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The plane motion of two rigid, straight articulated pipes conveying fluid is examined. In contrast to previous work, the flow rate is not taken as constant, but is allowed to have small periodic oscillations about a mean value, as would be expected in a pump-driven system. It is shown that in the presence of such disturbances, both parametric and combination resonances are possible. When the system can also admit loss of stabilty by static buckling or by flutter, it is found that the presence of small periodic disturbances constitutes a destabilizing effect. Floquet theory and converging infinite determinant expansions are used to illustrate a basic difference between systems which lose stability by divergence and those that lose stability by flutter. An algebraic criterion is obtained for the minimum amplitude of flow rate oscillation required for the system to be affected by the presence of small disturbances.

A numerical method which admits discontinuous variable coefficients in the wave equation is employed for handling the wave propagation in composite materials. The resulting algebraic eigenvalue system is simplified by a rank–one matrix modification that reduces the computing time by at least one order of magnitude. The first five modes are accurately and efficiently computed as a demonstration of the method presented.

This paper studies the attitude stability of a satellite consisting of n axisymmetrical bodies spinning at different rates about a common axis, each body containing a nutation damper which moves parallel to the spin axis. The resulting periodic-coefficient differential equations are solved by means of the asymptotic expansion method, thereby permitting a subsequent stability analysis. In the particular case of triple-spin satellites, a numerical parameter analysis based on Floquet theory is carried out to corroborate the analytical stability conditions.

The problem of helicopter mechanical instability is considered for the case where one blade damper is inoperative, and it is shown that if the hub is considered to be nonisotropic, the equations of motion have periodic coefficients which cannot be eliminated. The Floquet transition matrix method is shown to be an effective way of dealing with the nonisotropic hub and nonisotropic rotor situation. Time history calculations are examined and shown to be inferior to the Floquet technique for determining system stability. It is shown that instabilities which occur when one blade damper is inoperative may consist of nearly pure blade motion or they may be similar to the classical mechanical instability.

The transport properties associated with vortex motion in type-II superconductors are reformulated in terms of linear response theory (LRT). In particular, in the dirty limit we establish a simple relation between LRT and the time-dependent Ginzburg-Landau equation. In the more general case of a type-II superconductor with arbitrary electronic mean free path, we present the expressions for the transport coefficients in terms of retarded products. These expressions will be particularly useful in determining the transport properties of clean type-II superconductors.

A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this “variance equation” completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.

In a non-linear system, a disturbance away from a known periodic solution is expressed as μx, where μ is a small parameter. Perturbation of the governing equation in powers of μ leads to a sequence of equations, the first of which is the conventional variational equation for infinitesimal stability, amenable to Floquet theory. Solutions of the succeeding equations contain terms which are products of periodic functions and exponentials, as well as secular terms. The latter are inconsistent with a simple criterion of stability, but may be removed, as illustrated in an example related to the parametric excitation of a pendulum, by a procedure similar to that of Lindstedt. A comparison with the conventional analysis employing only one variational equation determines the magnitude of the disturbances within which infinitesimal stability may be assumed valid.

The stability of the vibrations of a particle constrained to a plane and restrained by two non-identical springs with initial stress is studied by using Floquet theory and perturbations. It is shown that unexpected regions of instability occur due to the non-identical properties of the springs.

This paper presents a new method for the analysis of sampling (or sampled-data) systems. Unlike methods currently in use, it is not necessary to assume that the sampling operations are synchronous, performed at constant rate, and representable by means of an impulse modulator. Several different types of sampling operations are considered in detail, with the analysis proceeding similarly in each case. The paper concludes with a brief study of the stability of sampling systems and a generalization of Floquet's theorem.

Application of Newton’s method to nonlinear vibration problems can lead to a sequence of nonhomogeneous ordinary differential equations with periodic coefficients. The form of the complementary solutions are known from Floquet theory. This paper suggests a method for avoiding “secular terms” that grow with time in the particular solution. The method consists of finding a single periodic solution of the complementary solutions and its adjoint. If the periodic solution exists, a frequency correction can be computed that eliminates secular terms. After the frequency correction, the rest of the particular solution is periodic and can be computed by the infinite determinant method or other numerical methods. In oversimplified terms, the procedure is to find the improved approximation to the period by variation of parameters and the next approximation to the amplitudes by undetermined coefficients which is a simpler computation than variation of parameters.

The allowed electromagnetic modes in the presence of a symmetric array of macroscopic bodies are investigated. They are systematically classified by their behaviour with respect to the different symmetry operations. In the presence of two bodies the modes are required to be even or odd, in the presence of lattices Floquet's theorem is applied, The van der Waals (vdW) energy of the array under consideration is calculated from the average quantum energy of the electromagnetic modes. Since finite boundary conditions are used, no difficulties regarding branch points and different Riemann surfaces are encountered. Closed expressions for the vdW energy in periodic lattices of spheres or cylinders are obtained which can be explicitly evaluated at a reasonable rate of effort.

In this paper, the definition of the primer vector is extended to include nonoptimal as well as optimal trajectories. With this definition, simple tests are developed which determine how a given trajectory can be improved (in terms of velocity requirements). This problem arose in the study of the use of impulsive trajectories to generate approximate adjoint initial conditions for finite thrust vehicles. To do this, the optimum fixed-time impulsive trajectory must be found. However, since many mission analyses are done on an impulsive basis, a wider application is foreseen. Necessary conditions are developed for when an additional impulse can improve the trajectory; how interior impulses of a multi-impulse trajectory can be moved so as to decrease the cost; and when initial and/or final coasts improve the trajectory. In the case of transfers between circular, coplanar orbits a geometric interpretation is given. For the case of an inverse-square gravitational field, the components of the primer vector can be calculated analytically. Using Floquet theory, a convenient form of this solution is presented.

SynopsisThe differential equationin N dimensions is considered, where q(x) is periodic. When N = 1, it is known that the conditional stability set coincides with the spectrum and that these also coincide with two other sets involving eigenvalues of associated eigenvalue problems. These results have been proved by means of the Floquet theory and the discriminant. Here an alternative proof is given which avoids the Floquet theory and which applies to the general case of N dimensions.

A method for the analysis of scattering from periodic structures based on the numerical solution of the integral equations is further developed. Using periodicity (Floquet's theorem), the range of the integral equations is reduced to a single period where the kernels are the Green's functions for periodic arrays. The numerical solution of the integral equations is obtained using the method of moments. Efficient numerical methods for the computation of the periodic Green's functions which allows their rapid evaluation with good accuracy are reported. A new treatment of the singularities which includes the effect of the surface curvature is given. Numerical results for the transverse electric scattering from a conducting surface with a sinusoidal height profile are presented, and several interesting physical phenomena are explored including Brewster angle effects and diffraction grating anomalies.

Bloch waves are special solutions of Schrodinger's equation with a periodic real potential. They are plane waves multiplied by periodic functions. In this paper we prove the existence and completeness of Bloch waves and of the related Kohn-Luttinger waves in unbounded domains for a class of partial differential equations which includes the Schrodinger equation. In addition, we discuss the dependence of these waves and the corresponding eigenvalues on the wave vector of the associated plane wave. The results may be interpreted as the analogs for certain partial differential equations of Floquet's theory for ordinary differential equations or as the determination of the spectral representation of certain periodic Hamiltonian operators.

A class of natural Lyapunov functions of the form \Sigma \lambda_{i} |x_{i}|, \lambda_{i} > 0 , is considered for the stability analysis of a general class of nonautonomous differential equations associated with many physical systems. It is shown that recent results on the existence of solutions to a class of linear inequalities directly yield necessary and sufficient conditions for the existence of a Lyapunov function of the form described.

The characteristic roots and exponents for the five equilibrium solutions in the elliptically restricted problem of three bodies are found numerically with Floquet theory. The method is essentially the same as that employed by Moulton in his classical study of the eighth satellite of Jupiter and recently by Danby in his study of the stability of the triangular points in the elliptic problem. The full range of mass ratio is considered and eccentricities as large as 0.975 are handled satisfactorily. Numerical difficulties that arose in grossly unstable situations were overcome by employing known symmetries of the characteristic equation. For the triangular points the results provide a refinement and extension of Danby's transition curves. The unstable region is found to be divided into three parts. The roots are of a different type in each of these parts. For the collinear points, as could be expected, no region of variational stability was found. For any value of mass ratio, the magnitudes of the real and imaginary parts of the characteristic exponent always increase with larger eccentricity.

An analysis is made of a surface wave traveling axially along the boundary of au infinitely long cylindrical surface that has small periodic variations superimposed upon it. By the application of Floquet's theorem, the acoustic pressure is expressed in terms of an infinite series. For waveguides that have acoustic reactances that are periodic with period L in the axial direction, an infinite system of equations is obtained. Unknown amplitude coefficients are determined by a method of infinite determinants. After the amplitude coefficients are known in terms of one coefficient, the phase variation for the surface wave along the cylindrical axis is calculated by Φ (z) = arctan (ImP/ReP). The general method is illustrated by considering a waveguide with a variable reactance given by χ = χ0 + x cos (2πz/L), where x≪χ0 and z is the axial coordinate. The effects of small changes in the parameters x, χ0, and L are analyzed for an arbitrary surface wave. It is shown that for a surface wave with wave number given ...

Equations of motion of elastic dumbbell satellite and two-mass spring-connected satellite in orbit, using energy and Floquet theory to investigate spinning motion stability

This chapter presents the periodic differential difference equations. It presents a scalar equation x ( t ) = ( t ) x ( t – Δ j ), where the p j are continuous real-valued periodic functions of period w and 0 = Δ 0 1 n . In the usual way, x t = x ( t, θ, ζ, τ ) for θ ∈ [ , 0] and t ≥ τ denotes an element of C [− Δ n , 0], and the translation map U is a compact map of C [− Δ n − 0], with the uniform norm, into itself if w ≥ Δ n . If w n , then a suitable power of U is compact and the results are essentially the same. As U is compact, there is for each characteristic multiplier γ an integer n ( γ ) = dim null space ( U− γl ) m for all m > m 0 ( γ ) and a basis Φ 1 ,…, Φ n ( γ ) for this null space. For any characteristic multiplier γ and associated basis element Φ , the solution x ( t, Φ, 0) is referred to as a solution of Floquet type.

Thin‐film dielectric waveguides with a periodic refractive index, a periodic substrate, or periodic surface are studied. The field is determined from Maxwell's equations using Floquet's theorem. The Brillouin diagram and the interaction regions are investigated. The bandwidth and the attenuation coefficients of the interaction regions are given as a function of the optical wavelength. A number of applications in active and passive integrated optics systems are discussed.

Neutron wave propagation is considered by multigroup diffusion theory in a heterogeneous multiplying system having periodic arrays of line fuel rods. The coupled equations are resolved into uncoupled equations by suitable linear transformation. By means of Green functions for a point or a line neutron source, the behavior of the neutron waves originating from a periodically varying neutron source are investigated in an infinite system. In such a system, the neutron fluxes are represented by means of a Floquet solution relevant to the buckling B which depends on the angular frequency φ of the external neutron source. Neutron waves propagate analogously to classic waves spreading in accordance with Huygens' & Fresnel's principles.

It is shown that in problems involving electron states in finite crystals it is necessary to consider the solutions at band edges more carefully than is usually done. A new form is given for the non-Floquet type solution at a band edge in a one dimensional crystal; it is the derivative in k space of a Floquet solution. Examples of this solution are given for the free-electron model, the Kronig-Penney delta well model, and the tight binding model.

also commonly known as the equation of the elliptic cylinder functions, is too well known to require any introduction. Its solutions govern problems of the greatest diversity in astronomy and theoretical physics, and have accordingly been the subjects of a vast number of investigations.t The differential equation as such depends upon two independent parameters, designated in the form written above by A and Q. In the present discussion these are to be taken real but are to be numerically unrestricted except that at least one is to be large. The variable will be permitted to range over the complex plane. Since the coefficient of the differential equation is an even simply periodic analytic function of z, it is known from Floquet's theory of such equations that the solutions are in general of the structure

1.1. This paper is a theoretical investigation in the real domain of the existence of subharmonic solutions of non-linear differential equations of the formwhere F is analytic and of least period 2π/ω in t; ε = (ε1, …, εn) is small; and F(x, ẋ, 0, t) is not linear in x and ẋ.

The necessary and sufficient conditions for the completeness of the system of Floquet solutions, the eigenfunctions and associated functions of the displacement operator, over a period of the coefficient in metrics C1 and C in the space of all solutions of the equationy (t)=q(t) y (t− nω)+ay (t−mω), are obtained. Here q(t) is a continuous function of the periodω.

It is shown that the Floquet factor ${e}^{\mathrm{ik}(E)a}$ is analytic in the upper half complex energy plane, thus enabling a set of four dispersion relations to be derived from this expression as a direct result of the application of Cauchy's theorem. These relations are characterized by their ability to relate the wave number $k$ at one energy to the wave number at all others. In particular, the imaginary part of the wave number ${k}_{i}$ in the forbidden gap may be equated to an integral of a function of the real part of the wave number ${k}_{r}$ over allowed energies. As an application of these dispersion relations a theorem regarding the location of the branch points has been established.

We determine all third order homogeneous linear differential equations with periodic coefficients and only periodic solutions. The method extends tonth order equations. As an application, we show that the Laguerre-Forsyth canonical form cannot be used for global investigations i projective differential geometry.

An analysis is given of an electromagnetic system composed of a rectangular waveguide in tandem with a corrugated waveguide which feeds a flat, corrugated surface of arbitrary length terminated by a ground plane, whose length is also arbitrary. An improved procedure of field determination is used which combines Floquet's theorem and the variational principle, thus revealing an additional requirement on the corrugation geometry. Factors influencing a match at the feed mouth, and satisfactory launching of the surface wave are discussed. The degree of suppression of the feed radiation is given in db as a function of the geometry of the system. Approximate radiation patterns are derived for two cases, (a) when the system is terminated by an infinite ground plane, and (b) when the system is terminated by a finite ground plane. For the latter case, an upper bound on the tilt angle of the main beam and a lower bound on its beamwidth result from an approximate theory. For both cases, the Hansen-Woodyard endfire relation is found to provide beam sharpening even when the feed radiation is considered. The presence of higher order surface modes, their effect, and their elimination are discussed. Comparison of the theory with experiment is reasonably good.

Introduction. In 1960, a paper by Hale and Stokes [2] established that a sufficient condition for the strong orbital stability of a nondegenerate (k + 1)parameter family of periodic solutions of an n-dimensional autonomous ordinary differential equation (ODE) was that n (k + 1) characteristic exponents of the related variational equation have negative real parts, uniformly in the parameters. By strong orbital stability is meant asymptotic stability with asymptotic amplitude and phase. And a nondegenerate family is one in which differentiation of the family with respect to the parameter generates k + 1 independent functions which span the generalized eigenspace corresponding to the characteristic exponent zero. In addition, a number of results were obtained which described the preservation of various types of stability in the presence of various classes of perturbations. An essential element in the proof was the existence of a special transformation, which provided a "natural" system of coordinates for the analysis of the behavior of solutions near the given family. The existence of this transformation required the "nondegeneracy" of the family and used the Floquet theory for linear ODE extensively. In 1963, these results were extended to functional differential equations (FDE) by Hale [1] (see also Kato [3]), provided that the equations were already in the "natural" system of coordinates for that family. But since as yet no "natural" system of coordinates has been introduced even in relation to a single limit cycle of an FDE, much less a (k + 1)-parameter family of periodic solutions, this result was not as generally applicable as were the earlier results on ODE. However, in 1963, Stokes [4] discussed the strong orbital stability of a limit cycle of an FDE, and in 1968, [5] extended these results to a family of periodic solutions of an FDE, using methods which did not require the introduction of a special coordinate system. These papers thus provide an extension of the theorems on sufficient conditions for strong orbital stability of periodic solutions of an ODE to similar theorems concerning periodic solutions of an FDE. The present paper then complements these, by discussing the behavior under various types of perturbations of a strongly orbitally stable family of solutions of an FDE. Again, the methods contained herein are elementary and do not use any coordinate transformations. Because of this, the presentation is in terms of a flow on a Banach space, rather than specifically in terms of an FDE. Thus, the results are applicable to any flow for which the basic hypotheses can be verified. In particular, this would include asymptotically stable time-indepen

1·1. Let van der Pol's equation be taken in the formwhere ε1, ε2, k1 and k2 are small, and ω ≠ 0 is a constant, rational or irrational, independent of them.

A stability criterion for feedback systems with a linear time-invariant forward loop and a periodically time-varying gain in the feedback loop is developed. As most recent frequency-domain stability criteria, it requires the existence of a multiplier having certain properties. A graphic technique is given to determine whether or not this multiplier exists. The proof of the criterion is based on the Floquet theory and the classical theory of Toeplitz forms. Examples to illustrate the criterion are included.