Operators of today’s and tomorrow’s air and space vehicles must understand clearly the terminology and physical principles relating to the motions of their craft so they can fly with precision and effectiveness. These crewmembers also must have a working knowledge of the structure and function of the various mechanical and electrical systems which comprise their craft, to help them understand the performance limits of their machines and to facilitate trouble-shooting and promote safe recovery when the machines fail in flight. So, too, must practitioners of aerospace medicine understand certain basic definitions and laws of mechanics so that they can analyze and describe the motional environment to which the flyer is exposed. In addition, the aeromedical professional must be familiar with the physiologic bases and operational limitations of the flyer’s orientational mechanisms. This understanding is necessary to enable the physician or physiologist to speak intelligently and credibly with aircrew about spatial disorientation and to enable them to contribute significantly to investigations of aircraft mishaps in which spatial disorientation may be implicated.
The basic parameter of linear motion is linear displacement. The other parameters–velocity, acceleration, jerk–are derived from the concept of displacement. Linear displacement, x, is the distance and direction of the object under consideration from some reference point; as such, it is a vector quantity, having both magnitude and direction. The position of an aircraft located at 25 nautical miles on the 150° radial of the San Antonio vortac, for example, describes completely the linear displacement of the aircraft from the navigational facility serving as the reference point. The meter (m), however, is the unit of linear displacement in the International System of Units (SI) and will eventually replace other units of linear displacement such as feet, nautical miles, and statute miles.
When linear displacement is changed during a period of time, another vector quantity, linear velocity, occurs. The formula for calculating the mean linear velocity, v, during a time interval, D t, is as follows:
where XI is the initial linear displacement and X2 is the final linear displacement. An aircraft that travels from San Antonio, Texas, to New Orleans, Louisiana, in 1 hour, for example, moves with a mean linear velocity of 434 knots (nautical miles per hour) on a true bearing of 086°. Statute miles per hour and feet per second are other commonly used units of linear speed, the magnitude of linear velocity; meters per second (m/sec), however, is the SI unit and is preferred. Frequently, it is important to describe linear velocity at a particular instant in time, that is, as t approaches zero. In this situation, one speaks of instantaneous linear velocity, x (pronounced “x dot”), which is the first derivative of displacement with respect to time, dx/dt.
When the linear velocity of an object changes over time, the difference in velocity, divided by the time required for the moving object to make the change, gives its mean linear acceleration, a. The following formula:
where v1 is the initial velocity, v2 is the final velocity, and Dt is the elapsed time, is used to calculate the mean linear acceleration, which, like displacement and velocity, is a vector quantity with magnitude and direction. Acceleration is thus the rate of change of velocity, just as velocity is the rate of change of displacement. The SI unit for the magnitude of linear acceleration is meters per second squared (m/sec2). Consider, for example, an aircraft that accelerates from a dead stop to a velocity of 100 m/sec in 5 seconds; the mean linear acceleration is (100 m/sec – 0 m/sec)/5 seconds, or 20 m/sec2. The instantaneous linear acceleration, (“x double dot”) or , is the second derivative of displacement or the first derivative of velocity, or ,respectively.
A very useful unit of acceleration is g, which for our purposes is equal to the constant go, the amount of acceleration exhibited by a free-falling body near the surface of the earth (9.81 m/sec2). To convert values of linear acceleration given in m/sec2 into go units, simply divide by 9.81. In the above example in which an aircraft accelerates at a mean rate of 20 m/sec2, one divides 20 m/sec2 by 9.81 m/sec2 to obtain 2.04 g.
A special type of linear acceleration–radial or centripetal acceleration–results in curvilinear, usually circular, motion. The acceleration acts along the line represented by the radius of the curve and is directed toward the center of the curvature. Its effect is a continuous redirection of the linear velocity, in this case called tangential velocity, of the object subjected to the acceleration. Examples of this type of linear acceleration are when an aircraft pulls out of a dive after firing on a ground target or flies a circular path during aerobatic maneuvering. The value of the centripetal acceleration, ac, can be calculated if one knows the tangential velocity, vt, and the radius, r, of the curved path followed:
For example, the centripetal acceleration of an aircraft traveling at 300 m/sec (approximately 600 knots) and having a radius of turn of 1500 m can be calculated. Dividing (300 m/sec)2 by 1500 m gives a value of 60 m/sec2, which, when divided by 9.81 m/sec2 per g, comes out to 6.12 g.
One can go another step in derivation of linear motion parameters by obtaining the rate of change of acceleration. This quantity, j, is known as linear jerk. Mean linear jerk is calculated as follows:
where a1 is the initial acceleration, a2 is the final acceleration, and t is the elapsed time. Instantaneous linear jerk is the third derivative of linear displacement or the first derivative of linear acceleration with respect to time, or , respectively. Although the SI unit for jerk is m/sec3, it is generally more useful to speak in terms of g-onset rate, measured in g’s per second (g/sec).
The derivation of the parameters of angular motion follows in a parallel fashion the scheme used to derive the parameters of linear motion. The basic parameter of angular motion is angular displacement. For an object to be able to undergo angular displacement it must be polarized; that is, it must have a front and back, so that it can face or be pointed in a particular direction. A simple example of angular displacement is seen in a person facing east. In this case, the individual’s angular displacement is 90° clockwise from the reference direction, which is north. Angular displacement, symbolized by the Greek letter theta, q, is generally measured in degrees, revolutions (1 revolution = 360°), or radians (1 radian = 1 revolution/2π, approximately 57.3°). The radian is a particularly convenient unit to use when dealing with circular motion (e.g., motion of a centrifuge) because it is necessary only to multiply the angular displacement of the system, in radians, by the length of the radius to find the value of the linear displacement along the circular path. The radian is the angle subtended by a circular arc the same length as the radius of the circle.
Angular velocity, ω, is the rate of change of angular displacement. The mean angular velocity occurring in a time interval, At, is calculated as follows:
where q1 is the initial angular displacement and q2 is the final angular displacement. As an example of angular velocity, consider the standard-rate turn of instrument flying, in which a heading change of 180° is made in 1 minute. Then ω = (180° – 0°)/60 seconds, or 3 degrees per second (degs/sec). This angular velocity also can be described as 0.5 revolutions per minute (rpm) or as 0.052 radians per second (rad/sec) (3°/sec divided by 57.3°/rad). The fact that an object may be undergoing curvilinear motion during a turn in no way affects the calculation of its angular velocity: an aircraft being rotated on the ground on a turntable at a rate of half a turn per minute has the same angular velocity as one flying a standard-rate instrument turn (3°/sec) in the air at 300 knots.
Because radial or centripetal linear acceleration results when rotation is associated with a radius from the axis of rotation, a formula for calculating the centripetal acceleration, ac, from the angular velocity, ω, and the radius, r, is often useful:
where ω is the angular velocity in radians per second. One can convert readily to the formula for centripetal acceleration in terms of tangential velocity (Equation 3) if one remembers the following:
To calculate the centripetal acceleration generated by a centrifuge having a 10-m arm and turning at 30 rpm, equation 6 is used after first converting 30 rpm to π rad/sec. Squaring the angular velocity and multiplying by the 10-m radius, a centripetal acceleration of 10 π2 m/sec2, or 10.1 g, is obtained.
The rate of change of angular velocity is angular acceleration, a. The mean angular acceleration is calculated as follows:
where ω1 is the initial angular velocity, ω2 is the final angular velocity, and Δt is the time interval over which angular velocity changes.
all can be used to symbolize instantaneous angular acceleration, the second derivative of angular displacement or the first derivative of angular velocity with respect to time. If a figure skater is spinning at 6 revolutions per second (2160°/sec, or 37.7 rad/sec) and then comes to a complete stop in 2 seconds, the rate of change of angular velocity, or angular acceleration, is (0 rad/sec -37.7 rad/sec)/2 seconds, or -18.9 rad/sec2. One cannot express angular acceleration in go units, which measure magnitude of linear acceleration only.
Although not commonly used in aerospace medicine, another parameter derived from angular displacement is angular jerk, the rate of change of angular acceleration. Its description is completely analogous to that for linear jerk, but angular rather than linear symbols and units are used.
Force, Inertia, and Momentum
Force and Torque
Force is an influence that produces, or tends to produce, linear motion or changes in linear motion; it is a pushing or pulling action. Torque produces, or tends to produce, angular motion or changes in angular motion; it is a twisting or turning action. The SI unit of force is the newton (N). Torque has dimensions of force and length because torque is applied as a force at a certain distance from the center of rotation. The newton meter (N m) is the SI unit of torque.
Mass and Rotational Inertia
Newton’s Law of Acceleration states the following:
where F is the unbalanced force applied to an object, m is the mass of the object, and a is linear acceleration.
To describe the analogous situation pertaining to angular motion, the following equation is used:
where M is unbalanced torque (or moment) applied to the rotating object, J is rotational inertia (moment of inertia) of the object, and ω is angular acceleration.
The mass of an object is thus the ratio of the force acting on the object to the acceleration resulting from that force. Mass, therefore, is a measure of the inertia of an object–its resistance to being accelerated. Similarly, rotational inertia is the ratio of the torque acting on an object to the angular acceleration resulting from that torque–again, a measure of resistance to acceleration. The kilogram (kg) is the SI unit of mass and is equivalent to 1 N/(m/sec2). The SI unit of rotational inertia is merely the N m/(radian/sec2).
Because F = ma, the centripetal force, Fc, needed to produce a centripetal acceleration, ac, of a mass, m, can be calculated as follows:
where vt is tangential velocity and ω is angular velocity.
Newton’s Law of Action and Reaction, which states that for every force applied to an object there is an equal and opposite reactive force exerted by that object, provides the basis for the concept of inertial force. Inertial force is an apparent force opposite in direction to an accelerating force and equal to the mass of the object times the acceleration. An aircraft exerting an accelerating forward thrust on its pilot causes an inertial force, the product of the pilot’s mass and the acceleration, to be exerted on the back of the seat by the pilot’s body. Similarly, an aircraft undergoing positive centripetal acceleration as a result of lift generated in a turn causes the pilot’s body to exert inertial force on the bottom of the seat. More important, however, are the inertial forces exerted on the pilot’s blood and organs of equilibrium because physiologic effects result directly from such forces.
At this point it is appropriate to introduce G, which is used to measure the strength of the gravitoinertial force environment. (Note: G, as used here, should not be confused with G, the symbol for the universal gravitational constant, which is equal to 6.70 x 10. 1 N m2/kg.) Strictly speaking, G is a measure of relative weight:
where w is the weight observed in the environment under consideration and wo is the normal weight on the surface of the earth. In the physical definition of weight,
where m is mass, a is the acceleratory field (vector sum of actual linear acceleration plus an imaginary acceleration opposite the force of gravity), and go is the standard value of the acceleration of gravity (9.81 m/sec2 ). Thus, a person having a mass of 100 kg would weigh 100 kg times 9.81 m/sec2 or 981 N on earth (although conventional scales would read “100 kg”). At some other location or under some other acceleratory condition, the same person could weigh twice as much–1962 N–and the scale would read “200 kg.” Our subject would then be in a 2-G environment, or in an aircraft, would be “pulling” 2 G. Consider also that since
G = w/wo = ma/mgo
Thus, the ratio between the ambient acceleratory field (a) and the standard acceleration (go) also can be represented in terms of G .
Therefore, g is used as a unit of acceleration (e.g., ac = 8 g), and the dimensionless ratio of weights, G, is reserved for describing the resulting gravitoinertial force environment (e.g., a force of 8 G, or an 8-G load). When in the vicinity of the surface of the earth, one feels a G force equal to 1 G in magnitude directed toward the center of the earth. If one also sustains a G force resulting from linear acceleration, the magnitude and direction of the resultant gravitoinertial force can be calculated by adding vectorially the 1-G gravitational force and the inertial G force. An aircraft pulling out of a dive with a centripetal acceleration of 3 g, for example, would exert 3 G of centrifugal force. At the bottom of the dive, the pilot would experience the 3-G centrifugal force in line with the 1-G gravitational force, for a total of 4 G directed toward the floor of the aircraft. If the pilot could continue his circular flight path at a constant airspeed, the G force experienced at the top of the loop would be 2 G because the 1-G gravitational force would subtract from the 3-G inertial force. Another common example of the addition of gravitational G force and inertial G force occurs during the application of power on takeoff or on a missed approach. If the forward acceleration is 1 g, the inertial force is 1 G directed backward. The inertial force adds vectorially to the 1-G force of gravity, directed downward, to provide a resultant gravitoinertial force of 1.414 G pointing 45° down from the aft direction, if the aircraft is traveling horizontally.
Just as inertial forces oppose acceleratory forces, so do inertial torques oppose acceleratory torques. No convenient derived unit exists, however, for measuring inertial torque; specifically, there is no such thing as angular G.
To complete this discussion of linear and angular motion, the concepts of momentum and impulse must be introduced. Linear momentum is the product of mass and linear velocity–mv. Angular momentum is the product of rotational inertia and angular velocity–Jω. Momentum is a quantity that a translating or rotating body conserves; that is, an object cannot gain or lose momentum unless it is acted on by a force or torque. A translational impulse is the product of force, F, and the time over which the force acts on an object, Δt, and is equal to the change in linear momentum imparted to the object. Thus:
where v1 is the initial linear velocity and v2 is the final linear velocity.
When dealing with angular motion, a rotational impulse is defined as the product of torque, M, and the time over which it acts, Δt. A rotational impulse is equal to the change in angular momentum. Thus:
where ω1 is the initial angular velocity and ω2 is the final angular velocity.
The above relations are derived from the Law of Acceleration, as follows:
Directions of Action and Reaction
Because space is three-dimensional, linear motions in space are described by reference to three linear axes and angular motions by three angular axes. In aviation, it is customary to speak of the longitudinal (fore-aft), lateral (right-left), and vertical (up-down) linear axes. And the roll, pitch, and yaw angular axes, as shown in Figure 1.
Most linear accelerations in aircraft occur in the vertical plane defined by the longitudinal and vertical axes because thrust is usually developed along the former axis and lift is usually developed along the latter axis. Aircraft capable of vectored thrust are now operational, however, and vectored-lift aircraft are currently being flight-tested. Most angular accelerations in fixed-wing aircraft occur in the roll plane (perpendicular to the roll axis) and, to a lesser extent, in the pitch plane. Angular motion in the yaw plane is common in rotating-wing aircraft, and it occurs during spins and several other aerobatic maneuvers in fixed-wing aircraft. Certainly, aircraft and space vehicles of the future can be expected to operate with considerably more freedom of both linear and angular motion than do those of the present.
Physiologic Acceleration and Reaction Nomenclature
Figure 2 depicts a practical system for describing linear and angular accelerations acting on man. This system is used extensively in aeromedical scientific writing. In this system, a linear acceleration of the type associated with a conventional takeoff roll is in the +ax direction; that is, it is a +ax acceleration. Braking to a stop during a landing roll results in -ax acceleration. Radial acceleration of the type usually developed during air combat maneuvering is +az acceleration–foot-to-head. The right-hand rule for describing the relationships among three orthogonal axes aids recall of the positive directions of ax, ay, and az accelerations in this particular system: if one lets the forward-pointing index finger of the right hand represent the positive x-axis, and the left-pointing middle finger of the right hand represent the positive y-axis, the positive z-axis is represented by the upward-pointing thumb of the right hand. A different right-hand rule, however, is used in another convention, one for describing vehicular coordinates. In that system, +ax is noseward acceleration, +ay is to the right, and +az is floorward; an inverted right hand illustrates that set of axes.
The angular accelerations, αx, αy, and αz, are roll, pitch, and yaw accelerations, respectively, in the system shown in Figure 2. Note that the relations among the positive x-axis, y-axis, and z-axis are identical to those for linear accelerations. The direction of positive angular displacement, velocity, or acceleration is described by another right-hand rule, wherein the flexed fingers of the right hand indicate the direction of angular motion corresponding to the vector represented by the extended, abducted right thumb. Thus, in this system, a right roll results from +αx acceleration, a pitch down results from +αy acceleration, and a left yaw results from +αz acceleration. Again, it is important to be aware of the inverted right-hand coordinate system commonly used to describe angular motions of vehicles. In that convention, a positive roll acceleration is to the right, positive pitch is upward, and positive yaw is to the right.
The nomenclature for the directions of gravitoinertial (G) forces acting on humans is also illustrated in Figure 2. Note that the relation of these axes to each other follows a backward, inverted, right-hand rule. In the illustrated convention, +ax acceleration results in +Gx inertial force, and +az acceleration results in +Gz force. This correspondence of polarity is not achieved on the y-axis, however, because +ay acceleration results in -Gy force. If the +Gy direction were reversed, full polarity correspondence could be achieved between all linear accelerations and all reactive forces, and that convention has been used by some authors. An example of the usage of the symbolic reaction terminology is: “An F-16 pilot must be able to sustain +9.0 Gz without losing vision or consciousness.”
The “eyeballs” nomenclature is another useful set of terms for describing gravitoinertial forces. In this system, the direction of the inertial reaction of the eyeballs when the head is subjected to an acceleration is used to describe the direction of the inertial force. The equivalent expressions, “eyeballs-in acceleration” and “eyeballs-in G force,” leave little room for confusion about either the direction of the applied acceleratory field or the resulting gravitoinertial force environment.
Inertial torques can be described conveniently by means of the system shown in Figure 2, in which the angular reaction axes are the same as the linear reaction axes. The inertial reactive torque resulting from +αx (right roll) angular acceleration is +Rx and +αz (left yaw) results in +Rz; however, +αy (downward pitch) results in -Ry. This incomplete correspondence between acceleration and reaction coordinate polarities again results from the mathematical tradition of using right-handed coordinate systems.
It should be apparent from this discussion that the potential for confusing the audience when speaking or writing about accelerations and inertial reactions is great, and it may be necessary to describe the coordinate system being used. For most applications, the “eyeballs” convention is perfectly adequate.