This chapter presents the elements of stochastic systems theory that we use in the analysis and synthesis of navigation and guidance systems. Roughly speaking, a stochastic phenomenon is randomly unpredictable but exhibits “statistical regularity,” in a sense defined subsequently. Consistent with our emphasis on linear systems, justified in Chapter 2, the purpose of this chapter is to quantify how linear systems respond to uncertain input signals.

We use stochastic models to account for uncertainty for at least two compelling reasons. The first one is purely pragmatic: stochastic models have proven extremely effective in modeling uncertainty in science and engineering and provide a rational basis for taking and optimizing decisions in the presence of the unknown. The second, deeper reason, appeals to the fundamental postulate of natural science [24], which states that “nature is governed by laws that are uniform.” Although this postulate leads us to expect that natural phenomena be highly and exactly repeatable, the uncertainty we observe in practice seems to violate this expectation. A way to resolve this dilemma is to relax the notion of exact repeatability into that of statistical regularity, leading seamlessly to stochastic systems theory. Hence stochastic systems theory affords a provision for uncertainty within the context of the fundamental postulate of natural science.

The motivation of this chapter is that many phenomena in navigation and guidance can be modeled as stochastic. They include measurement errors in components of acceleration, velocity and position, drift of gyroscopes, wind gusts, and so on, and the navigation and guidance errors that the aforementioned generate.

As explained at the end of Chapter 7, the computational vibroacoustic equation could be solved directly ω by ω, but, as explained in Chapter 1, it is recommended to use a reduced-order computational model, which is constructed as follows. The strategy used for constructing the reduced-order computational model consists in using the projection basis constituted of:

1. • The acoustic modes of the acoustic cavity with fixed boundary and without wall acoustic impedance. Two cases are considered. The first one is a closed acoustic cavity (the internal pressure varies with variation of the volume of the cavity), for which the boundary value problem has been defined in Section 6.2. For an almost closed cavity with a nonsealed wall, which are often encountered (the internal pressure does not vary with a variation of the volume of the cavity), an adapted procedure, derived from the closed cavity case, will be presented. The second one concerns an internal cavity filled with an acoustic liquid with a free surface for which the boundary value problem has been defined in Section 6.3.

2. • The elastic structural modes of the structure, taking into account a quasi-static effect of the internal acoustic fluid on the structure. The constitutive equation of the structure corresponds to an elastic material (see Eq. (5.35)) and consequently, the stiffness matrix of the structure has to be taken for ω = 0.

This chapter is devoted to the description of the equations in terms of the pressure field and the associated boundary conditions for the internal dissipative acoustic fluid of the vibroacoustic system. A wall acoustic impedance can be taken into account. In the last section, the case of a free surface for a compressible liquid is considered.

EQUATIONS IN THE FREQUENCY DOMAIN

As introduced in Section 2.3, the fluid is assumed to be homogeneous, compressible, and dissipative. In the reference configuration, the fluid is supposed to be at rest. The fluid is either a gas or a liquid and gravity effects are neglected (see Andrianarison and Ohayon, 2006, to take into account both gravity and compressibility effects for an inviscid internal fluid). Such a fluid is called a dissipative acoustic fluid. Generally, there are two main physical dissipations. The first one is an internal acoustic dissipation inside the cavity due to the viscosity and the thermal conduction of the fluid. These dissipation mechanisms are assumed to be small. In the model proposed, we consider only the dissipation due to the viscosity. This correction introduces an additional dissipative term in the Helmholtz equation without modifying the conservative part. The second one is the dissipation generated inside the wall viscothermal boundary layer of the cavity and is neglected here. We then consider only the acoustic mode (irrotational motion) predominant in the volume.

In this book, we are interested in the analysis of vibroacoustic systems, which are also called structural acoustic systems or fluid-structure interactions for compressible fluid (gas or liquid). Vibroacoustics concerns noise and vibration of structural systems coupled with external and/or internal acoustic fluids. Computational vibroacoustics is understood as the numerical methods solving the equations of physics corresponding to vibroacoustics of complex structures. Complex structures are encountered in many industries for which vibroacoustic numerical simulations play an important role in design and certification, such as the aerospace industry (aircrafts, helicopters, launchers, satellites), automotive industry (automobiles, trucks), railway industry (high speed trains), and naval industry (ships, submarines), as well as in energy production industries (electric power plants).

Since we are interested in the analysis of general complex structural systems in the sense of computational methods defined here, we do not consider analytical or semianalytical methods devoted to structures with simple geometry, asymptoticmethods mainly adapted to the high-frequency range (statistical energy analysis, diffusion of energy, etc.) and approaches that imply them. Concerning the latter, the coupling of the local dynamic equilibrium equation (finite element method) and power balances (implemented in the spirit of the statistical energy analysis) have been analyzed in Soize (1998); Shorter and Langley (2005); Cotoni et al. (2007).

UNCERTAINTY AND VARIABILITY

The designed vibroacoustic system is used to manufacture the real vibroacoustic system and to construct the nominal computational vibroacoustic model (also called the mean computational vibroacoustic model or sometime the mean model) using a mathematical-mechanical modeling process for which the main objective is the prediction of the responses of the real vibroacoustic system. This system can exhibit a variability in its responses due to fluctuations in the manufacturing process and due to small variations of the configuration around a nominal configuration associated with the designed vibroacoustic system. The mean computational model that results from a mathematical-mechanical modeling process of the designed vibroacoustic system has parameters (such as geometry, mechanical properties, and boundary conditions) that can be uncertain (for example, parameters related to the structure, the internal acoustic fluid, the wall acoustic impedance). In this case, there are uncertainties on the computational vibroacoustic model parameters, also called uncertainties on the system parameters. On the other hand, the modeling process induces some modeling errors defined as the model uncertainties. Figure 9.1 summarizes the two types of uncertainties in a computational model and the variabilities of a real system.