Static Aeroelasticity Analysis of Spinning Rocket for Divergence Speed -- Zeus Numerix

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Symposium on Applied Aerodynamics and Design of Aerospace Vehicles (SAROD-2018)
STATIC AEROELASTICITY ANALYSIS OF SPINNIG
ROCKET FOR DIVERGENCE SPEED
Sanjay Kumar1
, Subhas Mukane2
, P T Rojatkar3
and Gopal Shevare4
1
Zeus Numerix Pvt. Ltd., Pune, India
2&3 Armament Research & Development Establishment, Pune, India
4 Department of Aerospace Engineering, IIT-Bombay
Abstract: The performance of rockets in terms of speed, range, high maneuverability
and accuracy hitting the target is increasing day-by-day. However, this higher
performance is coming at a cost. The forces on the rocket are becoming larger and
more stringent analysis for structural integrity is becoming mandatory. Aero-elastic
analysis, normally considered less important for rockets, has now become an essential
requirement.
Rockets with WAF (wrap around fins) that spin during the flight are a popular
configuration as they offer an advantage of improved accuracy if there is a thrust
misalignment or mass imbalance. Thus, investigation of aero-elastic phenomena for
such rockets requires inclusion of spin in the mathematical model for better estimation
of divergence dynamic pressure.
This paper reports results from an updated methodology which models spin. Like the
existing currently practiced methodology, it uses equation of motion converted into
generalized coordinates in terms of free vibration mode shapes. RANS CFD is used
for finding out aerodynamic load distribution along the length of rocket. The
developed model is first validated against currently practiced model with no spin, then
the results are obtained with spinning motion modeled. The results from the proposed
methodology match very well with the results obtained from the existing methodology
(when spin is not modeled). The proposed methodology with spin modeled, shows
drastic reduction in divergence dynamic pressure/velocity.
The methodology is being used as a software application by designers of rockets for
predicting divergence dynamic pressure.
Keywords: Divergence, Aero-elasticity, Spinning Rocket

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1 Introduction to Aero-elasticity
Aero-elastic effects have significant influence during the design and flight performance of
flight vehicles. Since aero-elastic phenomena can lead to structural failure, structural
designs have to be heavier (the so-called aero-elastic penalty) in order to ensure structural
integrity. At worst, time consuming repeated aerodynamic shaping, changes to topology of
structural components and redistributions mass of equipments may be required. The root
cause for aero-elastic problems is the flexibility of structure. Since flexible structures
change their geometry and aerodynamic loads depend on geometry, any calculation of
deflections / stresses and aerodynamic loads in flexible structures requires a solution of
coupled structural and fluid dynamics or the solution of fluid-structure interaction or aero-
elasticity.
The current methodologies for solution of dynamic FSI problem (e.g. flutter) are highly
advanced [1]. They deploy high fidelity computational fluid dynamics simulations which
use arbitrary Lagrangian - Eulerian framework for coupling codes to computational
structural mechanics codes. Strategies for geometric conservation law, time advancement
schemes and wetted surface interface are used. To take care of vastly different time scales
of solids and fluid motions, reduced order models for unsteady aerodynamic load transfer
are employed [2]. When unsteady aerodynamic loads acting as forcing function leads to
increase in the amplitude of periodic motion of structure the structure is considered as
unstable.
In static instability (divergence) when the aerodynamic load (such as moment)
corresponding to free stream dynamic pressure equals the structural load (restoring
moment) a static instability is set. This dynamic pressure is called divergence dynamic
pressure. In principal, successive application of steady state CFD solution and static FEA
problem solves a static aero-elasticity problem. However, under the assumptions of linearity
a couple of CFD simulations and a simplified structural model are adequate to ascertain the
aero-elastic static stability of structure under the given dynamic pressure.
There is a plenty of literature on static aero-elasticity focusing on divergence velocity and
factor of safety estimation of rockets. Several studies can be found where divergence
velocity of lifting surfaces (like wing torsional divergence) or rocket and missile divergence
(missile bending divergence) have been analyzed [3, 4].
However, divergence speed of a spinning rocket which takes into account centrifugal force
due to spinning of rockets is rare. This paper gives a methodology to model spin of rockets
and the results of this new model on divergence speed.
1.1 Rocket with WAF fins
The free flight unguided rockets with WAF (Wrap around fins) that spin during the flight
are a popular rocket configuration. The rocket is spin stabilised due to gyroscopic effect
and that have an advantage of offering improved accuracy balancing deteriorating effects

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of thrust misalignment or mass imbalance. Unfortunately, this advantage is negated by
generation of lateral aerodynamic forces and moments at an angle of attack [7]. The spin
also brings in additional aerodynamic forces and moments, which in turn produces inertial
loads, additional bending of the central body, amplifying mass imbalance and finally
causing aero-elastic problems such as divergence or flutter.
Divergence is characterised by combined structural plus aerodynamic stiffness of central
body and lifting surface becoming zero. The occurrence of divergence not only affects
maximum dynamic head, but also affects the rigid body modes and their coupling, leading
to departure from intended flight trajectory and loss of accuracy.
Fig. 1. Rocket with WAF fins
2 Formulation of Static Aero-elasticity
Static aero-elasticity considers the non-oscillatory effects of aerodynamic loads acting on
the flexible vehicle. The flexible body deflects due to air loads and the air loads (normal
force) deform the structure further. This bi-directional coupling may lead to sever
deflections. In the case of rockets, static aero-elasticity of the structure may not lead to
failure but it may lead coupling of rigid body modes, which are equally detrimental and
unforgiving as much as structural failure [3,4].
Any aero-elastic phenomena can be considered as a dynamic system and modeled for
motion variable u by “aero-elastic equation of motion”:
𝑀𝑢̈ + 𝐶𝑢̇ + 𝐾𝑢 = 𝐹(𝑢, 𝑡) (1)
Where, the forcing function F represents aerodynamic loads as function of motion of
structure. M, C and K are mass, damping and stiffness of the structural system.
Ignoring damping terms and also the aerodynamic forces as function of motion velocity and
acceleration, the Equation of motion (1) becomes,
𝑀𝑢̈ + 𝐾𝑢 = 𝐹(𝑢, 𝑡) = 𝐺1(𝑢) + 𝐹2(𝑡) (2)
Since G1(u) is function of displacement, Equation (2) can rewritten as
𝑀𝑢̈ + (𝐾 − 𝐺1)𝑢 = 𝐹2(𝑡) (3)

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Note that this is similar structural dynamic problem, except that the new “stiffness” also
includes aerodynamic stiffness.
2.1 Numerical simulation method
For static aero-elasticity analysis mode summation method is used where equation of
motion is converted into normal coordinate using the free vibration mode shapes. The modal
equations are uncoupled equations that are much easier to solve than original dynamic
coupled equation. In mode summation method, only the first few mode shapes are used for
constructing the modal matrix that reduces the size of the matrix drastically. Only 10% of
the total mode shapes (Number of mode shapes are equal to the number of degrees of
freedom of the system) used for constructing the matrix. Modal equations are best suited
for problems where higher modes are not important like in the case of static aero-elasticity
[6].
The displacement variable ‘u’ is approximated as,
𝑢(𝑥, 𝑦, 𝑧, 𝑡) = [∅][𝑞] (4)
Here [] is modal matrix and {q} is principle coordinate matrix.
The Equation (3) can be converted into modal equation as follow
Mqq̈ + (Kq - Gq)q= Fq (5)
Where, Mq = T M , Kq = T K , Gq = T G  and Fq = T F
Mq, Kq , G and Fq are modal mass, modal stiffness, aerodynamic stiffness and modal
force respectively.
Modes shapes are orthogonal (independent of each other modes) and hence
∅i
T
M∅j = [
0, i ≠ j
1, i = j
] and ∅i
T
K∅j = [
0, i ≠ j
ω2
, i = j
]
Due to orthogonality, the mass matrix and stiffness matrix is a diagonal matrix. The
procedure converts aero-elastic motion equation into n modal equations. Each modal
equation represents motion of one mode. Note that only the first few modes are used as
higher modes do not contribute to energy in the motion. The equation of motion is
decoupled and each equation can be solved independently as a single degree of freedom
and sum of all the resulting solutions is the solution of the original Equation (3).

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2.2 FEA Modelling of Rocket
The rocket is divided into ‘N’ number of beam element connected at the node as shown in
Figure 2. The aerodynamic normal force distribution is estimated from CFD for given
extreme flight condition. The Mass and Stiffness matrix (M & K matrix) for beam element
is estimated using the assumed shape function that is standard [6]. Each node has two degree
of freedom that is one rotation and one translation as shown in Figure 3. The natural
frequency (λ) and mode shapes (Φ) of the rocket are obtained using the eigenvalue and
eigenvector of K and M matrix.
Fig. 2. Discretization of rocket for FEA
Fig. 3. Modeling of beam element
2.3 Stiffness model due to aerodynamic loads
Aerodynamic load (normal force) depends on dynamic pressure (Q= ½ V∞
2
), rocket
geometry and the angle of attack. The aerodynamic load of the rocket is given by:
𝐺1_𝑎𝑒𝑟𝑜𝑑𝑦𝑛𝑎𝑚𝑖𝑐(𝑥) = 𝐶 𝑁𝛼 𝛼 𝑄 𝑆 (6)

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Where CNα is the normal coefficient and α is angle of attack at location ‘x’ of the rocket.
CFD study is done for maximum dynamic pressure (i.e. at all-burnt-point) and pressure
distribution is obtained (Figure 4). The rocket is divided into ‘n’ section (200 strips) and for
each strip pressure on the surface is integrated to obtain normal force coefficient (𝐶 𝑁𝛼) of
each strip. The (𝐶 𝑁𝛼) is divided by length of each strip to obtain the normal force coefficient
per unit length of each strip (Figure 5). Assuming linear variation of CN with , CN can be
calculated for each strip/element at all dynamic pressure and for any deflection. Hence for
flexible vehicle, the deformation will change local angle of attack of the element and using
these coefficients the new load on the strip can be estimated.
Fig. 4. CFD simulation pressure distribution (scaled) (Mach 3.65, AOA 3deg, Alt = 2.3Km)
Fig. 5. Normal Force coefficient per unit length distribution along rocket length

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The aerodynamic load for each strip (G1_aerodyanmic (u), where u = α = local slope due to
deflection.) can be estimated using the following method. For a beam element with two
degrees of freedom at each node (Figure6), the transformation of distributed load to nodal
forces and moments of each element can be done as following equations.
Fig. 2. Transformation of distributed force to nodal force
𝐺1_𝑎𝑒𝑟𝑜𝑑𝑦𝑎𝑛𝑚𝑖𝑐
(𝑒)
=
[
𝑓1
(𝑒)
𝑚1
(𝑒)
𝑓𝑒
(2)
𝑚2
(𝑒)
]
=
[
𝑄𝑆𝐶 𝑁𝛼 (
𝐿
2
) (𝛼1 + 𝛼2)
𝐿2
12
) (𝛼1 + 𝛼2)
𝐿
2
) (𝛼1 + 𝛼2)
−𝑄𝑆𝐶 𝑁𝛼 (
𝐿2
12
) (𝛼1 + 𝛼2)]
(7)
Where α is the local angle of attack for deformed element (α1 and α2 are given by α1 = (u1-
u2)/L and α2 = (1-2)/L, where u is linear displacement and θ is angular displacement of
the nodes. Superscript indicates that the node numbers are local. The above equation is
used to represent G1 as a function of displacement as required in the modal equations: From
Equation (6) , G1_aerodynamic = Q S [CNα][u] = Q S [CNα] Φq
𝐺1_𝑎𝑒𝑟𝑜𝑑𝑦𝑛𝑎𝑚𝑖𝑐
(𝑒)
=
[
𝑓1
(𝑒)
𝑚1
(𝑒)
𝑓2
(𝑒)
𝑚2
(𝑒)
]
= 𝑄𝑆𝐶 𝑁∝
[
−1/2 −𝐿/2 1/2 𝐿/2
−𝐿/12
−1/2
𝐿/2
−𝐿2
/12 𝐿/12 𝐿2
/12
−𝐿/2 1/2 𝐿/2
𝐿2
/12 −𝐿/2 −𝐿2
/12]
[
𝑢1
𝜃1
𝑢2
𝜃2
] (8)
Where f and m are forces and moments at the nodes and u and  are the displacements and
rotations at the nodes. The matrix is estimated for each element and assembled for aero-
elastic analysis.

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2.4 Stiffness due to spinning of rocket
A deflected rocket spinning about its axis produces centrifugal force if mass distribution is
not evenly distributed [5]. This force is additional to aerodynamic force and it will lead to
early divergence. This centrifugal force (Fc) can be modeled as
Fci=mi ω2
δri, (9)
where, mi is mass of each element, ω is spin rate and δri is the radial distance between spin
axis and deflection of the ith
element.
Fig. 7. Centrifugal force application diagram
The spinning of rocket is provided by giving a cant angle to the fins. The spin rate is a
function of cant angle; as the free stream velocity increase the spin increases. At equilibrium
the spin is such that the net angle of attack of the fin is zero.
axismissileandfin
betweendistanceR
anglecant
raterollwhere
RV



 


 /)tan(
Fig. 8. Modeling of rocket spin
The centrifugal force matrix is estimated based on velocity, cant angle and deflection as
shown below (Equation 10). Note that in the matrix formulation, density is not playing a
separate role, because in aerodynamic matrix density is present in dynamic pressure. The
density is known at the given altitude.

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Here, M is the mass of element, δ is the cant angle of fin, V is free stream velocity, R is
the distance of centroid of the fin from the rocket axis.
2.5 Estimation of divergence velocity
Since divergence is a static phenomenon, in Equation (5), (ФT
M Ф 𝑞)̈ term can be set to
zero and analysis condition at which aerodynamic stiffness equals structural stiffness, i. e.
find out V at which the following expression becomes zero.
[∅] 𝑇
([𝐾] − (𝑉2
)[𝐺 𝑎𝑒𝑟𝑜 + 𝐹𝑐𝑒𝑛𝑡𝑟𝑖𝑓𝑢𝑔𝑎𝑙])[∅]𝑞 = 0 (11)
3 Results and Discussions
3.1 Divergence study and validation for non-spinning rocket
The above methodology is validated by comparing results for a WAF rocket with existing
methodology practiced in ARDE (which does not incorporate spin effects). The extreme
flight condition of the rocket is at maximum dynamic pressure, i.e. at all-burn-point. The
altitude is 2.3km, Mach no 3.65 and AOA of 3 deg. The dynamic pressure is 714886 Pa,
rocket mass is 171.3 kg and total normal force is 7520 N. The mass, flexural stiffness (EI)
and normal force coefficients are shown in Figure 9 and 10.
Fig. 9. Rocket flexural stiffness (EI) and Mass Distribution (upper figure - as calculated as per
existing practice, the lower figure – as found out from proposed methodology)

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Fig. 10. Comparison of linear displacement (upper figure - as calculated as per existing practice,
the lower figure – as found out from proposed methodology)
Fig. 11. Comparison of angular displacement

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The results from the proposed model are shown in Table-1 and Figures 10 and 11. The
comparison shows a good agreement with the data as obtained from the currently practiced
methodology.
This encourages usage of the proposed methodology for predicting divergence for a
spinning rocket.
Table 1. Comparison of divergence dynamic pressure
Parameter compared Existing methodology Proposed methodology
Max dynamic pressure (Pa) 7.14 x 105
7.14x 105
Divergence dynamic pressure (Pa) 1.67 x 107
1.74 x 107
Factor for divergence 23.4 24.4
3.2 Divergence study of rocket with spinning
The divergence study of spinning rocket is carried out for different fin cant angles. Note
that different cant angle produces different spin rates rocket for the same free stream
velocity.
The input and flight condition are same as the discussed above except the cant angle values
(50 min and 30 min.)
The results given in Table 2 shows that the spin has a significant effect on the divergence
velocity and dynamic pressure. The factor of safety is reduced from 24.4 for non-spinning
to 2.4 for spinning rocket. The deflection plot shows that the spinning has higher deflection
at the nose and tail due to higher mass distribution at these locations.
Table 2. Effect of spin on divergence dynamic pressure
Parameters (Without Spin)
With Spin
Cant angle
30 min
Cant angle
50 min
Spin @ all-burn-point (rpm) NA 570 1119
Velocity @ all-burn-point 1277 1277 1277
Mach @all-burn-point 3.65 3.65 3.65
Divergence Velocity (m/s) 8908 2908 1852
Mach @ divergence 25.8 8.8 5.6
Divergence dynamic pressure (Pa) 4.18 x 10
7
4.1 x 10
6 1.7 x 10
6
RPM @ Divergence NA 1615 1714
Factor of Safety (Div. Pressure) 24.4 5.8 2.4

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Fig. 12. Linear displacement for with and without spinning rocket
Fig. 13. Angular displacement for with and without spinning rocket
4 Conclusions
Static aeroelastic study of a spinning rocket is carried out modeling spinning motion of the
rocket probably for the first time. As expected, with spin modeled, the lateral deflections
of rocket is higher, which could be detrimental for divergence.
The study shows that divergence dynamic pressure drastically reduces if spinning motion
is modeled and the factor of safety is reduced from 24.4 to 2.4.

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Interaction, SANDIA REPORT, SAND2005-6573”
2. T. Lieu, C. Farhat, M. Lesoinne: Reduced-order fluid/structure modeling of a complete aircraft
conﬁguration, Comput. Methods Appl. Mech. Engrg. 195 (2006) 5730–574
3. Jan R.Wright, Jonathan E. Cooper: Introduction to aircraft aeroelasticity and loads, John Wiley
& Sons, Ltd, 2007
4. D. H. Hodges, G. A. Pierce: Introduction to structural dynamics and aeroelasticity, Cambridge
university, 2011
5. Livshits D. S., Yaniv S. And Karpel M.: Coupled aeroelasticity analysis of a free flight rocket,
IMI, Israel
6. Yijun Liu, “Introduction to finite element method: CAE Research Laboratory, University of
Cincinnati, USA, 2003
7. Ameer G. Mikhail, “Roll Damping for projectiles including wraparound, offset, and arbitrary
number of fins: Journal of Spacecraft & Rockets, Vol. 32 , No 6, pp 929-937, November-
December. 1995.
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Engineering Series, University of Minnesota, 1997

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