Eq. 4-2 expresses the basic relationship for rotational motion of a particle in an inertial (absolute) reference frame.
This equation is extended to rigid bodies by summing over all the particles (or incremental masses) of the body. If the moments acting on the missile are given, Eq. 4-2 is used to calculate the angular rate w of the missile. The angular rate is contained within the angular momentum vector h. In the general case with no constraints on the axis of rotation, the relationship between h and ω involves the inertia matrix (also called the inertia tensor) [I]. That relationship is given by
where
= x component in the body frame of the sum of gravitational forced on the missile, N
and let similar symbols with subscripts y and z represetn, respectively, the y and z components of the sums of the forces.
Then
coordinates of infinitesimal masses of the body, m
infinitesimal mass, kg.
Ixx Iyy and Izz are called moments of inertia, and Ixy Ixz, and Iyz are called products of inertia. A particular orientation of the reference frame axes relative to the body can always be chosen for which the products-of-inertia terms vanish, 4-18
giving
Since the double subscript is required only when products of inertia are involved, single subscripts are sufficient when the reference frame that causes products of inertia to vanish is selected. This particular orientation is such that the refer- ence frame axes are aligned with the principal axes of the body and is true for any rigid body, not only symmetrical ones. Selecting this particular reference frame greatly sim- plifies the equations of motion. Since the body frame axes meet this selection criterion, i.e., they are aligned with the principal axes of the body, there is considerable motivation to express the rotational equations of motion in the body frame. The mass distribution of a missile about its y-axis is often essentially the same as that about its z-axis. Therefore, the further simplification of setting IY equal to lZ is often possible; however, the distinction is retained here for gener- ality.
In some applications there may be other considerations that lead to selection of a reference frame other than the body frame, e.g., the wind coordinate frame. In that ease the products-of-inertia terms are, in general, nonzero, and they must be retained in the development of the rotational equa- tions of motion. The result is a very complex set of rota- tional equations of motion (Ref. 3). Even in that case, however, some of the products-of-inertia terms vanish if mass symmetries exist in the body about either the xz-plane or the xy-plane. In this handbook the body frame is selected, permitting use of the simpler diagonalized inertia matrix, i.e., all matrix elements are zero except on the diagonal.
Substituting Eq 4-38 into Eq. 4-2 and taking the deriva- tive with respect to time give
where
rate-of-change vector of angular momentum h, N-m
inertia matrix of a body, kg•m2
rate of change of inertia matrix, kg•m2/s angular rate vector of rotating reference frame relative to inertial frame, rad/s
angular acceleration vector, rad/s2.
If Eq. 4-40 were evaluated in an inertial reference frame, the moments of inertia about the frame axes would change as the body experienced rotational motion and result in non- zero values of [I]. However, if a reference frame fixed to the body were employed, the inertia matrix would not be changed by body motion and thus would provide another
motivation to select the body frame. During the operation of the propulsion system of a missile, there is another source of change in the moments of inertia that is not related to body motion. As the propellant mass is expelled from the missile, the moments of inertia change. This change in the value of [I] is usually updated continuously in a flight simulation, but• the time rate of change [I] is usually small enough to be neglected in Eq. 4-40. Thus selecting the body frame and assuming that the time rates of change of the moments of inertia caused by propellant expulsion are small cause [I] ω in Eq. 4-40 to vanish.
Since the angular momentum h is a vector quantity, its time rate of change relative to an inertial reference frame (as required by Eq. 4-2) is different from its rate of change rela- tive to a rotating reference frame. Again, this difference is taken into account by employing Eq. 4-9. Substituting h for B in Eq. 4-9 gives
where
Then, from
where
angular momentum vector of a body, N•m•s rate-of-change vector of angular momentum h relative to inertial reference frame, N•m rate-of-change vector of angular momentum h relative to (as viewed by an observer in) a rotat- ing reference frame, N•m
angular rate vector of rotating reference frame relative to inertial frame, rad/s.
Eq. 4-2
magnitude of the rate-of-change vector angular momentum h relative to (as viewed by an observer in) a rotating reference frame, N•m
total moment vector acting on the body, N•m.
If [I] is calculated relative to the body frame axes and ω is expressed in the body frame, the time rate of change of angular momentum relative to that frame (the first term on the right side of Eq. 4-42) is given by
The second term on the right side of Eq. 4-42 is
Substituting Eqs. 4-43 and 4-44 in Eq. 4-42 and rearranging give
where
moments of inertia (diagonal elements of inertia matrix when products of inertia are zero), kg•mz
components of total moment vector M expressed in body coordinate system (roll, pitch, and yaw, respectively), N•m
components of angular rate vector w expressed in body coordinate system (roll, pitch, and yaw, respectively), rad/s
components of angular acceleration w expressed in body coordinate sys- tem (roll, pitch, and yaw, respec- tively), rad/s2.
Eqs. 4-45 are the rotational equations of motion expressed in body frame coordinates. Integration of Eqs. 4- 45 yields the inertial angular rate w of the missile expressed in rotating body coordinates p, q, r.
4-5.2.1 Rotational Accelerations
Eqs. 4-45 give the absolute angular acceleration of the missile about its center of mass, expressed in the body refer- ence frame. These equations are used to calculate-the angu- lar acceleration when the moments on the missile are given.
into aerodynamic and propulsion components, this equation If the moment terms L, M, and N in Eqs 4-45 are separated
becomes
where
(4-46) components of aerodynamic moment vector MA expressed in body coordi- nate system (roll, pitch, and yaw, respectively, N•m
components of propulsion moment vector Mp expressed in body coordi- nate system (roll, pitch, and yaw, respectively), N.m.
4-5.22 Gyroscopic Moments
To evaluate the angular momentum vector h (Eq. 4-38) it is assumed that the missile is a single rigid body. If some portions of the missile mass are spinning relative to the body reference frame, e.g., spinning rotors, the additional angular momentum may be significant enough to be consid- ered in the equations of rotational motion. Each spinning mass has an angular momentum relative to the body axes.
This can be computed from Eq. 4-38 by interpreting the moments of inertia and products of inertia as those of the rotor with respect to axes parallel to the body axes and ori- gin at the rotor mass center. For this application the angular velocity in Eq. 4-38 is interpreted as that of the rotor relative to the body axes. Let the resultant relative angular momen- tum of all rotors be h´, assumed to be constant. It can be shown that the total angular momentum of a missile with spinning rotors is obtained simply by adding h´ to the h pre- viously obtained
where
angular momentum vector of body, N•m•s angular momentum vector of all rotors, N•m•s inertia matrix of a body, kg•m2
angular rate of rotating reference frame relative to inertial frame, rad/s.
As a result of adding h´ terms to the angular momentum equation (Eq. 4-38), certain extra terms, known as gyro- sc-epic couples (Ref. 2), appear in the rotational equations of motion. After adding these terms and solving for the angular rate components, the rotational equations of motion for a missile with spinning rotors are
(4-48) where
components of rotor angular momen- tum vector h´ expressed in the body coordinate system, N•m•s.
The angular momentum of a given rotor is
where
inertia matrix of a rotor relative to the body coordinate system, kg•m2
4-20
Ω = angular rate vector of the rotor relative to the body coordinate system, p´ib + q’jb + r´kb, rad/s.
For the general case the inertia matrix [I] for a rotor will include the rotor products of inertia; however, if the rotor axis is aligned with a missile body axis, the rotor products of inertia will vanish. The rotational equations of motion for a missile with a single rotor, and with the rotor axis aligned with a body axis, become
(4-50) where
I´x, I´y, I´z = p´, q´, r´ =
L,M,N =
p,q,r =
diagonal elements of rotor inertia matrix relative to body axes, kgžm2 components of rotor rate vector expressed in body coordinate system (roll, pitch, and yaw, respectively), rad/s
components of total moment vector M expressed in body coordinate system (roll, pitch, and yaw, respetively, N-m
components of the angular rate vector ω expressed in body coordinate sys- tem (roll, pitch, and yaw, respectively, rad/s
moments of inertia (diagonal elements of inertia matrix when products of inertia are zero), kg•m2.
Note that in Eqs. 4-50 if a hypothetical internal rotor has inertial and rotational rate components that are identical to those of the airframe without the rotor, i.e., primed terms=
unprimed terms, the terms representing the rotor are identi- cal with the terms representing the airframe without the rotor, as expected. This observation serves only to give an understanding of the additional rotor terms and to add credi-
bility to them.
4-5.2.3 Rate of Change of Euler Angles
As discussed in par. 4-2 and shown in Fig. 4-2, the orien- tation of the body reference frame is specified by the three Euler angles, Ψ, θ, and φ. As the missile changes its orienta- tion in space, the Euler angles change. The rates of change of the Euler angles are related to the angular rate w of the body frame. This relationship is given in terms of the com- ponents of ω in the body reference frame (p, q, and r) by Ref. 2
Euler angle rotation in elevation (pitch angle), rad (deg)
rate of change of 0, rad/s
Euler angle rotation in roll (roll angle), rad (deg)
rate of change of ø, rad/s
rate of change of Euler rotation in azimuth, rad/s.
The angular orientation of the missile at any time t is Obtained by integrating Eqs. 4-51 from launch to time t and adding to the initial orientation angles at launch.
4-6 APPLICATION OF EQUATIONS OF MOTION
The dynamic motion of a missile is calculated by using the equations of motion given in this chapter. The methods used to calculate the aerodynamic and propulsive forces and moments are given in Chapters 5 and 6, respectively. The gravitational force is determined from Table 4-1 or Eq. 4- 29. At each computational time step these forces and moments are substituted into Eqs. 4-37 and 4-46, respec- tively, to yield the translational and rotational accelerations of the missile. These are absolute accelerations expressed in the rotating body coordinates. Integration of these accelera- tions gives the translational and rotational absolute velocity components expressed in the rotating coordinates. These velocities are transformed to the fixed-earth (inertial) sys- tem by using the transformation equations in Appendix A.
The resulting absolute velocities are integrated in earth coordinates to yield missile position in earth coordinates.
The attitude of the missile in earth coordinates is obtained by integration of Eqs. 4-51.
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REFERENCES
C. D. Perkins and R. E. Hage, Airplane Performance Stability and Control, John Wiley & Sons, Inc., New
York, NY, 1949.
B. Etkin, Dynamics of Flight—Stability and Control, John Wiley & Sons, Inc., New York, NY, 1982.
Dynamics of the Airframe, BU AER Report AE-61-4 H, Bureau of Aeronautics, Navy Department, Washington, DC, 1952.
P. N. Jenkins, Missile Dynamics Equations for Guid- ance and Control Modeling and Analysis, Technical Report RG-84-17, Guidance and Control Directorate, US Army Missile Laboratory, US Army Missile Com- mand, Redstone Arsenal, AL April 1984.
R. W. Kolk Modem Flight Dynamics, Prentice-Hall, inc., Englewood Cliffs, NJ, 1961.
H. Goldstein, Classical Mechanics, Addison-Wesley Publishing Company, Inc., Reading, MA, 1965.
M. Rauscher, Introduction to Aeronautical Dynamics, John Wiley & Sons, Inc., New York NY, 1953.
J. L. Meriam and L. G. Kraige, Engineering Mechanics, Volume 2, Dynamics, Second Edition, John Wiley &
Sons, Inc., New York, NY, 1986.
D. Halliday and R. Resnick, Physics, Parts I and II, John Wiley & Sons, Inc., New York, NY, 1967.
R. c. Duncan, Dynamics of Atmospheric Entry, McGraw-Hill Book Company, Inc., New York, NY,
1%2.
R W. Wolverton, Flight Performance Handbook for Orbital Operations, John Wiley & Sons, Inc., New York NY, 1963.
K. R. Symon, Mechanics, Third Edition, Addison-Wes-