As discussed in Chapter 1, both structural dynamics and aeroelasticity are built on the foundations of dynamics and structural mechanics. Therefore, in this chapter, we review the fundamentals of mechanics for particles, rigid bodies, and simple structures such as strings and beams. The review encompasses laws of motion, expressions for energy and work, and background assumptions. The chapter concludes with a brief discussion of the behavior of single-degree-of-freedom systems and the notion of stability.

The field of structural dynamics addresses the dynamic deformation behavior of continuous structural configurations. In general, load-deflection relationships are nonlinear, and the deflections are not necessarily small. In this chapter, to facilitate tractable, analytical solutions, we restrict our attention to linearly elastic systems undergoing small deflections—conditions that typify most flight-vehicle operations.

However, some level of geometrically nonlinear theory is necessary to arrive at a set of linear equations for strings, membranes, helicopter blades, turbine blades, and flexible rods in rotating spacecraft. Among these problems, only strings are discussed herein. Indeed, linear equations of motion for free vibration of strings cannot be obtained without initial consideration and subsequent careful elimination of nonlinearities.

Introduction

When we wish to use Newton's laws to write the equations of motion of a particle or a system of particles, we must be careful to include all the forces of the system. The Lagrangean form of the equations of motion that we derive herein has the advantage that we can ignore all forces that do no work (e.g., forces at frictionless pins, forces at a point of rolling contact, forces at frictionless guides, and forces in inextensible connections). In the case of conservative systems (i.e., systems for which the total energy remains constant), the Lagrangean method gives us an automatic procedure for obtaining the equations of motion provided only that we can write the kinetic and potential energies of the system.

Degrees of Freedom

Before proceeding to develop the Lagrange equations, we must characterize our dynamical systems in a systematic way. The most important property of this sort for our present purpose is the number of independent coordinates that we must know to completely specify the position or configuration of our system. We say that a system has n degrees of freedom if exactly n coordinates serve to completely define its configuration.

EXAMPLE 1 A free particle in space has three degrees of freedom because we must know three coordinates—x, y, z, for example – to locate it.

Chapter 3 addressed the subject of structural dynamics, which is the study of phenomena associated with the interaction of inertial and elastic forces in mechanical systems. In particular, the mechanical systems considered were one-dimensional, continuous configurations that exhibit the general structural-dynamic behavior of flight vehicles. If in the analysis of these structural-dynamic systems aerodynamic loading is included, then the resulting dynamic phenomena may be classified as aeroelastic. As observed in Chapter 4, aeroelastic phenomena can have a significant influence on the design of flight vehicles. Indeed, these effects can greatly alter the design requirements that are specified for the disciplines of performance, structural loads, flight stability and control, and even propulsion. In addition, aeroelastic phenomena can introduce catastrophic instabilities of the structure that are unique to aeroelastic interactions and can limit the flight envelope.

Heat transfer is a result of the spatial variation of temperature within a medium, or within adjacent media, in which thermal energy may be stored, converted to or from other forms of energy and work, or exchanged with the surroundings. Heat transfer occurs in many natural and engineered systems. As an engineering discipline, heat transfer deals with the innovative use of the principles of thermal science in solving the relevant technological problems. This introductory textbook aims to provide undergraduate engineering students with the knowledge (principles, materials, and applications) they need to understand and analyze the heat transfer problems they are likely to encounter in practice. The approach of this book is to discuss heat transfer problems (in the search for innovative and optimal solutions) and engineering analyses, along with the introduction of the fundamentals and the analytical methods used in obtaining solutions. Although the treatment is basically analytical, empiricism is acknowledged because it helps in the study of more complex geometries, fluid flow conditions, and other complexities that are most suitably dealt with empirically.

A combination of descriptive and analytical discussions are used to enable students to understand and articulate a broad range of problems.

Chapter 8 is found on the Web site www.cambridge.org/kaviany. Chapter 8 is about heat transfer analysis and addresses control of heat transfer, in conjunction with energy storage and conversion, for innovative applications and optimized performance in thermal systems. This heat transfer can occur at various, cascading length scales within the system. The analysis is done by modeling the transport, storage, and conversion of thermal energy as thermal circuits. The elements of these circuits, i.e., mechanisms and models of resistances, storage, and energy conversion, have been discussed in Chapters 2 to 7. In this chapter we consider their combined usage in some innovative, significant, and practical thermal systems. We begin by addressing the primary thermal functions of heat transfer media and bounding surfaces. Then we summarize the elements of thermal engineering analysis. These include the need for assumptions and approximations for reduction of the physical models and conceptual thermal processes to thermal circuit models. Next we give five examples (Examples 8.1 to 8.5), in detail, to demonstrate the application of the fundamentals and relations developed in the text, along with the use of software. The examples are selected for their innovative potentials. There are also five end-of-chapter problems (Problems 8.1 to 8.5).

Chapter Five can be found on the text Web site www.cambridge.org/kaviany. In Chapter 5, we begin discussion of convection heat transfer by considering an unbounded fluid stream undergoing a temperature change along the stream flow direction, where this change is caused by an energy conversion. Convection heat transfer is the transfer of heat from one point to another by a net (i.e., macroscopic) fluid motion (i.e., by currents of gas or liquid particles) given by the fluid velocity vector. The fluid motion can be void of random fluctuations, which is called a laminar flow, or it can have these random fluctuations, which is called a turbulent flow. Convection heat transfer examines themagnitude, direction, and spatial and temporal variations of the convection heat flux vector in a fluid stream. In this chapter, we examine convective heat transfer within a fluid stream without directly considering the heat transfer between the fluid and a bounding solid surface (i.e., here we consider only intramedium conduction and convection of unbounded fluid streams). In addition to the occurrence of the intramedium (i.e., bulk) convection, in many practical applications involving cooling and heating of fluids, the review of the intramedium convection allows for an introductory treatment of laminar, steady-state, uniform, one-directional fluid flow without addressing the effect of the fluid viscosity emanating from the bounding surfaces, which would require inclusion of the momentum conservation equation.

A bounded fluid stream in thermal nonequilibrium with its bounding solid surface exchanges heat, and this alters its temperature across and along the stream. That this change occurs throughout the stream is what distinguishes it from the semi-bounded streams where the far-field temperature is assumed unchanged (i.e., the change was confined to the thermal boundary layer). We now consider the magnitude, direction, and spatial and temporal variations of qu in a bounded fluid stream (also called an internal flow) that is undergoing surface-convection heat transfer qku. As in Chapter 6, initially our focus is on the fluid and its motion and heat transfer (and this is the reason for the chapter title, bounded fluid streams); later we include the heat transfer in the bounding solid. We examine qu within the fluid and adjacent to the solid surface. As in Chapter 6, at the solid surface, where uf = 0, we use qku to refer to the surface-convection heat flux vector. We will develop the average convection resistance 〈Ru〉L, which represents the combined effect of surface-convection resistance and the change in the average fluid temperature as it exchanges heat with the bounding surface.

In this introductory chapter, we discuss some of the reasons for the study of heat transfer (applications and history), introduce the units used in heat transfer analysis, and give definitions for the thermal systems. Then we discuss the heat flux vector q, the heat transfer medium, the equation of conservation of energy (with a reformulation that places q as the central focus), and the equations for conservation of mass, species, and other conserved quantities. Finally we discuss the scope of the book, i.e., an outline of the principles of heat transfer, and give a description of the following chapters and their relations. Chart 1.1 gives the outline for this chapter. This introductory chapter is partly descriptive (as compared to quantitative) in order to depict the broad scope of heat transfer applications and analyses.

Applications and History

Heat transfer is the transport of thermal energy driven by thermal nonequilibrium within amedium or among neighboringmedia. As an academic discipline, it is part of the more general area of thermal science and engineering. In a broad sense, thermal science and engineering deals with a combination of thermal science, mechanics, and thermal engineering analysis and design. This is depicted in Chart 1.2. In turn, thermal science includes thermal physics, thermal chemistry, and thermal biology.