The eigenvalue problem associated with the determination of the natural frequencies, mode shapes and impulse response of a linear vibrating system is a classical and very important engineering problem. This dissertation presents a new technique for obtaining useful approximate, and in some cases, exact, solutions for such problems. The fundamental concepts on which the technique is based are: 1) The use of inertia (mass) as a perturbation parameter for developing series solutions for the response of a system to harmonic excitation. 2) The expansion of these series solutions as continued fractions. The series solutions are obtained by applying the classical perturbation technique, which assumes the solution for the governing differential equation (say, for the case of one independent and one dependent variable) can be expressed as w = w₀ + μ w₁ + μ² w₂ + μ³ w₃ + . . . . . where w is the dependent variable, and the wᵢ are unknown functions. µ is an inertia parameter that appears in the coefficients of the acceleration terms of the governing differential equation. The series for w is substituted in the governing differential equation, and the associated initial and boundary conditions. Since µ is arbitrary, the coefficients of like powers of µ are equated to zero. The result is an infinite set of differential equations, and boundary and initial conditions, each of which is (hopefully) easier to solve than the original problem. w₀ becomes the massless, or static, solution, in which the system responds instantaneously to, and in phase with, the applied excitation. The equation governing w₁ is the same as that for w₀ , except that some function of w₀ appears as a loading function, and, in general, the equation governing wᵢ will involve some function of wᵢ₋₁ as a loading function. There are two ways in which the problem can become a so-called singular perturbation problem. First, if the order of the equations governing wᵢ is not as high as that of the original equation, it may not be possible to accommodate all the initial and boundary conditions. However, initial conditions are not necessary for determining the eigenvalues of a linear vibrating system. The second way in which a singular perturbation may arise is the limiting of the range of validity of the series solution to small values of some combination an independent variable and the perturbation parameter. The range of validity of the series solution can be extended by truncating the series after some term and converting the truncated series to the quotient of two polynomials by means of continued fractions. The zeros of the denominator polynomial will correspond to resonant conditions. Lumped systems without damping are completely amenable to this method of solution, and a two-degree-of-freedom system with damping is solved. Approximate solutions for an axially loaded rod, a Timoshenko beam, and an Euler beam of variable cross-section illustrate the application of the method of analysis to continuous systems. / Doctor of Philosophy