Missile System Design

Performance

and

Stability and Control[1]

Contents

Contents 1

1 Introduction 1

1.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Mechanics of Manuevering . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Derivation for Normal Force Coefficient . . . . . . . . . . . . . . . 3

1.3 Stability and Control in Trim Condition . . . . . . . . . . . . . . . . . . 3

2 Stability 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Static Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Equilibrium Condition . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 Stability Condition when External Force Applied . . . . . . . . . 8

2.3.3 Static Stability Margin . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.4 Pitching Moment Coefficient Vs Angle of Attack . . . . . . . . . . 9

2.4 Dynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References 11

1

Chapter 1

Introduction

Performance, and stability and control of the airframe are two important factors that

determine the ability of the missile to carry a payload and also maneuver it into a

position so as to destroy the aircraft. However this performance relates only to the

maneuvering capabilities and motion characteristics of the missile and not to the overall

effectiveness as a weapon.

1.1 Performance

The performance of missiles is governed by the Newton’s Laws of Motion. Missiles

require manuevering forces to accelerate them in the required direction. Depending on

the medium in which they travel, missiles can be classified as

(a) Aerodynamic Vehicles: When they travel through air maneuvering is possible

through the mechanisms of aerodynamic forces i.e., lift and drag acting on the

vehicle.

(b) Space Vehicles: When they travel in outer space, maneuvering is possible thrust

vectoring.

In this report, only the aerodynamic vehicles will be considered.

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1.2 Mechanics of Manuevering

Assume that the missile is traveling with a fixed forward speed. Assuming that the

target is somewhat to the left of the missile, the mechanics by which the missile changes

its flight path toward a collision course can be discussed as follows:-

Figure 1.1: Mechanics of Maneuvering

(a) When the missile flight path has to be changed, the control surfaces are deflected

by the guidance and control system based on the required trajectory. This causes

misalignment of the vehicle or some of its surface (depending upon its design)

relative to its forward motion and thus to the relative airflow.

(b) This asymmetric condition or misalignment generates forces which accelerate the

missile in the required direction.

(c) The maneuvering force of an aerodynamic vehicle is given by the equation

N = CNqS (1.1)

where CN is the missile normal force coefficient which is chiefly a function of Mach

number and of wing planform, q is the dynamic pressure due to the flight speed

and air density, kg/m2 given by 12ρV 2 where ρ is the atmospheric air density and

2

V is the missile velocity in m/sec and S is the reference area(usually taken as the

free wing area)in m2.

The wing is the main lifting surface and develops or contribuites to, a major part of the

maneuvering force. The maneuvering force generated by the wing is modified by the

interference between the body and the wing. If the missile flies with a body angle of

attack, in that case, the body lift also contributes directly to the maneuvering force.

The principle function of the control surface is to maintain the desired angle of attack

or in other words, keep the missile aligned with the relative airstream. Depending on

their geometric configuration, control surfaces may either contribute to the maneuvering

force or detract from it. In case of wing-controlled missiles, the primary purpose of the

deflected wing is to develop direct lift.

1.2.1 Derivation for Normal Force Coefficient

The normal force coefficient for the complete missile is given by the equation

CN = CNW+ CNB

+ ∆CNBW+ ∆CNWB

+ CN ′T

(1.2)

where CNWis the normal-force coefficent for the wing alone CNB

is the normal-force

coefficent for the body alone ∆CNBWis the change in body normal-force coefficient due

to the presence of wing ∆CNWBis the change in wing normal-force coefficient due to the

presence of body CN ′T

is the normal-force coefficient of the tail in the presence of wing

and body.

1.3 Stability and Control in Trim Condition

The angle of attack, wing incidence and tail incidence are very important factors which

decide the stability and control characteristics of the missile. They cannot be arbitrarily

chosen as they must be compatible with one another stability-wise. Static stability of

a missile is defined as its inherent tendency to return to its trimmed angle of attack if

disturbed from this angle of attack by an external force. The trimmed angle of attack

is the result of complete equilibrium of all external aerodynamic forces acting upon the

missile at this particular angle of attack. Changing from one trimmed angle of attack

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to another requires a suitable control surface deflection and effectiveness. Thus in case

of a tail controlled missile, a deflection of the tail surface will change the angle of attack

and therefore the normal or maneuvering force.

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Chapter 2

Stability

2.1 Introduction

The stability of an aerodynamic missile operating in the atmosphere depends upon

its inherent static and dynamic stability characteristics. It also depends upon the nature

of the automatic control system employed.

2.2 Static Stability

Static stability of an aerodynamic vehicle is defined as its inherent tendency to align

itself with the direction of the relative airstream. There are three types of static stability.

(a) Statically Stable System: If any body is displaced from its equilibrium position

then if resulting forces or moments acting on that body tends to restore the system

to its original condition then the system is said to be Statically stable.

(b) Statically Unstable System: If resultant forces or moments are such that it trend

to cause the system to be displaced still further from its original condition, the

system is Statically unstable.

(c) Neutrally Stable System: If displacement cause no new forces or moments to be

created, so that the system has no tendency either to return to or to deviate further

from its original condition, then the condition is Neutral stable.

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Figure 2.1: Types of Static Stability

2.3 Longitudinal Stability

Compared to the single degree of freedom possessed by the weather vane, the num-

ber of degrees of freedom increases in the missile due to its free-flight condition. This

complicates the stability of the missile. If the rolling motions of the missile are ignored,

there are four degrees of freedom namely

(a) Angular pitching about the center of gravity,

(b) Translational motion of the center of gravity in the plane of the vertical wings,

(c) Angular yawing about the center of gravity and

(d) Translational motion of the center of gravity in the plane of the horizontal wings.

In a cruciform missile, the characteristics of the horizontal and vertical planes are

usually identical due to symmetry. Thus if the motions or moments of the missile of

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the vertical plane of the winds are calculated, these calculations can be related to the

horizontal plane of the wings. This reduces the number of degrees of freedom to two

when restricted to the vertical plane only namely

(a) Angular pitching about the center of gravity,

(b) Vertical translational motion of the center of gravity in the plane of the vertical

wings,

For the case of static stability, the vertical motions of center of gravity can be dis-

carded since such motions merely tend to alter the angle of attack with respect to the

relative airstream. Thus static stability is only involved with a single degree of freedom

namely the pitching moments developed as a function of angle of attack. This is called

longitudinal stability.

The figure below shows the dimensions and angles of various axes and the forces and

moments encountered in the longitudinal stability i.e., static stability in pitching of the

missile.

Figure 2.2: Forces,Moments and Angles - Longitudinal Stability

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2.3.1 Equilibrium Condition

When the missile is designed to be statically stable, the missile remains in equilibrium

at angles of attacks other than zero by changing the tail incidence i.e., deflecting the tail

and in effect changing the trim angle of attack. In other words, the algebraic sum of the

pitching moment contributions of each component of the missile i.e., the body, wing and

tail, is intentionally set to zero so that the missile satisfies the equilibrium condition.

Thus the pitching moment characteristics for the special case of trim is given by

CMBqSc+ CNWB

qScXc.p.

c+dCN

dα WiWXc.p.

cqSc− LT

c

ST

SCN ′

TqSc = 0

Dividing by qSc gives

CMB+ CNWB

Xc.p.

c+dCN

dα WiWXc.p.

c− LT

c

ST

SCN ′

T= 0 (2.1)

Thus every time the missile angle of attack changes, the tail incidence required to make

the net moment zero is calculated and applied as the required deflection.

2.3.2 Stability Condition when External Force Applied

Consider that the missile is made to pitch up due to an external force rather than the

tail deflection, initially. When the missile is pitched at an angle of attack (α) relative to

the airstream it develops a pitching moment (M) which determines the static stability.

In this case, the pitching moment will be the algebraic sum of the individual pitching

moment contributions of each component of the missile i.e., body, wing and tail. In

other words, the eqn.2.1 is modified by repacing the zero on the right hand side with

the pitching moment coefficient (CM) and rewritten as

CM = CMB+ CNWB

Xc.p.

c+dCN

dα WiWXc.p.

c− LT

c

ST

SCN ′

T(2.2)

One way of identifying whether the missile is statically stable or not is as follows: When

the angle of attack α is positive, if CM is positive, then the pitching moment can be seen

to be in a direction that increases the angle of attack further rather than reduce it thus

making the missile statically unstable.

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2.3.3 Static Stability Margin

The term Xc.p.

cwhich is the distance of the center of pressure in terms of the number

of wing chords (c is the wing chord) is a measure of static stability. If the location of

center of pressure at the neutral position is denoted by (Xc.p.)N , then the static stability

margin of the missile may be written as

dCM

dCNWB

=Xc.p. − (Xc.p.)N

c(2.3)

or

dCM

dCNWB

=−xc

(2.4)

where x is the distance of center of gravity forward of the neutral point. Thus if the

center of gravity is aft of the neutral, then x is negative and the eqn.2.4 gives a positive

value thus indicating static instability.

2.3.4 Pitching Moment Coefficient Vs Angle of Attack

Figure 2.3: Longitudinal Static Stability

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2.4 Dynamic Stability

Dynamic stability of an aerodynamic vehicle deals with the history of vehicles motion.

It is concerned with what actually happens to the mechanical system as a result of its

displacement from an equilibrium condition. Static stability is pre-requisite for dynamic

stability. A body is dynamically stable if, out of its own accord it eventually returns to

and remains at its equilibrium position over a period of time.

Figure 2.4: Types of Dynamic Stability

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References

[1] J. Jerger, Systems Preliminary Design, Principles of Guided Missile Design. Prince-

ton: D.Van Nostrand Inc, 1960.

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