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.
We shall discuss two types of physical
motion: linear motion or motion of translation, and angular motion or
motion of rotation. Linear motion can be further categorized as rectilinear,
meaning motion in a straight line, or curvilinear, meaning motion in
a curved path. Both linear motion and angular motion are composed of
an infinite variety of subtypes, or motion parameters, based on successive
derivatives of linear or angular position with respect to time. The
most basic of these motion parameters, and the most useful, are displacement,
velocity, acceleration, and jerk. Table 1 classifies linear and angular
motion parameters and their symbols and units, and serves as an outline
for the following discussions of linear and angular motion.
Linear Motion
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:
(1)
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:
(2)
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/sec2to 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:
(3)
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:
(4)
where a1 is the initial acceleration,
a2is the final acceleration, and t is the elapsed time.
Instantaneous linear jerk, , or , 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).
Angular Motion
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:
(5)
where q1 is the initial
angular displacement and q2 is the final angular displacement.
Instantaneous angular velocity is or . 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:
(6)
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:
(7)
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:
(8)
where ω1 is the initial
angular velocity, ω2 is the final angular velocity,
and Δt is the time interval over which angular velocity changes.
, , , and 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.
Generally speaking, the linear and angular
motions, by themselves, are not of physiologic importance. Forces and
torques that result in, or appear to result from, linear and angular
velocity changes are the entities that stimulate or compromise the crewmember's
physiologic mechanisms.
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:
(9)
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:
(10)
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:
(11)
Thus, from Equation 3:
(12)
or from Equation 6:
(13)
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:
(14)
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,
(15)
and
(16)
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
then
(17)
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.
Momentum
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:
(18)
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:
(19)
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:
A number of conventions have been used
in aerospace medicine to describe the directions of linear and angular
displacement, velocity, and acceleration, and of reactive forces and
torques. The more commonly used of those conventions will be discussed
in the following sections.
Vehicular Motions
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.
Figure 1.
Axes of linear and angular aircraft motions. Linear motions are
longitudinal, lateral, and vertical; angular motions are roll, pitch,
and yaw
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.
Figure 2.
Systems for describing accelerations and inertial reactions in humans. (Adapted from Hixson et al. 1)
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.
(1)
(2)
("x double dot") or 
, is
the second derivative of displacement or the first derivative of velocity, 
or 
,respectively.
(3)
(4)
, or 
, 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).
(5)
or 
. 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 
(6)
(7)
(8)
, 
, 
, and 
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.
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
