In the last decades the design process of engineering structures was characterized by an increasing demanding of the prediction capabilities at the design stage, which would have allowed to asses not only the performance and the integrity of the structure but also the vibro-acoustic performances. The more realistic is the model, the greater the opportunity to produce optimal design; this implies the need of using a model with a very high number of degrees of freedom. Predicting the response of a complex structural-acoustic system presents some difficulties. In fact, direct numerical solution of the governing equations of the system, while possible in principle, can be so computationally demanding as to be impractical. Furthermore, the application of deterministic methods of analysis to the prediction of its vibrational response would be not appropriate. In fact, in a complex system the presence of a high number of heterogeneous subsystems (beams, plates, acoustic cavities, etc.) and, above all, the high number of joints connecting the components are sources of a certain degree of variability of the system. However, geometric and fabrication tolerances, variation in material properties, structural irregularities, non uniform damping distribution are some of the cause of variability that lead to different dynamical behaviours of samples of a set of nominally identical manufactures. Moreover, many important engineering problems involve high frequency vibrations. In fact, especially in recent years, the use of light structures in aircraft and aerospace engineering, broad band excitations due to engines of increasing power and the increasing interest for the high speeds in ship, automotive and train engineering, created a general attention in the analysis of the high frequency vibration and of the acoustic problems. Approaching the high frequency problem both the difficulties above presented meet two serious limitations. In fact, high frequency problems involve vibrations characterized by short wavelengths compared with a typical length of the system. In conventional vibrational analysis based on a discretization of the continuum domains into small elements, being the size of these elements dependent on the minimum wavelength of interest, the computational time demanding could be so high that it would become prohibitive. Besides the problem of high time demanding, which could be in principle overcome with the growth of the computer power, the use of a very fine mesh also implies an accurate modelling of the system details. Obviously, such a modelling is difficult to obtain and it also introduces a certain degree of variability. On the other hand, at high frequency the response of the system becomes increasingly sensitive to small perturbations of its parameters: even small variations in geometrical component dimensions, material properties and assembly tolerances imply large variations in the mid- and high- frequency responses. In fact, while low order eigenvalues of the system are lightly affected by small variation of the system parameters, the more the order of the eigenvalues increases the more their values are modified, such that the behaviour of the structure becomes unpredictable. These considerations led to introduce a different way of tackling dynamic problems, based on a thermodynamic analogy. It is in fact in that frame that the problem related with system characterized by a high number of degrees of freedom (atoms) was studied for the first time. The necessity to reduce the computational cost has led to introduce global parameter as descriptor of the systems, rather than the local characterization, i.e. the punctual displacement. Further, to overcome the second problem, i.e. the inherent uncertainties of the system parameters, a statistical approach was favourite to a deterministic one, so that the attention is focused on an ensemble of similar system which characteristics are defined with a certain degree of variability. The Statistical Energy Analysis (SEA) represents in this scenario the most fruitful method inspired up to now. For the first time, some concepts reminiscent of thermodynamic were applied on the analysis of the dynamical behaviour of a mechanical system. In fact, SEA states an analogy between the energy flow in mechanical systems and the heat flow in thermal problems, such that the energy diffuses from one substructure to another at a rate proportional to the difference in "temperature", i.e. the modal energy, of the substructures and it is dissipated internally in each substructure at a rate proportional to the "temperature" of the substructure. Even if the analogy with thermodynamic problems brought important results in the energy analysis of vibrating structures, this analogy still present limits not yet completely understood. Moreover, the evaluation of the transient response to not steady excitation cannot be obtained by the classical SEA or other thermal-like approach, being their use limited at the steady state. Such an excitation is encountered in a large number of engineering structures: slamming load and breaking waves upon the bow of the ship hull, pyrotechnic shock in space vehicle structures, earthquake in civil engineering are only few examples of this. The impulsive excitation is particularly severe for the system, since can cause structural failure or unwanted noise. This work is concerned with the development of a predictive method for describing in the time domain the energy sharing among complex vibrating systems, affected by inherent uncertainty of their parameters. Starting from a new approach to the problem, based on a suitable combination of statistical thermodynamics and classical structural dynamics, the goal relies in a better understanding of both the initial transient energy sharing among the subcomponents and the long term energy response of the subcomponents. Before concluding this introduction, a brief description of the argument exposed in the following chapters is presented. In the next chapter, a critical analysis of the available method that, up to now, can be used for the analysis of complex systems is given. The attention is mainly focused upon the discernment of the advantages of each method, as well as upon the difficulties and the limitations of their application. In particular, the attention is directed to those methods that are partially inspired by a thermal analogy. In chapter 3 a new probabilistic energetic approach, (the Time Asymptotic Energy ensemble Average -TAEA), which has represented the foundation of the present work, is described. The originality of this methods lies in the development of an asymptotic expansion technique, that permits to evaluate the energy distribution between two subcomponents of a weakly coupled system, in both transient and steady state conditions with a low computational cost. As a particular result, this method also provides the long term energy responses of the two subsystems that, under some assumptions that will be later presented, expresses a condition reminiscent of the Energy Equipartition Principle (EEP) stated in Statistical Mechanics (SM). In the light of this results, it seemed interesting to go deep into the principle of energy equipartition as stated in SM, trying to understand the conditions under which it holds, the definitions introduced in SM and how these last could be translated to structural dynamics. This analysis, described in chapter 4, underlines that the EEP is not an obvious result in the analysis of dynamical systems. This is clear through the consideration that to approach the EEP in SM a number of assumptions are necessary, that are often not satisfied for engineering systems. This chapter is addressed investigating the conditions that can lead to the appearance of EEP and, as far as possible, a classification of the inhibiting or promoting factors for the reaching of energy equipartition conditions is presented. The analysis explores the field of linear and nonlinear vibrations, the effects of non-homogeneity and localization, the system dimension (i.e. its number of degrees of freedom), the weak or the strong coupling as well as the effect of the initial energy distribution among the subsystems. Chapter 5 is focused upon the theoretical improvement of TAEA theory to include an arbitrary strong coupling between the substructures and the interaction among more than two subsystems. These analyses has led to reformulate the theory introducing a new energetic parameter, which is the sum of two terms: the "blocked energy term", which represents the sum of the kinetic and the potential energy terms of the subsystem when the other subsystems are considered blocked and the "mixed energy term", which provide the energy contribution to the subsystem energy due to the motion of the others. This new formulation allows the determination of the energy sharing among two or more subsystems even in presence of strong coupling among them. In chapter 6 a validation of TAEA is presented through a comparison of the energy distribution among subcomponents of a system obtained applying the developed method and those obtained experimentally. Two different two-dimensional systems were analysed: the former is composed by two plates coupled by means of straps, while the second system is made up of three plates connected through straps among the subsystems. Despite a small difference between the theoretical results and the measured energies at the first instants, the agreement between them is very satisfactory in both cases. Chapters 7 and 8 close this thesis summarising the achieved results and showing some possible perspectives of the present study.
An investigation on energy dynamics of complex resonators
Francesca Magionesi
2007
Abstract
In the last decades the design process of engineering structures was characterized by an increasing demanding of the prediction capabilities at the design stage, which would have allowed to asses not only the performance and the integrity of the structure but also the vibro-acoustic performances. The more realistic is the model, the greater the opportunity to produce optimal design; this implies the need of using a model with a very high number of degrees of freedom. Predicting the response of a complex structural-acoustic system presents some difficulties. In fact, direct numerical solution of the governing equations of the system, while possible in principle, can be so computationally demanding as to be impractical. Furthermore, the application of deterministic methods of analysis to the prediction of its vibrational response would be not appropriate. In fact, in a complex system the presence of a high number of heterogeneous subsystems (beams, plates, acoustic cavities, etc.) and, above all, the high number of joints connecting the components are sources of a certain degree of variability of the system. However, geometric and fabrication tolerances, variation in material properties, structural irregularities, non uniform damping distribution are some of the cause of variability that lead to different dynamical behaviours of samples of a set of nominally identical manufactures. Moreover, many important engineering problems involve high frequency vibrations. In fact, especially in recent years, the use of light structures in aircraft and aerospace engineering, broad band excitations due to engines of increasing power and the increasing interest for the high speeds in ship, automotive and train engineering, created a general attention in the analysis of the high frequency vibration and of the acoustic problems. Approaching the high frequency problem both the difficulties above presented meet two serious limitations. In fact, high frequency problems involve vibrations characterized by short wavelengths compared with a typical length of the system. In conventional vibrational analysis based on a discretization of the continuum domains into small elements, being the size of these elements dependent on the minimum wavelength of interest, the computational time demanding could be so high that it would become prohibitive. Besides the problem of high time demanding, which could be in principle overcome with the growth of the computer power, the use of a very fine mesh also implies an accurate modelling of the system details. Obviously, such a modelling is difficult to obtain and it also introduces a certain degree of variability. On the other hand, at high frequency the response of the system becomes increasingly sensitive to small perturbations of its parameters: even small variations in geometrical component dimensions, material properties and assembly tolerances imply large variations in the mid- and high- frequency responses. In fact, while low order eigenvalues of the system are lightly affected by small variation of the system parameters, the more the order of the eigenvalues increases the more their values are modified, such that the behaviour of the structure becomes unpredictable. These considerations led to introduce a different way of tackling dynamic problems, based on a thermodynamic analogy. It is in fact in that frame that the problem related with system characterized by a high number of degrees of freedom (atoms) was studied for the first time. The necessity to reduce the computational cost has led to introduce global parameter as descriptor of the systems, rather than the local characterization, i.e. the punctual displacement. Further, to overcome the second problem, i.e. the inherent uncertainties of the system parameters, a statistical approach was favourite to a deterministic one, so that the attention is focused on an ensemble of similar system which characteristics are defined with a certain degree of variability. The Statistical Energy Analysis (SEA) represents in this scenario the most fruitful method inspired up to now. For the first time, some concepts reminiscent of thermodynamic were applied on the analysis of the dynamical behaviour of a mechanical system. In fact, SEA states an analogy between the energy flow in mechanical systems and the heat flow in thermal problems, such that the energy diffuses from one substructure to another at a rate proportional to the difference in "temperature", i.e. the modal energy, of the substructures and it is dissipated internally in each substructure at a rate proportional to the "temperature" of the substructure. Even if the analogy with thermodynamic problems brought important results in the energy analysis of vibrating structures, this analogy still present limits not yet completely understood. Moreover, the evaluation of the transient response to not steady excitation cannot be obtained by the classical SEA or other thermal-like approach, being their use limited at the steady state. Such an excitation is encountered in a large number of engineering structures: slamming load and breaking waves upon the bow of the ship hull, pyrotechnic shock in space vehicle structures, earthquake in civil engineering are only few examples of this. The impulsive excitation is particularly severe for the system, since can cause structural failure or unwanted noise. This work is concerned with the development of a predictive method for describing in the time domain the energy sharing among complex vibrating systems, affected by inherent uncertainty of their parameters. Starting from a new approach to the problem, based on a suitable combination of statistical thermodynamics and classical structural dynamics, the goal relies in a better understanding of both the initial transient energy sharing among the subcomponents and the long term energy response of the subcomponents. Before concluding this introduction, a brief description of the argument exposed in the following chapters is presented. In the next chapter, a critical analysis of the available method that, up to now, can be used for the analysis of complex systems is given. The attention is mainly focused upon the discernment of the advantages of each method, as well as upon the difficulties and the limitations of their application. In particular, the attention is directed to those methods that are partially inspired by a thermal analogy. In chapter 3 a new probabilistic energetic approach, (the Time Asymptotic Energy ensemble Average -TAEA), which has represented the foundation of the present work, is described. The originality of this methods lies in the development of an asymptotic expansion technique, that permits to evaluate the energy distribution between two subcomponents of a weakly coupled system, in both transient and steady state conditions with a low computational cost. As a particular result, this method also provides the long term energy responses of the two subsystems that, under some assumptions that will be later presented, expresses a condition reminiscent of the Energy Equipartition Principle (EEP) stated in Statistical Mechanics (SM). In the light of this results, it seemed interesting to go deep into the principle of energy equipartition as stated in SM, trying to understand the conditions under which it holds, the definitions introduced in SM and how these last could be translated to structural dynamics. This analysis, described in chapter 4, underlines that the EEP is not an obvious result in the analysis of dynamical systems. This is clear through the consideration that to approach the EEP in SM a number of assumptions are necessary, that are often not satisfied for engineering systems. This chapter is addressed investigating the conditions that can lead to the appearance of EEP and, as far as possible, a classification of the inhibiting or promoting factors for the reaching of energy equipartition conditions is presented. The analysis explores the field of linear and nonlinear vibrations, the effects of non-homogeneity and localization, the system dimension (i.e. its number of degrees of freedom), the weak or the strong coupling as well as the effect of the initial energy distribution among the subsystems. Chapter 5 is focused upon the theoretical improvement of TAEA theory to include an arbitrary strong coupling between the substructures and the interaction among more than two subsystems. These analyses has led to reformulate the theory introducing a new energetic parameter, which is the sum of two terms: the "blocked energy term", which represents the sum of the kinetic and the potential energy terms of the subsystem when the other subsystems are considered blocked and the "mixed energy term", which provide the energy contribution to the subsystem energy due to the motion of the others. This new formulation allows the determination of the energy sharing among two or more subsystems even in presence of strong coupling among them. In chapter 6 a validation of TAEA is presented through a comparison of the energy distribution among subcomponents of a system obtained applying the developed method and those obtained experimentally. Two different two-dimensional systems were analysed: the former is composed by two plates coupled by means of straps, while the second system is made up of three plates connected through straps among the subsystems. Despite a small difference between the theoretical results and the measured energies at the first instants, the agreement between them is very satisfactory in both cases. Chapters 7 and 8 close this thesis summarising the achieved results and showing some possible perspectives of the present study.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
