The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary differential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the post-processing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-difference technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

Decay of vibrations along deepwater mooring lines

Model tests for global design verification of deepwater floating structures cannot be made at reasonable scales. An overview of recent research efforts to tackle this challenge is given first, introducing the concept of line truncation techniques. In such a method the upper sections of each line are modelled in detail, capturing the wave action zone and all coupling effects with the vessel. These terminate to an approximate analytical model, that aims to simulate the remainder of the line. The rationale for this is that in deep water the transverse elastic waves of a line are likely to decay before they are reflected at the seabed. The focus of this paper is the verification of this rationale and the ongoing work, which is considering ways to produce a truncation model. Transverse dynamics of a mooring line are modelled using the equations of motion of an inextensible taut string, submerged in still water, one end fixed at the bottom the other assumed to follow the vessel response, which can be harmonic or random. Nonlinear hydrodynamic damping is included; bending and VIV effects are neglected. A dimensional analysis, supported by exact benchmark numerical solutions, has shown that it is possible to produce a universal curve for the decay of transverse vibrations along the line, which is suitable for any kind of line with any top motion. This has a significant engineering benefit, allowing for a rapid assessment of line dynamics - it is very useful in deciding whether a truncated line model is appropriate, and if so, at which point truncation might be applied. Initial efforts in developing a truncated model show that a linearized numerical solution in the frequency domain matches very closely the exact benchmark. Copyright © 2011 by ASME.

An integrated model of rolling tyre vibrations on a smooth road

We describe new results on the vibrations of rolling tyres, aimed at noise prediction for tyres of given design on a smooth road surface. This new approach incorporates our existing models, of smooth road-tyre interaction and belt vibration but includes additional features that are required for real tyre patterns. To this end, the model allows variable tread block size and grooves along the belt circumference; the density and angle of these grooves may also vary laterally. The key innovation is to treat the tyre belt as a laterally stacked series of rings, each of which is equipped with a set of viscoelastic springs around its circumference. It is shown how to use this construction to mimic the details of actual tyre patterns and, in conjunction with existing models, predict belt vibrations. The construction is applied to develop a ring discretisation for a real tyre that shows strong lateral variations. It is shown that the vibration amplitude is concentrated on a set of parallel lines in frequency-wavenumber space and that the tread pattern dictates the occurrence and spacing of these lines. Linkage to a boundary element calculation then allows quantification of the influence of tread parameters on radiated noise. Keywords: Vibration, tread pattern, tyre noise. Copyright © (2011) by the Institute of Noise Control Engineering.

Vibrations of coupled cylindrical-conical shells

The free vibrational characteristics of coupled conical-cylindrical shells is presented. The equations of motion for the cylindrical shell are solved using a wave approach while the equations of motion for the conical shells are solved using a power series solution. The use of both Donnell-Mushtari and Flügge equations of motion are investigated and their limitations are discussed. Results are presented in terms of natural frequencies for different boundary conditions and the purely torsional mode solution is described. The results from the analytical model presented are compared with those obtained from a finite element model solved with Nastran and other data available in literature.

The vibrations of bubbles and balloons

Bubbles and balloons are two examples of structures that feature a pressure difference across the skin, a thin, tensioned membrane, and a doubly curved interface surface. While mathematical models have been formulated for bubble vibrations, no such model exists for balloon vibrations. This paper reviews a model of bubble vibrations, and compares its predicted natural frequencies and modeshapes to those of a rubber balloon. It is shown that the bubble model consistently underpredicts the balloon's natural frequencies, and it is concluded that the nonlinear elasticity present in the balloon skin accounts for this result.

Chaotic behavior of pulsating heat pipes

Chaotic behavior of closed loop pulsating heat pipes (PHPs) was studied. The PHPs were fabricated by capillary tubes with outer and inner diameters of 2.0 and 1.20 mm. FC-72 and deionized water were used as the working fluids. Experiments cover the following data ranges: number of turns of 4, 6, and 9, inclination angles from 5 degrees (near horizontal) to 90, (vertical), charge ratios from 50% to 80%, heating powers from 7.5 to 60.0 W. The nonlinear analysis is based on the recorded time series of temperatures on the evaporation, adiabatic, and condensation sections. The present study confirms that PHPs are deterministic chaotic systems. Autocorrelation functions (ACF) are decreased versus time, indicating prediction ability of the system is finite. Three typical attractor patterns are identified. Hurst exponents are very high, i.e., from 0.85 to 0.95, indicating very strong persistent properties of PHPs. Curves of correlation integral versus radius of hypersphere indicate two linear sections for water PHPs, corresponding to both high frequency, low amplitude, and low frequency, large amplitude oscillations. At small inclination angles near horizontal, correlation dimensions are not uniform at different turns of PHPs. The non-uniformity of correlation dimensions is significantly improved with increases in inclination angles. Effect of inclination angles on the chaotic parameters is complex for FC-72 PHPs, but it is certain that correlation dimensions and Kolmogorov entropies are increased with increases in inclination angles. The optimal charge ratios are about 60-70%, at which correlation dimensions and Kolmogorov entropies are high. The higher the heating power, the larger the correlation dimensions and Kolmogorov entropies are. For most runs, large correlation dimensions and Kolmogorov entropies correspond to small thermal resistances, i.e., better thermal performance, except for FC-72 PHPs at small inclination angles of theta < 15 degrees.