A wave mechanics approach is used to solve the Timoshenko beam equation, revealing that two waves exist. One is called the s a -wave and the other the s b -wave. These two waves are found to be the basic constituent components of the mode shapes of the beam. An experiment was carried out and the measured mode shapes of a free-free beam are shown to consist of one s a -wave and one s b -wave in superposition for each of the modes. The measured s a -wave and s b -wave exhibit the Rayleigh-Lamb first (with anomalous dispersion) and Rayleigh-Lamb second (with normal dispersion) flexural modes, respectively. The issue of the second spectrum is addressed and it is shown that, within the measurable frequency range, Rayleigh-Lamb second flexural modes are present in the free-free beam. The s b -wave is identified as the secondspectrum mode. The role of shear deformation is also investigated in explaining the basic difference in the behaviour of the s a -and s b -waves. This paper also contributes to a physical interpretation of the hyperbolic functions in the classical solution of beam vibration problems.

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The dynamic stability of a two degrees-of-freedom system under bounded noise excitation with a narrowband characteristic is studied through the determination of moment Lyapunov exponents. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. For weak noise excitations, a singular perturbation method is employed to obtain second-order expansions of the moment Lyapunov exponents and Lyapunov exponents, which are shown to be in good agreement with those obtained using Monte Carlo simulation. The different cases when the system is in subharmonic resonance, combination additive resonance, and combined resonance in the absence of noise, respectively, are considered. The effects of noise and frequency detuning on the parametric resonance are investigated.

In the present study, the effect of Reynolds number (Re) on flow interference between two side-by-side stationary cylinders and the associated flow-induced forces are investigated using a finite element method. The pitch ratio chosen is T/D = 1.7, and Re is varied within the range of laminar flow regime, i.e., 60 < Re < 200. The method of continuous wavelet transform is used to analyze flow-induced forces, especially their time-variant features. Flow patterns in the form of vorticity plot are presented to demonstrate the underlying physics. It is found that flow interference initially occurs in the inner vortices shed from the two cylinders, and extends to the outer vortices with increment of Re. The flow behind two cylinders undergoes three regimes: Regime I - unbiased gap flow, Regime II - stable biased gap flow, and Regime III - unstable gap flow. Flow-induced forces show significant variations when the flow transits from one regime to another. In particular, during the transition from Regime II to III, the forces not only increase by amplitude, but also change their nature from deterministic to random, and show some non-stationary features. This is shown to be caused by the amalgamation of inner and outer vortices behind the two cylinders when the flow interference extends from inner vortices to outer vortices. Whenever possible, the present results are compared with experimental measurements and theoretical predictions. The numerical simulations are consistent with these other results.

Vortex-induced vibration of a single circular cylinder is a fundamental and significant case of flow-induced vibrations in both engineering practice and academic research. A force evolution model is developed for vortex-induced vibration of an elastic cylinder in a cross flow. In this model, the elastic cylinder is represented by the Euler-Bernoulli beam theory, and the vortex-induced force acting on the stationary cylinder is modeled by a bounded-noise process. Fluid-structure interaction is represented by an evolution process based on the concept of quasi-steady flow. A numerical iterative approach is used to simulate the evolution process and obtain time histories of cylinder vibration and vortex-induced force. The model is validated against available measurements. It is shown that the proposed model can reproduce the salient features observed in experiments. Furthermore, quantitative agreement with experimental measurements is obtained in general. However, when the vibration amplitude is very large due to intense fluid-structure interaction, the proposed model prediction is not quite satisfactory. This suggests that the concept of quasi-steady flow is only applicable for relatively weak fluid-structure interaction.