Definition of a linear equivalent model for a non-linear system with impacts

Modal characteristics of non-linear system are typically studied through response to harmonic excitation and using various definitions of non-linear modes. However, few results are available for systems under broadband excitation. The end objective sought here is to generate a linear system, in some sense equivalent to the non-linear system, whose modal characteristics evolve with a level of non-linearity. The considered application is the contact non-linearity found between the tubes of heat exchangers and their support plates. Such tubes, present in nuclear plants, participate to the nuclear safety and can be significantly excited by the fluid flow, so that their dynamic behavior is critical. The turbulent nature of the flow implies broadband excitation and the small gaps between the tubes and the support plate generate very significant non-linear behavior. The proposed equivalent linear system is based on a bilateral contact law whose stiffness and damping characteristics evolve with the amplitude of excitation. A non-linear model is first validated by correlation with experiments. It is then shown that three different indicators (bandwidth of main resonance, operational modal analysis of non-linear power spectral density and correlation of operational deflection shapes) lead to similar values of contact stiffness and damping in the equivalent linear model. This model is hus shown to be a very efficient tool to analyze the impact of the amplitude dependence of the non-linear behavior in the considered system

Definition of a linear equivalent model for a non-linear system with impacts

International audienceModal characteristics of non-linear system are typically studied through response to harmonic excitation and using various definitions of non-linear modes. However, few results are available for systems under broadband excitation. The end objective sought here is to generate a linear system, in some sense equivalent to the non-linear system, whose modal characteristics evolve with a level of non-linearity. The considered application is the contact non-linearity found between the tubes of heat exchangers and their support plates. Such tubes, present in nuclear plants, participate to the nuclear safety and can be significantly excited by the fluid flow, so that their dynamic behavior is critical. The turbulent nature of the flow implies broadband excitation and the small gaps between the tubes and the support plate generate very significant non-linear behavior. The proposed equivalent linear system is based on a bilateral contact law whose stiffness and damping characteristics evolve with the amplitude of excitation. A non-linear model is first validated by correlation with experiments. It is then shown that three different indicators (bandwidth of main resonance, operational modal analysis of non-linear power spectral density and correlation of operational deflection shapes) lead to similar values of contact stiffness and damping in the equivalent linear model. This model is hus shown to be a very efficient tool to analyze the impact of the amplitude dependence of the non-linear behavior in the considered system

Etude d'un système non linéaire à chocs sous excitation large bande : application à un tube de générateur de vapeur

The steam generator is a heat exchanger and participates to the nuclear safety. Energy is transferred from the primary to the secondary fluid through many U-tubes maintained vertically by support plates. A sludge deposit tends to modify the boundary conditions and the secondary fluid flow. A fluid-elastic instability can then occur and lead to quick tube ruin. This thesis seeks a better understanding of the effect of contact nonlinearity on the dynamics of a tube in-air intermittently impacting the support plates and its consequences in regards with instability. The use of discretized contact conditions with circular obstacles distributed over the thickness of the plates and the use of enriched reduction bases allow quick and relevant nonlinear numerical simulations. These simulations are well correlated with experimental measurements and valid even with strong nonlinearity or negative modal damping. The evolution of power spectral densities (PSD) with growing excitation amplitude is analyzed: padding of the anti-resonances, peak shift and spread. It is then shown that an apparent stiffness associated with a permanent bilateral contact is pertinent to describe these transitions. In the case of an unstable linear system, one demonstrates that the nonlinearity keeps the responses bounded or stabilised, thus paving the way for future work with real or simulated fluid flows.Le générateur de vapeur a un rôle d’échangeur thermique et de sûreté nucléaire. L’énergie du fluide primaire est transférée au circuit secondaire via un faisceau de tubes en U soutenus par des plaques entretoise. Un dépôt modifie les conditions de liaison et l’écoulement du fluide secondaire. Le tube peut alors entrer en instabilité fluide-élastique, qui aboutit à une ruine rapide. Cette thèse vise à mieux comprendre l’effet de la non-linéarité de contact sur la dynamique d’un tube en air rentrant en contact intermittent avec les entretoises et ses conséquences en présence d’une instabilité. La discrétisation des conditions de contact entre le tube et les plaques, par des obstacles circulaires répartis sur l’épaisseur, ainsi que l’utilisation de bases de réductions enrichies permettent des résolutions numériques non-linéaires fiables et rapides, numériquement valides pour de fortes non-linéarités et des amortissements modaux négatifs, et bien corrélées avec les mesures correspondantes. On analyse les évolutions du contenu spectral (DSP) en fonction de l’amplitude d’excitation : remplissage des anti-résonances, décalage et étalement des pics ; puis on met en évidence la pertinence d’une notion de raideur apparente d’un contact bilatéral permanent qui permet de décrire ces transitions. En présence d’un système libre instable, on montre enfin que la non-linéarité conduit à des réponses bornées ou stabilisées ouvrant ainsi la voie vers une extension des travaux réalisés vers des applications avec fluide réel ou simulé

Study of a nonlinear system with shocks under broadband excitation : application to a steam generator tube

Le générateur de vapeur a un rôle d’échangeur thermique et de sûreté nucléaire. L’énergie du fluide primaire est transférée au circuit secondaire via un faisceau de tubes en U soutenus par des plaques entretoise. Un dépôt modifie les conditions de liaison et l’écoulement du fluide secondaire. Le tube peut alors entrer en instabilité fluide-élastique, qui aboutit à une ruine rapide. Cette thèse vise à mieux comprendre l’effet de la non-linéarité de contact sur la dynamique d’un tube en air rentrant en contact intermittent avec les entretoises et ses conséquences en présence d’une instabilité. La discrétisation des conditions de contact entre le tube et les plaques, par des obstacles circulaires répartis sur l’épaisseur, ainsi que l’utilisation de bases de réductions enrichies permettent des résolutions numériques non-linéaires fiables et rapides, numériquement valides pour de fortes non-linéarités et des amortissements modaux négatifs, et bien corrélées avec les mesures correspondantes. On analyse les évolutions du contenu spectral (DSP) en fonction de l’amplitude d’excitation : remplissage des anti-résonances, décalage et étalement des pics ; puis on met en évidence la pertinence d’une notion de raideur apparente d’un contact bilatéral permanent qui permet de décrire ces transitions. En présence d’un système libre instable, on montre enfin que la non-linéarité conduit à des réponses bornées ou stabilisées ouvrant ainsi la voie vers une extension des travaux réalisés vers des applications avec fluide réel ou simulé.The steam generator is a heat exchanger and participates to the nuclear safety. Energy is transferred from the primary to the secondary fluid through many U-tubes maintained vertically by support plates. A sludge deposit tends to modify the boundary conditions and the secondary fluid flow. A fluid-elastic instability can then occur and lead to quick tube ruin. This thesis seeks a better understanding of the effect of contact nonlinearity on the dynamics of a tube in-air intermittently impacting the support plates and its consequences in regards with instability. The use of discretized contact conditions with circular obstacles distributed over the thickness of the plates and the use of enriched reduction bases allow quick and relevant nonlinear numerical simulations. These simulations are well correlated with experimental measurements and valid even with strong nonlinearity or negative modal damping. The evolution of power spectral densities (PSD) with growing excitation amplitude is analyzed: padding of the anti-resonances, peak shift and spread. It is then shown that an apparent stiffness associated with a permanent bilateral contact is pertinent to describe these transitions. In the case of an unstable linear system, one demonstrates that the nonlinearity keeps the responses bounded or stabilised, thus paving the way for future work with real or simulated fluid flows

Etude d'un système non linéaire à chocs sous excitation large bande. Application à un tube de générateur de vapeur,

Le générateur de vapeur a un rôle d échangeur thermique et de sûreté nucléaire. L énergie du fluide primaire est transférée au circuit secondaire via un faisceau de tubes en U soutenus par des plaques entretoise. Un dépôt modifie les conditions de liaison et l écoulement du fluide secondaire. Le tube peut alors entrer en instabilité fluide-élastique, qui aboutit à une ruine rapide. Cette thèse vise à mieux comprendre l effet de la non-linéarité de contact sur la dynamique d un tube en air rentrant en contact intermittent avec les entretoises et ses conséquences en présence d une instabilité. La discrétisation des conditions de contact entre le tube et les plaques, par des obstacles circulaires répartis sur l épaisseur, ainsi que l utilisation de bases de réductions enrichies permettent des résolutions numériques non-linéaires fiables et rapides, numériquement valides pour de fortes non-linéarités et des amortissements modaux négatifs, et bien corrélées avec les mesures correspondantes. On analyse les évolutions du contenu spectral (DSP) en fonction de l amplitude d excitation : remplissage des anti-résonances, décalage et étalement des pics ; puis on met en évidence la pertinence d une notion de raideur apparente d un contact bilatéral permanent qui permet de décrire ces transitions. En présence d un système libre instable, on montre enfin que la non-linéarité conduit à des réponses bornées ou stabilisées ouvrant ainsi la voie vers une extension des travaux réalisés vers des applications avec fluide réel ou simulé.The steam generator is a heat exchanger and participates to the nuclear safety. Energy is transferred from the primary to the secondary fluid through many U-tubes maintained vertically by support plates. A sludge deposit tends to modify the boundary conditions and the secondary fluid flow. A fluid-elastic instability can then occur and lead to quick tube ruin. This thesis seeks a better understanding of the effect of contact nonlinearity on the dynamics of a tube in-air intermittently impacting the support plates and its consequences in regards with instability. The use of discretized contact conditions with circular obstacles distributed over the thickness of the plates and the use of enriched reduction bases allow quick and relevant nonlinear numerical simulations. These simulations are well correlated with experimental measurements and valid even with strong nonlinearity or negative modal damping. The evolution of power spectral densities (PSD) with growing excitation amplitude is analyzed: padding of the anti-resonances, peak shift and spread. It is then shown that an apparent stiffness associated with a permanent bilateral contact is pertinent to describe these transitions. In the case of an unstable linear system, one demonstrates that the nonlinearity keeps the responses bounded or stabilised, thus paving the way for future work with real or simulated fluid flows.CHATENAY MALABRY-Ecole centrale (920192301) / SudocSudocFranceF

STABILIZATION EFFECT OF SHOCK NON-LINEARITY ON THE DYNAMICS OF A STEAM GENERATOR TUBE

International audienceIn Pressurized Water Reactors of nuclear power plants, steam generators act as heat exchangers between primary and secondary coolant fluids. They consist of a bundle of U-tubes in which flows the primary coolant fluid. Several support plates guide these tubes vertically. Secondary coolant fluid flows along the U-tubes and passes through the space between tubes and plates. This space, initially of a foliate shape, is filled with sludge deposits. Consequently, fluid flow is accelerated and the tubes are more excited. Moreover, the mechanical bonding between tube and plate is changed. The combination of these two phenomena can lead to dynamic instabilities and tube cracks. This paper focuses on the effects of shocks on tube instability. As we concentrate on a mechanical point of view, the study of the nonlinear dynamics is made in the following specific conditions: the tube is in air with no flowing fluid; an instability is generated by injecting a force proportional to the velocity of a point of the leg of the tube. The tube is modeled as an Euler-Bernoulli beam. The contact between tube and plate is computed assuming circular obstacles and a contact force that is linear with a gap. A reduced model is generated to represent the bandwidth of interest and the effect of contact forces. Time evolution is then computed using a nonlinear Newmark scheme. Numerical simulations show the effect of shock nonlinearities on the dynamics. Cases of unstable unconstrained tubes leading to bounded stable dynamics when shocks occur are analyzed. The instability in the unconstrained condition does not imply that the bilateral contact condition is unstable. Areas of the contact stiffness/feedback gain plane are shown to lead to stable modes of the bilateral model. The work then presents results on an experimental bench that includes the curved part of a tube and one Support Plate. The instability is generated by feeding back to a shaker the amplified signal of a velocity measurement performed with an integrated accelerometer. For great enough feedback gains, the vibration amplitudes grow until shocks occur. The permanent regime eventually found is analyzed. A good similarity between test and analysis is found

STABILIZATION EFFECT OF SHOCK NON-LINEARITY ON THE DYNAMICS OF A STEAM GENERATOR TUBE

International audienceIn Pressurized Water Reactors of nuclear power plants, steam generators act as heat exchangers between primary and secondary coolant fluids. They consist of a bundle of U-tubes in which flows the primary coolant fluid. Several support plates guide these tubes vertically. Secondary coolant fluid flows along the U-tubes and passes through the space between tubes and plates. This space, initially of a foliate shape, is filled with sludge deposits. Consequently, fluid flow is accelerated and the tubes are more excited. Moreover, the mechanical bonding between tube and plate is changed. The combination of these two phenomena can lead to dynamic instabilities and tube cracks. This paper focuses on the effects of shocks on tube instability. As we concentrate on a mechanical point of view, the study of the nonlinear dynamics is made in the following specific conditions: the tube is in air with no flowing fluid; an instability is generated by injecting a force proportional to the velocity of a point of the leg of the tube. The tube is modeled as an Euler-Bernoulli beam. The contact between tube and plate is computed assuming circular obstacles and a contact force that is linear with a gap. A reduced model is generated to represent the bandwidth of interest and the effect of contact forces. Time evolution is then computed using a nonlinear Newmark scheme. Numerical simulations show the effect of shock nonlinearities on the dynamics. Cases of unstable unconstrained tubes leading to bounded stable dynamics when shocks occur are analyzed. The instability in the unconstrained condition does not imply that the bilateral contact condition is unstable. Areas of the contact stiffness/feedback gain plane are shown to lead to stable modes of the bilateral model. The work then presents results on an experimental bench that includes the curved part of a tube and one Support Plate. The instability is generated by feeding back to a shaker the amplified signal of a velocity measurement performed with an integrated accelerometer. For great enough feedback gains, the vibration amplitudes grow until shocks occur. The permanent regime eventually found is analyzed. A good similarity between test and analysis is found

Ageing and air leakage assessment of a nuclear reactor containment mock-up:VERCORS 2nd benchmark

International audienceElectricité de France (EDF) operates a large fleet of nuclear reactors and is responsible for demonstrating the safety of facilities, including concrete containment buildings (CCB), which are non-replaceable components. The leak-tightness of CCBs is assessed every 10 years during integrated leak-rate tests (IRLT). For double-wall containments, which have no metallic liners, the leak-tightness is strongly influenced by the degree of cracking of concrete and opening of the cracks, which mostly depends on (a) the prestress decrease due to the delayed strains of concrete, and (b) the saturation degree of the concrete wall. Therefore, to optimize the maintenance programs on CCBs, it is important to predict the evolution of drying, creep shrinkage strains of concrete to be able to correctly assess the pre-stress losses, and finally the air leak-tightness at a structural level during pressure tests or under accidental loadings.To improve our understanding and identify the best modelling practices on this issue, a large experimental program called VERCORS was launched in 2014. VERCORS is a 1/3 mock-up of a 1300 MWe nuclear reactor CCB. It is widely instrumented, and its concrete thoroughly studied. A specific attention has been paid to ensure it is consistent with real CBBs features in EDF’s nuclear fleet.To complement its internal R&D efforts, EDF decided to associate external partners to this program. One of the means for this is the organization of benchmarks, where all teams are given data and information about the mock-up and are asked to forecast its behaviour. The present paper reports the organization and findings of the 2nd benchmark which was organized in 2018 and gathered several international teams around the same objective: improve the confidence in the modelling of structural behaviour as well as the leak-tightness of concrete in containment walls under pressure test loading