Inertial acceleration of a point on a rotating frame of reference
John Wallace, 21-Oct-2006
ax = u + q.w – r.v
ay = v + r.u – p.w
az = w + p.v – q.u
The equations for inertial acceleration of a point on a rotating frame of reference are used
in developing the full equations of motion of a rigid body. Cook (ref 1) simply states
these in equation 4.5, page 56, following the statement with a paragraph of explanation
that accounts for the rotary motion components by saying: “The accelerations arise from
the mutual interaction of the linear components of velocity with the components of
angular velocity.” The common practice in other modern texts on Flight Dynamics is to
use a vector explanation. For example, in equation 1.2-37, ref. 2 invokes the theorem of
Coriolis stating:
d
v B   v B  B  v B
dt A
and ref. 3 does much the same with the linear momentum in equation 3.2-35, then divides
by mass to get the linear acceleration expressed in body axes (eqn 3.3-36) as:
dv B
f
~ v
 B 
B B
dt
m(t )
~ is the cross product equivalent matrix of vector B). Unfortunately, neither of
(where 
B
these texts give any proof of the Coriolis effect, and the reader is left to go elsewhere to
find that proof. The former Princeton University text (ref 4, now out of print and
superceded by ref 3), has a very good explanation in section 4-3, with figures 4-2, 4-3,
and 4-4, and equations 4-9 to 4-18 which derives both the algebraic and vector equations
from first principles, with no reference to the name of Coriolis. The prevalent British text
from the 1960’s to the 1990’s (ref 5, now out of print), also has a good algebraic
explanation in appendix 2. Both of these are scanned as appendices to this note for
reference (but in separate word documents because they run to several megabytes each).
My own understanding of this is based on summing the direct effect of Newton’s second
law, F=m.a, with the effect of rotation on the velocities in body axes when no force is
applied. The first effect, expressed as acceleration as a function of mass, and expanded
into three dimensions, is:
ax = X/m
ay = Y/m
az = Z/m
If the body that is subject to these forces along its ox, oy, and oz axes is not rotating, the
accelerations are simply the derivative with respect to time of the velocity components
along each axis, i.e. U = ax, V = ay, W = az. The second effect, which considers the
rate of change of velocity in body axes of a body moving at constant velocity in an
inertial frame of reference is best seen by considering the effect of each of the three
rotation components separately. The three figures following each show a projection
looking along one of the three axes of a rigid body.
1) Looking along ox normal to the yz plane
P
o
V

y
VR
W
z
If there is no force applied, there is no acceleration, so the resultant velocity vector, VR,
in the yz plane is constant. VR has cartesian components [V, W], but can also be expressed
as a polar vector, [VR, ], where VR = |VR|, and  is the angle it makes with the oy axis
(measured positive clockwise from oy to VR). The cartesian values can be evaluated from
the polar values as:
V = VR.cos()
W = VR.sin()
At angular velocity, P,  reduces by P.t, in time t, so using the U’, V’ for the velocities
after t, we get:
V’ = VR.cos(-P.t)
W’ = VR.sin(-P.t)
Expanding these using the trig formulae:
sin(A+B)=sinA.cosB + cosA.sinB
cos(A+B) = cosA.cosB - sinAsinB
we get:
V’ = VR.(cos(.cos(-P.t) – sin().sin(-P.t))
W’ = VR.(sin(.cos(-P.t) + cos().sin(-P.t))
For very small angles, sin() =  (where  is in radians), and cos() = 1. Using this, we
see that as t approaches zero, V’ and W’ become:
V’ = VR.(cos(+ sin().P.t)
W’ = VR.(sin( - cos().P.t)
Subtracting the initial velocities, we get:
V = V’ – V
= VR.(cos(+ sin().P.t) - VR.cos()
= VR.sin().P.t
W = W’ – W
= VR.(sin( - cos().P.t) - VR.sin()
= -VR.cos().P.t)
Dividing through by t, taking the limit as t approaches zero, and substituting W for
VR.sin() and V for VR.cos(), we get:
dV/dt = V = P.W
dW/dt = W = -P.V
2) Looking along oy normal to the zx plane
Q
U
o
x

VR
W z
Once again, we can write the cartesian velocity components, [U, V] in terms of the polar
components, [VR, ] to give
W = VR.cos()
U = VR.sin()
The mathematical process to derive W and U is exactly the same as it was to derive V
and W from the variables on the previous figure, except that we must substitute W for V,
U for W, and Q for P. (note that  in this figure is oz to VR, instead of oy to VR as it was
in the previous diagram, but as  is eliminated, this is of no consequence). We can
therefore jump straight to the bottom two lines and make the substitutions to give:
dW/dt = W = Q.U
dU/dt = U = -Q.W
3) Looking along oz normal to the xy plane
R
o
U

x
VR
V
y
Again, we can write the cartesian velocity components, [U, V] in terms of the polar
components, [VR, ] to give
U = VR.cos()
V = VR.sin()
The mathematical process to derive U and V is exactly the same as it was to derive V
and W from the variables on the first figure, except that we must substitute U for V, V for
W, and R for P. (note that  in this figure is ox to VR, instead of oy to VR as it was in the
first diagram, but as  is eliminated, this is of no consequence). We can therefore jump
straight to the bottom two lines and make the substitutions to give:
dU/dt = U = R.V
dV/dt = V = -R.U
To get the effect of a 3d angular velocity vector with components [P, Q, R] on the
velocity components in body axes, in the absence of any external force, we simply sum
the effect of the rotation components about each axis obtained above, to get the full
expressions for U , V , and W . This gives:
U = R.V – Q.W
V = P.W – R.U
W = Q.U – P.V
To get the full expressions for U , V , and W , we must add the effect of inertial
acceleration due to an external force in the absence of rotation, U = ax, V = ay, W = az.
This gives us:
U = ax + R.V – Q.W
V = ay + P.W – R.U
W = az + Q.U – P.V
The expression for inertial acceleration of a point in body axes is obtained by rearranging the above equations to give:
ax = U + Q.W – R.V
ay = V + R.U – P.W
az = W + P.V – Q.U
Remember that in a gravity field, the inertial accelerations include a gravity component,
e.g.
ax = X/m + gx
ay =Y/m +gy
az = Z/m + gz
where gx, gy, and gz are the components of gravity acceleration in each axis.
References:
1)
2)
3)
4)
Flight Dynamics Principles, M.V. Cook (Arnold, 1997)
Aircraft Control and Simulation, B.L. Stevens & F.L. Lewis, (Wiley, 1992)
Flight Dynamics, R.F. Stengel (Princeton University Press, 2004)
Aircraft Dynamics & Control, McRuer, Ashkenas, & Graham (Princeton University
Press, 1973)
5) Aircraft Stability and Control, A.W. Babister (Pergamon Press, 1961)