The steady-state response of multidegree-of-freedom systems with a spatially localized nonlinearity

This thesis is concerned with the dynamic response of a General multidegree-of-freedom linear system with a one dimensional nonlinear constraint attached between two points. The nonlinear constraint is assumed to consist of rate-independent conservative and hysteretic nonlinearities and may contain a viscous dissipation element. The dynamic equations for general spatial and temporal load distributions are derived for both continuous and discrete systems. The method of equivalent linearization is used to develop equations which govern the approximate steady-state response to generally distributed loads with harmonic time dependence.

The qualitative response behavior of a class of undamped chainlike structures with a nonlinear terminal constraint is investigated. It is shown that the hardening or softening behavior of every resonance curve is similar and is determined by the properties of the constraint. Also examined are the number and location of resonance curves, the boundedness of the forced response, the loci of response extrema, and other characteristics of the response. Particular consideration is given to the dependence of the response characteristics on the properties of the linear system, the nonlinear constraint, and the load distribution.

Numerical examples of the approximate steady-state response of three structural systems are presented. These examples illustrate the application of the formulation and qualitative theory. It is shown that disconnected response curves and response curves which cross are obtained for base excitation of a uniform shear beam with a cubic spring foundation. Disconnected response curves are also obtained for the steady-state response to a concentrated load of a chainlike structure with a hardening hysteretic constraint. The accuracy of the approximate response curves is investigated.

Sound velocities and equation of state of iron-rich (Mg,Fe)O

Ultralow-velocity zones (ULVZs) are small structures at the base of the mantle characterized by sound velocities up to 30% lower than those of surrounding mantle. In this thesis, we propose that iron-rich (Mg,Fe)O plays a key role in the observed sound velocities, and argue that chemically distinct, iron-enriched structures are consistent with both the low sound velocities and the measured shapes of ULVZs.

Phase transitions from the solid state of monatomic elements

The principle aims of this thesis include the development of models of sublimation and melting from first principles and the application of these models to the rare gases.

A simple physical model is constructed to represent the sublimation of monatomic elements. According to this model, the solid and gas phases are two states of a single physical system. The nature of the phase transition is clearly revealed, and the relations between the vapor pressure, the latent heat, and the transition temperature are derived. The resulting theory is applied to argon, krypton, and xenon, and good agreement with experiment is found.

For the melting transition, the solid is represented by an anharmonic model and the liquid is described by the Percus-Yevick approximation. The behavior of the liquid at high densities is studied on the isotherms kT/∈ = 1.3, 1.8, and 2.0, where k is Boltzmann's constant, T is the temperature, and e is the well depth of the Lennard-Jones 12-6 pair potential. No solutions of the PercusYevick equation were found for ρσ³ above 1.3, where ρ is the particle density and σ is the radial parameter of the Lennard-Jones potential. The liquid structure is found to be very different from the solid structure near the melting line. The liquid pressures are about 50 percent low for experimental melting densities of argon. This discrepancy gives rise to melting pressures up to twice the experimental values.

The reduced-alpha-width of the lowest 1¯ state of ^(16)O

Cross sections for the reaction ¹²C(α,γ)¹⁶O have been measured for a range of center-of-mass alpha particle energies extending from 1.72 MeV to 2.94 MeV. Two 8"x5" NaI (Tℓ) crystals were used to detect gamma rays; time-of-flight technique was employed to suppress cosmic ray background and background due to neutrons arising mainly from the ¹³C(α,n)¹⁶O reaction. Angular distributions were measured at center-of-mass alpha energies of 2.18, 2.42, 2.56 and 2.83 MeV. Upper limits were placed on the amount of radiation cascading through the 6.92 or 7.12-MeV states in ¹⁶O. By means of theoretical fits to the measured electric dipole component of the total cross section, in which interference between the 1¯ states in ¹⁶O at 7.12 MeV and at 9.60 MeV is taken into account, it is possible to extract the dimensionless, reduced-alpha-width of the 7.12-MeV state in ¹⁶O. A three-level R-matrix parameterization of the data yields the width Θ_α,F² = 0.14^+0.10_-0.08. A "hybrid" R-matrix-optical-model parameterization yields Θ_α,F² = 0.11^+0.11_-0.07. This quantity is of crucial importance in determining the abundances of ¹²C and ¹⁶O at the end of helium burning in stars.