Complementary Functions Method Research Articles

Forced vibration of composite cylindrical helical rods

The dynamic behavior of composite cylindrical helical rods subjected to time-dependent loads is theoretically investigated in the Laplace domain. The governing equations for naturally twisted and curved spatial laminated rods obtained using Timoshenko beam theory are rewritten for cylindrical helical rods. The curvature of the rod axis, the anisotropy of the rod material, effect of the rotary inertia, axial and shear deformations are considered in the formulations. The material of the rod is assumed to be homogeneous, linear elastic and anisotropic. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem accurately. The solutions obtained are transformed to the time domain using an appropriate numerical inverse Laplace transform method. The free vibration is then taken into account as a special case of forced vibration. The results obtained in this study are found to be in a good agreement with those available in the literature.

Transient analysis of viscoelastic helical rods subject to time-dependent loads

The transient analysis of viscoelastic helical rods subject to time-dependent loads are examined in the Laplace domain. The governing equations for naturally twisted and curved spatial rods obtained using the Timoshenko beam theory are rewritten for cylindrical helical rods. The curvature of the rod axis, effect of rotary inertia and, shear and axial deformations are considered in the formulation. The material of the rod is assumed to be homogeneous, isotropic and linear viscoelastic. The viscoelastic constitutive equations are written in the Boltzmann–Volterra form. Ordinary differential equations in canonical form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem. The solutions obtained are transformed to the real space using an appropriate numerical inverse Laplace transform method. Numerical results for quasi-static and dynamic response of viscoelastic models are presented in the form of graphics.

Quasi-static and dynamic response of viscoelastic helical rods

In this study, the dynamic behaviour of cylindrical helical rods made of linear viscoelastic materials are investigated in the Laplace domain. The governing equations for naturally twisted and curved spatial rods obtained using the Timoshenko beam theory are rewritten for cylindrical helical rods. The curvature of the rod axis, effect of rotary inertia, and shear and axial deformations are considered in the formulation. The material of the rod is assumed to be homogeneous, isotropic and linear viscoelastic. In the viscoelastic material case, according to the correspondence principle, the material constants are replaced with their complex counterparts in the Laplace domain. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem. In the solutions, the Kelvin model is employed. The solutions obtained are transformed to the real space using the Durbin's numerical inverse Laplace transform method. Numerical results for quasi-static and dynamic response of viscoelastic models are presented in the form of graphics.

Forced Vibration of Cylindrical Helical Rods Subjected to Impulsive Loads

In this study, the forced vibration of cylindrical helical rods subjected to impulsive loads is theoretically investigated in the Laplace domain. The free vibration is then taken into account as a special case of forced vibration. The governing equations for naturally twisted and curved space rods obtained using Timoshenko beam theory are rewritten for cylindrical helical rods. The material of the rod is assumed to be homogeneous, linear elastic, and isotropic. The axial and shear deformations are also taken into account in the formulation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate exactly the dynamic stiffness matrix of the problem. The desired accuracy is obtained by taking only a few elements. The solutions obtained are transformed to the real space using the Durbin’s numerical inverse Laplace transform method. The free and forced vibrations of cylindrical helical rods are analyzed through various example. The results obtained in this study are found to be in a good agreement with those available in the literature.

In-Plane and Out-of-Plane Free Vibration Analysis of Archimedes-Type Spiral Springs

The in-plane and out-of-plane free vibration frequencies of Archimedes-type spiral springs are computed by the transfer matrix method. Taking into account the effects of the axial and the shear deformations and the rotary inertia, the overall dynamic transfer matrix is computed up to any desired numerical accuracy by the complementary functions method. Since there are no restrictions for the number of coils and for the form of the spring (close-coiled or open-coiled), the presented method is general. After having verified the soundness of the computer program devised, the effects of the number of coils, of the axial and shear deformations, of rotary inertia and of the boundary conditions on the frequencies are also investigated.

FREE VIBRATION ANALYSIS OF NON-CYLINDRICAL COIL SPRINGS BY COMBINED USE OF THE TRANSFER MATRIX AND THE COMPLEMENTARY FUNCTIONS METHODS

The transfer matrix method and the complementary functions method have been employed in the free vibration analysis of helical springs of irregular shape. The rotary inertia effects, the axial and shear deformations and cross-sections of different shapes having double symmetry have been included in the analysis. The formulation has yielded the exact free vibration frequencies of non-cylindrical coil springs. © 1997 John Wiley & Sons, Ltd.

FREE VIBRATION ANALYSIS OF NON‐CYLINDRICAL COIL SPRINGS BY COMBINED USE OF THE TRANSFER MATRIX AND THE COMPLEMENTARY FUNCTIONS METHODS

The transfer matrix method and the complementary functions method have been employed in the free vibration analysis of helical springs of irregular shape. The rotary inertia effects, the axial and shear deformations and cross-sections of different shapes having double symmetry have been included in the analysis. The formulation has yielded the exact free vibration frequencies of non-cylindrical coil springs. © 1997 John Wiley & Sons, Ltd.

The complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes

AbstractThe statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non‐circular helicoidal axis and cross‐section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first‐order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes. The stiffness matrix and the element load vector of a helicoidal bar with a non‐circular axis and arbitrary cross‐section are obtained taking into consideration both the presence of an elastic support and the effects of the axial and shear deformations. For helicoidal staircases, the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated. The developed model has been coded in Fortran‐77, which has been applied to various example problems available in the relevant literature, and the results have been compared.

A new method for the element stiffness matrix of arbitrary planar bars

The statical behaviour of a planar bar of an elastic and isotropic material having an arbitrary axis and a cross-section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. The Timoshenko beam theory is extended for three-dimensional curvilinear bars taking into consideration the effects of axial and shear deformation. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first order differential equations having variable coefficients is obtained for the spatial elements. From expressions developed in such manner for three-dimensional bars, the governing equations for the special case of a planar bar loaded within or perpendicular to its plane are derived, which are next solved by the complementary functions method. The stiffness matrix and the element load vector of a planar bar with an arbitrary axis are obtained taking into consideration both the presence of an elastic support and the effects of axial and shear deformations. The developed model has been coded in Fortran-77, which has been applied to various example problems available in the relevant literature, and the results have been compared.

On the method of complementary functions for linear and nonlinear two-point boundary value problems

On the method of complementary functions for linear and nonlinear two-point boundary value problems

Subsidence prediction by the use of influence functions

Abstract Surface subsidence occurs as a result of extraction of a mineral resource at some depth below the surface. Most coal is mined in this way in the United Kingdom and such coal workings are responsible for the great majority of subsidence and associated damage that occurs. Hence it has become necessary to develop methods of predicting the amount of subsidence likely to develop due to coal mining. Many methods have been advanced, dating back to the latter part of the last century. However, they can be separated into three groups, namely, the theoretical, the empirical and the semi-empirical methods. A number of these methods were reviewed with the object of selecting one which lent itself to the development of a relatively simple computer program that could be used on a microcomputer. The complementary influence function method was chosen and, although by no means a perfect method, a program was developed to predict a complete subsidence profile for a given set of circumstances. The concept of complementary influence functions considers not only the influence of the extracted elements on a surface point but also the influence of the material remaining after extraction. The subsidence of a surface point then is found by integrating the response of every element in the area of influence, whether mined or not. The mined region is divided into a number of mined and unmined elements. Summing the influence of all the elements on a particular point yields a picture of subsidence at that point. If this is done for a number of points across the proposed extraction area, then a complete picture of the subsidence over the area can be obtained.

Initial-value methods for discrete boundary value problems

Motivated by the work of Angel and Kalaba [S] and Golberg [8] on twopoint boundary value problems for difference equations, in this paper we shall study several initial-value methods for multipoint boundary value problems for linear and nonlinear difference systems. We generalize method of adjoints (Goodman and Lance [9]), method of complementary functions (Tifford [17]) d an method of particular solutions (Miele [ 111) proposed for two-point continuous problems to linear multipoint problems for difference equations. Next, we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems. To justify the results obtained here several examples are also given. The methods discussed here are similar to our earlier work [l-3] on continuous problems. We will take up the initial adjusting method of Ojika, Kasue, and Welsh [ 12-15, 18 ] for the discrete case later. Throughout the paper we shall assume the existence, uniqueness, and stability properties of solutions of the problems discussed and refer to [4] for some of these questions.

Convergence and Accuracy Properties of the Method of Quasilinearization

The paper deals with the method of quasilinearization as applied to the solution of two-point boundary-value (TPBV) problems associated with a system ${\bf y}' = {\bf f}(t,{\bf y})$ of second-order differential equations. It is supposed that the equations, linearized about an approximate solution ${\bf y}^{(k)} (t)$, are integrated numerically and that the linear TPBV problem is solved by a scheme such as the method of complementary functions. The principal concern is the effect that the approximate interpolation, used to represent ${\bf y}^{(k)} (t)$, has upon the order of convergence and the order of accuracy of the final converged solution. For example, it is shown that, if Runge-Kutta (RK) integration is employed, then there is an interpolation formula which gives second-order convergence. It is also proven that, if the order of the integration is q (suppose q is even), then the order of the interpolation need be only ${{(q - 2)} / 2}$ to give $(q - 1)$th order global accuracy in the converged solutio...

On the method of complementary functions for nonlinear boundary-value problems

We show that, like the method of adjoints, the method of complementary functions can be effectively used to solve nonlinear boundary-value problems.

A numerical method for linear two-point boundary-value problems using compound matrices

An initial-value method, based on the use of certain compound matrices, is presented for the treatment of mathematically unstable two-point boundary-value problems. A distinctive feature of the method is that the solution is obtained from an auxiliary equation which is mathematically stable and it thereby avoids the well-known difficulties associated with, for example, the method of complementary functions. The method is described in detail for fourth-order equations but it is also shown how the method can easily be adapted to deal with second- and third-order equations.

A comparative study of invariant imbedding and superposition

A comparison is made between a certain version of invariant imbedding and the method of superposition, as represented by the Goodman-Lance method of complementary functions, with regard to their computational efficiency and effectiveness for linear two-point boundary-value problems. The comparison is most complete for the class of problems termed normal. For such problems superposition appears to be preferable for problems with a sufficiently short underlying interval, on grounds of lesser effort, but the accuracy of super position degenerates more rapidly than that of invariant imbedding as the interval length increases, and consequently the latter method seems preferable for long problems. General areas of possible future investigation are identified.

General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions

In this paper, a general technique for solving nonlinear, two-point boundary-value problems is presented; it is assumed that the differential system has ordern and is subject top initial conditions andq final conditions, wherep+q=n. First, the differential equations and the boundary conditions are linearized about a nominal functionx(t) satisfying thep initial conditions. Next, the linearized system is imbedded into a more general system by means of a scaling factor α, 0≤α≤1, applied to each forcing term. Then, themethod of particular solutions is employed in order to obtain the perturbation Δx(t)=αA(t) leading from the nominal functionx(t) to the varied function $$\tilde x$$ (t); this method differs from the adjoint method and the complementary function method in that it employs only one differential system, namely, the nonhomogeneous, linearized system. The scaling factor (or stepsize) α is determined by a one-dimensional search starting from α=1 so as to ensure the decrease of the performance indexP (the cumulative error in the differential equations and the boundary conditions). It is shown that the performance index has a descent property; therefore, if α is sufficiently small, it is guaranteed that $$\tilde P$$ <P. Convergence to the desired solution is achieved when the inequalityP≤ɛ is met, where ɛ is a small, preselected number. In the present technique, the entire functionx(t) is updated according to $$\tilde x$$ (t)=x(t)+αA(t). This updating procedure is called Scheme (a). For comparison purposes, an alternate procedure, called Scheme (b), is considered: the initial pointx(0) is updated according to $$\tilde x$$ (0)=x(0)+αA(0), and the new nominal function $$\tilde x$$ (t) is obtained by forward integration of the nonlinear differential system. In this connection, five numerical examples are presented; they illustrate (i) the simplicity as well as the rapidity of convergence of the algorithm, (ii) the importance of stepsize control, and (iii) the desirability of updating the functionx(t) according to Scheme (a) rather than Scheme (b).

Method of particular solutions for linear, two-point boundary-value problems

The methods commonly employed for solving linear, two-point boundary-value problems require the use of two sets of differential equations: the original set and the derived set. This derived set is the adjoint set if the method of adjoint equations is used, the Green's functions set if the method of Green's functions is used, and the homogeneous set if the method of complementary functions is used. With particular regard to high-speed digital computing operations, this paper explores an alternate method, the method of particular solutions, in which only the original, nonhomogeneous set is used. A general theory is presented for a linear differential system ofnth order. The boundary-value problem is solved by combining linearly several particular solutions of the original, nonhomogeneous set. Both the case of an uncontrolled system and the case of a controlled system are considered.