A brief introduction to gas-turbine engines was presented in Chapter 1. A review of the different engines, included in this chapter, reveals that most of these engine components are composed of “lifting” bodies, termed airfoil cascades, some of which are rotating and others stationary. These are all, by necessity, bound by the hub surface and the engine casing (or housing), as shown in Figures. 2.1 through 2.5. As a result, the problem becomes one of the internal-aerodynamics type, as opposed to such traditional external-aerodynamics topics as “wing theory” and others. Referring, in particular, to the turbofan engines in Chapter 1 (e.g., Fig. 1.3), these components may come in the form of ducted fans. These, as well as compressors and turbines, can be categorically lumped under the term “turbomachines.” Being unbound, however, the propeller of a turboprop engine (Fig. 1.2) does not belong to the turbomachinery category.

The turbomachines just mentioned, however, are no more than a subfamily of a more inclusive category. These only constitute the turbomachines that commonly utilize a compressible working medium, which is totally, or predominantly, air. In fact, a complete list of this compressible-flow subfamily should also include such devices as steam turbines, which may utilize either a dry (superheated) or wet (liquid/vapor) steam mixture with high quality (or dryness factor). However, there exists a separate incompressible-flow turbomachinery classification, where the working medium may be water or, for instance, liquid forms of oxygen or hydrogen, as is the case in the Space Shuttle Main Engine turbopumps. This subcategory also includes powerproducing turbomachines, such as water turbines.

Definition

A gas turbine engine is a device that is designed to convert the thermal energy of a fuel into some form of useful power, such as mechanical (or shaft) power or a highspeed thrust of a jet. The engine consists, basically, of a gas generator and a power-conversion section, as shown in Figures 1.1 and 1.2.

As is clear in these figures, the gas generator consists of the compressor, combustor, and turbine sections. In this assembly, the turbine extracts shaft power to at least drive the compressor, which is the case of a turbojet. Typically, in most other applications, the turbine will extract more shaft work by comparison. The excess amount in this case will be transmitted to a ducted fan (turbofan engine) or a propeller (turboprop engine), as seen in Figure 1.2. However, the shaft work may also be utilized in supplying direct shaft work, or producing electricity in the case of a power plant or an auxiliary power unit (Fig. 1.1). The fact, in light of Figures 1.1 through 1.5, is that different types of gas-turbine engines clearly result from adding various inlet and exit components, to the gas generator. An always interesting component in this context is the thrust-augmentation devices known as afterburners in a special class of advanced propulsion systems (Fig. 1.4).

Gas-turbine engines are exclusively used to power airplanes because of their high power-to-weight ratio. They have also been used for electric-power generation in pipeline-compressor drives, as well as to propel trucks and tanks. In fact, it would be unwise to say that all possible turbomachinery applications have already been explored.

Closed-form solutions for layered media, or for homogeneous plates and strata with arbitrary boundary conditions, do not exist. Indeed, even the free–free plate – the so-called Mindlin plate – is ultimately intractable by purely analytical means. Thus, such problems must be solved with the aid of numerical tools. Among these, a widely used scheme is the propagator matrix or transfer matrix method of Haskell and Thomson. Nonetheless, we choose instead to present herein the related stiffness or impedance matrix method, which has the advantages over the propagator matrix method that are listed below and, at least in our judgment, no disadvantages in comparison with the latter. Thus, readers familiar with the propagator method are strongly encouraged to familiarize themselves with and switch to this superior method. Among the advantages of the stiffness matrix method are:

• Stiffness matrices are symmetric, while propagator matrices are not.

• Stiffness matrices involve half as many degrees of freedom as propagator matrices, so their bandwidth is only half as large. The former involve only displacements, whereas the state vector in propagator matrices contains both stresses and displacements.

• On account of the two previous items, the stiffness matrix method is nearly an order of magnitude faster than the propagator matrix method: The computational effort is smaller by a factor 2 due to symmetry, and a factor of more than 4 on account of bandwidth, which gives a total reduction of more than 8. Since the computations must be repeated for each frequency and each wavenumber, the savings are considerable.

• […]

We present in this work a collection of fundamental solutions, or so-called Green's functions, for some classical or canonical problems in elastodynamics. Such formulas provide the dynamic response functions for transient point sources acting within isotropic, elastic media, in both the frequency domain and the time domain, and in both two and three dimensions. The bodies considered are full spaces, half-spaces, and plates of infinite lateral extent, while the sources range from point and line forces to torques, seismic moments, and pressure pulses. By appropriate convolutions, these solutions can be extended to spatially distributed sources and/or sources with an arbitrary variation in time.

These fundamental solutions, as their name implies, constitute invaluable tools for a large class of numerical solution techniques for wave propagation problems in elasticity, soil dynamics, earthquake engineering, or geophysics. Examples are the Boundary Integral (or element) Method (BIM), which is often used to obtain the solution to wave propagation problems in finite bodies of irregular shape, even while working with the Green's functions for a full space.

The solutions included herein are found scattered throughout the literature, and no single book was found to deal with them all in one place. In addition, each author, paper, or book uses sign conventions and symbols that differ from one another, or they include only partial results, say only the solution in the frequency domain or for some particular value of Poisson's ratio. Sometimes, published results are also displayed in unconventional manners, for example, taking forces to be positive down, but displacements up, or scaling the displays in unusual ways or using too small a scale, and so forth.