Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

Influence of Axial Excitations and of Boundary Conditions on the Parametric Instability of a Beam

In this paper the stability of the lateral dynamic behavior of a pinned-pinned, clamped-pinned and clamped-clamped beam under axial periodic force or torque is studied. The time-varying parameter equations are derived using the Rayleigh-Ritz method. The stability analysis of the solution is based on Floquet’s theory and investigated in detail. The Rayleigh-Ritz results are compared to those of a finite element modal reduction. It shows that the lateral instabilities of the beam depend on the forcing frequency, the type of excitation and the boundary conditions. Several experimental tests enable the validation of the numerical results.

Gear Damage Detection Using Chaotic Dynamics Techniques: A Preliminary Study

This paper is a preliminary study applying nonlinear time series analysis to crack detection in gearboxes. Our investigations show that the vibration signal emerging from a gearbox is chaotic. Appearance of a crack in a gear tooth alters this response and hence the chaotic signature. We used correlation dimension and Lyapunov exponents to quantify this change. The main goal of this study is to point out the great potential of these methods in detection of cracks and faults in machinery.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

An Experimental Study of the Forced Response of Pre- and Post-Critical Plates

Experimental results are presented which characterize the dynamic response of homogeneous, fully clamped, rectangular plates to narrow band acoustic excitation and uniform thermal loads. Using time series, pseudo-phase projections, power spectra and auto-correlation functions, small amplitude vibrations are considered about both the pre- and post-critical states. These techniques are then employed to investigate the snap-through response. The results for snap-through suggest that the motion is temporally complex and a Lyapunov exponent calculation confirms that the motion is chaotic. Finally, a snap-through boundary is mapped in the (ω, SPL) parameter space separating the regions of snap-through and no snap-through.

On Integrating Factors and a Perturbation Method

In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

A Quasiperiodic Mathieu Equation

In this work we investigate the following quasiperiodic Mathieu equation: x ¨ + ( δ + ϵ cos ⁡ t + ϵ cos ⁡ ω t ) x = 0 We use numerical integration to determine regions of stability in the δ–ω plane for fixed ϵ. Graphs of these stability regions are presented, based on extensive computation. In addition, we use perturbations to obtain approximations for the stability regions near δ=14 for small ω, and we compare the results with those of direct numerical integration.

A Harmonic Tracking Technique for Recovery of Gear Motion From Non-Stationary Vibration Signals

A method termed harmonic tracking is developed to recover time dependent gear motion from machine casing vibration. The harmonic tracking method uses short-time spectral generation and a subsequent set of algorithms to locate and track gear meshing frequencies as functions of time. The meshing frequencies are then integrated with respect to time to obtain the rotation of individual gears. More specifically, spectral generation is performed using the discrete Fourier transform, and the locating and tracking algorithms involve locating tones in each short-time spectrum and tracking them through successive spectra to recover gear meshing harmonics. The harmonic tracking method is found to be more robust than demodulation-based methods in the presence of measurement noise and signal distortion from the structural transfer function between gears and the casing. The harmonic tracking method is tested, both through simulation and experiments involving motor-operated valves (MOV’s) as part of the development of a diagnostic system for MOV’s. In all cases, the harmonic tracking method is found to recover gear motion with sufficient accuracy to perform diagnostics. The harmonic tracking method should be generally applicable to situations in which a non-invasive technique is required for determining the time-dependent angular speeds and displacements of gearbox input, intermediary, and output shafts.

On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

Numerical Study of an Impact Oscillator

Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.