The parametric pendulum is a fruitful dynamical system manifesting some of the most interesting phenomena of nonlinear dynamics, well-known to exhibit rather complex motion including period doubling, fold and pitchfork bifurcations, let alone the global bifurcations leading to chaotic or rotational motion. In this thesis, the potential of establishing rotational motion is studied considering the bobbing motion of ocean waves as the source of excitation of a oating pendulum. The challenge within this investigation lies on the fact that waves are random, as well as their observed low frequency, characteristics which pose a broader signi cance within the study of vibrating systems. Thus, a generic study is conducted with the parametric pendulum being excited by a narrow-band stochastic process and particularly, the random phase modulation is utilized. In order to explore the dynamics of the stochastic system, Markov-chain Monte-Calro simulations are performed to acquire a view on the in uence of randomness onto the parameter regions leading to rotational response. Furthermore, the Probability Density Function of the response is calculated, applying a numerical iterative scheme to solve the total probability law, exploiting the Chapman-Kolmogorov equation inherent to Markov processes. A special case of the studied structure undergoing impacts is considered to account for extreme weather conditions and nally, a novel design is investigated experimentally, aiming to set the ground for future development.

The subject of this thesis is to establish the basic vibration characteristics of shrouded blades on infinitely rigid and flexible disks using the finite element method. In the first part of the thesis vibrations of blade packets on rigid disks with infinite radius are considered. It is found that there are many more natural frequencies than those indicated by previous studies. The existence of these extra frequencies is established in an original way by drawing the frequency inference diagrams. A comprehensive method of predicting the multi-bladed packet behaviour from a two-bladed packet is introduced. To investigate the behaviour of blade packets on smaller disks the shroud is modified by using curved beam finite elements. A critical survey of finite curved beam elements is presented and a very efficient curved beam element based on the simple strain function concept is developed. The effect of shroud curvature on the frequency of vibration is studied in detail. Finally, the effect of the disk flexibility on the vibration characteristics of the shrouded blades is considered. Some of the available disk finite elements are critically discussed and compared. The finite element method of wave propagation technique with the cyclic symmetry of the system is used to reduce the size of an otherwise very large eigenvalue problem. The resulting eigenvectors from the wave propagation analysis is interpreted for the first time and the calculation of phase difference between the bending and torsional modes is illustrated. The analysis is also extended to include the shroud. A series of tests are conducted on a twenty-bladed disk assembly. Experimentally observed mode shapes are compared with the theoretical mode shapes. The same procedure is repeated with the inclusion of the shroud. Experimental results showed good agreement with the theoretical results.

The analysis of torsional vibrations in the running gear of reciprocating engine systems is normally carried out by neglecting the variation in inertia torques of the system arising from the motion of the reciprocating parts. When the variable inertia effect is allowed for the equation of motion taking into account the effect is non-linear. Assuming small displacements, the equation can be linearized to predict important characteristics of the motion. Such an equation when solved by numerical methods using a digital computer predicts the regions of instability and the manner in which the amplitude and frequency vary with the speed of rotation of the engine. The responses of the system show a modulation of amplitude and frequency at definite rotational speeds. The occurrence of such a modulation in amplitude and frequency is established by use of the process given by Wentzel, Kramers, Brillouin, and Jeffreys generally known as the WKBJ approximation. The method of variation of parameters and the WKBJ approximation are used to determine the time responses of the system as an independent verification of the numerical methods although this method is applicable only in a limited range of the engine speed for the specific ratio of the equivalent inertia of the reciprocating parts to the total equivalent inertia of the system. The first order term in the equation, the forcing term which represents the outer impulse from the reciprocating parts and the variable part of the elastic term are investigated for their effect on the waveforms of the responses of the system. Further investigations of the effect of external excitations on the characteristics of the motion are carried out which explain the cause of failure in some multi-cylinder engines due to secondary resonance. Various other characteristics of the variable inertia system and the complete solutions of the equation of motion including damping are explored in the present work. Theoretical results are compared with solutions of the equation obtained from an analogue computer. A discussion of results obtained by measurements on an engine in service with suspected secondary resonance and some actual cases of crankshaft failures in practice is included.

The extinction of second order subharmonic vibration response of a single degree of freedom system is investigated for the case of damped centrifugal excitation. The degree of the restoring force asymmetry resulting from the effect of gravity is expressed in terms of the parameter of static deflection. The resonance under gravity effects is analysed theoretically for a wide range of physical conditions to determine the behavioural characteristics of the subharmonic components. The inherently coupled algebraic equations are obtained by the approximate energy method of Ritz-Galerkin and by the method of harmonic balance. These two methods are not bounded to any degree of non-linearity. As there is no exact solution for this investigation and because of the dissipative forces inevitably introducing the problem of stability, the actual existence of the approximate solution over the frequency band-width is ascertained. There are no real roots in the instability region. The algebraic polynomial expressions cannot be satisfied simultaneously because of the accumulative effect in an accompanying harmonic of the vibratory motion. The build-up oscillation occurs in the second order region, having a frequency the same as that of the main component of subharmonic motion. The stability criterion is derived from comparing the characteristic exponent of solution to the variational equation with damping coefficient of the system. The response characteristics are then investigated where the polynomial equations are simplified through justifying the approximation of the fundamental harmonic as the effective amplitude of the disturbing force. The results are of comparable accuracy for cases in which gravity effects do not increase the effective non-linearity with resonance. The approximation, however, is applicable whatever tha physical characteristic behaviour of the non-linearity in the region of the critical state of subharmonic extinction. The subharmonic motion in the process of analysis is shown to exist in two opposite phases, differing by pi radians. The resulting phase of periodic vibration depends upon initial conditions. The isocline graphical method is used to depict the transient motion. The effective non-linearity is determined to be governed by the influence of gravity effects on the equilibrium of moition. The pronounced subharmonics are of the order one half, and the extinction conditions for the resonance predominant over the higher orders ere investigated through expressing the limiting inequalities in terms of the system parameters. In the critical state, complete suppression of the subharmonics is achieved. The limiting condition is then examined where damping is fixed at a convenient minimum value and the corresponding optimum limit of gravity effect tolerable can be evaluated for which the amplitude of excitation has no influence on the effective non-linearity as regard to exciting the resonance. As inequality is also presented from which the limiting frequencies of the subharmonic vibration can be predicted with reasonable accuracy. In these investigations the limiting inequalities are not dependant on the resulting variables in the non linear phenomena. An experimental test-rig is designed to demonstrate the subharmonic response. The values recorded from it compare favourably with the approximated theoretical results and with the experimental results obtained from the electronic analogue computer.

Deviations from isochronism for the osciliating system of spiral hairspring and balance wheel are investigated. Only those deviations caused by non-linearity of the hairspring are considered. It is shown that, in the case of springs manufactured from wire of a circular cross section, the deviations are related to the displacement of the outer end point of the hairspring, where the latter is arranged such that this end point is free to move in the plane of the spring, while its tangent is held constant in direction. Such an arrangement, proposed by Bouasse (12), is called a "free end" spring. Hairsprings of the more common flat strip are subject to the influence of additional effects that modify the above relationship. The modifications are investigated experimentally. The influence of manufacturing errors, and certain modifications to the spring shape are considered for round wire hairsprings, and approximate formulae derived that predict the deviations from isochronism of the oscillator in terms of the spring parameters and values of introduced errors. The theoretical predictions are verified experimentally. The formulae describing the deviation from isochronism are stated such that they can be used in the design of balance wheel and hairspring oscillators. A design example is included.