Galerkin's procedure for nonlinear periodic systems

Urabe, Minoru

doi:10.1007/bf00284614

Cited by 307 publications

(146 citation statements)

References 2 publications

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“…In relative time it correlates to a period of free oscillations. Upon slow transition (time of transition makes some periods of cross oscillations), at a small deviation of excitement frequency from the resonant one and at low external friction the solution of the equation (2) is searched by an averaging method, also known as a projective method, in the form of the sum of harmonicas with slowly changing coefficients [4]:…”

Section: Fig2 -Stiffness Characteristic Of a Magnetic Springmentioning

confidence: 99%

Method of the unbalanced rotor critical frequency overcoming by means of the operated magnetic spring

Лукин

Popov

Skubov

et al. 2014

MATEC Web of Conferences

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Abstract.When passing critical rotational frequency of a rotor with significant imbalance there can be crossoscillations with large amplitudes. The algorithm of critical rotational frequency overcoming which allows reducing these oscillations considerably is described.It is offered to use a differential magnetic spring as an elastic support with operated stiffness.When passing critical rotational frequency of a rotor with significant imbalance there can be cross-oscillations with large amplitudes. The algorithm of critical rotational frequency overcoming which allows reducing these oscillations considerably is offered. For this aim it's proposed that elastic supports have two stiffness coefficients: working and increased, with the corresponding two values of critical frequency. It is offered to use a differential magnetic spring as an elastic support with operated stiffness. The magnetic spring consists of the permanent magnets connected by magnetic conductors. The schematic diagram of a differential magnetic spring is shown in Fig. 1.Its important advantage is capability to change easily the equivalent stiffness of an elastic support. It allows to operate the value of stiffness during the rotor acceleration so that to achieve reduction of oscillations amplitude, and, therefore, stresses in the supports up to the acceptable values even at big imbalance and to provide the acceleration to the over-critical frequency of rotation. Technically it is quite possible to change (to reduce) stiffness of a magnetic spring (to increase an initial gap) quickly. If to make fastening of magnets by means of a latch, atits release magnets can be moved apart under the force of their interaction, thereby increasing an initial gap for rather small time.The magnetic support (spring) contains three permanent magnets, side magnets are fixed and the middle one, which is connected with the shaft train, can move in the vertical direction, and is itself established on the amortized support. Dependences of force ‫ܨ‬ሺ‫ݔ‬ሻ acting on a middle permanent magnet, on vertical displacement ‫ݔ‬ at various values of an initial gap ݀ are given in Fig. 2. These dependences are received by analysis of a stationary magnetic field in system of magnets with the subsequent calculation of force of their interaction. Fig.2 -Stiffness characteristic of a magnetic spring Varying the size of an initial gap between extreme magnets, it is possible to change the size of equivalent stiffness (Fig. 2), thereby changing a skeletal curve. The established oscillations of a platform at the fixed position of magnets and, therefore, amplitude-frequency characteristics can be found by solving the shaft train dynamics equations by a harmonic balance method. In Fig. 3 dependence of a platform oscillations amplitude on the rotation frequency is shown at various values of an initial gap in a magnetic spring (amplitude-frequency characteristics).

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Section: Fig2 -Stiffness Characteristic Of a Magnetic Springmentioning

confidence: 99%

Method of the unbalanced rotor critical frequency overcoming by means of the operated magnetic spring

Лукин

Popov

Skubov

et al. 2014

MATEC Web of Conferences

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“…The method described here has been justified mathematically for isolated periodic solutions (see Urabe (1965) and Stokes (1969)), but further work is needed for singular solutions (periodic orbits which are members of a natural family) as in this study. The indetermi.nancies are removed here by applying constraints.…”

Section: Comparison With Numerical Integrationmentioning

confidence: 99%

“…good discussion of the method, as applied to isolated periodic solutions has been given by Urabe (1965). Numerical examples are given by Urabe and Reiter (1966).…”

Section: Introductionmentioning

confidence: 99%

Periodic Orbits in Trigonometric Series

Carpenter

1970

Periodic Orbits, Stability and Resonances

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A method is given for the study of families of periodic motion using trigonometric series to represent the individual solutions. The continuous deformations of a periodic orbit along a family are represented by the variations of the trigonometric coefficients with respect to a parameter.For the applications the series are truncated, while the coefficients and their variations are determined numerically. In this form, the continuation with respect to the parameter is given by a mapping f: R" R" of the space of coefficients into itself. The values of the coefficients are then improved by another mapping which is a contraction operator in some neighborhood of the fixed point representing the solution.The method is applied to the natural families defined by Wintner and to the families of the first and second kinds of Poincare in the restricted problem of three bodies.

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“…The development of computer-assisted proofs to two-point boundary value problems (one-dimensional case) has pioneered by Kantorovich [8] and Urabe [27]. The works of McCarthy and Tapia [14] and Kedem [9] followed.…”

Section: Introductionmentioning

confidence: 99%

Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains

Takayasu

Liu

Oishi

2013

NOLTA

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Abstract:In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the NewtonKantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

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Galerkin's procedure for nonlinear periodic systems

Cited by 307 publications

References 2 publications

Method of the unbalanced rotor critical frequency overcoming by means of the operated magnetic spring

Method of the unbalanced rotor critical frequency overcoming by means of the operated magnetic spring

Periodic Orbits in Trigonometric Series

Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains

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