Figure 4 - uploaded by Richard H. Rand
Content may be subject to copyright.
Growth and overlap of the primary resonance bands in the nonlinear QP Mathieu equation for δ = 0 . 20, ω 1 = 1, and ω 2 = 1 . 65. The numerically generated solutions are strobed at times t n = n 2 π /ω 1 and t n = n 2 π/ω 2
Source publication
In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + [ + (cos 1 t + cos 2 t)] x + x3 = 0.We assume that 1 and that the coefficient of the nonlinearterm, , is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the u...
Similar publications
Background
This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator.
Objective
The goal is to determine (a) the system par...
Citations
... Recent developments for single as well as cou-pled Mathieu equations can be found in [8], also utilizing symplectic properties [27,28]. The quasi-periodic Mathieu equation, the simplest time-periodic system with multiple frequency excitation, was investigated in [31,45], showing very rich dynamics. In the present contribution dealing with a set of coupled Mathieu equations, such rich dynamics were not observed, which is why the treatment by singular perturbation leads to still compact expressions. ...
Introducing a time-periodicity into a system parameter leads to parametric excitation, which in general, may cause a parametric resonance with exponentially increased vibration. Applying a parametric excitation but carefully tuning its frequencies to multiple parametric anti-resonance frequencies is investigated here. The parametric excitation here is realized by an open-loop control at the system boundary that allows for an energy flow into or from the system. A parametric anti-resonance successfully triggers an energy transfer between specific vibration modes of the system and occurs in systems with at least two degrees of freedom. Such an energy transfer increases the overall dissipation of kinetic energy of a lightly damped system. This contribution presents an approach to accelerate the mitigation of transient vibrations by applying a multi-frequency parametric excitation with two or more parametric anti-resonance frequencies. The potential application in a MEMS sensor arrangement consisting of two and more coupled flexible beams exemplifies the method. Starting from the minimum system with two degrees of freedom, the averaging method is applied to analyze the transient slow flow, leading to an analytical approximation of the transition time response of a pulsed multi-frequency parametric excitation system. For a specific example, a reduction of 96.7% of the transient vibrations is achievable.
... Recent developments for single as well as coupled Mathieu equations can be found in [6], also utilizing symplectic properties [24,25]. The quasi-periodic Mathieu equation, the simplest time-periodic system with multiple frequency excitation, was investigated in [28,40], showing very rich dynamics. In the present contribution dealing with a set of coupled Mathieu equations, such rich dynamics were not observed, which is why the treatment by singular perturbation leads to still compact expressions. ...
Introducing a time-periodicity into a system parameter leads to parametric excitation, which in general, may cause a parametric resonance with exponentially increased vibration. Applying the same parametric excitation but carefully tuning its frequencies to multiple parametric anti-resonance frequencies is investigated here. The parametric excitation is an open-loop control at the system boundary that allows for an energy flow into or from the system. A parametric anti-resonance successfully triggers an energy transfer between specific vibration modes of the system and occurs in systems with at least two degrees of freedom. Such an energy transfer increases the overall dissipation of kinetic energy of a lightly damped system. This contribution presents an approach to accelerate the mitigation of transient vibrations by applying a multi-frequency parametric excitation with two or more parametric anti-resonance frequencies. The potential application in a MEMS sensor arrangement consisting of two and more coupled flexible beams exemplifies the method. Starting from the minimum system with two degrees of freedom, the averaging method is applied to analyze the transient slow flow of the underlying physical system. This procedure allows for an analytical approximation of the transition time of a pulsed multi-frequency parametric excitation. For a specific example, a reduction of 96.7% of the transient vibrations is achievable.
... The damped nonlinear QP Mathieu equation contains two harmonic excitation frequencies, the parametric excitation frequency ω and small excitation frequency ω d , which are incommensurate. Zounes and Rand [19] used Lie transform with elliptic functions to investigate the global behavior of the undamped equation of Eq. (4). The same equation with 2:2:1 resonance was studied by Abouhazim et al. [20] using the double-step MMS. ...
... K A is the tangent matrix, R is the corrective vector, and R ω , R ωd , R α , and R β are vectors due to unit changes of Δω, Δω d , Δα, and Δβ, respectively. The number of incremental unknowns (ΔA, Δω, Δω d , Δα, Δβ) is greater than the number of equations in Eq. (19). If one wants to perform frequency response studies for a given excitation frequency ω d and given constant parametric excitation magnitudes α and β, i.e., ω d , α, and β are fixed as constants, which implies Δω d = 0, Δα = 0, and Δβ = 0, respectively, then Eq. (19) is reduced to ...
Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu's equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method with two time scales. The damped nonlinear QP Mathieu's equation contains two incommensurable harmonic excitation frequencies, one is a small frequency while the other nearly equals to twice the linear natural frequency. It is found that Fourier spectra of QP solutions of the equation consist of uniformly spaced sidebands due to cubic nonlinearity. The IHB method with two time scales, which relates to the two excitation frequencies, is adopted to trace solution curves of the equation in an automatical way and all frequencies of solutions and their corresponding amplitudes. Effects of parametric excitation are studied in detail. The stability of a QP solution is evaluated from the Floquet theory via examining the perturbation superposing on the QP solution. Three types of QP solutions can be obtained from the IHB method, which agree very well with the results from numerical integration. However, the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions since the ratio of the small frequency to the linear natural frequency of the first reduced-modulation equation is nearly 1 in the second perturbation procedure, while the other two types of QP solutions from the IHB method do not need the ratio. Furthermore, the results from the double-step MMS are different from those from numerical integration and the IHB method.
... In [20] the prerequisites for existing of at least one periodic solution for the Mathieu-Duffing equation has been presented. And, in [21][22][23][24][25], different combinations of perturbation methods are used in other existing studies to find the quasi-periodic solutions for these equations. Although non-homogenous Mathieu equations have been solved analytically before, as in [26], yet, the more general form of these equations which can be used in wider range form of problems, and has both consistent and sinusoidal term in the non-homogenous part, to the best of our knowledge, has not been tracked before. ...
Purpose Spinal cord injury (SCI) subjects use passive orthosis to assist them with the postural balance in daily life. Vertical vibrations are used to improve the postural balance of theses patients, mainly in terms of whole-body-vibrations, as in therapies. It is noteworthy to understand in what situation these patients are able to tolerate vertical vibrations while wearing orthosis. The main focus of this paper is to model vertical vibrations in the postural balance of arm free standing of the paraplegic subjects who use passive orthosis, and, mathematically understand the stability limits for these vibrations.
To find the stable amplitudes and frequencies, equations of motions of the system are translated into non-homogeneous Mathieu equations, which are then analytically solved and the results are numerically plotted. In addition, the effect of shoe stiffness on the amplitude and frequency of the vibrations which violate stability are studied. The results show to be sure of the stability of the paraplegics in vertical vibrations, the shoe sole of the orthosis should have high values of stiffness, and this will guarantee bigger ranges of stable vibrations and safer stance for the patients. Using the results, one can interpret how a paraplegic subject with passive orthosis may react to vertical perturbations, meaning, what ranges of frequencies and amplitudes of perturbations is tolerable without losing stability. This study is one of the necessities in designing new passive orthosis with more stability when exposed to perturbations.
... This is because Eq. (125) is linear. The effect of adding nonlinear terms to Eq. (125) was considered in Ref. [48], where the following nonlinear quasiperiodic Mathieu equation was studied: ...
... In Ref. [48], a perturbation scheme using Jacobi elliptic functions was combined with Kolmogorov-Arnold-Moser theory to determine the global behavior of Eq. (128) ...
This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalisations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.
... In this paper, the effects of asphericity and orbital eccentricity on the widespread chaos in the rotation of the secondary asteroid in the binary systems due to the overlapping of the first order resonance is investigated comprehensively. This problem is studied using the well-known analytical and numerical methods: Chirikov [14,15] and Poincare section [16] methods, respectively. This paper is organized as follows: In the next section, rotational dynamics of the secondary asteroid is derived in Hamiltonian form. ...
The chaotic behavior of the secondary asteroid in a system of binary asteroids due to the asphericity and orbital eccentricity is investigated analytically and numerically. The binary asteroids are modeled with a sphere-ellipsoid model, in which the secondary asteroid is ellipsoid. The first order resonance is studied for different values of asphericity and eccentricity of the secondary asteroid. The results of the Chirikov method are verified by Poincare section which show good agreement between analytical and numerical methods. It is also shown that asphericity and eccentricity affect the size of resonance regions such that beyond the threshold value, the resonance overlapping occurs and widespread chaos is visible.
... Specific resonances are examined in Rand and Morrison [7] and Rand et al [6]. In Sah et al [9] a form of the quasiperiodic Mathieu equation is derived and examined in the context of the stability of motion of a particle constrained by a set of linear springs, and finally in Rand and Zounes [14] a nonlinear version of the quasiperiodic Mathieu equation is studied. In these papers a combination of numerical and analytical techniques are used to examine how resonances between the internal driving frequencies lead to a loss of stability in solutions, and it is very much in the spirit of these papers that we present the following work. ...
In the following we consider a 2-dimensional system of ODE's containing
quasiperiodic terms. The system is proposed as an extension of Mathieu-type
equations to higher dimensions, with emphasis on how resonance between the
internal frequencies leads to a loss of stability. The 2-d system has two
`natural' frequencies when the time dependent terms are switched off, and it is
internally driven by quasiperiodic terms in the same frequencies. Stability
charts in the parameter space are generated first using numerical simulations
and Floquet theory. While some instability regions are easy to anticipate,
there are some surprises: within instability zones small islands of stability
develop, and unusual `arcs' of instability arise also. The transition curves
are analyzed using the method of harmonic balance, and we find we can use this
method to easily predict the `resonance curves' from which bands of instability
emanate. In addition, the method of multiple scales is used to examine the
islands of stability near the 1:1 resonance.
... In [20], the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation has been investigated. Belhaq and fahsi [21] showed that in the vicinity of the 2 : 1 and 1 : 1 resonances in a fast harmonically excited Van Der Pol-Mathieu-Duffing oscillator, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. ...
... The small parameter techniques of generalized averaging and multiple-scale perturbation are employed to obtain a solution. Rand and his associates [20,27,28] analyze a linear homogeneous quasiperiodic Mathieu equation via several methods such as numerical integration, Lyapunov exponents, regular perturbation, Lie transform perturbation and harmonic balance. ...
This paper describes a comprehensive non-linear multiphysics model based on the Euler–Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one.
... The present work involves quasiperiodic parametric excitation, that is, systems which involve differential equations with quasiperiodic coefficients. Here the system which is comparable to Mathieu's equation (1) has been called the quasiperiodic Mathieu equation [8][9][10][11][12][13][14][15][16]: ...
... Although many of the regions are too small to show up in Fig. 3, which was obtained by numerical integration, the method of harmonic balance is able in principle to reveal infinitely many such regions, in the limit as the truncation N goes to infinity. Each instability region may be associated with a distinct resonance between the x and y motions (10), (11) which force the stability Equation (14)- (15) and the z equation (9). This may be seen most clearly in the perturbation approach which results in a study of Eq.(50). ...
The dynamics of an autonomous conservative three degree of freedom system which exhibits autoparametric quasiperiodic excitation is investigated. The system is a generalization of a classical system known as the “particle in the plane”. The system exhibits a motion, the z=0 mode, whose stability is governed by a linear second order ODE with quasiperiodic coefficients. The behavior of the latter ODE is studied by using three different methods: numerical integration, harmonic balance and perturbation methods.
... There are several publications dealing with single or coupled differential equations having a time-periodic coefficient, i.e. [4], [34], [12], [5], [23], and the literature cited therein, and more recent studies [25], [36], [15] or [26]. The main focus of these works is the destabilising effect of a parametric excitation. ...
Stability investigations on vibration cancelling employing the concept of actuators with general stiffness elements are presented.
Systems with an arbitrary number of degrees of freedom with linear spring- and damping elements are considered, that are subject
to self-excitation as well as parametric excitation by stiffness variations with arbitrary phase relations. General conditions
for full vibration suppression are derived analytically by applying a singular perturbation technique. These conditions naturally
lead to the terms of parametric resonance and anti-resonance and enable a stability classification with respect to the parametric
excitation matrices and their symmetry properties. The results are compared to former investigations of systems with a single
or synchronous stiffness variation in time and geometrical interpretations are given. These basic results obtained can be
used for design of a control strategy for actuators with periodically actuated stiffness elements and arbitrary phase relations.
