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Uses a real jet transport the Boeing for many numerical and worked-out examples. Table of Contents Static Stability and Control 1. Static Stability and Control 2. Unit vectors along x, y, and z will be denoted i, j, and k, respectively, and subscripted appropriately. If this is not the case then the lack of subscripts will be taken to mean a generic system. The location of the origin and orientation of the axes may be arbitrary within certain restrictions, but once selected may not be changed. Following are the main coordinate systems of interest.
For the purposes of this book the origin is at the Great Galactic Center. This coordinate system obviously rotates with the Earth. In some applications in which the CG will change appreciably e. Determination of the orientation of the axes is as follows see Figure 2. If the aircraft has a plane of symmetry and we will assume in this book that they all do then xB and zB lie in that plane of symmetry. Figure 2. The axes intersect at the airplane center of gravity. The author sits astraddle the xB-axis. Source: US Navy. Source: NASA. Principal axes, FP For every rigid body an orthogonal coordinate system may be found in which cross- products of inertia are zero.
By the assumption of a plane of symmetry geometric and mass symmetry two of these axes will lie in the plane of symmetry. These are named xP and zP , and the typical longish nature of aircraft will permit one of these axes to be selected to be toward the nose, and this is xP. Zero-lift body-axis system, FZ Assuming the relative wind lies in the plane of symmetry no sideslip then there is a direction of the wind in this plane for which the net contribution of all surfaces of the aircraft toward creating lift is zero. The xZ axis is selected to be into the relative wind when lift is zero.
This is usually toward the nose of the aircraft, permitting the other axes to be chosen as described. The xS axis is taken as the projection of the velocity vector of the aircraft relative to the air mass into the aircraft plane of symmetry. Normally this is toward the nose of the aircraft, permitting the other axes to be chosen as for other vc xS yS zSxS, vc yS vc xS Figure 2.
Aircraft Flight Dynamics and Control
The zW axis is chosen to lie in the plane of symmetry of the aircraft and the yW axis to the right of the plane of symmetry. Note that xW need not lie in the plane of symmetry. Omission of a superscript may usually be taken to mean we are referring to an inertial reference frame. Examples: rE p : Position vector of the point p relative to the origin of some FE. Vectors can exist notionally as just described, but to quantify them they must be repre- sented in some coordinate system.
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In general these components will be different in two different coordinate systems, unless the two coordinate systems have parallel axes. The omission of a superscript suggests the inertial reference frame. If the meaning is unclear, then the user of such an expression should be asked to specify how the system was chosen. This leaves the z-axis positive up the negative of our zB-axis.
Waterlines are measured along the z-axis from this plane. Fuselage stations coordinates along the y-axis are numbered beginning at the nose of the aircraft, not the origin. Thus, a component may be mounted on a bulkhead at Fuselage Station FS , meaning the bulkhead is inches back from the nose. The origin of the system is not generally the center of gravity of the aircraft, but is often close to it.
Sideslip is positive when the relative wind is from the right of the plane of symmetry, as shown in Figure 2. Aircraft plane of symmetry b vc A yB Figure 2. The axis yB and velocity vector vA c lie in the plane of the page. It is positive when the relative wind is from below the xB axis as shown in Figure 2. Proj vA c is the projection of vA c onto the plane of symmetry.
Where does yEC intersect the surface of the Earth? The gravity vector usually does not have subscripts or superscripts as these will be suggested by context. Use the same assumptions as in Problem 2. Describe the relationship between the axes of FE and FH at the instant the aircraft is at zero longitude and zero latitude, and again at the instant it is at deg east longitude and zero latitude. Answer true or false, and explain: a All body-carried axis systems have a common x-axis.
What is the angle of the relative wind to xz when the net lift on the aircraft is zero no sideslip? Is this always the case? References Etkin, B. Liming, R. Now we address the subject of how these coordinate systems are related to one another. For instance, when we begin to sum the external forces acting on the aircraft, we will have to relate all these forces to a common reference frame. First we must characterize the relationship between two coordinate systems at some frozen instant in time.
The instantaneous relationship between two coordinate systems will be addressed by determining a transformation that will take the representation of an arbitrary vector in one system and convert it to its representation in the other. Next we will discuss Euler angles, by far the most common approach but one with a potentially serious problem.
Finally we will examine Euler parameters, an elegant solution to the problem found with Euler angles. Transfor- mations such as T2,1 are called similarity transformations. The transformations involved in simple rotations of orthogonal reference frames have many special properties that will be shown. The order of subscripts of T2,1 is such that the left subscript goes with the system of the vector on the left side of the equation and the right subscript with the vector on the right.
We claim to know the representation of v in F1. The vector v is the vector sum of the three components vx1 i1, vy1 j1, and vz1 k1 so we may replace the vector by those three components, shown in Figure 3. Now, the projection of v onto x2 is the same as the vector sum of the projections of each of its components vx1 i1, vy1 j1, and vz1 k1 onto x2, and similarly for y2 and z2. Coordinate System Transformations 19 v x1 y1 z1 x2 y2 z2 Figure 3.
The vector v may be represented in either reference frame. It is therefore the same no matter what we choose for v. When the rows or columns of the direction cosine matrix are viewed as vectors, this means the rows or the columns form orthogonal bases for three-dimensional space. In principle all nine may be derived given any three that do not violate a constraint.
One means of determining these variables is by use of a famous theorem due to the Swiss mathematician Leonhard Euler — The order of selection of axes in these rotations is arbitrary, but the same axis may not be used twice in succesion. The rotation sequences are usually denoted by three numbers, 1 for x, 2 for y, and 3 for z. The 12 valid sequences are , , , , , , , , , , , and The angles through Since the rotation was about z1, z is parallel to it but neither of the other two primed axes are.
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The next rotation Figure 3. We will do this by seeing how the arbitrary vector Clearly the relationships among the two sets of angles are non-trivial. Euler parameters are based on the observation that any two coordinate systems are instantaneously related by a single rotation about some axis that has the same representation in each system.
A given set of Euler parameters, however, conveys almost no information about how the systems are related.
Dynamics of Flight: Stability and Control (3rd Edition)
The direct approach to convert a set of Euler parameters to the corresponding set of Euler angles is to equate corresponding elements of the two representations of the transformation matrix. Most software libraries have the two-argument arctangent function which avoids this problem and helps keep track of the quadrant. It may be done in a somewhat similar manner to equating elements of the transformation matrix. Frequently in the literature Euler parameters are referred to as quaternions. The algebra of quaternions is useful for proving theorems regarding Euler parameters, but will not be used in this book.
The reason for doing Such transformations may be very useful. Problems 1.
Use this information to write down TB2,B1 using the direction cosine matrix. Use this information to write down TB2,B1 using the Euler angle representation of the direction cosine matrix. Use this information to write down TB2,B1 using the Euler parameter representation of the direction cosine matrix. Show all work. Consider two left-handed coordinate systems.
Coordinate System Transformations 29 b How does this result differ from the direction cosine matrix for right-handed systems?