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Vibrational Stability Effects in Rotorcraft Flight Dynamics
Umberto Saetti
Assistant Professor
Department of Aerospace Engineering
University of Maryland
College Park, MD 20740
Zhouzhou Chen
Graduate Research Assistant
Joseph F. Horn
Professor
Department of Aerospace Engineering
Pennsylvania State University
University Park, PA 16802
Tom Berger
Flight Controls Group Lead
U.S. Army Combat Capabilities Development
Command Aviation & Missile
Moffett Field, CA 94035
ABSTRACT
This paper investigates vibrational stabilization effects in rotorcraft flight dynamics. This study is motivated
by the fact that eigenvalues of the rotorcraft flight dynamics identified from flight test often differ from those
computed with physics-based simulations, and that some commonly observed mismatches may be ascribed to
vibrational stability effects due to rotor blade imbalance or other periodic disturbance on the rotorcraft. Start-
ing from a simple example involving an inverted pendulum, the paper demonstrates the use of the harmonic
decomposition method for the study of vibrational stabilization effects. The concept is then extended to analyze
the effect of blade imbalance on the flight dynamics of a helicopter. Additionally, vibrational stablization of a
slung load in forward flight is investigated using small-amplitude and disturbances on an active cargo hook.
Results show that while vibrations induced by rotor blade imbalance do not stabilize the hovering dynamics of
a helicopter, these vibrations still have a significant effect on the hovering dynamics. Rotor blade imbalance
results in a symmetric effect on the roll and pitch axes, in that it tends to decrease the frequency of the sub-
sidence modes of the hovering cubic, while the unstable oscillatory modes tend to increase in frequency and
decrease in damping. On the other hand, the yaw/heave dynamics are relatively unaffected compared to the
lateral and longitudinal axes. Moreover, small-amplitude oscillations of an active cargo hook were shown to
significantly decrease the amplitude of the limit cycle oscillation of a slung load in forward flight and to stabilize
its dynamics. This constitutes a practical solution to the semi-active control of a suspended load.
INTRODUCTION
Vibrational stabilization is a phenomena where the unstable
dynamics of a system about an equilibrium point of interest
can be stabilized via periodic forcing of the system at a high-
enough forcing frequency (Refs. 1,2). Examples of this phe-
nomena include, but are not limited to, those involving an in-
verted pendulum with vibrating suspension point (Ref. 3) and
hovering insects where their hovering cubic (Ref. 4) is stabi-
lized by the wing periodic flapping motion (Ref. 5, 6). In the
latter case, the wing flapping motion inducing a vibrational
stabilization mechanism that increases the pitch damping and
stiffness while reducing the speed stability. This results in
stabilization of the pitch oscillatory mode and thus of the lon-
gitudinal hovering cubic.
Presented at the 49th European Rotorcraft Forum, B¨
uckeburg, Ger-
many, September 5–7, 2023.
Distribution Statement A. Approved for public release; distribution
is unlimited.
Despite apparent differences in shape and dimension, the dy-
namics of flapping-wing flyers is indeed mathematically sim-
ilar to those of rotary-wing vehicles such as helicopters. This
stems from the time-varying aerodynamic and inertial loads
due to the wing/blade periodic motion. In fact, the dynam-
ics of both flapping- and rotary-wing flight is described by
nonlinear time-periodic (NLTP) systems of the coupled rigid-
body and complex interactional aerodynamics between the
wings/blades, body, and the self-induced wake. However, the
relative importance of the time-periodic dynamics (i.e., those
dynamics with natural frequencies that are multiples of the
fundamental frequency of the system) and the overall dynam-
ics of the system (i.e., averaged + time-periodic dynamics) is
significantly higher for flapping-wing flyers than it is for heli-
copters – approximately 7% for helicopters (Ref. 7) and up to
50% for flapping-wing flyers (Ref. 8, 9).
In light of these similarities, vibrational stabilization mech-
anisms that are observed for flapping-wing flight might also
extend to rotary-wing vehicles. One indication of this comes
1
from anecdotal evidence suggesting that eigenvalues of the
rotorcraft flight dynamics identified from flight test often dif-
fer from those computed with physics-based simulations. In
fact, these commonly observed mismatches may be ascribed
to vibrational stability effects due to rotor blade imbalance or
other periodic disturbance on the rotorcraft. Another instance
of vibrational stabilization comes from from slung load car-
riage, where previous research has shown that slung loads can
exhibit better stability /lower amplitude oscillations when sub-
jected to periodic disturbances at the hook (Ref. 10). As such,
the objective of this paper is to investigate vibrational stabi-
lization effects on the flight dynamics of rotorcraft.
One major challenge in the analysis of vibrational stabiliza-
tion lies in the stability analysis of NLTP systems. While the
stability analysis of nonlinear time-invariant (NLTI) systems
is readily assessed by linearizing the dynamics about an equi-
librium point and performing eigenanalysis, or by means of
Lyapounov theory, the stability analysis of NLTP systems is
typically a more challenging task. This is because the equi-
librium solution of NLTP systems is typically represented by
a periodic orbit rather than by a fixed point. Two main ap-
proaches exist for determining the stability of NLTP systems
(Fig. 1), as articulated in Ref. 11: the first is based on Floquet
theory (Refs. 12–16) and the is second based on averaging
methods (Refs. 11, 17–25). The first approach is articulated
in the following four major steps: (i) a periodic orbit is found
by solving the dynamic equations, (ii) the dynamic equations
are linearized about that periodic orbit to yield a linear time-
periodic (LTP) system, (iii) the LTP system is transformed
into a linear time-invariant (LTI) system via Floquet transfor-
mation/decomposition, and (iv) stability is assessed by check-
ing the eigenvalues of the LTI system. The second approach
to determine stability of NLTP systems leverages averaging
methods to transform the NLTP system into an equivalent
NLTI system where the periodic orbit of the original system
collapses to a single point in the state space. Stability is thus
assessed by linearizing the equivalent NLTI system about its
equilibrium point and by performing spectral analysis (Ref.
17).
Historically, the only methods available for transforming the
LTP dynamics into approximate higher-order LTI systems
were the Lyapounov-Floquet method (Ref. 26) and frequency
lifting methods (Ref. 27), which both suffered from the
common disadvantage of the need for state transition ma-
trices. State transition matrices constitute a particular chal-
lenge in that their computation can be numerically intensive
and/or very sensitive to tuning parameters which require prior
knowledge on the system’s dynamics. However, this limi-
tation was recently relaxed by the harmonic decomposition
method that originated from the rotorcraft community (Refs.
28–30). Within the context of rotorcraft, harmonic decom-
position models have been used to: (i) study the interference
effects between higher-harmonic control (HHC) and the air-
craft flight control system (AFCS) (Refs. 28, 31–33); (ii) de-
sign load alleviation control (LAC) laws (the PI’s efforts in
Refs. 34–36); and (iii) prediction and avoidance of flight en-
velope limits (Refs. 36–38). A survey by the PI on the use
Fig. 1: Illustration of the two main approaches to stability
analysis of NLTP systems: averaging methods (left), Floquet
theory and harmonic decomposition (right).
of harmonic decomposition models in the rotorcraft field can
be found in Ref. 39. When coupled with a harmonic balance
scheme, harmonic decomposition can also be used to solve for
(stable and unstable) periodic solutions (Ref. 40) and com-
pute open-loop higher-harmonic control (HHC) inputs that at-
tenuate arbitrary state/output harmonics (Ref. 9). Because
harmonic decomposition (i) relaxes all limitations associated
with the Floquet-based approach and (ii) can be used to com-
pute unstable periodic orbits (note that the unaugmented hover
dynamics of rotorcraft is unstable, see Ref. 4), it is used to an-
alyze vibrational stability of rotorcraft in this study.
The paper starts from a simple example involving an inverted
pendulum to demonstrate the use of the harmonic decompo-
sition method (Ref. 28, 29) for the study of vibrational sta-
bilization effects. The concept is then extended to analyze
the effect of blade imbalance on the flight dynamics of a he-
licopter, as well as vibrational stablization of a slung load in
forward flight using small-amplitude and disturbances on an
active cargo hook. Final remarks summarize the overall find-
ings of the study and future developments are identified.
METHODOLOGY
Mathematical Background
Consider a nonlinear time-periodic (NLTP) system in first-
order form representative of the rotorcraft flight dynamics:
˙
x
x
x=f
f
f(x
x
x,u
u
u,t)(1a)
y
y
y=g
g
g(x
x
x,u
u
u,t)(1b)
where x
x
x∈Rnis the state vector, u
u
u∈Rmis the control in-
put vector, y
y
y∈Rlis the output vector, and tis the dimen-
sional time in seconds. The nonlinear functions f
f
fand g
g
gare
2
T-periodic in time such that:
f
f
f(x
x
x,u
u
u,t) = f
f
f(x
x
x,u
u
u,t+T)(2a)
g
g
g(x
x
x,u
u
u,t) = g
g
g(x
x
x,u
u
u,t+T)(2b)
Note that the fundamental period of the system is T=2π
Ωsec-
onds, where Ωis the angular speed of the main rotor in rad/s.
Let x
x
x∗(t)and u
u
u∗(t)represent a periodic solution of the system
such that x
x
x∗(t) = x
x
x∗(t+T)and u
u
u∗(t) = u
u
u∗(t+T). Then, the
NLTP system can be linearized about the periodic solution.
Consider the case of small disturbances:
x
x
x=x
x
x∗+∆x
∆x
∆x(3a)
u
u
u=u
u
u∗+∆u
∆u
∆u(3b)
where ∆x
∆x
∆xand ∆u
∆u
∆uare the state and control perturbation vectors
from the candidate periodic solution. A Taylor series expan-
sion is performed on the state derivative and output vectors.
Neglecting terms higher than first order results in the follow-
ing equations:
f
f
f(x
x
x∗+∆x
∆x
∆x,u
u
u∗+∆u
∆u
∆u,t) = f
f
f(x
x
x∗,u
u
u∗,t) + F
F
F(t)∆x
∆x
∆x+G
G
G(t)∆u
∆u
∆u
(4a)
g
g
g(x
x
x∗+∆x
∆x
∆x,u
u
u∗+∆u
∆u
∆u,t) = g
g
g(x
x
x∗,u
u
u∗,t) + P
P
P(t)∆x
∆x
∆x+Q
Q
Q(t)∆u
∆u
∆u
(4b)
where:
F
F
F(t) = ∂f
f
f(x
x
x,u
u
u)
∂x
x
xx
x
x∗,u
u
u∗,G
G
G(t) = ∂f
f
f(x
x
x,u
u
u)
∂u
u
ux
x
x∗,u
u
u∗(5a-b)
P
P
P(t) = ∂g
g
g(x
x
x,u
u
u)
∂x
x
xx
x
x∗,u
u
u∗,Q
Q
Q(t) = ∂g
g
g(x
x
x,u
u
u)
∂u
u
ux
x
x∗,u
u
u∗(5c-d)
Note that the state-space matrices in Eq. (5) have T-periodic
coefficients such that:
F
F
F(t) = F
F
F(t+T),G
G
G(t) = G
G
G(t+T)(6a-b)
P
P
P(t) = P
P
P(t+T),Q
Q
Q(t) = Q
Q
Q(t+T)(6c-d)
Equations (4a) and (4b) yield a linear time-periodic (LTP) ap-
proximation of the NLTP system of Eq. (1) as follows:
∆
∆
∆˙
x
x
x=F
F
F(t)∆x
∆x
∆x+G
G
G(t)∆u
∆u
∆u(7a)
∆
∆
∆y
y
y=P
P
P(t)∆x
∆x
∆x+Q
Q
Q(t)∆u
∆u
∆u(7b)
Hereafter, the notation is simplified by dropping the ∆in
front of the linearized perturbation state and control vectors
while keeping in mind that these vectors represent perturba-
tions from a periodic equilibrium. Next, the state, input, and
output vectors of the LTP systems are decomposed into a finite
number of harmonics of the fundamental period via Fourier
analysis:
x
x
x=x
x
x0+
N
∑
i=1
x
x
xic cos2πit
T+x
x
xis sin2πit
T(8a)
u
u
u=u
u
u0+
M
∑
j=1
u
u
ujc cos 2πjt
T+u
u
ujs sin 2πjt
T(8b)
y
y
y=y
y
y0+
L
∑
k=1
y
y
ykc cos 2πkt
T+y
y
yks sin 2πkt
T(8c)
As shown in Ref. 28, the harmonic decomposition method-
ology can be used to transform the LTP model into an ap-
proximate higher-order linear time-invariant (LTI) model in
first-order form:
˙
X
X
X=A
A
AX
X
X+B
B
BU
U
U(9a)
Y
Y
Y=C
C
CX
X
X+D
D
DU
U
U(9b)
where the augmented state, control, and output vectors X
X
X∈
Rn(2N+1),U
U
U∈Rm(2M+1), and Y
Y
Y∈Rl(2L+1), respectively, are
given by:
X
X
XT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns (10a)
U
U
UT=u
u
uT
0u
u
uT
1cu
u
uT
1s... u
u
uT
Mc u
u
uT
Ms(10b)
Y
Y
YT=y
y
yT
0y
y
yT
1cy
y
yT
1s... y
y
yT
Lc y
y
yT
Ls(10c)
with A
A
A∈Rn(2N+1)×n(2N+1),B
B
B∈Rn(2N+1)×m(2M+1),C
C
C∈
Rl(2L+1)×n(2N+1), and D
D
D∈Rl(2L+1)×m(2M+1). Closed-form
expressions for these matrices can be found in Ref. 28. It
is worth noting that harmonic decomposition does not rely
on state transition matrices, which makes the methodology
more computationally efficient and less numerically sensitive
than other approaches such as the Lyapounov-Floquet method
(Ref. 26) and frequency lifting methods (Ref. 27).
Model-Order Reduction
Because one of the objective of this paper is to investigate
the change in stability derivatives that results from vibrational
stabilization, it is desired to reduce the order of the harmonic
decomposition models down to a model that is representative
of rigid-body dynamics. Ideally, these reduced-order models
do not include the higher harmonic states but still retain part
of the higher-harmonic response characteristics. This can be
achieved through residualization, a portion of singular pertur-
bation theory that pertains to LTI systems (Ref. 41). Assum-
ing that one or more states of the system have stable dynam-
ics which are faster than that of the remaining states, the state
vector in Eq. (10a) can be partitioned into fast X
X
Xfand slow X
X
Xs
components:
X
X
XT=X
X
XT
sX
X
XT
f(11)
Then, the system in Eq. (9a) can be re-written as:
˙
X
X
Xs
˙
X
X
Xf=A
A
AsA
A
Asf
A
A
Afs A
A
AfX
X
Xs
X
X
Xf+B
B
Bs
B
B
BfU
U
U(12)
By neglecting the dynamics of the fast states (i.e.,˙
X
X
Xf=0) and
performing a few algebraic manipulations, the equations for
a reduced-order system with the state vector composed of the
slow states may be found:
˙
X
X
Xs=ˆ
A
A
AX
X
Xs+ˆ
B
B
BU
U
U(13)
where:
ˆ
A
A
A=A
A
As−A
A
AsfA
A
Af−1A
A
Afs (14a)
ˆ
B
B
B=B
B
Bs−A
A
AsfA
A
Af−1B
B
Bf(14b)
3
Note that A
A
Afmust be invertible. This is guaranteed if A
A
Afis
non-singular, i.e., no eigenvalue has a real-part that is equal
to zero. Additionally, to apply residualization, A
A
Afmust be is
asymptotically stable, i.e., all eigenvalues have their real part
that is strictly negative. In this study, the slow states are cho-
sen as the zeroth harmonic rigid-body states with the excep-
tion of the position states as the flight dynamics are invariant
with respect to these (42), whereas the fast states are taken as
the higher harmonics of the rotor states:
X
X
XT
s=u0v0w0p0q0r0φ0θ0ψ0(15a)
X
X
XT
f=x
x
xT
R0x
x
xT
R1c··· x
x
xT
RNs (15b)
where x
x
xRare the rotor states. The rotor states are gen-
erally asymptotically stable, unless there is some aerome-
chanic/aeroelastic instability, which is not modelled herein
anyway. This guarantees A
A
Afto be asymptotically stable. Note
that the higher harmonics of the rigid-body states were trun-
cated from the fast state vector. This is because the higher har-
monics of the rigid-body states are generally unstable at hover
and low-to-moderate speeds (their eigenvalues are shifted on
the imaginary axis with respect to the zeroth-harmonic eigen-
values by kΩ, with k=1,...,N). This would cause A
A
Afnot to
be asymptotically stable. The truncation of these states is jus-
tified by the fact that the higher harmonics of the rigid body
states have negligible contribution to the overall flight dynam-
ics (Ref. 7). An alternative reduction that yields a 10-state
model involves retaining the longitudinal and lateral flapping
angles as part of the slow state vector, such that:
X
X
XT
s=u0v0w0p0q0r0φ0θ0ψ0β1cβ1s(16a)
X
X
XT
f=ˆ
x
x
xT
R0x
x
xT
R1c··· x
x
xT
RNs (16b)
where ˆ
x
x
xR0is the zeroth harmonic of the rotor state vector with
the longitudonal and lateral flapping states removed. The 10-
state model will model the regressive flap mode and be accu-
rate, at least for the UH-60 helicopter, up to about 10 rad/s,
compared to approximately the 4 rad/s of the 8-state model
retaining the rigid-body dynamics only (Refs. 43, 44). Both
methods are used in this paper.
To ensure the residualized model retains information about
the influence of the residualized dynamics on both the zeroth
harmonics and the higher output harmonics of the output, con-
sider partitioning the output equations in Eq. (9b) as:
Y
Y
Y=C
C
CsC
C
CfX
X
Xs
X
X
Xf+D
D
DU
U
U(17)
Then, it can be shown that the residualized output equations
are:
˙
Y
Y
Y=ˆ
C
C
CX
X
Xs+ˆ
D
D
DU
U
U(18)
where:
ˆ
C
C
C=C
C
Cs−C
C
CfA
A
Af−1A
A
Afs (19a)
ˆ
D
D
D=D
D
D−C
C
CfA
A
Af−1B
B
Bf(19b)
If now the augmented output vector is selected to coincide
with the harmonics of the output vector in Eq. (20), such that:
Y
Y
YT=y
y
yT
0y
y
yT
1cy
y
yT
1s... y
y
yT
Lc y
y
yT
Ls(20)
then, the residualized model will predict the influence of the
residualized dynamics on the zeroth and higher-harmonic of
the output.
Vibrational Stabilization
The overall methodology used for this investigation is ex-
plained through a simple example involving an inverted pen-
dulum (Ref. 3, 45) with a vibrating suspension point, shown
in Fig. 2. Consider the dynamics of such pendulum:
L¨
θ−(g+A)θ=0 (21)
where Lis the pendulum length, Athe acceleration resulting
from the periodic displacement of the point of suspension, and
gis the gravitational acceleration.
Fig. 2: Inverted pendulum with the point of suspension being
vibrated.
Assume that the displacement of the point of suspension is
given by:
D=asinψ(22)
where ψ=Ωt. Then, the acceleration Aof the suspension
point is:
A=¨
D=−aΩ2sinψ(23)
The system in Eq. (21) can be reformulated as a system of
ordinary differential equations (ODEs) such that:
˙
x
x
x=0g
L−a
LΩ2sinψ
0 1 x
x
x=F
F
F(ψ)x
x
x(24)
where x
x
xT=˙
θ θ. Note that F
F
F(ψ) = F
F
F(ψ+ΩT), where T=
2π/Ω. Thus, the system in Eq. (24) is a linear time-periodic
(LTP) system.
Consider now decomposing the state vector into harmonics of
the fundamental frequency Ω, such that:
x
x
x=x
x
x0+
N
∑
n=1
[x
x
xnc cos(nψ) + x
x
xns sin(nψ)] (25)
Then, it can be shown (Ref. 28, 29) that the system in Eq.
(24) is approximated with a higher-order linear time-invariant
(LTI) system of the form:
˙
X
X
X=A
A
AX
X
X(26)
4
where X
X
XT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns is the augmented state
vector and:
A
A
A=
H
H
H0F
F
FH
H
H0F
F
F1cH
H
H0F
F
F1s···
H
H
H1cF
F
FH
H
H1cF
F
F1s−Ω+H
H
H1cF
F
F1s···
H
H
H1sF
F
FΩ+H
H
H1sF
F
F1sH
H
H1sF
F
F1s···
.
.
..
.
..
.
....
H
H
HNcF
F
FH
H
HNcF
F
F1sH
H
HNcF
F
F1s···
H
H
HNsF
F
FH
H
HNsF
F
F1sH
H
HNsF
F
F1s···
H
H
H0F
F
FNc H
H
H0F
F
FNs
H
H
H1cF
F
FNc H
H
H1cF
F
FNs
H
H
H1sF
F
FNc H
H
H1sF
F
FNs
.
.
..
.
.
H
H
HNcF
F
FNc −NΩ+H
H
HNcF
F
FNs
NΩ+H
H
HNsF
F
FNc H
H
HNsF
F
FNs
(27)
The A
A
Amatrix coefficients are given by:
H
H
H0M
M
M=1
2πZ2π
0
F
F
F(ψ)dψ(28a)
H
H
HicM
M
M=1
πZ2π
0
F
F
F(ψ)cos(nψ)dψ(28b)
H
H
HisM
M
M=1
πZ2π
0
F
F
F(ψ)sin(nψ)dψ(28ba-c)
H
H
H0F
F
Fnc =1
2πZ2π
0
F
F
Fnc(ψ)dψ(28c)
H
H
HicF
F
Fnc =1
πZ2π
0
F
F
Fnc(ψ)cos (nψ)dψ(28d)
H
H
HisF
F
Fnc =1
πZ2π
0
F
F
Fnc(ψ)sin (nψ)dψ(28dd-f)
H
H
H0F
F
Fns =1
2πZ2π
0
F
F
Fns(ψ)dψ(28e)
H
H
HicF
F
Fns =1
πZ2π
0
F
F
Fns(ψ)cos (nψ)dψ(28f)
H
H
HisF
F
Fns =1
πZ2π
0
F
F
Fns(ψ)sin (nψ)dψ(28fg-i)
where:
F
F
Fnc(ψ) = F
F
F(ψ)cosψ(29a)
F
F
Fns(ψ) = F
F
F(ψ)sinψ(29b)
Because the periodicity in Eq. (24) is limited to frequencies
of one per forcing cycle, it is sufficient to retain up to the first
harmonic in the harmonic decomposition of state vector in Eq.
(25). By doing so, the LTP system in Eq. (24) is transformed
into an equivalent LTI system with a system matrix:
A
A
A=
0g
L0 0 0 Ω2a
2L
1 0 0 0 0 0
0 0 0 g
L−Ω0
0 0 1 0 0 −Ω
0Ω2a
LΩ0 0 g
L
0 0 0 Ω1 0
(30)
and where x
x
xT=˙
θ0θ0˙
θ1cθ1c˙
θ1sθ1sis the augmented state
vector. The stability of the system will be determined by the
eigenvalues of the A
A
Amatrix. The eigenvalues are given by:
λ1,2=±c1(31a)
λ3,4=±c2(31b)
λ5,6=±rc3+c4−c5
c3
(31c)
where:
c1=sc4−c3
2+c5
2c3−1
2√3c3
c5
c3
i(32a)
c2=sc4−c3
2+c5
2c3
+1
2√3c3
c5
c3
i(32b)
c3=(c6+v
u
u
t"c6+6L2g−4L3Ω23
216L9−c7#2
+c3
5
+6L2g−4L3Ω23
216L9−c7)1/3(32c)
c4=6L2g−4L3Ω2
6L3(32d)
c5=2L3Ω4−LΩ4a2+6Lg2
6L3−6L2g−4L3Ω22
36L6(32e)
c6=2L2Ω4g−LΩ6a2+4LΩ2g2−Ω4a2g+2g3
4L3(32f)
c7=6L2g−4L3Ω22L3Ω4−LΩ4a2+6Lg2
24L6(32g)
It can be shown (Ref. 3) that the inverted pendulum is strongly
stable (i.e., stable for a high-enough forcing frequency) if:
a<π2
32 L(33)
Consider an inverted pendulum where g=9.81 m/s2,L=1
m, and a=1
2π2
32 L. Then, this pendulum will be stable
for forcing frequencies Ω⪆40.59 rad/s. For instance, the
eigenvalues for the case where Ω=0 rad/s (i.e., no periodic
forcing) are:
λ1=3.1321,λ2=−3.1321 (34)
whereas the eigenvalues for the case where Ω=50 rad/s are:
λ1,2=±4.5314i,λ3,4=±47.4166i,λ5,6=±51.9779i
(35)
The order of the system in Eq. (26) can be reduced by means
of residualization. As such, the augmented state vector is par-
titioned into fast and slow components, such that the slow
states are chosen as the zeroth harmonic states, whereas the
fast states are taken as the higher harmonics:
X
X
Xs=x
x
x0(36a)
X
X
XT
f=x
x
xT
1cx
x
xT
1s](36b)
5
Then, the submatrices of Eq. (12) become:
A
A
As=0g
L
1 0(37a)
A
A
Asf =000Ω2a
2L
0 0 0 0 (37b)
A
A
Af=
0g
L−Ω0
1 0 0 −Ω
Ω0 0 g
L
0Ω1 0
(37c)
A
A
Afs =
0 0
0 0
0Ω2a
L
0 0
(37d)
Applying Eq. (14) yields the following residualized dynam-
ics:
ˆ
A
A
A="0g
L−Ω4a2
2L(LΩ2+g)
1 0 #(38)
The eigenvalues of this matrix are:
λ1,2=±√2p−Ω4a2+2LΩ2g+2g2
2√LpLΩ2+g(39)
Setting ℜ(λ1,2) = 0, solving for Ω, and discarding any non-
physical solution yields:
Ω=±1
aqLg +pL2g2+2a2g2+a2
≈28.89 rad/s
(40)
Note that this is only an approximation to the minimum fre-
quency that yields neutral stability.
SIMULATION MODELS
Helicopter Flight Dynamics Model
The nonlinear flight dynamics of a utility helicopter are mod-
eled using PSUHeloSim (Ref. 46), a MATLAB®implementa-
tion of the General Helicopter (GenHel) flight dynamics simu-
lation model (Ref. 47) with improved rotor, trimming, and lin-
earization routines. PSUHeloSim is representative of a utility
helicopter similar to a UH-60. The model contains a 6-degree-
of-freedom rigid-body dynamic model of the fuselage, nonlin-
ear aerodynamic lookup tables for the fuselage, rotor blades,
and empennage, rigid flap and lead-lag rotor blade dynamics,
a three-state Pitt-Peters inflow model (Ref. 48), and a Bailey
tail rotor model (Ref. 49). The state vector is:
x
x
xT=uvwpqrφ θ ψ x y z β0β0Dβ1cβ1s˙
β0˙
β0D˙
β1c˙
β1s
ζ0ζ0Dζ1cζ1s˙
ζ0˙
ζ0D˙
ζ1c˙
ζ1sλ0λ1cλ1sλ0T
(41)
where:
u,v,ware the longitudinal, lateral, and vertical velocities
in the body-fixed frame,
p,q,rare the roll, pitch, and yaw rates,
φ,θ,ψare the Euler angles,
x,y,zare the positions in the North-East-Down (NED)
frame,
β0,, β0D,β1c,β1sare the flapping angles in multi-blade
coordinates,
ζ0,ζ0D,ζ1c,ζ1sare the lead-lag angles in multi-blade co-
ordinates,
λ0,λ1c,λ1s,are the main rotor induced inflow ratio har-
monics, and
λ0Tis the tail rotor induced inflow ratio.
The control vector is:
u
u
uT=δlat δlon δcol δped(42)
where δlat and δlon are the lateral and longitudinal cyclic in-
puts, δcol is the collective input, and δped is the pedal input.
External Slung Load
A non-linear model of dual-point suspend external slung load
was developed in Ref. 10. This simulation was used to model
a 6x6x8 ft, 2400 lbs, CONEX cargo container carried in an in-
verted V configuration as shown in Fig. 3. The hook geometry
imitates a CH-47 Chinook with a spacing of 13.3 ft between
the forward and rear cargo hooks. A conventional rigging pro-
cedure with four equal-length cables was applied with cable
length equal to 16 ft. In the simulation, the dynamics of the
CONEX container are isolated, i.e., assumed to be not coupled
to the rotorcraft flight dynamics, but subject to aerodynamic
forces due to the forward flight condition of the rotorcraft.
The simulation model includes non-linear look-up tables of
aerodynamic coefficients based on static wind tunnel tests, a
transfer function representing certain unsteady aerodynamic
forces, a non-linear model of the forces due to cable stretch-
ing, and a complete non-linear 6-DOF dynamic model using
representative mass and inertia properties of the load. The
simulation model was partially validated against wind tunnel
tests of a scale model in Ref. 10.
The dynamics of this load are known to exhibit unstable os-
cillations over certain airspeed ranges. The instability was
observed in both simulation and wind tunnel tests in Ref. 10
and has been observed for similar dual-point suspended loads
in flight tests. During instability events, the load will typi-
cally enter a prescribed periodic motion primarily involving
roll and yaw oscillations but with some pitch motion as well.
The trajectory is representative of a Limit Cycle Oscillation
(LCO). The LCO orbits an unstable, static equilibrium point
(a stationary point). If the load is trimmed at the static equi-
librium, any small disturbance will cause the load to depart
and enter the LCO orbit. The amplitude of the LCO is often
large enough to be objectionable to the pilot or even present a
hazard to the rotorcraft.
In Ref. 10, a load stablization controller was investigated us-
ing a laterally active forward cargo hook (ACH). The con-
troller used feedback of cable angle and/or inertial measure-
ments of the external load to drive the forward cargo hook
6
Fig. 3: Simulated slung load model expressed in load-fixed
(L) and inertial (E) coordinate systems
lateral position in such a way that the load stabilizes at the
static equilibrium point. The objective of this part of the study
is to investigate the feasibility of stabilizing the load using an
ACH oscillation of fixed frequency and amplitude without use
of feedback, i.e., use the ACH to provide vibrational stability
of the load.
RESULTS
Vibrational Stabilization due to Blade Imbalance
Consider now an example involving the flight dynamics of
a helicopter. Since helicopters are subjected to a variety of
vibrations, it is feasible that vibrational effects might affect
their stability characteristics. Traditionally, linear time invari-
ant models of helicopter flight dynamics neglect vibrational
effects, but through the use of harmonic decomposition (Ref.
28) time periodic terms can be retained in the LTI models.
In general, the flight dynamics of this model are non-linear
time-periodic (NLTP), such that:
˙
x
x
x=f
f
f(x
x
x,u
u
u,ψ)(43)
where ψ=Ωtis the azimuth angle of a reference main rotor
blade. Blade imbalance is modeled by assuming one of the
four rotor blades to have a different mass with respect to the
others. Given this difference in mass, first and second flap-
ping moments are varied accordingly. Because blade imbal-
ance results in a periodic forcing at one-per-rotor-revolution
(1/rev), trim at hover will no longer be represented a single
point in the state space but rather by a periodic orbit such that
the trim state and control vectors are x
x
x∗(ψ) = x
x
x∗(ψ+ΩT)and
u
u
u∗(ψ) = u
u
u∗(ψ+ΩT).
To solve for this periodic orbit, the modified harmonic bal-
ance algorithm of Ref. 40 is used. Figure 4 shows the periodic
trim acceleration at the center of gravity (CG) over one rotor
revolution. In this figure, it is shown how rotor blade imbal-
ance results in 1/rev vibrations with amplitudes of up to 0.5 g
in the lateral and vertical acceleration for the 10% blade im-
balance case. For reference, these vibrations are in line with
those encountered by fixed-wing aircraft in severe turbulence
(Refs. 50–53).
0 50 100 150 200 250 300 350
-0.02
0
0.02
ax [g]
0 % 2 % 4 % 6 % 8 % 10 %
0 50 100 150 200 250 300 350
-0.5
0
0.5
ay [g]
0 50 100 150 200 250 300 350
Azimuth angle, [deg]
-0.5
0
0.5
az [g]
Fig. 4: Periodic trim acceleration at the CG over one rotor
revolution for increasing blade imbalance.
Next, the flight dynamics are linearized about this periodic
orbit to yield an LTP system. The LTP system is approxi-
mated with a higher-order LTI system by retaining up to the
fourth harmonic of the fundamental frequency (i.e., the angu-
lar speed of the main rotor Ω) in the states and output vectors
(i.e.,N=4 and L=4), while retaining only the zeroth har-
monic of the control input vector (i.e.,M=0). While the
dominant oscillations at hover will be the 1/rev oscillations
due to blade imbalance, small 4/rev oscillations may also be
present, although these become significant only in forward
flight. Thus, the choice of retaining up to the fourth harmonic
in the states and outputs. Retaining only the zeroth harmonic
in the control input relies on the assumption that no higher-
harmonic control (HHC) is employed, which is typically the
case for conventional rotorcraft.
The order of the higher-order LTI model is subsequently re-
duced to a 10-state model representative of the rigid-body dy-
namics and of the regressive flap mode. An 8-state model is
also obtained, which is representative of the rigid-body dy-
namics only. Because this model does not retain the flap-
ping states used to predict the regressive flap mode, its ac-
curacy will be limited to frequencies up to approximately 4
rad/s. To validate the linearized models obtained via the pro-
posed method, the on-axis frequency responses of these mod-
els are compared to frequency responses extracted from fre-
quency sweeps (Ref. 54). Because the dynamics of rotorcraft
are unstable at hover, a control law similar to that in Ref. 46
was used to stabilize the hover dynamics during the frequency
sweeps. Figure 5 shows the bare-airframe roll rate due to lat-
eral input frequency responses for varying blade imbalance.
Figure 5a corresponds to no blade imbalance, whereas 5b cor-
responds to a 10% blade imbalance. These figures show that
7
the frequency responses of high-order LTI model match those
obtained from frequency sweeps for both case, validating the
LTI model. The 8-state model is shown to provide good ac-
curacy up to about 4 rad/s, whereas the accuracy of the 10-
state model extends up to about 10.5 rad/s. These results sug-
gest that the linearization, harmonic decomposition, and suc-
cessive model-order reduction are suitable for obtaining lin-
earized models representative of rotor blade imbalance.
10-1 10 0101102
-80
-60
-40
-20
Mag [dB]
High-Order LTI Sweep LTI 10 LTI 8
10-1 10 0101102
0
200
400
Phase [deg]
10-1 10 0101102
Frequency, [rad/s]
0.4
0.6
0.8
1
Coh
(a) 0% blade imbalance.
10-1 10 0101102
-100
-50
0
Mag [dB]
High-Order LTI Sweep LTI 10 LTI 8
10-1 10 0101102
0
200
400
Phase [deg]
10-1 10 0101102
Frequency, [rad/s]
0.4
0.6
0.8
1
Coh
(b) 10% blade imbalance.
Fig. 5: Comparison between the roll rate due to lateral input
frequency responses, p
δlat
(s), obtained from linearized
models and frequency sweeps.
The effect of blade imbalance on the flight dynamics can
be assessed via spectral analysis of the reduced-order system
thus obtained. Figure 6 shows the eigenvalues of the 10-state
model for blade imbalance varying from zero to 10%. Figure
6a shows all ten eigenvalues and the regressive flapping eigen-
values are clearly shown to migrate to the left of the complex
plane for increasing imbalance, eventually converging to fixed
values. Figure 6b shows a detail of the rigid-body dynamics
eigenvalues, where both the roll and pitch subsidence mode
eigenvalues move toward the origin for increasing blade im-
balance. This means that their frequency decreases with in-
creasing blade imbalance. Moreover, both the roll and pitch
oscillation eigenvalues move away from the origin along the
imaginary axis, which is indicative of higher frequency and
lower damping. One difference, though, is that the pitch oscil-
lation eigenvalues move slightly to the right whereas the roll
oscillatory eigenvalues move slightly to the left. The poles of
the coupled yaw-heave mode are relatively unaffected com-
pared to the other modes. These results are reported quantita-
tively in Table 1 for two cases: zero blade imbalance and 10%
blade imbalance. Thus, it is concluded that rotor blade imbal-
ance has a symmetric effect on roll and pitch axes, in that it
tends to decrease the frequency of the subsidence modes of
the hovering cubic (Ref. 4), while the oscillatory modes tend
to increase in frequency and decrease in damping.
Figure 7 shows the lateral (Fig. 7a) and longitudinal (Fig. 7b)
stability derivatives of the 8-state model for varying blade im-
balance. The magnitude of the roll and pitch axes damping
derivatives Lpand Mqis shown to decrease with increasing
blade imbalance. On the other hand, while lateral speed sta-
bility (Yb) becomes more negative with increasing blade im-
balance, longitudinal speed stability (Xu) decreases in mag-
nitude. The roll and pitch moment derivatives due to lateral
and logitudinal speed, respectively, show similar trennds in
that both become more negative. These results indicate that,
indeed, rotor blade imbalance affects the eigenvalues of a ro-
torcraft at hover.
In light of these results, the dynamics identified for a heli-
copter with an imbalanced rotor may indeed differ from those
obtained from simulations where the rotor is perfectly bal-
anced. A more in-depth explanation follows. When per-
forming system/parametric identification from flight test data,
which inevitably corresponds to some rotor imbalance on the
real aircraft, what is identified are, in fact, dynamics equiv-
alent to the residualized dynamics from the high-order LTI
model. These can be referred to as the “true” dynamics. On
the other hand, when performing flight dynamics predictions
from physics-based simulations, any periodic component in
the flight dynamics due to blade imbalance is not modeled in
that rotor blades are typically assumed as perfectly balanced.
Even if blade imbalance was modeled, flight dynamics pre-
dictions adopting linearized models only typically consider
the averaged (or zeroth harmonic) dynamics. This is equiv-
alent to performing eigenanalysis on the averaged portion of
the high-order LTI system in harmonic decomposition form.
Or, in other terms, it corresponds to performing eigenanaly-
sis on a reduced-order system obtained by truncating all of
the higher-harmonic states from the high-order LTI system.
Thus, any harmonic component is ignored. It is worth not-
ing that the averaged dynamics of all cases presented with ro-
tor imbalance is in fact the same. This concept is illustrated
in Fig. 8, which features the frequency response of the ze-
8
Table 1: Hover modal characteristics for varying blade imbalance.
Mode Imbalance [%] Eigenvalues [rad/s] Natural Frequency, ωn[rad/s] Damping Ratio, ζ
Roll Subsidence 0−5.5645 - 1
10 −3.9270 - 1
Pitch Subsidence 0−1.7763 - 1
10 −1.5681 - 1
Roll Oscillation 0 0.4980 ±0.4364i0.6622 -
10 0.4624 ±0.6305i0.7819 -
Pitch Oscillation 0−0.2348 ±0.4856i0.5394 0.4353
10 −0.1106 ±0.6601i0.6693 0.1730
Coupled Yaw-Heave 0−0.2924 ±0.0470i0.2962 0.9873
10 −0.1106 ±0.6601i0.2994 0.9799
-10 -8 -6 -4 -2 0 2
Real
-8
-6
-4
-2
0
2
4
6
8
10
12
Imag
0 %
1 %
2 %
3 %
4 %
5 %
6 %
7 %
8 %
9 %
10 %
Regressive Flap Mode
Roll
Subsidence
Pitch Subsidence Pitch Oscillation
+
Heave Mode
Roll Oscillation
(a) Rigid-body and regressive flap dynamics.
-6 -5 -4 -3 -2 -1 0 1
Real
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Imag
0 %
1 %
2 %
3 %
4 %
5 %
6 %
7 %
8 %
9 %
10 %
Roll
Subsidence
Pitch
Subsidence
Heave
Mode
Pitch
Oscillation
Roll Oscillation
(b) Rigid-body dynamics.
Fig. 6: Hover eigenvalues of the 10-state residualized model
for varying blade imbalance.
roth harmonic of the roll rate to lateral inputs, i.e.,p0
δlat
(s), for
012345678910
-0.08
-0.06
-0.04
Yv [1/s]
012345678910
-0.05
0
0.05
Lv [1/(ft-s)]
012345678910
Blade imbalance [%]
-5
-4.5
-4
Lp [1/s]
(a) Lateral stability derivatives.
012345678910
-0.03
-0.02
-0.01
Xu [1/s]
012345678910
-2
0
2
Mu [1/(ft-s)]
10-3
012345678910
Blade imbalance [%]
-0.7
-0.6
-0.5
Mq [1/s]
(b) Longitudinal stability derivatives.
Fig. 7: Hover stability derivatives for varying blade
imbalance.
the high-order LTI model in harmonic decomposition form
representative of a 10% rotor blade imbalance, the averaged
dynamics, and the 8- and 10-state residualized models corre-
9
sponding to a 10% rotor blade imbalance. The 8-state resid-
ualized model constitutes a good approximation of the high-
order LTI up to approximately 4 rad/s. In fact, the 8-state
model shows reduced phase roll-off after 4 rad/s. This is be-
cause it does not include the longitudinal and lateral flapping
states needed to predict the regressive flap mode. The 10-state
model is accurate up to approximately 10.5 rad/s. The average
dynamics, however, fails to predict the response in low-to-mid
frequencies of interest to flight dynamics (i.e., 0.3 to 30 rad/s).
These differences will cause discrepancies in the estimation
of the stability derivative Lp. The same concept applies to all
other frequency responses of interest for the identification of
reduced-order models of the rotorcraft flight dynamics.
10-1 10 0101102
-100
-80
-60
-40
-20
Mag [dB]
High-Order LTI
Averaged LTI
LTI 10
LTI 8
10-1 10 0101102
Frequency, [rad/s]
0
100
200
300
400
Phase [deg]
Fig. 8: Frequency response of the zeroth harmonic of the roll
rate to lateral inputs, p0
δlat
(s).
Vibrational Stabilization of an External Slung Load
This analysis focused on the dual-point external load dynam-
ics at 78 knots forward airspeed. In this flight condition, the
external load exhibits a lateral-directional LCO as illustrated
in Fig. 9. In this simulation the external load starts in a static
equilibrium with a slight pitch attitude due to the drag forces
on the load. A ±0.1 ft perturbation is applied to the forward
cargo hook at 10 seconds, which initiates a divergence (since
the static equilibrium is unstable) until the load reaches a sta-
ble LCO with an amplitude of about ±12 deg in yaw and ±8
deg in roll. The roll-yaw trajectory of the LCO, shown in Fig.
10, is not a simple elliptical orbit but a fairly complex trajec-
tory due to the non-linear load dynamics.
Through trial-and-error it was found that with small amplitude
excitation of the ACH at 7 rad/sec, the simulated load stayed
closer to the static equilibrium point and remained in a small
amplitude orbit (much smaller amplitude oscillations than the
LCO shown in Fig. 10). The time history response is shown in
Fig. 11, where a ±0.04 ft (1.2 cm), 7 rad/sec sine wave is ap-
plied to the forward cargo hook. At 10 seconds, the same per-
turbation ±0.1 ft perturbation used in Fig. 9 is super-imposed
on the cargo hook. The response shows that the response of
the external load remains very small, less than 0.3 deg roll
and yaw oscillations. The roll-yaw trajectory is shown in Fig.
12. If there is no perturbation applied to the system, a small
amplitude roll-yaw oscillation is observed in red. This oscil-
lation scales with the amplitude of the ACH input down to
a certain limit (for amplitudes much lower than ±0.04 ft the
load will depart to the large amplitude LCO). Following the
perturbation we see the load response remains small over a
long period of time, but there is a lower frequency oscillation
superimposed on the roll response. Nonetheless, the simula-
tions indicate the forced response is at least marginally stable
and the small oscillation of the ACH prevents the load from
entering the large amplitude LCO.
It should be noted that there were some discrepancies be-
tween the wind tunnel tests and simulation model presented
in Ref. 10, but the qualitative behavior was similar between
test and simulation. The wind tunnel tests showed a somewhat
higher amplitude LCO at an airspeed of 78 knots full-scale (12
m/s model-scale), with +15/−20 deg oscillations in yaw and
+10/−8 oscillation in roll, but with similar trajectory shape.
Frequency sweeps were performed on the ACH at 78 knots
full-scale airspeed, and significant reductions in load oscilla-
tions were observed around 4.1 rad/sec ACH inputs, possibly
indicating vibrational stability at a somewhat lower driving
frequency.
0 20 40 60 80 100 120
Time (s)
-20
0
20
L (deg)
0 20 40 60 80 100 120
Time (s)
-20
0
20
L (deg)
0 20 40 60 80 100 120
Time (s)
-20
0
20
L (deg)
0 20 40 60 80 100 120
Time (s)
-0.5
0
0.5
Yfwd (ft)
Fig. 9: Time history of external load LCO at 78 knots
forward airspeed.
CONCLUSIONS
This paper investigated vibrational stabilization effects in ro-
torcraft flight dynamics. Starting from a simple example in-
volving an inverted pendulum, the use of the harmonic de-
composition method for the study of vibrational stabilization
effects was demonstrated. The concept was then extended to
10
-10 -5 0 5 10
L (deg)
-10
-8
-6
-4
-2
0
2
4
6
8
10
L (deg)
Fig. 10: External load LCO trajectory at 78 knots forward
airspeed.
0 20 40 60 80 100 120
Time (s)
-1
0
1
L (deg)
0 20 40 60 80 100 120
Time (s)
-2
-1
0
L (deg)
0 20 40 60 80 100 120
Time (s)
-1
0
1
L (deg)
0 20 40 60 80 100 120
Time (s)
-0.2
0
0.2
Yfwd (ft)
Fig. 11: Time history of external load at 78 knots dorward
airspeed with vibrational stability.
analyze the effect of blade imbalance on the flight dynamics
of a helicopter, as well as vibrational stablization of a slung
load in forward flight using small-amplitude and disturbances
on an active cargo hook. Based on this work, the following
conclusions can be reached:
1. While vibrations induced by rotor blade imbalance did
not stabilize the hovering dynamics of a helicopter, these
vibrations still have a significant effect on the hover-
ing dynamics. Increasing rotor blade imbalance results
in a symmetric effect on the roll and pitch axes, in
that it tends to decrease the frequency of the subsidence
modes of the hovering cubic, while the unstable oscilla-
tory modes tend to increase in frequency and decrease in
damping. To the contrary, the yaw/heave dynamics are
-0.2 -0.1 0 0.1 0.2
L (deg)
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
L (deg)
Oscillation with perturbation
Oscillation without perturbation
Fig. 12: External load trajectory at 78 knots forward airspeed
with vibrational stability.
relatively unaffected compared to the lateral and longitu-
dinal axes.
2. Commonly observed mismatches between the flight
identified dynamics and those predicted via physics-
based models may indeed be ascribed vibrational stabil-
ity effects due to rotor blade imbalance. This is because
when performing system/parametric identification from
flight test data, which inevitably corresponds to some ro-
tor imbalance on the real aircraft, what is identified are,
effectively, dynamics equivalent to the residualized dy-
namics obtained from the linearized time-periodic dy-
namics. On the other hand, when performing flight dy-
namics predictions from physics-based simulations, any
periodic component in the flight dynamics due to blade
imbalance is not modeled in that rotor blades are typi-
cally assumed as perfectly balanced. Moreover, typical
flight dynamics predictions only adopt the averaged lin-
earized dynamics. As such, any time-periodic effect that
may have an effect on the flight dynamics is typically ig-
nored in analyses from simulations while periodicity will
show up in models identified from flight test data.
3. A small-amplitude oscillation of the active cargo hook
(ACH) at 7 rad/s resulted in the simulated load to stay
closer to the static equilibrium point and to remain in an
orbit significantly smaller (less than 0.3 deg roll and yaw
oscillations) than the limit cycle it incurred in with no
ACH periodic forcing (±12 deg in yaw and ±8 deg in
roll oscillations). The simulated load oscillation ampli-
tude scales with the amplitude of the ACH input down to
a certain limit.
4. Simulations indicate the forced response is stable and the
small oscillation of the ACH prevents the load from en-
tering the high amplitude LCO. These results are in line
11
with previous research showing that slung loads can ex-
hibit better stability /lower amplitude oscillations when
subjected to periodic disturbances at the hook. This po-
tentially constitutes a practical solution to the semi-active
control of a suspended load.
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