The importance of fluid-elastic coupling forces in tube bundle vibrations is well documented and can hardly be over-emphasized, in view of their damaging potential. Even when adequate tube supports ...are provided to suppress fluid-elastic instabilities, the flow-coupling forces still affect the dynamical tube responses and remain a significant issue, in particular concerning the vibro-impact motions of tubes assembled using clearance supports. Therefore, the need remains for more advanced models of fluid-elastic coupling, as well as for experimental flow-coupling coefficients to feed and validate such models. In this work, we report an extensive series of experiments performed at CEA-Saclay leading to the identification of stiffness and damping fluid-elastic coefficients, for a 3×5 square tube bundle (D = 30 mm, P/D = 1.5) subjected to single-phase transverse flow, in a water test-loop. The bundle is rigid, except for the central tube which is mounted on a flexible suspension (two parallel steel blades) allowing for translation motions of the tube in the lift direction. The system is thus single-degree of freedom, allowing fluid-elastic instability to arise through a negative damping mechanism. The flow-coupling stiffness and damping coefficients, Kf(Vr) and Cf(Vr), are experimentally identified as functions of the reduced velocity Vr. The maximum value of the Reynolds number ranged from 105 to 2.16 105 (based on the maximum pitch velocity), according to the tested configuration. Identification is achieved on the basis of changes in tube vibration frequency and reduced damping as a function of flow velocity, assuming a constant fluid added mass. In the present experiments, coefficient identification is performed well beyond the instability boundary, by using active control, thereafter allowing exploration of a significant range of flow velocity. The modal frequency and the modal mass of the system are respectively modified by changing the tube suspension stiffness, and/or by adding a mass to the system. We can thus assert how the fluid-elastic coefficients change, for this configuration, with these two system parameters, all other parameters being kept constant. The results obtained from the configurations tested suggest that formulations for coefficient reduction may be improved, in order to better collapse the identified data.
The modal identification of dynamical systems under operational conditions, when subjected to wide-band unmeasured excitations, is today a viable alternative to more traditional modal identification ...approaches based on processing sets of measured FRFs or impulse responses. Among current techniques for performing operational modal identification, the so-called blind identification methods are the subject of considerable investigation. In particular, the SOBI (Second-Order Blind Identification) method was found to be quite efficient. SOBI was originally developed for systems with normal modes. To address systems with complex modes, various extension approaches have been proposed, in particular: (a) Using a first-order state-space formulation for the system dynamics; (b) Building complex analytic signals from the measured responses using the Hilbert transform. In this paper we further explore the latter option, which is conceptually interesting while preserving the model order and size. Focus is on applicability of the SOBI technique for extracting the modal responses from analytic signals built from a set of vibratory responses. The novelty of this work is to propose a straightforward computational procedure for obtaining the complex cross-correlation response matrix to be used for the modal identification procedure. After clarifying subtle aspects of the general theoretical framework, we demonstrate that the correlation matrix of the analytic responses can be computed through a Hilbert transform of the real correlation matrix, so that the actual time-domain responses are no longer required for modal identification purposes. The numerical validation of the proposed technique is presented based on time-domain simulations of a conceptual physical multi-modal system, designed to display modes ranging from normal to highly complex, while keeping modal damping low and nearly independent of the modal complexity, and which can prove very interesting in test bench applications. Numerical results for complex modal identifications are presented, and the quality of the identified modal matrix and modal responses, extracted using the complex SOBI technique and implementing the proposed formulation, is assessed.
Flow-induced vibrations of heat-exchanger tubes are extensively studied in the nuclear industry for safety reasons. Adequate designs, such as anti-vibration bars in PWR steam generators, prevent ...excessive vibrations provided the tubes are well supported. Nevertheless, degraded situations where the tube/support gaps would widen, must also be considered. In such a case, the tubes become loosely supported and may exhibit vibro-impacting responses due to both turbulence and fluid–elastic coupling forces induced by the cross-flow. This paper deals with the predictive analysis of such a nonlinear situation, given the necessity of taking into account both the strong impact nonlinearity due to the gap and the linearized fluid–elastic forces. In time-domain numerical simulations, computation of flow-coupling forces defined in the frequency-domain is a delicate problem. We recently developed an approach based on a hybrid time–frequency method. In the present paper a more straightforward and effective technique, based on the convolution of a flow impulse response pre-computed from the frequency-domain coefficients, is developed. Illustrative results are presented and discussed, in connection with the previous hybrid method and with experiments. All results agree in a satisfactory manner, validating both computational methods, however the convolutional technique is faster than the hybrid method by two orders of magnitude. Finally, to highlight the subtle self-regulating frequency effect on the stabilization of such system, additional demonstrative computations are presented.
A potential theory is presented for the problem of two moving circular cylinders, with possibly different radii, large motions, immersed in an perfect stagnant fluid. We show that the fluid force is ...the superposition of an added mass term, related to the time variations of the potential, and a quadratic term related to its spatial variations. We provide new simple and exact analytical expressions for the fluid added mass coefficients, in which the effect of the confinement is made explicit. The self-added mass (resp. cross-added mass) is shown to decrease (resp. increase) with the separation distance and increase (resp. decreases) with the radius ratio. We then consider the case in which one cylinder translates along the line joining the centers with a constant speed. We show that the two cylinders are repelled from each other, with a force that diverges to infinity at impact. We extend our approach to the case in which one cylinder is imposed a sinusoidal vibration. We show that the force on the stationary cylinder and the vibration displacement have opposite (resp. identical) axial (resp. transverse) directions. For large vibration amplitudes, this force is strongly altered by the nonlinear effects induced by the spatial variations of the potential. The force on the vibrating cylinder is in phase with the imposed displacement and is mainly driven by the added mass term. The results of this paper are of particular interest for engineers who need to understand the essential features associated with the vibration of a solid body in a still fluid.
In this paper we address the issue of generating, from the spectral and spatial parameters of turbulent flow excitations, time-domain random excitations suitable for performing representative ...nonlinear numerical simulations of the dynamical responses of flow-excited tubes with multiple clearance supports. The new method proposed in this work, which is anchored in a sound physical basis, can effectively deal with non-uniform turbulent flows, which display significant changes in their spatial excitation properties. Contrary to the classic technique developed by Shinozuka and coworkers, which generates a large set of correlated physical forces, the proposed method directly generates a set of correlated modal forces. Our approach is particularly effective leading to a much smaller number of generated time-histories than would be needed using physical forces to simulate the turbulence random field. In the case of strongly non-uniform flows, our approach allows for a suitable decomposition of the flow velocity profile, so that the spectral properties of the turbulence excitation are modeled in a consistent manner. The proposed method for simulating turbulence excitations is faster than Shinozuka׳s technique by two orders of magnitude. Also, in the framework of our modal computational approach, nonlinear computations are faster, because no modal projection of physical turbulent forces is needed. After presenting the theoretical background and the details of the proposed simulation method, we illustrate it with representative linear and nonlinear computations performed on a multi-supported tube.
In this paper the problem of computing the nonlinear vibro-impact responses of loosely supported heat-exchanger tubes subjected to fluidelastic coupling forces, as well as to the turbulence ...excitation from transverse flows, is addressed. Emphasis is on the fluidelastic modeling within a time domain nonlinear framework, as well as on the stabilizing effect of impacts on the fluidelastic coupling forces. Theoretical computations of the linear and vibro-impacting regimes of a flow-excited flexible cantilever test tube, within a rigid 3×5 square bundle, are based on the experimentally identified fluidelastic coupling force coefficients and turbulence spectrum. Computations are then compared with the experimental vibratory responses, enabling a full validation of the modeling approach. Furthermore, interesting conclusions are drawn, concerning (a) the energy balance between sources and sinks, for a vibro-impacting tube subjected to fluidelastic forces and (b) the dependence of the vibration response frequency on impacts at the loose supports, and their effect on the nonlinear restabilization of fluidelastically unstable tubes. Details on the following aspects are reported in the paper: (1) numerical modeling of the fluidelastic coupling forces for the time domain computations; (2) experimental identification of the fluidelastic coupling coefficients; (3) computations and experiments of both linear and vibro-impacting responses under the combined action of turbulence and fluidelastic coupling and (4) energy aspects of the vibro-impacting fluidelastically coupled tube responses.
► Computation of the nonlinear response of loosely supported tube subjected to cross flow. ► Emphasis is on the fluidelastic modeling within a time domain nonlinear framework. ► Experiments and computations were performed for a large range of flow velocity. ► The results suggest that our approach is consistent with the experimental data.
Most systems consist on dynamical substructures connected at a number of constraining points. Moreover, constraints often display intermittent contact phenomena, such as arising from clearance ...supports. A significant difficulty when computing time-domain responses is the manner to enforce such coupling constraints. Here, we explore the Udwadia-Kalaba (U-K) formulation, which has been very seldom used in this context. By extending the basic U-K analytical framework, we address continuous flexible subsystems modelled by their unconstrained modes and coupled through the highly nonlinear intermittent point-constraints. For continuous flexible systems, a modal U-K formulation is implemented such that the constraint is applied when contact is detected at the clearance location. A crucial aspect is that constraint violations must be prevented, not only at the acceleration level, but also at the velocity and displacement levels, in order to avoid computational drift. This is achieved through a constraint violation correction method. For single gap-constraints, a convenient formulation is obtained, in which the constraint matrix is pre-computed prior to the simulation time-loop and applied whenever an intermittent contact is detected, leading to an efficient computation of vibro-impact responses. For systems with several intermittent constraints, an essential difficulty within the context of the proposed formulation is that every possible combination of contact/non-contact conditions is expressed by a different constraint matrix. A pragmatic solution is to keep track of the current system contact configuration and rebuild the constraint matrix whenever a change in the constraint state is detected. We formulate and illustrate such computational strategy, as applied to random-excited multi-supported beams with a significant number of clearance supports. Results are compared with dynamical computations performed using a classic penalty technique for enforcing the nonlinear support constraints, emphasizing the viability of the proposed technique for performing predictive analysis of flexible structures with multiple clearance supports.
In this paper, we propose analytical and numerical straightforward approximate methods to estimate the unknown terms of incomplete spectral or correlation matrices, when the cross-spectra or ...cross-correlations available from multiple measurements do not cover all pairs of transducer locations. The proposed techniques may be applied whenever the available data includes the auto-spectra at all measurement locations, as well as selected cross-spectra which implicates all measurement locations. The suggested methods can also be used for checking the consistency between the spectral or correlation functions pertaining to measurement matrices, in cases of suspicious data. After presenting the proposed spectral estimation formulations, we discuss their merits and limitations. Then we illustrate their use on a realistic simulation of a multi-supported tube subjected to turbulence excitation from cross-flow. Finally, we show the effectiveness of the proposed techniques by extracting the modal responses of the simulated flow-excited tube, using the SOBI (Second Order Blind Identification) method, from an incomplete response matrix 11This paper is an extended version of the work presented at the International Conference on Condition Machinery in Non-Stationary Operations (CMMNO13), May 8–10, 2013, Ferrara, Italy..
•We propose methods to estimate unknown cross-spectra in incomplete measurement sets.•The first two reconstruction methods are stated through explicit formulae.•The third method is a constrained iterative technique which always converged.•Convincing results were obtained for a vibrating tube subjected to flow turbulence.•These estimation techniques currently apply to systems with non-degenerate modes.
In some degraded situations of heat-exchangers, tubes may be loosely supported while subjected to intense cross-flow generating both turbulence and fluid-elastic forces, and so have vibro-impacting ...responses. This paper aims at providing a better understanding of the dynamics of such systems in which the conjunction of stops with gaps, broadband random excitation, and fluid-elastic coupling forces, either stabilizing or destabilizing, produces some rather amazing effects linked to a general capacity for the system to be auto-adjusted by the impacts. It is especially shown, (i) how a non-linear gap-system “escapes from instability” by reinforcing the sequence of impacts, (ii) the almost perfect time-history mirror symmetry between the works of turbulence and fluid-elastic forces in the energy balance; (iii) the dissipative aspect of turbulence in some cases. Moreover the paper deals with the relative weight of turbulence and fluid-elastic forces in the tube response that notably depends on the gap size. For this, turbulence is considered as a perturbation of limit cycles obtained when the fluid-elastic forces are the only acting fluid-forces. The study is based on configurations mainly 1-DOF, like the one that has been experimentally tested, but extrapolations for real tube bundle are evoked in conclusion.
Having gone through the specificity of nuclear energy beforehand, this article deals with the issue of Fluid Structure Interaction (FSI) in nuclear reactors. In this respect, it makes use of some ...cases taken out of the French electro-nuclear project. Bearing these examples in mind, it ends with a prospective elaboration of the needs of FSI in 10 years time.
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