Academic literature on the topic 'Subharmonic mode'

The general theory for analysis of subharmonic oscillations at a frequency of ω/3 in three-phase ferroresonance circuits is presented. The occurrence and existence of ferroresonance oscillations at subharmonic frequencies in power transmission lines and power supply systems is highly undesirable, since they cause overvoltages at various frequencies. At the same time, there is an extensive class of nonlinear electrical circuits in which the excitation of autoparametric oscillations at the frequency of subharmonics forms the basis of phase-discrete frequency converting devices serving as secondary power sources. To study the regularities of excitation and maintaining of subharmonic oscillations at a frequency of ω/3 in three-phase ferroresonance circuits, theoretical and experimental studies of an equivalent model of a three-phase circuit with nonlinear inductance were carried out. A generalized nonlinear differential equation for a three-phase circuit with mixed connection of its elements is derived. The steady-state mode of subharmonic oscillations at a frequency of ω/3 is analyzed using the small parameter (averaging) method, which made it possible to determine their existence domains and circuit critical parameters. A mathematical model and algorithm for calculating autoparametric oscillations have been developed to study the subharmonic oscillation excitation processes at a frequency of ω/3 in three-phase ferroresonance circuits depending on the initial conditions, circuit parameters and input voltage. The theoretical study results have been confirmed experimentally.

An instability-wave analysis is presented to describe the spatial evolution of a fundamental mode and its subharmonic on an inviscid parallel mixing layer. It incorporates explicitly the weakly nonlinear interaction between the two modes. The computational finding that the development of the subharmonic, leading eventually to pairing or shredding, crucially depends on its phase relation with the fundamental is fully confirmed. Furthermore it is shown that a critical fundamental amplitude has to be reached before the (spatial) subharmonic becomes phase locked with the fundamental and exhibits a modified growth rate. Then the analysis is exploited to explain the occurrence of amplitude modulations in ‘natural’ mixing layers and to estimate the width of the subharmonic spectral peaks. Also, the case of oblique subharmonic waves is briefly touched upon. In the last part, ways are explored to model non-parallel effects, i.e. to handle the saturation of the rapidly growing subharmonic. Using this wave description, the role of mode interaction in the ‘vortex pairing’ and ‘shredding’ process is assessed.

Experiments have been conducted to study the development of flow behind a surface mounted rib under different phase controlled excitation. Single mode excitation and multi-mode excitation with different relative phases are studied. The results presented include the coherent and random components of the turbulent energy and shear stresses, the energy exchange with the mean flow and between the modes, and the phase decorrelation of the coherent components. The fundamental-subharmonic excitation does not provide any significant improvements in the shear layer growth over the fundamental excitation. The shear layer growth correlates with the subharmonic mode development. The large scale structures are significant even after the reattachment region as evident from the magnitude of the coherent components of the turbulent energy and the shear stress. The binary exchange terms are significant in the near-field region whose sign is phase dependent, i.e., it reverses its sign based on the phase difference between the fundamental and 1st subharmonic mode. The location of the fundamental and subharmonic peaks are different from the peak location of their respective energy exchange with the mean flow; this is attributed to the significance of the binary energy exchange between the fundamental and the subharmonic mode in this region. The excitation regularizes the flow leading to low phase jitter in the near field region. The origin and development of phase decorrelation is attributed primarily to the subharmonic instability.

Experiment have shown that the well-known vortex pairing process may take place first in defect regions where the fundamental structure is weakest. A model is introduced here to describe this defect-induced pairing process. The model is constructed in such a way that, in a certain parameter range, a stable fundamental mode and a stable subharmonic mode may coexist. The numerical simulation demonstrates that, when initial conditions consist of a dominant fundamental with one or more defects, the subharmonic component is preferentially generated in the cores of these defects. Moreover, the results also indicate that pairing may first commence wherever the fundamental mode is weakest provided the white-noise level of the subharmonic is high enough. The numerical results are in good agreement with experimental observations.

Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

The nature of instability occurring in a differentially heated infinite slot under steady gravity depends only on the Prandtl number of the contained Boussinesq fluid. For fluids with Pr < 12.5, the instability is shear dominated and onsets in a steady convection mode; for fluids with Pr > 12.5, the instability is buoyancy dominated and onsets in an oscillatory mode. In this paper, we examine the effect of gravity modulation on the stability characteristics of convection in an infinite slot with both kinds of fluids, in particular, Pr = 1 and Pr = 25. Using the method of Sinha & Wu (1991), we are able to obtain accurate results without excessive numerical integration in the linear stability analysis by Floquet theory. Results show that, for Pr = 1, at a non-dimensional oscillation frequency ω = 20, the critical state alternates between the synchronous and subharmonic modes. At higher frequencies, ω > 100, all critical states occur in the synchronous mode. For Pr = 25, with a modulation amplitude ratio of 0.5, resonant interaction occurs in the neighbourhood of ω = 2σc, where σc is the oscillation frequency of the instability at the critical state under steady gravity. This resonant interaction is destabilizing, with the critical Grashof number being reduced by approximately 20% from that at steady gravity. It is due to the presence of a detached subharmonic branch of the marginal stability curve. In frequency ranges where the detached subharmonic branch is absent, the critical state is in the quasi-periodic mode consisting of two waves of different oscillation frequencies whose sum is the forcing frequency. An analysis of the rate of change of the perturbation kinetic energy shows that, for Pr = 1, the instability is shear dominated regardless of the mode of oscillation, synchronous or subharmonic. Similarly, for Pr = 25, the instability is buoyancy dominated whether it is in the quasi-periodic or subharmonic mode. The mode switching is a response to the forcing and is independent of the dominant mechanisms of instability.

In this paper, we have identified a new mechanism which can promote rapid growth of three-dimensional disturbances. The mechanism involves the interaction between a planar mode and an oblique mode, or a pair of oblique modes, which are phase-locked in the sense that they have the same phase speed. This allows a powerful nonlinear interaction to take place within the common critical layer(s). The disturbance is not required to form a subharmonic resonant triad, and hence the mechanism operates under much less restrictive conditions than does subharmonic resonance (although it is somewhat less powerful). We show that the quadratic interaction between the planar mode and the oblique modes drives an exceptionally large forced mode with the difference frequency, which in turn interacts with the planar mode to contribute the dominant nonlinear effect. This interaction can cause the oblique modes to grow super-exponentially provided that their magnitude is sufficiently small. As a result of the super-exponential growth, the oblique mode may soon become strong enough to produce a feedback effect on the planar mode, so that the interactions eventually become fully coupled. This subsequent stage takes slightly different forms depending on whether a single or a pair of oblique modes is present. Both cases are investigated. Particular attention is paid to symmetric plane shear layers, e.g. a planar wake or jet, for which subharmonic resonance of sinuous modes is inactive.

Диссертация на соискание научной степени кандидата технических наук по специальности 05.02.09 – динамика и прочность машин. – Национальный технический университет "Харьковский политехнический институт", Харьков, 2016. Диссертация посвящена разработке подходов, методов и моделей для исследования динамических процессов в виброударных системах с учетом изменяемой во времени массы технологического груза и нелинейной жесткости упругих опор. В данной работе изложен новый подход к учету влияния переменной массы технологического груза на характер динамических процессов в виброударной системе. Разработаны и предложены зависимости, математические модели, которые описывают характер изменения массы технологического груза во времени, как на основе экспериментальных исследований (эмпирический закон), так и в зависимости от количества диссипированной энергии. Установлено, что вследствии нелинейности жесткости упругих опор и в зависимости от их конструктивного исполнения, становится возможной реализация субгармонических режимов, что приводит к росту силы ударного взаимодействия. Для оценки влияния параметров упругих опор (варианты конструктивного исполнения, жесткость пружин) проведены многовариантные исследования, по результатам которых выделены характерные субгармонические режимы и их характеристики (фазовые портреты, временные распределения). Установлена степень влияния изменения различных параметров упругих опор на характер колебательного процесса в ходе их многовариантного варьирования. На базе корпуса усовершенствованной вибромашины, функционирующей в составе автоматизированной линии для выбивки крупного вагонного литья, проведен натурный эксперимент с целью подтверждения точности и адекватности полученных ранее результатов численных исследований.
The thesis on competition of a scientific degree of Candidate of technical Science on a specialty 05.02.09 – dynamic and strength of machines. – National Technical University "Kharkiv polytechnical institute", Kharkiv, 2016. This thesis is devoted to the development of approaches, methods and models for investigation of dynamic processes in vibroimpact systems with variable mass of the weight and nonlinear rigidness of elastic supports. In this paper new approach to the accounting of the variable mass influence on the character of dynamical processes in vibroimpact systems was presented. The dependences, that describes the mass change character, based on experimental researches and dependence from dissipated energy were proposed. It was found that the realization of sybharmonical modes became possible because of elastic supports nonlinearity and design features. The realization of subharmonical modes lead to growth of impact interaction force. The approach, based on corresponding values for phase variables at the beginning and the end of period was proposed to periodic solutions search. The criteria for tuning from resonance frequencies that can appear on perturbing force with multiple (partite) frequency were formulated. The machine body designed with taking to the account previously formulated recommen-dations was created. The investigation of stress-strain status was performed and found that improved machine body satisfies strength requirements. The comparison of numerical and experimental data was done. The accuracy and authenticity of numerical investigations was confirmed.

Дисертація на здобуття наукового ступеня кандидата технічних наук за спеціальністю 05.02.09 – динаміка та міцність машин. – Національний технічний університет "Харківський політехнічний інститут", Харків, 2016. Дисертація присвячена розробці підходів, методів і моделей для дослідження динамічних процесів у віброударних системах з урахуванням змінної у часі маси технологічного вантажу та нелінійної жорсткості пружних опор. У роботі викладений новий підхід до урахування впливу змінної маси технологічного вантажу на характер динамічних процесів у віброударній системі. Запропоновані залежності, що описують характер зміни маси як на основі експериментальних досліджень, так і залежно від дисипації енергії при ударі. Встановлено, що внаслідок нелінійної жорсткості пружних опор та залежно від їх конструктивного виконання, можливою є реалізація субгармонійних режимів, що призводить до зростання сили ударної взаємодії. Для пошуку періодичних розв'язків був запропонований новий підхід, що базується на відповідності фазових змінних на початку та в кінці періоду процесу. На основі результатів досліджень сформульовані критерії для відлаштування від резонансу, що може виникнути на кратних (дольних) частотах збудження. З урахування рекомендацій створена удосконалена конструкція корпусу віброударної машини. Проведені дослідження напружено-деформованого стану удосконаленого корпусу вібромашини та встановлено, що він відповідає вимогам міцності. Проведене порівняння отриманих числових результатів досліджень з експериментальними даними. Підтверджена точність і достовірність проведених числових досліджень.
The thesis on competition of a scientific degree of Candidate of technical Science on a specialty 05.02.09 – dynamic and strength of machines. – National Technical University "Kharkiv polytechnical institute", Kharkiv, 2016. This thesis is devoted to the development of approaches, methods and models for investigation of dynamic processes in vibroimpact systems with variable mass of the weight and nonlinear rigidness of elastic supports. In this paper new approach to the accounting of the variable mass influence on the character of dynamical processes in vibroimpact systems was presented. The dependences, that describes the mass change character, based on experimental researches and dependence from dissipated energy were proposed. It was found that the realization of sybharmonical modes became possible because of elastic supports nonlinearity and design features. The realization of subharmonical modes lead to growth of impact interaction force. The approach, based on corresponding values for phase variables at the beginning and the end of period was proposed to periodic solutions search. The criteria for tuning from resonance frequencies that can appear on perturbing force with multiple (partite) frequency were formulated. The machine body designed with taking to the account previously formulated recommen-dations was created. The investigation of stress-strain status was performed and found that improved machine body satisfies strength requirements. The comparison of numerical and experimental data was done. The accuracy and authenticity of numerical investigations was confirmed.

The natural dynamics are not ideal or linear. To understand their complex behavior, we needs to study the nonlinear dynamics in more simple models. This thesis is consist of two main setups. Both setups are simplified models for the behavior occurs in the complex systems. We studied in both systems the same nonlinear dynamics such as higher-harmonics, sub-harmonics, solitary waves,...etc. In Chapter (2), the propagation of nonlinear waves in a lattice of repelling particles is studied theoretically and experimentally. A simple experimental setup is proposed, consisting in an array of coupled magnetic dipoles. By driving harmonically the lattice at one boundary, we excite propagating waves and demonstrate different regimes of mode conversion into higher harmonics, strongly in influenced by dispersion. The phenomenon of acoustic dilatation of the chain is also predicted and discussed. The results are compared with the theoretical predictions of FPU equation, describing a chain of masses connected by nonlinear quadratic springs. The results can be extrapolated to other systems described by this equation. We studied theoretically and experimentally the generation and propagation of kinks in the system. We excite pulses at one boundary of the system and demonstrate the existence of kinks, whose properties are in very good agreement with the theoretical predictions, that is the equation that approaches, under the conditions of our experiments, the one corresponding to full model describing a chain of masses connected by magnetic forces. The results can be extrapolated to other systems described by this equation. Also, In the case of a lattice of finite length, where standing waves are formed, we report the observation of subharmonics of the driving wave. In chapter (3), we studied the propagation of intense acoustic waves in a multilayer crystal. The medium consists in a structured fluid, formed by a periodic array of fluid layerswith alternating linear acoustic properties and quadratic nonlinearity coefficient. We presents the results for different mathematicalmodels (NonlinearWave Equation,Westervelt Equation and Constitutive equations). We show that the interplay between strong dispersion and nonlinearity leads to new scenarios of wave propagation. The classical waveform distortion process typical of intense acoustic waves in homogeneous media can be strongly altered when nonlinearly generated harmonics lie inside or close to band gaps. This allows the possibility of engineer a medium in order to get a particular waveform. Examples of this include the design of media with effective (e.g. cubic) nonlinearities, or extremely linear media. In chapter (4), the oscillatory behavior of a microbubble is investigated through an acousto-mechanical analogy based on a ring-shaped chain of coupled pendula. Observation of parametric vibration modes of the pendula ring excited at frequencies between 1 and 5 Hz is considered. Simulations have been carried out and show spatial mode, mixing and localization phenomena. The relevance of the analogy between a microbubble and the macroscopic acousto-mechanical setup is discussed and suggested as an alternative way to investigate the complexity of microbubble dynamics.
La dinámica natural no es ideal ni lineal. Para entender su comportamiento complejo, necesitamos estudiar la dinámica no lineal en modelos más simples. Esta tesis consta de dos configuraciones principales. Ambas configuraciones son modelos simplificados de el comportamiento que se produce en los sistemas complejos. Estudiamos en ambos sistemas la misma dinámica no lineal como son la generación de armónicos superiores, los sub-armónicos, las ondas solitarias, etc. En elCapítulo (2), se estudia, tanto teórica comoexperimentalmente, la propagación de ondas no lineales en sistemas periodicos de partículas acopladas mediante fuerzas repulsivas. Se propone una configuración experimental simple, que consiste en una matriz de dipolos magnéticos acoplados. Inyectando armónicamente la señal en un extremo, excitamos ondas de propagación y demostramos diferentes regímenes de conversión de modos en armónicos, fuertemente influenciados por la dispersión. También se predice y se discute el fenómeno de dilatación acústica de la cadena. Los resultados se comparan con las predicciones teóricas de la ecuación FPU, describiendo una cadena de masas conectadas por muelles cuadráticos no lineales. Los resultados pueden ser extrapolados a otros sistemas descritos por esta ecuación. Estudiamos también teórica y experimentalmente la generación y propagación de kinks. Excitamos pulsos en la frontera del sistema y demostramos la existencia de kinks cuyas propiedades están en muy buen acuerdo con las predicciones teóricas, es decir, con la ecuación que aproxima bajo las condiciones de nuestros experimentos la correspondiente al modelo completo que describe un cadena de masas conectadas por fuerzas magnéticas. Los resultados pueden ser extrapolados a otros sistemas descritos por esta ecuación. Además, en el caso de una red finita, donde se forman ondas estacionarias, describimos la observación de subarmónicos del armónico principal. En el capítulo (3), estudiamos la propagación de ondas acústicas intensas en un cristal multicapa. El medio consiste en un fluido estructurado, formado por un conjunto periódico de capas fluidas con propiedades acústicas lineales alternas y coeficiente de no linealidad cuadrática. Presentamos los resultados de diferentes modelos matemáticos (ecuación de ondas no lineal, ecuación de Westervelt y ecuaciones constitutivas). Mostramos que la interacción entre la fuerte dispersión y la no linealidad conduce a nuevos escenarios de propagaciónde ondas. El proceso de distorsión de la onda clásica, típico de las ondas acústicas intensas en medios homogéneos, puede ser alterado de forma importante cuando los armónicos generados no linealmente se encuentran dentro o cerca del gap. Esto permite la posibilidad de diseñar un medio con el fin de obtener una forma de onda en particular. Ejemplos de esto incluyen el diseño demedios con no linealidad efectiva (por ejemplo, cúbica) o medios extremadamente lineales. En el capítulo (4), el comportamiento oscilatorio de una microburbuja se investiga a través de una analogía acusto-mecánica basada en una cadena en forma de anillo de péndulos acoplados. Se estudian los modos de vibración paramétrica del anillo pendular excitado a frecuencias entre 1 y 5 Hz. Se han llevado a cabo simulaciones que muestran la presencia de modos espaciales, mixtos y fenómenos de localización. Se discute la relevancia de la analogía entre una microburbuja y la configuración macroscópica acústico-mecánica y se sugiere como una vía alternativa para investigar la complejidad de la dinàmica de microburbujas.
La dinàmica natural no és ideal ni tampoc lineal. Per entendre el seu comportament complex, es necessita estudiar la dinàmica no lineal dels models més simples. Aquesta tesi consisteix en l'estudi de dues configuracions principals, que són models simplificats del comportament que es produeix en els sistemes complexos. Estudiem en ambdós sistemes la mateixa dinàmica no lineal, com és la generació d'harmònics superiors, sub-harmònics, ones solitàries, etc. En el capítol (2), estudiem, tant teòrica com experimentalment, la propagació de les ones no lineals en sistemes periòdics de partícules acoblades mitjançant forces repulsives. Es proposa una configuració experimental simple, que consisteixen en una matriu de dipols magnètics acoblats. En conduint harmònicament la xarxa en un límit, excitemla propagació de les ones i demostrem diferents règims de conversió de modes en harmònics més alts, força influenciada per la dispersió. El fenomen de la dilatació acústica de la cadena també es prediu i es discuteix. Els resultats es comparen amb les prediccions teòriques que descriu una cadena de masses conectades per molls quadràtics no lineals. Els resultats es poden extrapolar a altres sistemes descrits per aquesta equació. Hem estudiat teòrica i experimentalment la generació i propagació de Kinks. Excitem polsos a la frontera del sistema i demostrem l'existència d'Kinks, les propietats desl quals estan en molt bon acord amb les prediccions teòriques, és a dir, de l'equació que aproxima sota les condicions dels nostres experiments la corresponent al model complet que descriu un cadena demasses connectades per forcesmagnètiques. Els resultats es poden extrapolar a altres sistemes descrits per aquesta equació. A més, en el cas d'una xarxa finita, on es formen ones estacionàries, descrivim l'observació de subarmónicos de l'harmònic principal. En el capítol (3), s'estudia la propagació d'ones acústiques intenses en un medi multicapa. El medi consisteix en un fluid estructurat, format per una matriu periòdica de capes de fluid amb l'alternança de propietats acústiques lineals i coeficient de no linealitat de segon grau. Es presenten els resultats per a diferents models matemàtics no lineals (equació d'ones no lineal, equació de Westervelt i les equacions constitutives). Es demostra que la interacciò entre la forta dispersió i no linealitat condueix a nous escenaris de propagació de l'ona. El procés de distorsió en formad'ona clàssica, típica de les ones acústiques intenses en medis homogenis, es pot alterar de manera significativa quan els harmònics generats de forma no lineal es troben dins o a prop del gap. Això obri la possibilitat de dissenyar unmedi per tal d'obtenir una forma d'ona particular. Exemples d'això inclouen el disseny delsmedis amb una no linealitat efectiva (per exemple cúbica), o medis extremadament lineals. En el capítol (4), el comportament oscilatori d'una micro-bombolla és investigat a través d' una analogia acústica-mecànica basada en una cadena en forma d'anell de pèndols acoblats. Es considera l'observació dels modes de vibració paramètriques de l'anell pendular excitat amb freqüències entre 1 i 5 Hz. S'han dut a terme simulacions que mostren la presència de moes espacilas, mixtes i fenòmens de localització. Es discuteix la relevància de l'analogia entre les microbambolles i la configuració macroscòpica acústica-mecànica i es suggereix una formaalternativa per a investigar la complexitat de la dinàmica demicrobombolles.
Mehrem Issa Mohamed Mehrem, A. (2017). Nonlinear acoustics in periodic media: from fundamental effects to applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/80289
TESIS

The first objective of this Dissertation is to present a methodology to calculate analytically the mode shapes and corresponding natural frequencies and determine critical buckling loads of mechanically coupled microbeam resonators with a focus on micromechanical filters. The second objective is to adopt a nonlinear approach to build a reduced-order model and obtain closed-form expressions for the response of the filter to a primary resonance. The third objective is to investigate the feasibility of employing subharmonic excitation to build bandpass filters consisting of either two sets of two beams coupled mechanically or two sets of clamped-clamped beams. Throughout this Dissertation, we treat filters as distributed-parameter systems. In the first part of the Dissertation, we demonstrate the methodology by considering a mechanical filter composed of two beams coupled by a weak beam. We solve a boundary-value problem (BVP) composed of five equations and twenty boundary conditions for the natural frequencies and mode shapes. We reduce the problem to a set of three linear homogeneous algebraic equations for three constants and the frequencies in order to obtain a deeper insight into the relation between the design parameters and the performance metrics. In an approach similar to the vibration problem, we solve the buckling problem to study the effect of the residual stress on the static stability of the structure. To achieve the second objective, we develop a reduced-order model for the filter by writing the Lagrangian and applying the Galerkin procedure using its analytically calculated linear global mode shapes as basis functions. The resulting model accounts for the geometric and electric nonlinearities and the coupling between them. Using the method of multiple scales, we obtain closed-form expressions for the deflection and the electric current in the case of one-to-one internal and primary resonances. The closed-form solution shows that there are three possible operating ranges, depending on the DC voltage. For low DC voltages, the effective nonlinearity is positive and the filter behavior is hardening, whereas for large DC voltages, the effective nonlinearity is negative and the filter behavior is softening. We found that, when mismatched DC voltages are applied to the primary resonators, the first mode is localized in the softer resonator and the second mode is localized in the stiffer resonator. We note that the excitation amplitude can be increased without worrying about the appearance of multivaluedness when operating the filter in the near-linear range. The upper bound in this case is the occurrence of the dynamic pull-in instability. In the softening and hardening operating ranges, the adverse effects of the multi-valued response, such as hysteresis and jumps, limit the range of the input signal. To achieve the third objective, we propose a filtration technique based on subharmonic resonance excitation to attain bandpass filters with ideal stopband rejection and sharp rolloff. The filtration mechanism depends on tuning two oscillators such that one operates in the softening range and the other operates in the hardening range. Hardware and logic schemes are necessary to realize the proposed filter. We derive a reduced-order model using a methodology similar to that used in the primary excitation case, but with all necessary changes to account for the subharmonic resonance of order one-half. We observe that some manipulations are essential for a structure of two beams coupled by a weak spring to be suitable for filtration. To avoid these complications, we use a pair of single clamped-clamped beams to achieve our goal. Using a model derived by attacking directly the distributed-parameters problem, we suggest design guidelines to select beams that are potential candidates for building a bandpass filter. We demonstrate the proposed mechanism using an example.
Ph. D.

This paper describes and investigates the dynamics of a new model of biventricular model for artificial heart which consists of two ferromagnetic masses fixed on springs and subjected to a common electromagnet powered by sinusoidal and square wave voltage sources. The effects of the following control parameters are considered: the external voltage source frequency and amplitude, the ratio between the stiffness coefficients, and the one between the masses. As results in both cases of external excitations, subharmonic oscillations, hysteresis, coexistence of limit cycle and point attractor, permanent bounding are found. The optimal voltage and the ratio between the stiffness coefficients are obtained. Bursting oscillations usable for pulsatile pumping are observed for low frequencies under square wave voltage.

We present a novel micromechanical filter exploiting the subharmonic resonance of order one-half to obtain a center frequency twice the fundamental frequency of the primary resonators, an ideal stopband, and a sharp roll-off. The filter is made up of two clamped-clamped microbeam resonators connected by a coupling beam. We discretize the distributed-parameter system using the Galerkin procedure to obtain a reduced-order model composed of two nonlinear coupled ODEs. It accounts for geometrical and electrical nonlinearities as well as the coupling between these two fields. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We use these equations to determine closed-form expressions for the static and dynamic deflections of the filter. The basis functions in the discretization are the linear undamped global mode shapes of the unactuated filter. The filtering mechanism is based on the exploitation of the interval where the trivial response to subharmonic excitations is unstable. We found criteria to tune the effective nonlinearities of the filter to realize a bandpass filter of an ideal stopband rejection and a sharp roll-off. When these criteria are not met, multivalued responses appear and distort the filter performance.

We study the feasibility of employing subharmonic resonance of order one-half to create a bandpass filter using two clamped-clamped microbeam resonators connected by a weak coupling beam. We discretize the distributed-parameter system using the Galerkin procedure to obtain a reduced-order model composed of two nonlinear coupled ODEs. It accounts for geometric and electric nonlinearities as well as the coupling between these two fields. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We use these equations to determine closed-form expressions for the static and dynamic deflections of the structure. The basis functions in the discretization are the linear undamped global mode shapes of the unactuated structure. We found that we can not produce a single-valued response for small excitation amplitudes. So that, it is impractical to use a single structure made of two mechanically coupled beams excited subharmonically in filtration. But we can use a pair of structures to build a bandpass filter by operating one in the softening domain and the other in the hardening domain and, more importantly, implementing processing logic and hardware schemes. However, the complications brought about by mechanically coupling of two microbeams can be avoided by using a pair of uncoupled beams. This makes the fabrication and modeling processes much easier. Using subharmonic excitation with mechanically uncoupled microbeams to realize bandpass filters is the subject of the next work.

Abstract In the first part of this work an analysis is presented on the dynamics of a two degree of freedom nonlinear mechanical oscillator. The model consists of a rigid body which rests on a foundation with nonlinear stiffness. This body can exhibit both vertical and rocking motions, which are coupled through the nonlinearities only. In the present study, attention is focused on the response of the system under external harmonic excitation of the vertical translation only, leading to conditions of subharmonic resonance of order three. Also, the model parameters are chosen so that its two linear natural frequencies are almost identical (1:1 internal resonance). For this case, the method of multiple time scales is first applied and a set of four coupled odes is derived, governing the amplitudes and phases of approximate motions of the system. Then, determination of approximate periodic steady state response of the oscillator is reduced to solving a set of four nonlinear algebraic equations. It is shown that besides linear and nonlinear single-mode response, two-mode response is also possible, due to the internal resonance. In addition, the stability of the various single- and two-mode periodic responses of the system is analyzed. In the last part of the work, the analytical findings are verified and complemented by numerical results. The main interest lies on identifying the effect of system parameters on the existence and stability of the predicted motions. The results of this study reveal patterns of appearance of these motions, which provide valuable help in the efforts to eliminate them. Finally, direct integration of the original equations of motion reveals the existence of other more complex motions, which coexist with the analytically predicted motions within the frequency ranges of interest.

Abstract An elastic rotating shaft supported by ball bearings may have nonlinear spring characteristics due to clearance and internal damping due to friction between the shaft and the bearings. In such a system, self-excited oscillations appear in the post critical region and nonlinear forced oscillations appear at various resonance points. In this paper, a phenomenon in the neighborhood of the critical speed of the subharmonic oscillation of order 1/3 of a forward whirling mode is discussed. It is clarified that, similar to the case of the subharmonic oscillation of order 1/2 of a forward whirling mode, an entrainment phenomenon appears due to the interplay between self-excited oscillations and forced oscillations.