Chapter 2 dealt with 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 has been observed in Chapter 3, 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.

Aeroelasticity is the term used to denote the field of study concerned with the interaction between the deformation of an elastic structure in an airstream and the resulting aerodynamic force. The interdisciplinary nature of the field can be best illustrated by Fig. 1.1, which depicts the interaction of the three disciplines of aerodynamics, dynamics, and elasticity. Classical aerodynamic theories provide a prediction of the forces acting on a body of a given shape. Elasticity provides a prediction of the shape of an elastic body under a given load. Dynamics introduces the effects of inertial forces. With the knowledge of elementary aerodynamics, dynamics, and elasticity, the student is in a position to look at problems in which two or more of these phenomena interact. One of those areas of interaction is the field of flight mechanics, which most students have studied in a separate course by the senior year. The present text will consider the three remaining areas of interaction:

• between elasticity and dynamics (structural dynamics),

• between aerodynamics and elasticity (static aeroelasticity), and

• among all three (dynamic aeroelasticity).

Because of their importance to aerospace system design, these are appropriate for study in an undergraduate aeronautics/aeronautical engineering curriculum. In aeroelasticity one finds that the loads depend on the deformation (aerodynamics), and the deformation depends on the loads (structural mechanics/dynamics); thus one has a coupled problem. Consequently, prior study of all three constituent disciplines is necessary before a study in aeroelasticity can be undertaken.

The field of static aeroelasticity is the study of flight vehicle phenomena associated with the interaction of aerodynamic loading induced by steady flow and the resulting elastic deformation of the lifting surface structure. These phenomena are characterized as being insensitive to the rates and accelerations of the structural deflections. There are two classes of design problems that are encountered in this area. The first and most common to all flight vehicles is the effect of elastic deformation on the airloads associated with normal operating conditions. These effects can have a profound influence on the performance, handling qualities, flight stability, and structural load distribution. The second class of problems involves the potential for static instability of the structure that will result in a catastrophic failure. This instability is often termed “divergence” and can impose a limit on the flight envelope.

The material presented in this chapter provides an introduction to some of these static aeroelastic phenomena.

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 shall proceed to derive 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 (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 on some 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.

A senior-level undergraduate course entitled “Vibration and Flutter” was taught for many years at Georgia Tech under the quarter system. This course dealt with elementary topics involving the static and/or dynamic behavior of structural elements, both without and with the influence of a flowing fluid. The course did not deal with the static behavior of structures in the absence of fluid flow, because this is typically considered in courses in structural mechanics. Thus, the course essentially dealt with the fields of “Structural Dynamics” (when fluid flow is not considered) and “Aeroelasticity” (when it is).

As the name suggests, structural dynamics is concerned with the vibration and dynamic response of structural elements. It can be regarded as a subset of aeroelasticity, the field of study concerned with interaction between the deformation of an elastic structure in an airstream and the resulting aerodynamic force. Aeroelastic phenomena can be observed on a daily basis in nature (e.g., the swaying of trees in the wind, the humming sound Venetian blinds make in the wind, etc.). The most general aeroelastic phenomena include dynamics, but static aeroelastic phenomena are also quite important. The course has been expanded to cover a full semester, and the course title has been appropriately changed to “Introduction to Structural Dynamics and Aeroelasticity.”

Aeroelastic and structural dynamic phenomena can result in dangerous static and dynamic deformations and instabilities and, thus, have very important practical consequences in many areas of technology.