Newton’s second law of motion maybe considered equiv- alently as a fundamental postdate or as a definition of force and mass (Ref. 6). Newton’s laws are basic and cannot be derived because they are simply the result of observation.
Their validity can be accepted on the basis of 300 yr of fruitless attempts to find them fallacious, at least for veloci- ties that are small compared to the speed of light (Ref. 7).
For a single particle the correct and most fundamental form of Newton’s second law of motion is
and a direct extension of this law to rotational motion gives
4-6
where
vector sum of forces acting on the particle, N angular momentum vector of the particle, N•m•s
total moment (torque) vector acting on the particle, N•m
linear momentum vector of the particle, N-s time,s.
Eq. 4-1 states that the force acting on a particle in a given direction equals the time rate of change of the momentum of the particle in that direction. Eq. 4-2 states that the moment of force (torque) on a particle about a given axis equals the time rate of change of the angular momentum of the particle about that axis. In the application of these laws, the linear and angular rates of momentum must be measured relative to an inertial reference frame.
Rigid bodies are composed of individual particles that do not move relative to each other. The equations of motion for a rigid body are developed by summing (integrating) the equations for individual particles (or incremental masses) over all the constituent particles. Performing the differentia- tion indicated in Eqs. 4-1 and 4-2 and integrating over all particles in the body yields the equations of motion of a rigid body as given in scalar form in Eqs. 3-1 and 3-2 and repeated here in vector form:
where
vector sum of forces acting on the body, N inertia matrix of the body relative to the axis of rotation (See subpar. 4-5.2.), kg•m2 total moment vector acting on the body, N•m mass of the body, kg
absolute acceleration vector of center of mass of a body, m/s2
absolute angular acceleration vector of the body, rad/s2.
To qualify the variables of Eqs. 4-3 and 4-4 further, F is the vector sum of all forces acting on the missile, M is the vec- tor sum of all moments acting on the missile, is the angu- lar rate of the missile about an axis through the center of mass, and the moment of inertia matrix [I] is taken with respect to the axis of rotation. The vectors in Eqs. 4-3 and 4- 4 can be expressed as components in any reference frame, but the accelerations represented by and must be mea- sured relative to an inertial frame.
Derivation of Eqs. 4-3 and 4-4 is based on the assumption
that the mass m and the inertia matrix [I] are constants, but these parameters are not constant in a missile during the operation of the propulsion system because of the consump- tion of propellant. Fortunately, as is shown in subpar. 4- 4.2.1, a derivation that rigorously takes into account the change in mass arrives at the same result given in Eq. 4-3, i.e., Eq. 4-3 is also applicable to a missile that has variable mass due to the burning of propellant. Rigorous accounting for a variable moment of inertia is not easy, but as discussed in subpar. 4-5.2, the time rate of change of the moment of inertia due to the burning of propellant is usually small enough that terms expressing this rate can be neglected. By this assumption, Eq. 4-4 also is applicable to missile fright simulations. In a simulation the values of m and [I] substi- tuted into the equations are continuously updated during the time propellant is being consumed.
These equations of motion are expanded in the para- graphs that follow to account for rotating reference frames and moments of inertia that are specified relative to the axes of reference frames rather than the specific axis of rotation.
4-3.2 ROTATING REFERENCE FRAMES In considering rotating reference frames, it is important to make a clear distinction between two different relationships that exist between a vector and a reference frame. The first is that a vector can have components expressed in any given reference frame; the second relationship is that a vector can change in magnitude and direction over the relative to the given reference frame. These changes can result from changes in the vector, from motion of the reference frame, or both. Vector components in the first relationship are eas- ily transformed from one reference frame to another, how- ever, considerations of vector rates of change in the second relationship are more complicated. Any given vector can be resolved into components in any reference frame, i.e., it can be expressed in terms of the unit vectors that define the axes of any reference frame. Thus the same vector can be expressed in either a rotating reference frame or an inertial reference frame, but its rate of change with respect to time as viewed by observers in the two systems is very different.
A vector resolved in a given reference frame is said to be
“expressed" in that frame (some authors use the term
“referred to” (Ref. 4)). The rate of change of a vector, as viewed by an observer fixed to and moving with a given ref- erence frame, is said to be “relative to” or “with respect to”
that reference frame. These terms are used many times in this chapter with the very specific meanings given here.
Many references do not emphasize the distinction between these terms; this leads to confusion on the part of the reader.
The point to be made here, and by the example in par. 4- 3.2.1, is: to be applied to Newton’s equations of motion, the rate of change of a vector must be relative to an inertial ref- erence frame, but it can be expressed in any reference frame.
4-3.2.1 Time Derivative of a Vector
An instantaneous vector, for example, a force on the mis- sile, can be transformed between inertial and rotating frames by application of the transformation equations in Appendix A. In such a transformation the vector is unchanged; only its components are changed. Great care must be taken, however, when any vector representing the rate of change of a vector is transferred between reference frames.
To illustrate, consider an example in which some arbi- trary vector B has constant magnitude B and is always directed along the xb-axis of the body reference frame as shown in Fig. 4-4(A). The vector B is expressed in the body coordinate frame by
where
Figure 4-4. Time Rate of Change of Vector B
B = magnitude of general vector B
ib, jb, kb = unit vectors along the xb-, yb-, and Zb-axes, respectively, of the body frame, dimension- less.
Assume that the body frame is rotating about the Zb-axis at a rate of ω rad/s, but to simplify the example, assume that at a given instant of time the angular orientation of the rotat- ing body coordinate frame happens to coincide with that of the fixed-earth (inertial) coordinate frame as shown in Fig.
4-4(B). At that instant the coordinates of B expressed in the earth frame are the same as the coordinates in the rotating the by the assumption that ib, jb, and kb momentarily coincide with ie, je, and ke, respectively
where
B = arbitrary general vector B = magnitude of general vector B
ie, je, ke = unit vectors along the xe-, ye-, and Ze- axes, respectively, of the earth coordinate system, dimensionless.
However, when we consider the absolute rate of change of vector B, we also must consider the rate of change of the rotating reference frame. Since we have assumed that B has constant magnitude and always points along the xb-axis, the rate of change of B with respect to the rotating body frame is zero. That is
where
B = arbitrary general vector
ib, jb., kb = unit vectors along the xb-, yb-, and zb- axes, respectively, of the body frame,-dimensionless t = time,s.
But, from elementary mechanics, as shown in Fig. 4-4(B), the rate of change of B with respect to the fixed-earth (iner- tial) frame at that instant is
where
ω = magnitude of angular rate of rotating frame rel- ative to inertial frame, rad/s (deg/s).
In general, when the derivative (or incremental change) of a vector is calculated using components in a given refer- ence frame, the resulting rate of change of the vector is rela- 4-8
tive to that particular reference frame. If that reference frame is not an inertial one, the rate of change is not an absolute one as required by Newton’s laws. In the example the rate of change of B calculated by finding the rates of change of its components in the rotating frame (zero) is not an absolute rate of change. A mathematical procedure is required to convert the rate-of-change vector to one that is relative to an inertial frame. The general mathematical equation for calculating the rate of change of any vector rel- ative to an inertial frame when the rate of change of that vector is known relative to a rotating frame is (Ref. 8)
(expressed in any frame) (4-9) where
any vector
time rate of change of B relative to inertial frame
time rate of change of B relative to rotating frame
angular rate vector of rotating frame relative to inertial frame, rad/s (deg/s).
Here x B represents the difference between the time derivative of the vector as measured in an inertial reference frame and its time derivative as measured in the rotating ref- erence frame. It is important to note that B is the same vec- tor in both the inertial and the rotating reference frames, but the vector representing the time rate of change of B as seen by an observer in the moving system is not the same as the vector representing the absolute time rate of change.
By applying Eq. 4-9 to the example, (dB/dt)rot is zero, and w x B has magnitude and is in the direction of the yb-axis Substitution into Eq. 4-9 yields the vector (dB/dt)inrtl expressed in the rotating frame. The final result is identical with Eq. 4-8 and shows that the general equation gives the same result as the analysis based on elementary mechanics (Fig. 4-4) as expected.
4-3.2.2 Acceleration in a Rotating Frame
We now extend the discussion of rotating reference frames to include the handling of acceleration vectors and to consider the motion of particles or bodies located at posi- tions other than the origin of the rotating frame. Applica- tions could be, for example, the motion of a mechanical linkage within the missile as the missile experiences rota- tional motion or the motion of an object with respect to the rotating earth. Although most applications of this type are beyond the scope of this handbook the equations are pre- sented in this paragraph (1) as a basis for discussion in sub- par. 4-4.3 of the acceleration due to gravity and (2) to reinforce the understanding that the equations employed in subpars. 4-5.1 and 4-5.2 are a subset of the overall analysis
of rotating reference frames.
Let P be the instantaneous position vector of a particle (or a point), and let w be the absolute angular rate of a rotating reference frame. Given the rate of change of position rela- tive to the rotating frame Prot, the rate of change of position of the particle relative to an inertial frame Pinrtl is obtained by substituting P for B in Eq. 4-9
where
instantaneous position vector of a particle or point, m
rate of change of position vector P relative to inertial reference frame, m/s
rate of change of position vector P relative to (as viewed by an observer in) a rotating refer- ence frame, m/s
angular rate vector of rotating reference frame relative to inertial frame, rad/s.
In the same way the rates of change of velocity with respect to the two reference frames are related by
absolute linear velocity vector of a body, m/s (It is equivalent to Pinrtl in Eq. 4-10.)
acceleration vector of a body relative to an inertial reference frame, m/sz
acceleration vector of a body relative to (as viewed by an observer in) a rotating reference frame, m/s2
angular rate vector of rotating reference frame relative to inertial frame, rad/s.
Substituting Eq. 4-10 into 4-11 leads to the general expres- sion that yields the acceleration of a particle with respect to an inertial reference frame for a given position and motion of the particle measured with respect to a rotating reference frame (Ref. 4)
m/s2 where
Ainrtl = A rot =
P=
v rot =
(4-12)
absolute acceleration vector of a particle, i.e., relative to an inertial reference frame, m/s2
acceleration vector of a particle relative to (as viewed by an observer in) a rotating reference frame, m/s2
instantaneous position vector of a particle or point, m
velocity vector of a body relative to (as viewed
by an observer in) a rotating reference frame, In/s
angular acceleration vector of a body, rad/s2 angular rate vector of rotating reference frame relative to inertial frame, rad/s.
Eq. 4-12 gives the absolute acceleration of a particle as a function of the position, velocity, and acceleration of the point in a rotating reference frame for a given angular rate and angular acceleration of the rotating frame. The term on the left Ainrtl is the acceleration appropriate for use in New- ton’s equations. The first term on the right Arot is the accel- eration of the particle as viewed by an observer in the rotating frame. The variable Vrot is the velocity of the parti- cle as viewed by an observer in the rotating frame. The sec- ond term on the right x P results from the angular acceleration of the rotating frame; this term vanishes when the rotating frame rotates at a uniform rate, such as a frame fixed to the rotating earth. The negative of the third term on the right -203 x Vrot is called the Coriolis acceleration, and the negative of the last term −ω x (ω x P) is called the cen- trifugal acceleration.
Multiplication of Eq. 4-12 by the mass of the body and setting the result equal to the sum of forces on the body is a way of applying Newtonian mechanics to a rotating refer- ence frame. After multiplication by mass, the various terms in Eq. 4-12 have the units of force, and indeed, to an observer in a rotating reference frame, objects behave as if they have been acted upon by forces even when no external forces have been applied. This behavior is explained by the fact that bodies in motion relative to inertial space retain that motion until acted upon by an external force (Newton’s first law), and points on the rotating reference frame accel- crate away from the steady, straight-line path of the body.
To an observer located on the rotating frame, this relative acceleration produces the appearance that some force is act- ing on the body to cause the relative acceleration. Further- more, the observed relative motion can be accurately predicted by introducing forces called pseudoforces (or inertia.? forces) into Newton’s equations of motion. The pseudoforces are so named because they cannot be associ- ated with any particular body or agent in the environment of the body on which they act (Ref. 9). When multiplied by mass, the Coriolis and centrifugal acceleration terms in Eq.
4-12 become pseudoforces and are respectively called the Coriolis force and the centrifugal force. When viewed from an inertial reference frame, the pseudoforces disappear.
These pseudoforces simply provide a technique that permits the application of Newtonian mechanics to events that are viewed from an accelerating reference frame.
Thus for mechanical problems dealing with rotational motion, there are two choices: (1) select an inertial frame as a reference frame and consider only “real” forces, i.e., forces that can be associated with definite agents in the
environment or (2) select a noninertial frame as a reference frame and consider not only the “real” forces but also suit- ably defined pseudoforces. Although the first alternative leads to a clearer understanding of the problem, the second is often employed because other aspects of the problem cause it to be the simplest approach, especially in the treat- ment of moments of inertia. Both approaches are com- pletely equivalent, and the choice is a matter of convenience (Ref. 9).