Nonlinear Dynamic Responses of Tension Leg Platform with Slack-Taut Tether

The slack-taut state of tether is a particular Averse circumstance, which may influence the normal operation stale of tension leg platform (TLP). The dynamic responses of TLP with slack-taut tether are studied with consideration of several nonlinear factors introduced by large amplitude motions. The time histories of stresses of tethers of a typical TLP in slack-taut state are given. In addition, the sensitivities of slack to stiffness and mass are investigated by varying file stiffness of tether and mass of TLP. It is found that slack is sensitive to the mass of TLP. The critical culled surfaces (over which indicates the slack) for the increase of mass are obtained.

Random response of nonlinear continuous systems

This thesis presents a technique for obtaining the stochastic response of a nonlinear continuous system. First, the general method of nonstationary continuous equivalent linearization is developed. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. Next, a numerical implementation is described which allows solution of complex problems on a digital computer. In this procedure, the linear replacement system is discretized by the finite element method. Application of this method to systems satisfying the one-dimensional wave equation with two different types of constitutive nonlinearities is described. Results are discussed for nonlinear stress-strain laws of both hardening and softening types.

Exact solutions and transformation properties of nonlinear partial differential equations from general relativity

Various families of exact solutions to the Einstein and Einstein-Maxwell field equations of General Relativity are treated for situations of sufficient symmetry that only two independent variables arise. The mathematical problem then reduces to consideration of sets of two coupled nonlinear differential equations.

The physical situations in which such equations arise include: a) the external gravitational field of an axisymmetric, uncharged steadily rotating body, b) cylindrical gravitational waves with two degrees of freedom, c) colliding plane gravitational waves, d) the external gravitational and electromagnetic fields of a static, charged axisymmetric body, and e) colliding plane electromagnetic and gravitational waves. Through the introduction of suitable potentials and coordinate transformations, a formalism is presented which treats all these problems simultaneously. These transformations and potentials may be used to generate new solutions to the Einstein-Maxwell equations from solutions to the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a tool for generation of similarity solutions and generalized similarity solutions. It is further used to find the invariance group of the equations; this in turn leads to various finite transformations that give new, physically distinct solutions from old. Some of the above results are then generalized to the case of three independent variables.

Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Topics in linear and nonlinear dispersive waves

The various singularities and instabilities which arise in the modulation theory of dispersive wavetrains are studied. Primary interest is in the theory of nonlinear waves, but a study of associated questions in linear theory provides background information and is of independent interest.

The full modulation theory is developed in general terms. In the first approximation for slow modulations, the modulation equations are solved. In both the linear and nonlinear theories, singularities and regions of multivalued modulations are predicted. Higher order effects are considered to evaluate this first order theory. An improved approximation is presented which gives the true behavior in the singular regions. For the linear case, the end result can be interpreted as the overlap of elementary wavetrains. In the nonlinear case, it is found that a sufficiently strong nonlinearity prevents this overlap. Transition zones with a predictable structure replace the singular regions.

For linear problems, exact solutions are found by Fourier integrals and other superposition techniques. These show the true behavior when breaking modulations are predicted.

A numerical study is made for the anharmonic lattice to assess the nonlinear theory. This confirms the theoretical predictions of nonlinear group velocities, group splitting, and wavetrain instability, as well as higher order effects in the singular regions.

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

Some modified bifurcation problems with application to imperfection sensitivity in buckling

The branching theory of solutions of certain nonlinear elliptic partial differential equations is developed, when the nonlinear term is perturbed from unforced to forced. We find families of branching points and the associated nonisolated solutions which emanate from a bifurcation point of the unforced problem. Nontrivial solution branches are constructed which contain the nonisolated solutions, and the branching is exhibited. An iteration procedure is used to establish the existence of these solutions, and a formal perturbation theory is shown to give asymptotically valid results. The stability of the solutions is examined and certain solution branches are shown to consist of minimal positive solutions. Other solution branches which do not contain branching points are also found in a neighborhood of the bifurcation point.

The qualitative features of branching points and their associated nonisolated solutions are used to obtain useful information about buckling of columns and arches. Global stability characteristics for the buckled equilibrium states of imperfect columns and arches are discussed. Asymptotic expansions for the imperfection sensitive buckling load of a column on a nonlinearly elastic foundation are found and rigorously justified.

Some problems in nonlinear elasticity

Two separate problems are discussed: axisymmetric equilibrium configurations of a circular membrane under pressure and subject to thrust along its edge, and the buckling of a circular cylindrical shell.

An ordinary differential equation governing the circular membrane is imbedded in a family of n-dimensional nonlinear equations. Phase plane methods are used to examine the number of solutions corresponding to a parameter which generalizes the thrust, as well as other parameters determining the shape of the nonlinearity and the undeformed shape of the membrane. It is found that in any number of dimensions there exists a value of the generalized thrust for which a countable infinity of solutions exist if some of the remaining parameters are made sufficiently large. Criteria describing the number of solutions in other cases are also given.

Donnell-type equations are used to model a circular cylindrical shell. The static problem of bifurcation of buckled modes from Poisson expansion is analyzed using an iteration scheme and pertubation methods. Analysis shows that although buckling loads are usually simple eigenvalues, they may have arbitrarily large but finite multiplicity when the ratio of the shell's length and circumference is rational. A numerical study of the critical buckling load for simple eigenvalues indicates that the number of waves along the axis of the deformed shell is roughly proportional to the length of the shell, suggesting the possibility of a "characteristic length." Further numerical work indicates that initial post-buckling curves are typically steep, although the load may increase or decrease. It is shown that either a sheet of solutions or two distinct branches bifurcate from a double eigenvalue. Furthermore, a shell may be subject to a uniform torque, even though one is not prescribed at the ends of the shell, through the interaction of two modes with the same number of circumferential waves. Finally, multiple time scale techniques are used to study the dynamic buckling of a rectangular plate as well as a circular cylindrical shell; transition to a new steady state amplitude determined by the nonlinearity is shown. The importance of damping in determining equilibrium configurations independent of initial conditions is illustrated.

Nonlinear dispersive waves in nonlinear optics

A study is made of solutions of the macroscopic Maxwell equations in nonlinear media. Both nonlinear and dispersive terms are responsible for effects that are not taken into account in the geometrical optics approximation. The nonlinear terms can, depending on the nature of the nonlinearity, cause plane waves to focus when the amplitude varies across the wavefront. The dispersive terms prevent the singularities that nonlinearity alone would produce. Solutions are found which de scribe periodic plane waves in fully nonlinear media. Equations describing the evolution of the amplitude, frequency and wave number are generated by means of averaged Lagrangian techniques. The equations are solved for near linear media to produce the form of focusing waves which develop a singularity at the focal point. When higher dispersion is included nonlinear and dispersive effects can balance and one finds amplitude profiles that propagate with straight rays.

I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Nonlinear dispersive wave problems

The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonlinearity is retained but in which the scale of the nonuniformity introduces a small parameter. The first order results agree with the results that Whitham obtained by averaging methods. The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differential equations is analogous to the "multiple time scale" methods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of geometrical optics to nonlinear problems.

To apply the expansion method to the classical water wave problem, it is crucial to find an appropriate variational principle. It was found in the present investigation that a Lagrangian function equal to the pressure yields the full set of equations of motion for the problem. After this result is derived, the Lagrangian is compared with the more usual expression formed from kinetic minus potential energy. The water wave problem is then examined by means of the expansion procedure.

Bifurcation theory of nonlinear boundary value problems

The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered.

The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented. Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described.

The theory is applied to the problem of the axisymmetric buckling of thin spherical shells. Results are obtained which confirm recent numerical computations.