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7/27/2019 441 Lecture 10
1/19
Spacecraft and Aircraft Dynamics
Matthew M. Peet
Illinois Institute of Technology
Lecture 10: Linearized Equations of Motion
7/27/2019 441 Lecture 10
2/19
Aircraft DynamicsLecture 10
In this Lecture we will cover:
Linearization of 6DOF EOM
Linearization of Motion
Linearization of Forces Discussion of Coefficients
Longitudinal and Lateral Dynamics
Omit Negligible Terms
Decouple Equations of Motion
M. Peet Lecture 10: 2 / 19
7/27/2019 441 Lecture 10
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Review: 6DOF EOM
FxFyFz
= m
u + qw rvv + ru pww + pv qu
and
L
M
N
=
Ixx p Ixz r qpIxz + qrIzz rqIyyIyy q+ p
2Ixz prIzz + rpIxx r2Ixz
Ixz p + Izz r + pqIyy qpIxx + qrIxz
M. Peet Lecture 10: 3 / 19
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Linearization
Consider our collection of variables:
X, Y, Z, p, q, r, L, M, N, u, v, w... also dont forget ,, .
To Linearize:Step 1: Choose Equilibrium point:X0, Y0, Z0, p0, q0, r0, L0, M0, N0, u0, v0, w0.
Step 2:Substitute.
u(t) = u0 + u(t) v(t) = v0 + v(t) w(t) = u0 + w(t)
p(t) = p0 + p(t) q(t) = q0 + q(t) r(t) = r0 + r(t)
X(t) = X0 + X(t) Y(t) = Y0 + Y(t) Z(t) = Z0 + Z(t)
L(t) = L0 + L(t) M(t) = M0 + M(t) N(t) = N0 + N(t)
Step 3: Eliminate small nonlinear terms. e.g.
u(t)2 = 0, u(t)r(t) = 0, etc.
M. Peet Lecture 10: 4 / 19
7/27/2019 441 Lecture 10
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LinearizationStep 1: Choose Equilibrium
For steady flight, let
u0 = 0 v0 = 0 w0 = 0
p0 = 0 q0 = 0 r0 = 0
X0 = 0 Y0 = 0 Z0 = 0
L0 = 0 M0 = 0 N0 = 0
Other Equilibrium Factors
Weight
Weight: Pitching Motion changes forcedistribution.
X0 = mg sin 0
Z0 = mg cos 0
M. Peet Lecture 10: 5 / 19
7/27/2019 441 Lecture 10
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LinearizationStep 2: Substitute into EOM
We use trig identities and small angle approximations ( small):
sin(0 + ) = sin 0 cos + cos 0 sin= sin 0 + cos 0
cos(0 + ) = cos 0 cos sin 0 sin= cos 0 sin 0
M. Peet Lecture 10: 6 / 19
7/27/2019 441 Lecture 10
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LinearizationStep 2: Substitute into EOM
Substituting into EOM, and ignoring 2nd order terms, we get
u + g cos 0 =X
m
v g cos 0 + u0r =Y
m
w + g sin 0 u0q =Z
m
p =Izz
IxxIzz I2xzL +
Ixz
IxxIzz I2xzN
q =M
Iyy
r =Ixz
IxxIzz I2xzL +
Ixx
IxxIzz I2xzN
Note these are coupled with , .
M. Peet Lecture 10: 7 / 19
7/27/2019 441 Lecture 10
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LinearizationStep 2: Substitute into EOM
We include expressions for , .
= q
= p + r tan 0
= r sec 0
For steady-level flight, 0 = 0, so we can simplify
= q
= p
= r
which is what we will mostly do.
M. Peet Lecture 10: 8 / 19
7/27/2019 441 Lecture 10
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7/27/2019 441 Lecture 10
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LinearizationForce Contribution
Out of (u,v,w, u, v, w ,p,q,r, a, e, r, T), we make the restrictive
assumptions on form (Why?): X(u, w, e, T)
Y(v, p, r, r)
Z(u, w, w, q, e, T)
L(v, p, r, r, a) M(u, w, w, q, e, T)
N(v, p, r, r, a)
where we have following new variables
T - Throttle control input.
e - Elevator control input.
a - Aileron control input.
r - Rudder control input.
It could be worse ( , , ). Reality is worse.
M. Peet Lecture 10: 10 / 19
7/27/2019 441 Lecture 10
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LinearizationForce Contribution
To linearize the forces/moments we use first-order derivative approximations:
X =X
uu +
X
ww +
X
ee +
X
TT
Y =Y
vv +
Y
pp +
Y
rr +
Y
rr
Z =Z
uu +
Z
ww +
Z
w w +
Z
qq+
Z
ee +
Z
TT
L =L
vv +
L
pp +
L
rr +
L
rr +
L
aa
M =M
u u +M
w w +M
w w +M
q q+M
ee +
M
TT
N =N
vv +
N
pp +
N
rr +
N
rr +
N
aa
M. Peet Lecture 10: 11 / 19
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7/27/2019 441 Lecture 10
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Longitudinal Dynamics
Although we now have many equations, we notice that some of them decouple:
u + g cos 0 =
X
m
w + g sin 0 u0q =Z
m
q =M
Iyy
= q
where 1
mX = Xuu + Xww + Xee + XTT
1
mZ = Zuu + Zww +
Z
w w + Zqq+
Z
e
e + ZTT
1
IyyM = Muu + Mww + Mw w + Mqq+ Mee + MTT
and also, x = u cos 0 u0 sin 0 + w sin 0
z = u sin 0 u0 cos 0 + w cos 0
M. Peet Lecture 10: 13 / 19
7/27/2019 441 Lecture 10
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Simplified Longitudinal Dynamics
Note = q
u + g cos 0 = Xm
w + g sin 0 u0 =Z
m
=M
Iyy
where
1
m
X = Xuu + Xww + Xee + XTT
1
mZ = Zuu + Zww + Zw w + Zq + Zee + ZTT
1
IyyM = Muu + Mww + Mw w + Mq + Mee + MTT
M. Peet Lecture 10: 14 / 19
7/27/2019 441 Lecture 10
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Simplified Longitudinal Dynamics
Combining:
u + g cos 0 = Xuu + Xww + Xee + XTT
w + g sin 0
u0
= Zuu + Zww + Zw w + Zq
+ Zee + ZT T
= Muu + Mww + Mw w + Mq + Mee + MTT
Homework: Put in state-space form. Hint: Watch out for w.
M. Peet Lecture 10: 15 / 19
7/27/2019 441 Lecture 10
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Lateral Dynamics
The rest of the equations are also decoupled:
v g cos 0 + u0r =
Y
m
p =Izz
IxxIzz I2xzL +
Ixz
IxxIzz I2xzN
r =Ixz
IxxIzz I2xzL +
Ixx
IxxIzz I2xzN
= p + r tan 0
where 1
mY = Yvv + Ypp + Yrr + Yrr
1
Izz L = Lvv + Lpp + Lrr + Lrr + Laa
1
IxxN = Nvv + Npp + Nrr + Nrr + Naa
and alsoy = u0 cos 0 + v
M. Peet Lecture 10: 16 / 19
Al i R i
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Alternative Representation
M. Peet Lecture 10: 17 / 19
C l i
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Conclusion
Today you learned
Linearized Equations of Motion:
Linearized Rotational Dynamics
Linearized Force Contributions New Force Coefficients
Equations of Motion Decouple
Longitudinal Dynamics
Lateral Dynamics
M. Peet Lecture 10: 18 / 19
C l i
7/27/2019 441 Lecture 10
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Conclusion
Next class we will cover:
Longitudinal Dynamics:
Finding dimensional Coefficients from non-dimensional coefficients Approximate modal behaviour
short period mode phugoid mode
M. Peet Lecture 10: 19 / 19