RICERCA ITALIANA     

Dynamic modeling and control of complex mechanical structures with uncertain parameters

Università degli Studi di Roma "La Sapienza"

Abstract

The purpose of the research proposed is threefold:
- to study of dynamic effects of complex structural models with uncertain parameters;
- to create dynamic models of complex structures that consist of elements which are difficult or even impossible to characterize for lack of information on their construction details and/or on the materials used;
- to control the dynamic response of systems with uncertain parameters.
All these items can be considered in the general topic "Modeling of uncertain systems", that is becoming a major interest in Industry and, especially, in the transportation sector. In fact, increasingly the interest of industry is not to address the design or validation of a single unit but instead to estimate how a set of such units behaves when assembled to form a complex system. Since any component differs from the others because of unavoidable variations due to working tolerances, enviromental conditions, material non-uniformities, etc., it is necessary to evaluate the way such variations affect the whole system. Moreover, some components are too complex to model and, thus, neither more powerful computers nor more specific information on the components' features can lead to a confident description of them, at least with the standard techniques (typically FE). On the other hand, such components often have a relevant role in the general behavior of the whole system (vibration and vibroacoustic), so that they require a particular modeling approach. Finally, when dealing with uncertainties or with components with manufacturing non-uniformities or working in very different enviromental conditions, it is also necessary to have appropriate control systems, robust while adaptive, so that they can automatically fit conditions different from those originally designed for.
Therefore, the project will address different but intrinsically related activities: first, the effect of uncertainties by probabilistic and non-deterministic techniques (interval algebra and fuzzy mathematics) will be studied to estimate in which cases and under which conditions the former are preferrable to the latter; in this framework the project will also propose new developments that will produce significant contributions to the non-deterministic analysis from computation and reliability points of view. These studies are also important for the design of control systems. Then, techniques of "robust modeling" will be developed to describe complex systems: this will be obtained by avoiding a local description of the system response while looking for a global description, e.g. in terms of energy. Such model should be used efficiently both to assemble the component to the main system and to also represent the behavior of components of the same population. Finally, attention will be focused on control techniques:
- the effect of uncertainties on the dynamic characteristics of the system to be controlled will be analyzed;
- methods based on energy models will be developed with the aim of reducing the dynamics of complex assembled structures to a few parameters;
- adaptive control systems will be studied and developed. <<<

Principal Investigator

Aldo SESTIERI Università degli Studi di ROMA "La Sapienza"

Research Objectives

The present project is aimed to provide a significant contribution to describe and control the dynamic behavior of complex structures affected by stochastic uncertainties and variability in the system parameters. The subject is very important for the transportation industry where many components, with uncertain characteristics, play a crucial role in the vibrational and vibroacoustic comfort of passengers.
In recent years the industry has paid increasing attention and devoted important efforts to the dynamic characterization of the vehicle and its components. In the low frequency range (specifically in the range where the wavelengths involved are comparable with the characteristic dimensions of the system considered), the finite element method (FEM) is the most common procedure and is used with increasing confidence, although an updating procedure is often necessary to obtain results comparable with the experimental data. The high frequency range, important in the analysis of vibroacoustic problems, is more and more addressed by the Statistical Energy Analysis (SEA) and/or alternative techniques that produce, in general, a response averaged in space and frequency (octave or third-octave bandwidths). However, excluding SEA whose explicit goal is that of solving the problem in statistical terms with reference to a population of similar systems, none of the other methods, either for low or high frequencies, address directly the modeling of components in presence of uncertainties. To overcome this limitation, in the field of structural mechanics great attention was given to probabilistic and/or non-deterministic methods to investigate the way uncertainties can be introduced in the design or modeling phase of a structure. The most studied procedures are the Monte Carlo method, the stochastic finite elements, the polynomial chaos, the stochastic perturbation methods (probabilistic methods), the interval algebra, the fuzzy arithmetic (non-deterministic methods).
However, so far the modeling of components with uncertain parameters is generally unreliable, either because some of these techniques are still at an early stage of development or because the computational burden is still too heavy for the present computers. This implies the inability for the designer to obtain an acceptable prediction of the comfort performances before the prototype phase of the vehicle. Consequently, in general a trial and error procedure is adopted when the prototype or a pre-series vehicle is built, leading to iterative modifications with high costs and, quite often, unsatisfactory results. The immediate consequence of this approach is a real difficulty to integrate the vibration and vibroacoustic performances of the vehicle into the process of virtual prototyping.
The proposed project addresses specifically these concepts. The research units participating in the project all have extensive experience in vibration and vibroacoustic modeling and most of them have recently paid increasing attention to problems with stochastic uncertainties and design variabilities. It is worthwhile to underline that the groups involved in the project have already worked together in a previous PRIN (2003) "Energy models for the analysis and control of high frequency vibration amd vibroacoustic problems". Moreover, the research partners belong to different disciplinary groups: Florence is represented by people working on Machine Design, Naples on Aeronautical Constructions, the two groups in Rome work on Applied Mechanics (Vibration) and Theory of Elasticity, respectively. This indicates that the background of the different groups, though they have a common denominator born of industrial requirements, are quite dissimilar. This, in turn, can have very positive effects on the project development. The proposal of a common project that links together these groups could, on one hand, give an important contribution to the items considered in the project, by creating better coordinated developments, while providing, on the other hand, tested procedures that can be easily transferred to industrial applications. This would also create a stronger interaction between Industry and Academia in a field of increasing importance for its implications on comfort.
This goal will be pursued in the following way:
1. application of some probabilistic and non-deterministic techniques, e.g. stochastic perturbation, asymptotic probability, interval algebra, and fuzzy arithmetic, to simple structures of engineering interest. This phase is aimed to evaluate, for any system, the applicability and reliability of the above techniques. A specific benchmark will be designed and constructed: it is expected to be complex enough but still "academic", with uncertain elements such as a damping layer and a bolted joint connecting three plates, and will be delivered to all the groups that will perform numerical and experimental tests based on it, and will compare the results and procedures;
2. analysis and developments on the interval and fuzzy methods with relation to the estimate of reliability of models describing uncertain systems and with reference to the study of coupled structures in the modal domain that would be able to reduce drastically the computational time when using a fuzzy description of uncertainties;
3. development of "robust modeling" for the analysis of some real vehicle components that account for the uncertainties and parameters variability;
4. development of methods of robust control that ensure the control system performances for moderate variations of the structural characteristics are kept stable; design of systems of adaptative control that will provide optimal performances even when there is a lack of information on the dynamic characteristics of the structure and capable of evolving to track changes due to internal and/or external factors;
5. strict interaction among the different groups of the project, with on-going comparison of the methods developed and on the quality of the obtained results. <<<

Timescale

24 months

National and international background

The dynamic characterization of complex systems is a problem that has not a simple solution and represents an open challenge for the research community. Several items contribute to such difficulty.
- Some vehicle components are difficult to model and, yet, play a significant role in the vibroacoustis characterization of vehicles: for example the seats, the dash-board, the internal trimmings, etc. are themselves complex systems each of which consists of a large number of different elements and different materials. The complexity of such systems precludes the analyst from the possibility of using standard models for their description. Moreover, even when appropriate models are developed to represent joints and particular elements, they introduce quite often systematic modeling errors, that are amplified by their respective number when assembling the whole system. A possible strategy to circumvent such difficulty is the development of robust models. A robust model should simplify considerably the dynamics of the considered system by reducing the information of the output. More precisely, one sacrifices the information related to the local details while keeping a global, but still significant, information on the system response. An example of this is represented by those methods describing only the average energy (in frequency, space or in statistical sense) of the subsystems that make up the system under consideration. The word "robust" is appropriate in that the quantity considered in the model (e.g. the energy) is not affected by the complex construction details of the system: in other words, systems with different construction components respond in the same way in terms of the quantity considered in the model.
- The complex systems considered above introduce large stochastic uncertainties when modeling the system. For example one has, in general, a very small knowledge on the large number of plastic materials that are used in many components, on the foams, on the clamping characteristics of hundred of elements (welded, screwed, bolted), on the surface properties of several materials subjected to friction, etc. These are the elements that produce the largest differences between the estimated response and those determined experimentally a posteriori, making many classic deterministic models unreliable for this purpose. Therefore, within this framework, such deterministic models must be necessarily abandoned in favour of probabilistic methods or methods based on the interval algebra and/or fuzzy arithmetic (non-deterministic). The last are more efficient computationally but overestimate the range of the uncertain response. However, their relevant computational advantage pushes to investigate on algorithms to reduce such overestimate. In some recent references it is possible to find interesting applications in structural dynamics devoted to this end.
- It is useful to underline that, beyond the uncertainties described above, the possibility for the designer to vary some system parameters must be considered. In fact, at the design stage, the analyst can play with different design parameters that have not necessarily a unique value so that he may choose within a limited range of values. Under these conditions, it is necessary to determine the response range when varying the input parameters in their admissible intervals. These design variables have a different nature with respect to the stochastic uncertainties. While the last ones belong to the class of problems that are traditionally studied by the theory of probability, the first ones are not easily set in this context and instead they can be tackled more appropriately with the interval algebra or the fuzzy mathematics.
- The numerical prediction of the dynamic behavior, and, to a larger extent, of the vibroacoustic response of complex structures needs often a very large mesh because the wavelengths involved are much smaller that the dimensions of the structure considered. This implies a very long computation time that becomes often prohibitive at the design stage. Moreover, because of the small wavelengths, the dynamic and vibroacoustic response of a large number of components is characterized by a high modal density (a large number of modes per frequency band). Then, one easily understands that the vibroacoustic field is intrinsically characterized by a high sensitivity to small perturbations of the used parameters [1]. Therefore, for dynamic problems at medium and high frequencies, the uncertainties produce an important disagreement between the actual response and the response predicted deterministically with assigned reference parameters.
- The dynamic and vibroacoustic problem is strictly related to the problem of the response control. The stochastic uncertainties imply the need to design a robust and/or adaptive control, able to meet the lack of crisp (fixed) reference parameters. Thus, the problem needs the use of control techniques that auto-adapt over a given population of similar systems. In this way, the system control must be capable to adapt itself to the vibroacoustic characteristics of the particular sample on which it is installed.
Historically, the analysis of dynamic stochastic problems originates at the end of 1800 with the field of statistical physics (Boltzman, Maxwell, Gibbs) [2] and, at the beginning of the twentieth century, with the study of Brownian movements (Langevin, Einstein) [3,4]. The study of Brownian movements by Langevin outlines some elements that are common to the modern analysis of stochastic dynamics. The innovative feature of this theory was the derivation of the motion equation for a solid particle immersed in a fluid under the action of two forces: one deterministic, representing the friction effect of the fluid, the second stochastic, representing the random shocks of the fluid molecules. Starting from this pioneering work, efforts were made to develop a general theory for this class of differential equations. It is worthwhile to point out that, from a mathematical point of view, the case of a random force acting on a deterministic system has been widely analyzed and is now a classic part of the theory of linear systems [5]. On the contrary, the analysis of stochastic systems is much more complex and requires the solution of differential equations with stochastic coefficients [6,7]: these problems were addressed only starting in the '40s, with the work of Rice on the sound propagation in a medium with a randomly varying refraction index.
An important contribution to the study of dynamics of uncertain systems was given by applications where the uncertainty has a direct relationship with the construction of corresponding structures, as in civil and aerospace engineering. Using a statistical approach to the problem, the first step is represented by the statistical description of the solution as long as the statistics of the uncertainties characterizing the system is known, generally given by the probability density function (PDF). This can be determined theoretically by the Fokker-Plank-Kolmogorov equation, developed in the field of kinetic physics [2,3]. However, the enormous difficulty of this partial differential equation does not justify its use, especially when much simpler problems must be solved.
Considering the difficulty to model uncertain systems as compared with deterministic systems with random excitation, the methods available are complex and often not well established. Among the applicable techniques, the Monte Carlo simulation is one of the most commonly used methods [8]. However, its solution may be often prohibitive because the number of simulations to obtain significant statistical results is generally very high. Another technique to analyze uncertain systems is the stochastic perturbation method [6,9,10], consisting of expanding the random parameters in Taylor series around their mean values: the validity of this method, however, is restricted to random elements presenting a small fluctuation around their mean value.
Starting in the ‘60s, a new procedure for the analysis of complex and uncertain engineering structures was developed, inspired by the statistical mechanics: the Statistical Energy Analysis (SEA) [11-14]. The basic idea of the method assumes that the mechanical energy exchanged between dynamic systems behaves as does the temperature in thermal systems, and establishes a proportionality relationship between the energy flow and the energy of each subsystem. In SEA and in other techniques that refer to it (e.g. the Energy Flow Analysis [15,16]), only the principle of energy conservation is used to express the energy balance between subsystems, while other important theoretical elements, related to the entropic evolution, are neglected [17]. As a consequence, for an efficient application of all these techniques, a large number of assumptions, often not clearly defined, must be satisfied.
Another interesting approach developed recently is the asymptotic probabilistic energy model [18]. The word asymptotic implies that the analysis holds for sufficiently long times starting from the beginning of the energy interaction between the considered structures. Such a method, developed for linear elastic systems (conservative or not) made of a set of coupled subsystems in which a stochastic variability of the system eigenfrequencies is introduced, can determine the average energy of a set of subsystems. Actually the method is capable of determining the subsystems' energy under steady-state conditions, but it can also provide the time energy trend in the transient phase.
Among new proposed probabilistic methods, the Smooth Integral Formulation [19] also deserves mentioning. It is based on a boundary element formulation with uncertainties in the physical parameters of the system. The presence of uncertainties provides a solution that is representative of a population of similar structures. The method seems to be computationally efficient in the range of medium and high frequencies.
In parallel with the probabilistic methods, the uncertainties in structural dynamics can be dealt with using other mathematical approaches that refer directly to the concept of interval. Their use is appropriate either when it is not possible to determine the probability density of the uncertain variables or, more generally, when it is only necessary to determine the range of variation of the response resulting frpm variations of the uncertain parameters. From a computational point of view, these methods have considerable advantage over the traditional Monte Carlo methods. Although the mathematical application of intervals dates back probably to Archimedes who, in 212 a.C. defined pi as a number included between 3+10/71 and 3+1/7 so that he was the first to introduce the concept of a number ranging between two values [120], the first who introduced the concept of function with values ranging between two values was Young (1908), and Sunanga and Moore were those who first introduced the interval algebra and interval analysis. Around 1959 Moore, in studying the errors in digital computing, introduced a method based on the interval arithmetic [21]. The problem was related to the representation of numbers. Since the number of bits in a digital computer is limited, the numbers must be represented with reference to this limitation. This implies an approximation when representing the real numbers that, with reference to a set of operations, leads to errors on the final result [22,23]. An interval is a subset of real numbers bounded between two extremes, a lower one and an upper one. The interval arithmetic is a set of rules defining the operations between intervals. This arithmetic is now more and more used to study uncertainties in vibrating systems. The first problem considered was the natural frequency problem [20]. The mass and stiffness matrices of a structure with parameters defined within intervals induce an eigenvalue problem whose solution is a set of natural frequencies and vibration modes ranging within intervals. Starting from these matrices, the corresponding forced problem is considered and, of course, the response is determined in an interval. More recently, the interval algebra was used to develop optimization techniques and to solve inverse problems [24].
An important evolution of the interval algebra, that is now becoming a established theoretical and applicative discipline, is the fuzzy mathematics that was originally developed in a work on fuzzy sets by L.A. Zadeh [25,26]. Recently, some techniques related to fuzzy logic received an important lift from engineering applications with uncertain parameters, when the probability distribution of such parameters is unknown [27]. With respect to the interval algebra, the fuzzy logic needs a characteristic function discriminating the possible values in the interval range, by defining a membership level. <<<