The fixed-point iteration algorithm is turned into a quadratically convergent scheme for a system of nonlinear equations. Most of the usual methods for obtaining the roots of a system of nonlinear equations rely on expanding the equation system about the roots in a Taylor series, and neglecting the higher order terms. Rearrangement of the resulting truncated system then results in the usual Newton-Raphson and Halley type approximations. In this paper the introduction of unit root functions avoids the direct expansion of the nonlinear system about the root, and relies, instead, on approximations which enable the unit root functions to considerably widen the radius of convergence of the iteration method. Methods for obtaining higher order rates of convergence and larger radii of convergence are discussed.

The nonlinear response of a composite laminated panel that is suddenly exposed to a heat flux is examined using the finite element method. The panel is cantilevered onto a rigid hub, the rotation motion of which is either fully or partially restrained. The panel elastic deformations are assumed large and are modeled via the von Ka´rma´n strain-displacement relationship while the rigid-body angular rotation, for the case of a rotating rigid hub, is assumed small. The system of nonlinear governing equations is solved by the Newton-Raphson method in conjunction with the Newmark time integration scheme. The panel deformation is observed to be sensitive to the motion of the base.

This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation of Newton’s method. The proposed technique is also applicable for arbitrary nonlinearities via a cubicization procedure over the interval T. Upon determining the Fourier coefficients, estimates of the response power spectral density matrix are constructed by averaging their squared moduli over the samples ensemble. Examples of application prove the reliability of the technique by comparison with digital simulation data.

In this paper a mathematical formulation and a numerical algorithm for the analysis of impact of rigid bodies against rigid obstacles are developed. The paper concentrates on three-dimensional motion using a direct approach where the impenetrability condition and Coulomb’s law of friction are formulated as equations, which are not differentiable in the usual sense, and solved together with the equations of motion and necessary kinematical relations using Newton’s method. An experiment has also been performed and compared with predictions of the algorithm, with favorable results. [S0021-8936(00)01402-1]

Problems associated with viscoelastic membrane structures have been documented, e.g., dynamic wrinkling and its effects on fatigue analysis and on snap loading. In the proposed analysis method, the constitutive equation is approximated by a finite difference equation and embedded within a nonlinear finite element spatial discretization. Implicit temporal integration and a modified Newton-Raphson method are used within a time increment. The stress-strain hereditary relation is formally derived from thermodynamic considerations. Use of modified strain-energy and dissipation functions facilitates the description of wrinkling during the analysis. Applications are demonstrated on an inflated cylindrical cantilever and on a submerged cylindrical membrane excited by waves.

A geometrical nonlinear theory of composite laminated beams is derived with the effect of transverse shear deformation taken into account. The theory is based on a high-order kinematic model, with the nonlinear differential equations solved by Newton’s method and a special finite-difference scheme. A parametric study of the shear effect involving several kinematic approaches was carried out for isotropic and anisotropic beams.

An iterative numerical solution is given for the case of a thin elastic rod whose inputs are simultaneously the positions of the two ends and an applied moment at one end. The development begins by considering the real rod as an end section of a longer fictitious rod loaded with end forces only. Newton’s method is then used to obtain both the shape of the real road and its vector restoring force. The results show that both the magnitude and the direction of the restoring force are changed considerably from the zero-moment case, especially when the percent deflection of the elastica is small. Such a model is a useful alternative to a pure force-deflection one because it accounts not only for the direct effect of the applied moment on the reaction rigid body but the indirect contribution to the reaction force as well.

A periodic, Fourier series solution is presented for the nonlinear dynamic response of an impulsively loaded, elastic cylindrical shell. This result has been obtained by a new application of the quadratic form of Newton’s method; in this modification the frequency is considered to be the basic independent variable, and analytic corrections are obtained when solutions of the linear variational equations are nonperiodic. The solution presented here is shown to consist of a slowly varying function modulating a rapidly varying function, and is compared with results obtained by direct numerical integration.

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 numerical solution of problems of elastic stability through the use of the iteration method of Newton is examined. It is found that if the equations of equilibrium are completed by a simple auxiliary equation, problems governed by a snapping condition can, in principle, always be calculated as long as the problem at hand is properly formulated. The effectiveness of the proposed procedure is demonstrated by means of an elementary example.

Numerical solutions for the time history of motion of a shallow viscoelastic spherical shell are presented under large deformations. Axisymmetric motion is considered due to uniform pressure with arbitrary time variation. The dynamic viscoelastic buckling load is established by comparing long-time equilibrium states due to small changes in the applied pressure. The problem is formulated as a coupled pair of nonlinear integro-differential equations which is solved by Newton’s method.

A modification of Newton’s method is applied to the solution of the nonlinear differential equations for clamped, shallow spherical caps under uniform pressure. The linear form of Newton’s method or quasi-linearization breaks down at limit points of the differential equations. A simplified “quadratic form” is derived in the paper and shown to be satisfactory for continuing the solution past the limit point and into the postbuckling region. Results for the buckling pressures defined by the limit points agree with published results for perfect caps. New results are presented for imperfect caps that check experiment.

A modification of Newton’s method is suggested that provides a practical means of continuing solutions of nonlinear differential equations through limit points or bifurcation points. The method is applicable when the linear “variational” equations for the problem are self-adjoint. The procedure is illustrated by examples from the field of elastic stability.

Approximate solutions are obtained, by Newton’s method, for shells subjected to uniform and/or point loads in the prebuckled or postbuckled configurations. Comparison of results with numerical solutions shows maximum deviations of 5 percent for clamped shells subjected to uniform loads in the prebuckled configuration. Similar comparisons for other cases show larger deviations. A characteristic load is developed which compares favorably with numerical buckling loads.

Many problems in mechanics are formulated as nonlinear boundary-value problems. A practical method of solving such problems is to extend Newton’s method for calculating roots of algebraic equations. Three problems are treated in this paper to illustrate the use of this method and compare it with other methods.