INTRODUCTION

The equations in this report are the coordinate transformation and motion equations used in the various tasks associated with free-flight and wind-tunnel data reduction and analysis. These tasks range from reducing flight data to calculating the motions on a digital or analog computer and to applying various techniques for analyzing the data, such as in references 1 and 2.

While many publications contain a number of these equations, no one contains all that are usually needed in a complicated aerodynamic analysis; even the more nearly complete reports (refs. 3 and 4, for example) omit the equations for transferring aerodynamic stability derivatives from one moment reference to another. Moreover, in most cases the equations are simplified, when they are presented, by assumptions such as small angles of attack, zero sideslip, and small perturbation motions. Expanded forms of many of the equations, on the other hand, are needed in special problems that may arise. For example, parawing vehicles, which have their center of gravity located well below the wing surface, require the expanded forms of the axes transformations when data measured about a point on the wing are to be transferred to the center of gravity; reentry motion studies sometimes involve large-amplitude motions so the complete forms of the transformations, without the assumptions of small angles of attack or sideslip, are needed. The engineer working on any of these special problems usually has to derive these equations himself, and this can be time consuming.

The purpose of this report is to provide the basic equations from which many of the equations needed in a particular analysis can be generated. A comprehensive summary of the basic axes transformation and motion equations is included, with most of these given in their expanded, most general forms. Once these expanded forms are available, the simpler forms can be written out fairly easily, and yet the general forms are here when needed for special cases.

The general forms presented include axes transformations that enable transfer back and forth between any of the five axes systems that are encountered in aerodynamic analysis. Equations of motion are presented that enable calculation of motions anywhere in the vicinity of the earth. Special problems are also considered; since flight instruments, such as accelerometers or rate gyros, are not always alined along mutually perpendicular axes, the procedure for correcting instrument readings for nonorthogonal alinements is outlined.

In addition to these general forms, many of the simplified forms used frequently in practical applications are summarized in appendix A.

Other relationships are presented in appendixes B to F. For example, appendix C summarizes methods used to measure moments of inertia of models and full-scale vehicles; appendix E discusses the use of the direction-cosine and the quaternion methods, often used in place of Euler angles in specifying vehicle alinement; appendix F discusses the scaling parameters used in model testing. However, throughout this paper, the emphasis is on providing the basic equations. For discussions of their development and of the procedures used in their application, the user should turn to general published works on flight-motion analysis. A comprehensive bibliography of these works is provided and includes textbooks and reports dealing with stability, control, and performance as well as reports discussing various techniques for extracting stability derivatives from flight data.