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Spacecraft and Aircraft Dynamics

Matthew M. Peet

Illinois Institute of Technology

Lecture 10: Linearized Equations of Motion

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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

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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

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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.

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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

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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

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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 , .

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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.

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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.

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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

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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

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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

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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.

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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

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Al i R i

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Alternative Representation

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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

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C l i

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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

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