<p> The frequency pulling of the van dcr Pol nonlinear oscillator due to an external forcing signal is investigated. The nonlinearity is of the zero-memory symmetric-cut-off type following a cube law.</p> <p> An experimental oscillator was built, and curves of the frequency shift of the oscillator fundamental against the magnitude of the input forcing signal are shown, both for a sinusoidal input and for a narrow band noise input. An empirical result is derived.</p> <p> The case of the sinusoidal input is examined theoretically. The importance of harmonic and intermodulation frequencies in the oscillator output is shown, and relations giving the oscillator frequency shift are given for small forcing amplitudes and for large amplitudes when the oscillator is nearly synchronized.</p> / Thesis / Master of Engineering (MEngr)

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Understanding static and dynamic nonlinear behavior of pipes and risers is crucial for the design aspects in offshore engineering fields. In this work, we examine two nonlinear problems in offshore engineering field: vortex Induced vibration of straight horizontal pipes, and boundary layer static solution of inclined risers. In the first study, we analyze the effect of the internal velocity of straight horizontal pipe and obtain the vortex induced vibration forces via coupling the pipe equation of motion with the recently modified Van Der Pol oscillator governing the lift coefficient. Our numerical results are obtained for two different pipe configurations: hinged-hinged, and clamped- clamped. The results show that the internal velocity reduces the vibration and the oscillation amplitudes. Also, it is shown that the clamped-clamped pipe configuration offers a wider range of internal velocities before buckling instability occurs. The results also demonstrate the effect of the end condition on the amplitudes of vibration. In the second study, we develop a boundary layer perturbation static solution to govern and simulate the static behavior of inclined risers. In the boundary layer analysis, we take in consideration the effects of the axial stretch, applied tension, and internal velocity. Our numerical simulation results show good agreement with the exact solutions for special cases. In addition, our developed method overcomes the mathematical and numerical limitations of the previous methods used before.

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Orientador: Masayoshi Tsuchida / Banca: José Manoel Balthalzar / Banca: Adalberto Spezamiglio / Resumo: 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. / Abstract: 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. / Mestre

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.

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.