Complex Variable Equations Research Articles

Wind-induced vibration energy harvesters under base excitation: Analytical and numerical analysis

In this paper, analytical analysis of a wind-induced vibration energy harvester used for the bacterial disinfection of water is mainly considered. The complexification-averaging (CA) method is utilized to analytically calculate the results of the harvester, the model equations of the harvester satisfied are reduced into a set of complex variable equations. The relationship between the displacement amplitude and voltage amplitude is derived under the steady-state case. The amplitudes and phases of both the displacement and voltage are obtained. The analytical solutions of the harvester are then verified by direct numerical simulation. The response mechanism under different parameters is discussed. The frequency island and multi-valued phenomena are found. Meanwhile, it is found that a higher wind speed corresponds to a larger displacement amplitude and voltage. In addition, the RMS voltage and average power are studied. Both have the same variation rules under different excitation amplitudes. Under the same coupling coefficient, load resistance and piezoelectric patch capacitance, the lower the frequency, the easier it is to obtain the larger voltage output. However, the load resistance has the opposite effect on both of them. Finally, to obtain maximum output in a wider frequency region, the optimization and choice of the parameters is studied, and the critical boundary can be identified.

Retractable Undercarriage Design for a Quadcopter

In the current paper, design of a foldable undercarriage for quadcopter is suggested and investigated. The deploying/retracting mechanism comprises a four-bar linkage. The mechanism lowermost configuration forms a kinematic lock. A solution to complex variable equations has been worked out to put together the mechanism geometry. In addition, a 3D CAD model has been developed to visualize the design better.

Design of $$C^2$$ C 2 algebraic-trigonometric pythagorean hodograph splines with shape parameters

In this paper, we construct \(C^2\) Algebraic-Trigonometric Pythagorean Hodograph (ATPH) splines by solving a non-linear system of equations in complex variables. We compare these splines, which depend on several shape parameters, with their polynomial PH counterpart as well as with the well-known \(C^2\) cubic B-splines. We finally present criteria for choosing the free shape parameters based on the minimization of certain fairness functionals.

Product Vectors in the Ranges of Multi-Partite States with Positive Partial Transposes and Permanents of Matrices

In this paper, we consider a system of homogeneous algebraic equations in complex variables and their conjugates, which arise naturally from the range criterion for separability of PPT states. We examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions. This gives us possible upper bounds of ranks of PPT entangled edge states and their partial transposes. We will focus on the multi-partite cases, which are much more delicate than the bi-partite cases. We use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion.

On two functional equations originating from number theory

Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91–98 and in Appl. Math. Lett. 21 (2008) 974–977 to equations in complex variables and quaternions, we find general solutions of the equations. We also obtain the stability of the equations.

Complex variable equations and the numerical solution of harmonic problems for piecewise-homogeneous media

Complex variable boundary integral equations are derived using of holomorphicity theorems for plane harmonic problems concerning unit structures with inclusions, pores and lines of discontinuity of the potential and/or the flow. Unlike the method of analytical elements, the equations cover problems in which discontinuities in the potential, flow and conductance can simultaneously be encountered at the contact points. Versions of the equations are given for connected half planes and for periodic and biperiodic problems. Formulae are obtained which determine the effective impedance tensor of the equivalent homogeneous medium in cases when the unit structure is biperiodic or when the representative volume of a structured medium is identified with the basic cell of a biperiodic system. Recurrence quadrature formulae are proposed which enable one to solve the resulting equations effectively using the complex variable boundary element method. They indicate the computational advantages of using the complex variable method compared with the real variable method: the three integrals appearing in the resulting equations are evaluated analytically in the case of linear elements (regular and singular) with the densities approximated using algebraic polynomials of arbitrary degree. In the case of elements (regular and singular) in the form of an arc of a circle, only one integral requires numerical integration when the densities are approximated using complex trigonometrical polynomials of arbitrary degree. It is emphasized that the combination of the linear and curved boundary elements which have been developed enables the smooth part of a contour to be approximated while retaining the continuity of the tangent and avoiding the complications which arise when the smoothness of the approximation of a contour is ensured using conformal mapping. Examples are presented which illustrate the computational merits of the method developed. They show a sharp increase in accuracy (by orders of magnitude) when curved elements are used for the curvilinear parts of a contour and when terminal elements are used to calculate the flow intensity coefficient at singular points (the crack tips the vertices of angular notches and the common vertices of the units of the medium).

Dynamical stability of finite anisotropic panels with elliptical cutouts and cracks

An approximate analysis for dynamical stability of anisotropic finite panels with centrally located elliptical cutouts is presented. The analysis is divided into two parts: a plane stress analysis and a stability analysis. The plane stress distribution is determined by using Lekhnitskii’s complex variable equations of plane elastostatics combined with a Laurent series approximation constructed by the conformal mapping and a boundary collocation method. Its solutions satisfy the conditions along the interior boundary and at a discrete number of points along the exterior panel ones. The stability analysis is conducted by using the differential equations which result from the Hamilton’s principle and the classical plate theory. The relation of vibration frequency, load parameter and stability of panels is investigated by solving the fundamental equations using separation of variables, so as to obtain the critical loads. Finally, comparisons with documented experimental results and finite element analysis are made. Results of a parameter study are presented.

Shear and compression buckling analysis for anisotropic panels with elliptical cutouts

An approximate analysis for buckling of biaxial- and shear-loaded anisotropic panels with centrally located elliptical cutouts is presented. The analysis is composed of two parts: a prebuclling analysis and a buckling analysis. The prebuckling solution is determined using Lekhnitskii's complex variable equations of plane elstostatics combined with a Laurent series approximation and a boundary collocation method. The buckling solution is obtained using the principle of minimum potential energy. A byproduct of the minimum potential energy equation is an integral equation that is solved using Gaussian quadrature

Using the method of dual quadratic solutions to solve systems of polynomial equations in the complex domain

In general, the dual solutions are applied in the branch and bound scheme to solve the optimization problem. Of special interest, however, are problems that can be reduced to a quadratic problem with {Omega}-property. In this paper, we consider one such problem, namely the problem of solving a system of polynomial equations in complex variables.

Zur Darstellung von Lösungen bei Systemen partieller Differentialgleichungen, insbesondere be! isolierten Singularitäten

For specific systems of elliptic partial differential equations and the associated systems of formally hyperbolic equations in complex variables the solutions can be given using certain differential operators. These operators map holomorphic functions into the set of solutions of the system. In this article we consider the relation between a solution and the holomorphic functions generating it. In particular we determine the generators of the zero solution. Finally we give a general representation theorem for the solutions defined in the neighbourhood of an isolated singularity.

Doubly fed induction machine state variables model and dynamic response

A state variables model in terms of d-q component displacements is deduced from linearized complex variable equations describing small-signal dynamic performance of the doubly fed induction machine. A state variables model in terms of magnitude and phase displacements of both stator and rotor voltages is deduced and studied. Transfer functions are also described. Study of the dynamics shows that the regions of instability are influenced by the variations of speed, of RMS value, and of phase angle of rotor applied voltage. This state variables model in polar coordinates eliminates the computations for the intermediate d-q components. >

Equation in complex variable of axisymmetrical deformation problems for a general shell of revolution

In this paper, the equation of axisymmetrical deformation problems for a general shell of revolution is derived in one complex variable under the usual Love-Kirchhoff assumption. In the case of circular ring shells, this equation may be simplified into the equation given by F.Tolke(1938)[3], R.A.Clark(1950)[4]and V.V.Novozhilov(1951)[5]. When the horizontal radius of the shell of revolution is much larger than the average radius of curvature of meridian curve, this equation in complex variable may be simplified into the equation for slander ring shells. If the ring shell is circular in shape, then this equation can be reduced into the equation in complex variable for slander circular ring shells given by this author (1979)[6]. If the form of elliptic cross-section is near a circle, then the equation of slander ring shell with near-circle ellipitic cross-section may be reduced to the complex variable equation similar in form for circular slander ring shells.

Approximated-asymptotic solution of the complex variable equation of toroidal shells under axial symmetric loads

The series-approach and the asymptotic-approach are usually used to solve the complex variable equation of the toroidal shells under axial symmetric loads. As is known, the convergence of the series-solution is good only for small values of\(\mu = \sqrt {12(1 - v^2 )} a^2 /(r_0 h)\). On the other hand, the convergence of the asymptotic solution is good only for large values of μ. In this paper, based on an earlier work[1], a new approach which may be called the approximated-asymptotic solution has been developed and it is valid for both small and large values of μ. It is shown that the results of the approximated-asymptotic solution for toroidal shell with μ=0.5 coincide very well with those of the series-solution, while the results of the asymptotic solution for this value of μ are not as good, and the results of the approximated-asymptotic solution for μ=15 agree with those of the asymptotic solution.

A graphics typesetting language

We present a new programming language, IDEAL, in which two-dimensional figures can be expressed. The language is intended to work with existing text-formatting systems so that text and figures can be typeset at the same time. The building block for IDEAL programs is called a “box” it shares some features with procedures and some with records in general-purpose programming languages. A box includes a system of constraints (in this incarnation, equations in complex variables) that declares the relative positions of its significant points and requests for actions to be performed at those points. A box is called by adding enough constraints to the system in its definition that its significant points can be solved for uniquely. A box call may also include additional actions to be performed during the call. The notions of drawing a line using a pattern and texturing a polygonal area follow directly from the mechanism for defining and calling boxes. Users may also ask for a box to be “opaque” so that it blots out pieces of picture that it covers. Finally, two commands embody the idea of sketching several pictures on different parallel planes, then merging them into a single picture. We use these when erasing to create one part of a picture would destroy another part that we want. We discuss good algorithms for implementing IDEAL when the constraints are expressed as a special kind of nonlinear system and the pictures are composed of straight lines and circular arcs. The language has been implemented, and was used to produce the paper.

A graphics typesetting language

We present a new programming language, IDEAL, in which two-dimensional figures can be expressed. The language is intended to work with existing text-formatting systems so that text and figures can be typeset at the same time. The building block for IDEAL programs is called a “box” it shares some features with procedures and some with records in general-purpose programming languages. A box includes a system of constraints (in this incarnation, equations in complex variables) that declares the relative positions of its significant points and requests for actions to be performed at those points. A box is called by adding enough constraints to the system in its definition that its significant points can be solved for uniquely. A box call may also include additional actions to be performed during the call. The notions of drawing a line using a pattern and texturing a polygonal area follow directly from the mechanism for defining and calling boxes. Users may also ask for a box to be “opaque” so that it blots out pieces of picture that it covers. Finally, two commands embody the idea of sketching several pictures on different parallel planes, then merging them into a single picture. We use these when erasing to create one part of a picture would destroy another part that we want. We discuss good algorithms for implementing IDEAL when the constraints are expressed as a special kind of nonlinear system and the pictures are composed of straight lines and circular arcs. The language has been implemented, and was used to produce the paper.

Induction machine transfer functions and dynamic response by means of complex time variables

The symmetry of the induction machine can be exploited to obtain general closed form expressions for the small signal transfer functions describing speed, voltage, frequency, or load perturbations by utilizing the complex time variables introduced by Ku and Lyon in the 1950's. After a brief introduction to complex variables, the linearlzed complex variable equations describing small signal dynamic performance are presented. These equations are used to obtain transfer functions in which the effects of excitation level are isolated in the gain factors. The speed and frequency dependence of the poles and zeros is expressed in closed form employing a useful non-dimensional parameter system. To illustrate the application of these results, the dynamic behavior of the induction machine without feed-back control is analyzed. It is shown that the general dynamic response can be characterized by the non-dimensional loop gain and stator frequency. A set of general non-dimensional root loci are presented which permit rapid estimation of the relative stability (dominant eigenvalues) and the frequency of minimum damping of any specific machine. The application of the transfer functions to cases involving feedback control of the machine is also discussed.