On the Dynamic Stability of a Rocket Under Constant
Thrust
Melahat Cihan
Metin Orhan Kaya
Civil Aviation Research and Application Center
Anadolu University
Eskişehir, Turkey
melahatcihan@anadolu.edu.tr
Faculty of Aeronautics and Astronautics
İstanbul Technical University
İstanbul, Turkey
kayam@itu.edu.tr
Abstract— Dynamic stability of a free flight aerospace vehicle
is investigated in this paper. Firstly, the slender rocket body is
modeled as a classic uniform beam that subjected to constant end
rocket thrust. The one-dimensional free-free beam under
follower force is established for structural model to discover
dynamic stability. Equations of motion of vehicle is derived by
applying extended Hamilton’s principle for non-conservative
systems. Natural frequencies of rocket are determined and
critical thrust is obtained by using finite element method. It is
noted that, transverse vibrational modes differ by thrust value.
Secondly, natural frequencies of non-homogeneous beam are
discussed by considering that rocket has different types and
number of stages. Numerical results for both cases are
represented.
Keywords—dynamic stability; missile; rocket; free vibration;
natural frequency
I. INTRODUCTION
Dynamic stability of a launch vehicles has been carried out
using varies beam theories by many researchers up to now. A
missile or a rocket under thrust can be modeled as free-free
beam is subjected to follower force. Beal [1] investigated
dynamic stability of a flexible missile under constant and
pulsating thrust by considering with and without control
system. He figured up critical thrust magnitude and bending
frequencies under both conditions with Galerkin method. Wu
[2] studied stability behavior of a flexible missile using finite
element technique with unconstrained variational approach.
He carried out numerical results for a uniform free-free beam
with different concentrated mass under a constant thrusting
force; also there exists a nonzero mode, which represents a
divergent stability. Yoon and Kim [4] also studied stability of
a spinning beam with a concentrated mass subjected to
pulsating thrust. They considered a concentrated mass at
arbitrary location and modeled the beam as a Timoshenko
beam and solved by applying finite element method.
Recently, effect of conservative and non-conservative
forces on dynamic stability of slender launch vehicles has
studied in many publications. Constant and pulsating thrust,
aerodynamic forces, gravitational forces and internal loads
effects on dynamic and aeroelastic stability of rocket have
been examined [5-18]. Trikha et al. [5, 6] studied on problem
of structural instability in slender aerospace vehicles. They
used a one dimensional beam model to find rigid-body modes,
longitudinal and transverse vibrational modes by considering
aerodynamic pressure and propulsive thrust of the vehicle.
Two different finite element methods were applied and
numerical results were demonstrated. Also, Elyada [7]
investigated the aeroelastic divergence of rocket
configurations in closed form approach. In that research,
various expressions that have importance in instability of
rocket, were derived such as acceleration, angle of attack,
aerodinamic loads and elastic bending for perfectly aligned
vehicle at divergence and misaligned vehicle at predivergence.
Rockets, missiles or launch vehicles can be modeled
structurally as a uniform or non-uniform beam or shell
according to the complexity of the system. Joshi [19] modeled
the rocket as non-uniform beam and investigated the vibration
characteristics under thrust and acceleration. Also, Trikha [6]
gave an example for three-stage rocket to find numeric
solutions. In this paper, natural frequencies of a rocket will be
carried out.
II. FORMULATION
A. Structural Model
Structure of typical slender aerospace vehicles (missile or
rocket) can be simplified as a uniform free-free beam. In
general, rockets are subjected to aerodynamic forces and thrust
forces that cause engines at the end of vehicle. According to
the altitude of rocket, effect of these forces on the stability of
vehicle can change. In this work, we assume that a rocket has
thrust force and not heavy machine and any fin; also the flight
is above atmosphere region accordingly no aerodynamic force.
A generic rocket body that moves in x-direction, forces
and coordinate system are demonstrated in Fig.1. Here, l is the
length of rocket, and A is cross-sectional area. The rocket has
constant bending rigidity EI and constant mass per unit length
𝜌𝐴. The thrust force, P denotes the follower force at the end of
rocket and axial displacement of the beam is discarded.
Displacements and finite element discretization is shown in
figure, also. The rocket in this study is modeled as an EulerBernoulli beam by using classical beam theory.
In this work, energy expressions are used to determine
governing equation of motion of the rocket, which are given
by;
Kinetic energy:
1
𝑑𝑤 2
𝒯 = ∫ 𝜌𝐴 ( ) 𝑑𝑥
2
𝑑𝑡
Non-dimensional parameters are used to simplify the
equations. Because of this idea following non-dimensional
forms are introduced;
𝑙
𝑥
𝑃𝑙 2
𝜔2 𝑙 4 𝜌𝐴
𝑄=
𝜆2 =
𝑙
𝐸𝐼
𝐸𝐼
So, the governing differential equation is rewritten as:
Strain energy:
𝜉=
2
𝒰=
1
𝑑2𝑤
∫ 𝐸𝐼 ( 2 ) 𝑑𝑥
2
𝑑𝑥
𝑙
𝑤 ′′′′ + (𝑄𝑤 ′ )′ + 𝜆2 𝑤 = 0
Work done by follower force for 𝑃(𝑥) = 𝑃. 𝑥 ⁄𝑙 :
𝒲𝑐 =
1
𝑑𝑤 2
1
𝑥 𝑑𝑤 2
∫ 𝑃(𝑥) ( ) 𝑑𝑥 = ∫ 𝑃 ( ) 𝑑𝑥
2
𝑑𝑥
2
𝑙 𝑑𝑥
𝑙
𝑙
Work done by non-conservative force:
𝑑𝑤
𝛿𝒲𝑛𝑐 = −𝑃 ( ) 𝛿𝑤 |𝑥=𝑙
𝑑𝑥
The Extended Hamilton’s Principle
conservative systems can be represented as:
𝑡2
in
the
non-
𝑡2
𝛿 ∫ (𝒯 − 𝒰 + 𝒲𝑐 ) 𝑑𝑡 + ∫ 𝛿𝒲𝑛𝑐 𝑑𝑡 = 0
𝑡1
𝑡1
By using Extended Hamilton’s Principle, we drive the
governing differential equation and boundary condition below
respectively. To rearrange the differential equation, here we
set 𝑤 = 𝑤(𝑥) 𝑒 𝑖𝜔𝑡 .
𝛿𝑤:
(𝐸𝐼 𝑤 ′′ )′′ + (𝑃(𝑥)𝑤 ′ )′ + 𝜔2 𝜌𝐴𝑤 = 0
𝑃 𝑤 ′ |𝑥=𝑙 = 0
B. Finite Element Formulation
The finite element model of the beam is shown in Fig.1, c.
The finite element formulation to obtain stiffness and mass
matrices of beam is derived using the Lagrange Equation.
Transverse displacement of beam in terms of nodal
displacement is defined as 𝑤(𝑥, 𝑡) = 𝑁(𝑥)𝑞(𝑡). Nodal
displacements 𝑞(𝑡) are deflections and rotations for both end
of one-dimensional finite element that can be find in finite
element methods books. N(x) is the shape function matrix of
one dimensional flexural element. As represented in figure,
beam is subdivided to Ne that is the number of finite elements;
L is the length of ith finite element.
𝑁(𝑥) = [ 1 − 3𝜉 2 + 2𝜉 3 𝐿(𝜉 3 − 2𝜉 2 + 𝜉) 3𝜉 2 − 2𝜉 3 𝐿(𝜉 3 − 𝜉 2 )]
Lagrange Equation:
𝜕 𝜕𝒯
𝜕𝒯 𝜕𝒰 𝜕𝒲𝑐
( )−
+
+
=𝐹
𝜕𝑡 𝜕𝑑̇
𝜕𝑑 𝜕𝑑
𝜕𝑑
𝑁𝑒
𝐿
𝜕 𝜕𝒯
( ) = ∑ ∫ 𝜌𝐴[𝑁 𝑇 𝑁] 𝒒̈ 𝑑𝑥
𝜕𝑡 𝜕𝑑̇
𝑖=1 0
𝑁𝑒
𝐿
𝜕𝒰
𝑇
= ∑ ∫ 𝐸𝐼[𝑁 ′′ 𝑁 ′′ ] 𝒒𝑑𝑥
𝜕𝑑
𝑖=1 0
𝑁𝑒
𝐿
𝜕𝒲𝑐
𝑃 𝑥 ′𝑇 ′
= ∑∫
[𝑁 𝑁 ] 𝒒𝑑𝑥
𝜕𝑑
𝑙
𝑖=1 0
1
Here we can assume 𝑥 = (𝜉 + 𝑖 − 1) that mentioned in
𝐿
[2].
When all these terms replace in the Lagrange equation, in
general form, below equation of motion for free vibration is
obtained;
[𝑀𝐺 ]𝒒̈ + [𝐾𝐺 ]𝒒 = 𝟎
Fig. 1. Structural model of a rocket body, deflections, finite element
model
(∗)
Here, [𝑀𝐺 ] is general mass matrix, [𝐾𝐺 ] is general stiffness
matrix that consists of [𝐾𝐸 ] is bending stiffness matrix and
also [𝐾𝑞1 ], [𝐾𝑞2 ], [𝐾𝑞3 ] are stiffness matrices resulting from
follower force, Q.
1
𝑴𝑮 = ∫ 𝜌𝐴 [𝑁(𝜉)𝑇 𝑁(𝜉)] 𝑑𝜉
0
𝑲𝑮 = 𝑲𝑬 + 𝑲𝒒𝟏 + 𝑲𝒒𝟐 + 𝑲𝒒𝟑
1
𝑇
𝑲𝑬 = ∫ 𝐸𝐼 [𝑁(𝜉)′′ 𝑁(𝜉)′′ ]𝑑𝜉
0
1
𝑇
𝑲𝒒𝟏 = ∫ 𝑄 𝜉[𝑁(𝜉)′ 𝑁(𝜉)′ ] 𝑑𝜉
0
1
the thrust value increases, natural frequency of the beam
changes. Two lowest non-zero eigenvalues for different values
of the non-dimensional thrust are demonstrated in Table-1 and
also Fig.2. As seen in figure, while increasing thrust, both
non-dimensional natural frequencies get close to each other.
When 𝑄 = 11.1138 𝜋 2 , both natural frequencies coincide in
one mode. For this mode, thrust gets the critical value, 𝑄𝑐𝑟 .
For critical thrust, the non-dimensional coupled modes merge
about 23.10.
In addition, free vibration modes for increasing thrust are
given in Table-1 comparatively with [6].
𝑇
𝑲𝒒𝟐 = ∫ 𝑄 (𝑖 − 1)[𝑁(𝜉)′ 𝑁(𝜉)′ ] 𝑑𝜉
0
1
𝑇
𝑲𝒒𝟑 = ∫ 𝑄 [𝑁(𝜉)′ 𝑁(𝜉)′ ] 𝑑𝜉
0
Natural frequencies of the system can be determined by
transforming the equation (*) to generalized eigenvalue
problem. The deflection of the beam as mentioned before is in
the form of 𝑤 = 𝑤(𝑥) 𝑒 𝑖𝜔𝑡 . Due to the fact that this
formulation, eigenvalues of the system 𝜆 are complex number
in general, which are consist of real and imaginary parts in the
form of;
𝜆 = 𝜆𝑅 + 𝑖𝜆𝐼
The stability problem of the non-conservative systems
considering in this paper, in the case of vehicle is subjected to
follower force specially is discussed in [3, 22]. As mentioned
in references, characteristics of the eigenvalues are specified
the attitude of the deflection w(x). The system is recognized
stable when the real part of 𝜆 is zero or negative. In the other
case, w(x) increases with time and divergence or flutter
instability can begins according to the characteristics of the 𝜆𝐼 .
If the 𝜆𝐼 = 0, the system is unstable and divergence occurs.
Also, if the 𝜆𝐼 ≠ 0, the system is unstable and flutter occurs.
When both real and imaginary parts of 𝜆 are zero, the system
behaves like a rigid body.
The eigenvalues 𝜆 of the system will be discussed on the
following section.
III. RESULTS
A. Vibration of a uniform Rocket
Structural stability has the great importance for aerospace
vehicle design. Any changes in stability characteristics of the
system can cause the changes of design, mission or trajectory.
In this paper, vibration of a slender aerospace vehicle under
constant thrust has been carried out. Rocket/missile type
vehicles are modeled as a homogeneous free-free beam under
follower force and its equation of motion and natural
frequencies are obtained. Vibration characteristics of the
vehicle are discussed and verified with some paper in
literature.
Natural frequencies of the rocket body according to
constant thrust are obtained by considering the eigenvalues of
the equation of motion (*). Free vibration of the beam that has
free ends in the case of thrust is zero (𝑄 = 0), is found. When
Fig. 2. First two non-dimensional natural frequencies of rocket by
increasing thrust
TABLE I.
TWO LOWEST NATURAL FREQUENCIES OF ROCKET UNDER
CONSTANT THRUST
Q/𝝅𝟐
…
0
1
2
5
Mode1
22.37
20.99
19.49
15.11
23.08
Mode2
61.67
59.79
57.79
51.26
23.08
Present Mode1
work
Mode2
22.37
20.99
19.57
15.12
23.10
61.67
59.79
57.83
51.27
23.10
Ref-6
11.11
B. Two-stage Rocket
There can be different number of stages in rockets and
missiles according to mission. Stages can have different
physical and geometrical properties because of parts of
warheads, fuselage, tail and fins, and ammunition.
Accordingly, beams that modeled structurally can have
various mass, cross-sectional areas and bending stiffness. Here
we can assume that rockets have variable bending stiffness
𝐸𝐼(𝑥) and variable mass per unit length 𝜌𝐴(𝑥) for different
stages. In this section of the work, natural frequencies of these
types of rocket are discussed. Here, a two-staged body sketch
is demonstrated in Fig.3.
[2]
[3]
[4]
[5]
Fig. 3. Structural model of a two-stage body
[6]
We use the example-1 in [20] to verify our finite element
code for non-homogeneous beam. In our example there are
two parts in body and its dimensions and material properties
that given in Table.2. The first three natural frequencies of this
body are represented in Table.3 to confirm solutions with the
reference. In this regard, dynamic and aeroelastic instability of
vehicle can be solved numerically notwithstanding the
physical properties of the rocket.
TABLE II.
L
EI
𝝆𝑨
[7]
[8]
[9]
MATERIAL PROPERTIES OF PARTS
Part-1
254
1049.00
1.37
[10]
Part-2
140
25.11
0.39
[11]
[12]
TABLE III.
Ref-20
Present
work
NATURAL FREQUENCIES
Mode1
292.44
Mode2
1181.3
Mode3
1804.1
292.42
1181.3
1804.1
IV. CONCLUSION
Numerical results in order to specify rocket/missile
stability characteristics under thrust force are discussed in this
paper. Here, we can say that stability characteristics differ by
variation in the physical condition or design of rocket. When
thrust, mass or rigidity of rocket changes, stability of the
system changes. Primarily, the aim of the study is to determine
the effect of thrust force on rocket stability. The results for our
model obviously indicate that, when thrust increase natural
frequencies reduce. Critical thrust value where the flutter
occurs is stated and the system is defined as instable. Besides,
for the further work, the effects of variations in aerodynamic
condition on rocket instability will be discussed. Additionally,
vibration characteristics for multi-stage rocket without thrust
are examined to deal with the complexity of the system.
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