confère des capacités de pompage énergétique large bande, c'est-à-dire pour un large intervalle de fréquences de sollicitation, ce qui représente un avantage significatif en comparaison des absorbeurs comme l'amortisseur à masse accordée. Le but de ce manuscrit est d'étudier le couplage de chaîne d'oscillateurs non linéaires à des systèmes dynamiques linéaires soumis à des sollicitations harmoniques et d'analyser d'une part le comportement global du système, et d'autre part les potentialités de contrôle passif de telles chaînes. Une méthodologie analytique générale est présentée, puis appliquée à des exemples où des absorbeurs à non-linéarités cubiques à un, puis à N degrés de liberté sont attachés à un oscillateur linéaire. Une variation de cette méthodologie adoptant une vision continue de la chaîne est ensuite proposée. Enfin, un dispositif expérimental étudie le comportement d'un modèle réduit de bâtiment à un étage couplé à une chaîne de huit oscillateurs non linéaires. / Nonlinear light oscillators can be used for performing vibratory passive control of structures undergoing unacceptable oscillations in terms of comfort and safety. The study of Nonlinear Energy Sinks (NES) has been especially subject to an important research effort since the beginning of the 2000s. Its essential nonlinearity enables it to achieve large-band energy pumping, which is a significant advantage in comparison with classical Tuned Mass Dampers. In this manuscript, nonlinear chains of oscillators coupled to linear systems under harmonic excitation are studied. The main goal is to understand the behavior of the whole system and find evidence of passive control abilities of such chains. First of all, a general analytical methodology is presented and applied to examples where single and multi-degree-of-freedom absorbers with cubic nonlinearities are linked to a linear oscillator. A modification of this approach by considering the chain in the form of a continuous approximation is then proposed. Finally, an experimental device composed of a single storey reduced-scale building coupled to a chain of eight nonlinear oscillators is investigated.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-06-20T18:27:52Z No. of bitstreams: 2 Dissertação - Rosane Gonçalves Ferreira - 2016.pdf: 4272548 bytes, checksum: a5f44a1be60a4ace1d85167dc75c33c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Cláudia Bueno (claudiamoura18@gmail.com) on 2017-07-07T19:47:39Z (GMT) No. of bitstreams: 2 Dissertação - Rosane Gonçalves Ferreira - 2016.pdf: 4272548 bytes, checksum: a5f44a1be60a4ace1d85167dc75c33c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-07-07T19:47:39Z (GMT). No. of bitstreams: 2 Dissertação - Rosane Gonçalves Ferreira - 2016.pdf: 4272548 bytes, checksum: a5f44a1be60a4ace1d85167dc75c33c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-03-04 / Vibrations of mechanical systems have a wide field of research, where many work have been dedicated. Such importance is due to the fact that most human activities involve vibrations. It is worth noting that many device can suffer or produce vibrations, such as, machines, structures, motors, turbines. Vibratory systems, generally can produce complex behavior, thus the analysis of such dynamics behavior needs to use sophisticated mathematical tools. The mathematical model assigns important features of real processes with respect to linear and non-linear differential equations. In this work we are interested in the analysis of behavior of delayed mechanical systems. Time delayed can compromise the performance of controls even adding instability in the systems. On the other hand, write choose of delays can improve its performance. Systems with time delay, similar to ordinary systems, are molded by ordinary and/or partial differential equations, but, unlikely ordinary differential equations, delayed differential equations, also known as functional differential equations, are molded on Banach spaces with infinite dimension, which introduce serious difficulty in analysis of stability, since that, the spectra of solution semi-group associated with the linear part of the model can presents infinite eigenvalues. Thus, our contribution of the study of dynamics behavior of such systems will be in two directions. In the first one, we apply the perturbation method of multiple scales in themodel of differential equations, since that the system shows nonlinear vibrations. It is worth noting that the differential analysis used in the stage regarding differential equations in Banach spaces, which has infinite dimension, this approach differ substantially from standards approaches. Then we obtain numerical solutions for the amplitude at steady state using the Newton Raphson method and then we made a numerical analysis of the model of stability with delay and without delay to different parameters, using the Runge-Kuttamethod. / As vibrações possuem um campo extenso de estudos, ao quais trabalhos inteiros têm sido dedicados. Tamanha importância deve-se ao fato de que a maioria das atividades humanas envolve vibrações. Muitos sistemas construídos sofrem ou produzem vibração, tais como máquinas, estruturas, motores, turbinas e sistemas de controle. Umsistema vibratório geralmente apresenta comportamento complexo, assim a análise do comportamento dinâmicos envolve o uso de ferramentas matemáticas sofisticadas. O modelo matemático incorpora os aspectos importantes do processo real, em termos de equações diferenciais lineares ou não lineares. Neste trabalho nosso objetivo é analisar o comportamento de um modelo de sistemas mecânicos. Os tempos de atrasos quando presentes em controladores e atuadores podem ser motivo de ineficiência ou mesmo causar a instabilidade do sistema. Porém, o controle adequado desses atrasos pode melhorar o desempenho de sistemas mecânicos. Os sistemas com tempo de atraso, assim como os sistemas ordinários, são modelados por equações diferenciais ordinárias ou parciais, mas diferentemente das equações ordinárias, equações com tempo de atraso, também conhecidas como equações funcionais, são modeladas em espaços de dimensão infinita, o que dificulta enormemente a análise de estabilidade, uma vez que, o espectro do semigrupo solução associado à parte linear do modelo pode apresentar infinitos autovalores. Assim, nossa contribuição ao estudo do comportamento dinâmico de tais sistemas foi feito em duas partes. Na primeira, aplicamos o método de perturbação das múltiplas escalas no sistema de equações diferenciais do modelo, uma vez que o sistema apresenta vibrações não lineares. Nesta parte, é importante ressaltar que a análise diferencial usada foi em um espaço de dimensão infinita, também conhecido como espaço de Banach; esta análise difere substancialmente daquela usada no caso ordinário. Em seguida obtemos soluções numéricas para a amplitude em estado estacionário usando o método de Newton Raphson e depois fizemos uma análise numérica da estabilidade do modelo com atraso e sem atraso para diferentes parâmetros, usando o método de Runge- Kutta.

No description available.

In this thesis a number of statistical methods have been developed and applied to habitat suitability modelling for rare and threatened species. Data available on these species are typically limited. Therefore, developing these models from these data can be problematic and may produce prediction biases. To address these problems there are three aims of this thesis. The _rst aim is to develop and implement frequentist and Bayesian statistical modelling approaches for these types of data. The second aim is develop and implement expert elicitation methods. The third aim is to apply these novel approaches to Australian rare and threatened species case studies with the intention of habitat suitability modelling. The _rst aim is ful_lled by investigating two innovative approaches for habitat suitability modelling and sensitivity analysis of the second approach to priors. The _rst approach is a new multilevel framework developed to model the species distribution at multiple scales and identify excess zeros (absences outside the species range). Applying a statistical modelling approach to the identi_cation of excess zeros has not previously been conducted. The second approach is an extension and application of Bayesian classi_cation trees to modelling the habitat suitability of a threatened species. This is the _rst `real' application of this approach in ecology. Lastly, sensitivity analysis of the priors in Bayesian classi_cation trees are examined for a real case study. Previously, sensitivity analysis of this approach to priors has not been examined. To address the second aim, expert elicitation methods are developed, extended and compared in this thesis. In particular, one elicitation approach is extended from previous research, there is a comparison of three elicitation methods, and one new elicitation approach is proposed. These approaches are illustrated for habitat suitability modelling of a rare species and the opinions of one or two experts are elicited. The _rst approach utilises a simple questionnaire, in which expert opinion is elicited on whether increasing values of a covariate either increases, decreases or does not substantively impact on a response. This approach is extended to express this information as a mixture of three normally distributed prior distributions, which are then combined with available presence/absence data in a logistic regression. This is one of the _rst elicitation approaches within the habitat suitability modelling literature that is appropriate for experts with limited statistical knowledge and can be used to elicit information from single or multiple experts. Three relatively new approaches to eliciting expert knowledge in a form suitable for Bayesian logistic regression are compared, one of which is the questionnaire approach. Included in this comparison of three elicitation methods are a summary of the advantages and disadvantages of these three methods, the results from elicitations and comparison of the prior and posterior distributions. An expert elicitation approach is developed for classi_cation trees, in which the size and structure of the tree is elicited. There have been numerous elicitation approaches proposed for logistic regression, however no approaches have been suggested for classi_cation trees. The last aim of this thesis is addressed in all chapters, since the statistical approaches proposed and extended in this thesis have been applied to real case studies. Two case studies have been examined in this thesis. The _rst is the rare native Australian thistle (Stemmacantha australis), in which the dataset contains a large number of absences distributed over the majority of Queensland, and a small number of presence sites that are only within South-East Queensland. This case study motivated the multilevel modelling framework. The second case study is the threatened Australian brush-tailed rock-wallaby (Petrogale penicillata). The application and sensitivity analysis of Bayesian classi_cation trees, and all expert elicitation approaches investigated in this thesis are applied to this case study. This work has several implications for conservation and management of rare and threatened species. Novel statistical approaches addressing the _rst aim provide extensions to currently existing methods, or propose a new approach, for identi _cation of current and potential habitat. We demonstrate that better model predictions can be achieved using each method, compared to standard techniques. Elicitation approaches addressing the second aim ensure expert knowledge in various forms can be harnessed for habitat modelling, a particular bene_t for rare and threatened species which typically have limited data. Throughout, innovations in statistical methodology are both motivated and illustrated via habitat modelling for two rare and threatened species: the native thistle Stemmacantha australis and the brush-tailed rock wallaby Petrogale penicillata.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-16Bitstream added on 2014-06-13T19:34:53Z : No. of bitstreams: 1 santos_jm_me_sjrp.pdf: 407078 bytes, checksum: 96bda75a3b280db0c6b8bdd488530e5a (MD5) / Osciladores eletromecânicos podem ser modelados matematicamente através da equação de Duffing ou equação de Van der Pol, mesmo que sejam sistemas de escala nanomética. Nesta dissertação analisamos um oscilador forçado sujeito a um amortecimento não-linear, que é representado pela equação de Duffing - Van der Pol. Em geral, não é fácil obter solução analítica exata para esta equação, então a análise é feita utilizando a teoria de perturbações para obter uma solução analítica aproximada. Para isso consideramos certos parâmetros do problema como sendo pequenos parâmetros, e obtemos a solução na forma de expansão direta. Devido o fato da frequência natural do sistema dinâmico depender do pequeno parâmetro, essa expansão é não uniforme, ou seja, apresenta termos seculares mistos (termos de Poisson), e além disso possui pequenos divisores. Essas inconveniências são eliminadas aplicando o método das múltiplas escalas e o método da média. Inicialmente os pequenos parâmetros são escolhidos de modo que o problema não perturbado se reduz a um oscilador harmônico forçado, e na escolha posterior o problema não perturbado é um oscilador linear amortecido e forçado. / Electromechanical oscillators can be mathematically modeled by a Du±ng equation or a Van der Pol equation, even if they are nanometric systems. In this work we studied a forced oscillator having nonlinear damping, that is represented by a Du±ng - Van der Pol equation. In general, it is not easy to get the exact analytical solution for this equation, then the analysis is done using the perturbation theory to get an approximate analytical solution. For this reason we considered that certain parameters of the problem are small parameters and we obtain the solution in the form of straightforward expansion. Due to the fact that natural frequency of the dynamic system depends on the small parameter, this expansion is not uniform, i.e. presents secular terms (Poisson terms) and also small-divisors. These inconveniences are eliminated using the method of multiple scales and the aver- aging method. Initially the small parameters are chosen so that the unperturbed problem is reduced to a forced harmonic oscillator, and in the subsequent choice the unperturbed is a forced oscillator having linear damping.