ERF91-76 
SEVENTEENTH EUROPEAN ROTORCRAFf FORUM 
Paper No. 91 - 76 
COUPLED ROTOR/FUSELAGE VIBRATION REDUCTION 
USING MULTIPLE FREQUENCY BLADE PITCH CONTROL 
I. PAPAVASSILIOU, P.P. FRIEDMANN, C.VENKATESAN 
MECHANICAL. AEROSPACE AND NUCLEAR ENGINEERING DEPARTMEJ';1 
UNIVERSITY OF CALIFORNIA AT LOS ANGELES 
LOS ANGELES, CA 90024-1597, U.S.A 
SEPTEMBER 24 - 26, 1991 
BERLIN, GERMANY 
Deutsche Gesellschaft fur Luft- und Raumfahrt e.v. (DG LR) 
Gedesberger Allee 70, 5300 Bonn 2, Germany 
OPGENOMENIN 
GE/\UTG:;:ATISEERDE 
CA","/~LC:3US 
ERF91-76 
COUPLED ROTOR-FUSELAGE VIBRATION REDUCTION 
WITH MULTIPLE FREQUENCY BLADE PITCH CONTROL 
· I.Papavassiliou 1 , 
P.P.Friedmann2 and C.Venkatesan3• 
Mechanical, Aerospace, and Nuclear Engineering Department 
University of California, Los Angeles, CA 90024 
Abstract 
A nonlinear coupled rotor/flexible fuselage analysis has been developed and 
used to study the effects of higher harmonic blade pitch control on the vibratory 
hub loads and fuselage acceleration levels. Previous results, obtained with this 
model have shown that conventional higher harmonic control (HHC) inputs aimed 
at hub shear reduction cause an increase in the fuselage accelerations and vise­
versa. It was also found that for simultaneous reduction of hub shears and fuselage 
accelerations, a pitch input representing a combination of two higher harmonic 
components of different frequencies was needed. Subsequently, it was found that 
this input could not be implemented through a conventional swashplate. This pa­
per corrects a mistake originally made in the representation of the multiple fre­
quency pitch input and shows that such a pitch input can be only implemented in 
the rotating reference frame. A rigorous mathematical solution is found, for the 
pitch input in the rotating reference frame, which produces simultaneous reduction 
of hub shears and fuselage acceleration. New insight on vibration reduction in 
coupled rotor/fuselage systems is obtained from the sensitivity of hub shears to the 
frequency and amplitude of the open loop HHC signal in the rotating reference 
frame. Finally the role of fuselage flexibilty in this class of problems is determined. 
Nomenclature 
a Rotor blade lift curve slope 
Acx4c• Acx4s, Acy4c 4/rev components of the fuselage C.G 
Acy4s, Acz4c, Acz4s accelerations 
AP Amplitude of pitch input in Eq. (41) 
b Blade semichord 
Blade drag coefficient 
Weight coefficient 
Blade damping constants 
e Hinge offs et 
1 Postdoctoral Fellow 
3 Professor 
2 Associate Research Engineer 
91 - 76.1 
Column vectors of blade, fuselage rigid body, fuselage 
elastic and inflow equations for equilibrium 
F Hx4c• F Hx4s• F Hy4c 4/rev components of the vibratory hub 
F Hy4s• F Hz4c• F Hz4s shears 
FoHz4c• FoHz4s 4/rev components of the baseline vertical hub shear 
J Performance index, Eq. (35) 
Number of blades 
Transformation matrix, Eq. (8) 
Vector of the degrees of freedom of blade, fuselage rigid 
body modes, fuselage elastic modes and trim variables 
Harmonic components of blade response 
Harmonic components of fuselage rigid body response 
Harmonic components of fuselage elastic response 
R Dimensional rotor radius 
Elastic coupling coefficient 
[T], [TE], [T RJ MFPC Transfer matrices 
[WzJ Weighting matrix, Eq. (35) 
X and Z position of the fuselage aerodynamic center 
measured from point M on the helicopter 
X and Z position of the fuselage center of mass 
measured from point M on the helicopter 
X and Z position of the rotor hub center measured 
from point M on the helicopter 
{Z}, {ZA}, {ZF} Vectors of vibratory response 
__ ..,._. . 
{Zo} Vector of baseline vibrations 
Fuselage attitude in pitch 
n-th harmonic cosine and sine components of flap 
91 - 76.2 
response of the blade 
The k-th blade rotating flap, lead-lag and torsional 
degrees of freedom 
}' Lock number 
Blade pitch settings for equilibrium 
Blade twist distribution 
Control pitch angle of kth blade 
Higher harmonic pitch input of kth blade 
Blade pitch input or vector of pitch inputs 
Vector of pitch control inputs 
Vector of pitch control inputs 
Amplitudes of cosine inputs in collective, 
lateral and longitudinal HHC 
Amplitudes of sine inputs in collective, 
lateral and longitudinal HHC 
8Bc,8Bs,f3~c Components of pitch input control vector, 
e~s, e~c, e~s Eq. (5) 
ec1, er-1, e~ Components of pitch input control vector for 
e~, e~1, er1 frequencies p-1,p and p + 1/ rev respectively 
l Total inflow 
µ Advance ratio 
i-th generalized degree of freedom in flap, 
lag and torsion for the elastic blade 
p Density of the air 
. . . 2Nbb 
(J Sohd1ty rat10 = n 
cpP Phase angle of pitch input in Eq. (41) 
i/1 k Azimuth angle of the k-th blade 
91 - 76.3 
Rotating first flap, lag, and 
torsional blade frequencies 
Frequency of the HHC input 
n Rotor R.P.M 
. Overbars indicate 
dimensional quantities 
1. Introduction 
Vibration reduction is one of the central problems in modern helicopter design. 
Among the various schemes available for vibration reduction [1,2] vibration re­
duction using higher harmonic control (HHC) appears to have considerable prom­
ise. The higher harmonic blade pitch control can be implemented either through 
the use of actuators in the nonrotating frame (i.e. below the the swashplate) or in 
the rotating frame, with actuators between the swashplate and the rotor blade. 
The second approach based on actuators in the rotating system is denoted 
Individual-Blade-Control (IBC) [3]. With the constraint that all the blades in the 
rotor must perform identical motion, the use of actuators in the nonrotating frame 
imposes limitations on the frequencies of the higher harmonic blade pitch angle 
which can be implemented in practice. These restrictions can be removed by using 
actuators in the rotating frame [4]'. 
Vibration reduction using HHC has been demonstrated by analytical simulation 
[5-10] ,wind tunnel tests [11-13] and flight tests [14-16]. The analytical stud.ies 
and wind tunnel tests have shown that under a fixed hub condition, the use of high 
frequency blade pitch inputs (HHC) reduces hub loads. It should be noted that the 
purpose of the analytical and wind tunnel studies was not only to assess the effe c­
tiveness of various control algorithms for HHC but also to demonstrate the tech­
nical feasibility of the approach. On the other hand, flight tests have demonstrated 
fuselage vibration ( usually acceleration levels at the pilot seat ) reduction by using 
HHC inputs to the main rotor. In some flight tests it was observed that reduction 
of acceleration components at the pilot seat was accompanied by increases in hub 
and blade loads from their baseline values. . 
In a number of recent studies [17-19] it was shown that for a coupled 
rotor/flexible fuselage model, shown schematically in Fig. 1, conventional single 
frequency higher harmonic pitch control applied through a conventional 
swashplate was capable of reducing either the hub loads or the fuselage acceler­
ations but not both simultaneously. A simultaneous reduction of both hub shears 
and fuselage accelerations could be obtained only when assuming that the fuselage 
was rigid. 
In an attempt to obtain simultaneous reduction of hub shears fuselage acceler­
ations for a flexible fuselage a pitch input consisting of two different frequencies 
was considered. To distinguish between this input and convetional HHC, in Refs. 
17-19 this input was denoted as Multiple Higher Harmonic Control(MHHC). This 
approach was based on employing two higher harmonic pitch inputs \Vith fre­
quencies of (Nb -1) /rev and (Nb) /rev for a rotor having Nb blades. Subsequently 
91 - 76.4 
the authors found that this pitch input used, in the previous studies [17 - 19], 
was incorrect; in the sense that it could not be mechanically implemented through 
a conventional swashplate which uses actuators in the nonrotating reference frame, 
As will be shown in this paper, the pitch input found in Refs. 17-19 can be im­
plemented by using actuators in the rotating reference frame, and therefore its 
practical implementation can be categorized as individual blade control(IBC). 
Furthermore to avoid any misconception created in our previous studies, the use 
of pitch control inputs which consist of more than one frequency in the rotating 
reference will be denoted in this paper as Multiple Frequency Pitch Control 
(MFPC). 
It turned out that the use of such multiple frequency pitch inputs, in the open 
loop mode, has very interesting properties, which enhance our UI1derstanding of 
vibration reduction in rotorcraft using HHC or any other type of actively con­
trolled pitch input. A fairly detailed study was conducted to analyie the vibration 
reduction capability of such pitch inputs, using a nonlinear coupled rotor/flexible 
fuselage model of a helicopter in forward flight which was developed in Refs. 
17-19. The mathematical model for the system schematically shown in Fig. I, was 
derived using computer algebra implemented on a symbolic computing facility and 
the details of the derivation can be found in Refs. 17-19. 
The main objectives of this study are: 
1. To correct the error made in the previous studies [17-19] associated with the 
application of multiple frequency pitch control inputs to the coupled 
rotor/flexible fuselage system; 
2. To provide an improved understanding of the effect of the open loop HHC 
inputs on a coupled rotor/fexible fuselage system by studying the sensitivity 
of such a system to higher harmonic blade pitch inputs, applied in the ro­
tating system, one frequency at a time; 
3. To undestand the fundamental mechanism of simultaneous reduction of hub 
shears and fuselage accelerations using MFPC ; 
4. To study the influence of fuselage modeling on the capability of MFPC to 
produce simultaneous reduction in hub shears and fuselage accelerations. 
2. Coupled Rotor/Flexible Fuselage Model 
The first step in studying the vibration problem in helicopters is the formulation 
of the nonlinear differential equations of motion representing the dynamics of the 
coupled rotor-flexible fuselage system in forward flight. Due to the complexity of 
the problem, certain simplifying assumptions have been made in the idealization 
of the rotor-fuselage system. 
A schematic diagram of the couriled rotor-fuselage system is shown in Fig. I. 
The mathematical model \Vhich hc1s :--een developed can accomodate two different 
blade models: (a) the offset hinged spring restrained blade model and (b) the fully 
elastic hingeless blade model. For both cases, the blades have fully coupled flap­
lag- torsional dynamics. The fuselage is idealized as a uniform beam having 
bending deformations in the vertical and horizontal planes and elastic torsion 
about_the x. 1 axis. In addition to the elastic deformations, the fuselage has five rigid 
body degrees of freedom namelly, pitch, roll and three translations. The rotor sys­
tem is connected to the flexible beam through a rigid shaft at point "D". 
91 - 76.5 
The equations of motion of the coupled rotor-flexible fuselage system are de­
rived using force and moment equilibrium conditions. For the offset hinged, spring 
restrained blade case the rotor blade equations are obtained by enforcing moment 
equilibrium at the root of the blade in flap lag and torsion. For the elastic blade 
case the equations are the nonlin_ear partial differential equations of an elastic 
beam. These equations are transformed to a system of ordinary nonlinear differ­
ential equations using Galerk.in's method to eliminate the spatial variable. The final 
system of equations of motion describing the coupled flap-lag-torsional motion of 
the elastic blade consists of three flap equations corresponding to the first three 
bending modes in flap; two lag equations corresponding to the first two bending 
modes in lead-lag; and one torsional equation corresponding to the fundamental 
torsional mode. The rigid body equations of motion of the fuselage are obtained 
using force and moment equilibrium at the center of gravity(C.G) of the fuselage; 
and the elastic mode equations of the fuselage are formulated using generalized 
force and moment equilibrium for the various generalized modes representing the 
elastic deformation of the fuselage. 
The details of the derivation of the equations on a symbolic computing facility 
can be found in Refs . 17 -19. 
3. Blade Pitch Representation for Open Loop Control 
The total pitch angle in the rotating frame consists of two contributions; those 
needed to trim the helicopter and the higher harmonic pitch inputs used for vi­
bration reduction. The pitch angle of the k-th rotor blade in the rotating frame can 
be expressed as: 
8pk = 80 + 81c cos tf;k + 815· sin t/Jk + 8HHk (I) 
where tf;k is the blade azimuth angle of the k-th blade: 
2n 
t/Jk = t/1 + Nb (k - 1) ; k = 1 , 2 , ... , Nb (2) 
Where 80, 81c,and 815 are the collective and cyclic pitch inputs required for trim, 
and 8HHk the higher harmonic pitch input. For HHC through a conventional 
swash plate, the pitch input in the rotating frame can be written: 
(3) 
The expressions inside the bracketts are the collective, lateral ap.d l~mgitudinal 
HHC inputs corresponding to translation, lateral tilting and longitudinal tilting of 
the stationary swashplate. To prevent the blades from going out of track, wHH in 
91 - 76.6 
Eq. (3) has to be a multiple of the number of the rotor blades Nb, which results in 
a pitch input signal containing three frequencies, namely 
(Nb - I )/rev, NJrev and (Nb+ I )/rev, in the rotating frame. For a four bladed 
rotor, w8 H = 4 and the signal in the rotating frame contains only 3/rev, 4/rev and 
5/rev harmonics. This imposes certain limitations in the domain of search for the 
signal which minimizes the vibrations. In Refs. [17-19] the HHC signal was er­
roneously represented by: 
eHHk = [8os sin WHH!pk + 8oc cos WHHl/lk] 
(4) 
where 1/1 was erroneously replaced by 1/Jk in the expressions inside the square 
bracketts in Eq. (3). When the frequency wHH is a multiple of the number of blades 
Nb , Eqns. (3) and (4) are mathematically identical. If wHH is not a multiple of Nb, 
the signal given by .Eq. (4) cannot be practically implemented through a conven­
tional swashplate using actuators in the nonrotating frame. However it can be me­
chan.ically implemented by using actuators located in the rotating reference frame. 
The practical implementation of such a system is currently being considered by 
MBB [4]. 
For a given integer value of wHH = p/rev the the signal given by Eq. (4) can be 
written as a vector with six elements: 
(5) 
The subscript E stands for "Error", to indicate that the input vector represented 
by Eq. (5) corresponds to the input signal, given by Eq. (4). Expanding Eq. (4) 
using trigonometric relations and collecting the harmonic contents of the signal in 
the rotatin_g frame, for wHH = p/rev, yields: 
+ [.~ 8tc -+ e~s Jc os(p + I )1/Jk + [ T8 ts + Te ~c Js in(p + I )1/Jk (6) 
Therefore, the pitch input represented by Eq. (4), with wHH = p/rev, is equivalent 
to a pitch input consisting of three frequencies, namely (p-1 )/rev, p/rev and 
91 - 76.7 
(p + 1) /rev, in the rotating frame. The cosine and sine components of this signal, 
given by Eq. (6) , can be also represented by a vector denoted as 
(7) 
where the subscript R stands for "Rotating", to indicate that the components of the 
vector in Eq. (7) represent inputs provided in the rotating frame. Equation (6) 
provides the relationship between tl;le vectors {8E} and {8R} , ·1Nhich can be written 
in matrix form: 
(8) 
where the transformation matrix is given by: 
0 0 .5 0 0 .5 
0 0 0 .5 -.5 0 
0 0 0 0 0 
[PER] - (9) 
0 I 0 0 0 0 
0 0 .5 0 0 -.5 
0 0 0 .5 .5 0 
and the inverse transformation matrix is given by: 
0 0 I 0 0 0 
0 0 0 I 0 0 
0 0 0 I 0 
[PRE] - [PER]-1 - ( I 0) 
0 I 0 0 0 I 
0 -1 0 0 0 I 
0 0 0 -1 () 
Equations (8) through (10) , imply a one-to-one correspondence between the com­
ponents of the vectors {8E} and {8R} . This means that the six independent quan­
tities, 80 c, 805 , 8cc, 8cs, 85c, 855 , in Eq. (4) are associated with six independent 
physical quantities which represent cosine and sine components of blade pitch in-
91 - 76.8 
puts 'in the rotating frame, as represented by Eq. (7). When two pitch inputs of the 
form given by Eq. (4), with two different frequencies, w 88 = p/rev and q/rev are 
combined, the control input vector {8E} will have a total of 12 elements, \\'ith six 
elements corresponding to each of the two frequencies w 88 = p/rev and q/rev re­
spectively. When formulating the c;ontrol vector {8R} in the rotating frame, using 
Eq. (6) and (7), the total number of elements in the vector {8R} depends on the 
values of the frequencies p and q/rev. If Ip - q I < 2, then there is a frequency 
overlap in the rotating frame corresponding to p/rev and q/rev. Therefore there is 
no one-to-one correspondence between the two pitch control vectors {8E} and {8R} 
, implying that the total number of elements in the vector {8R} is less than that in 
the vector {8E} . However, if Ip - q I > 2 , then both vectors {8d and {8R} will 
contain 12 elements. For a four bladed rotor, the combination of two inputs given 
by Eq. (4), with w 88 equal to 3/rev and 4/rev respectively, produces a pitch input 
with four different frequencies, namely 2/rev, 3/rev, 4/rev and 5/rev, in the rotating 
frame. Note that from Eq. (6), the frequency w88 = 3/rev will produce the fre­
quencies 2,3 and 4/rev in the rotating frame; and w 88 = 4/rev will provide the 
frequencies 3,4 and 5/rev in the rotating frame. After combining the terms corre­
sponding L: :he common frequencies ( namely 3 and 4/rev in this case) , the pitch 
input in the rotating frame will consist of four different harmonics which are 2, 3, 
4, and 5/rev. In this case the vector {8E} will have 12 elements: 
{8E} = {8bc ets e~c e~s e~c 8§s I etc ets et:c et:s e~c ets }T  (II) 
and the vector {8R} will have 8 elements: 
{BR} = {e~ e~ e~ 8§ et: e; et 8§ }T  (12) 
The transformation matrix [PER] is an 8x12 matrix: 
0 0 .5 0 0 .5 0 0 0 0 0 0 
0 0 0 .5 -.5 0 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 .5 0 0 .5 
0 1 0 0 0 0 0 0 0 .5 -.5 0 
[PER] - (13) 
0 0 .5 0 0 -.5 I 0 0 0 0 0 
0 0 0 .5 .5 0 0 I 0 0 0 0 
0 0 0 0 0 0 0 0 .5 0 0 -.5 
0 0 0 0 0 0 0 0 0 .5 .5 0 
91 - 76.9 
4. Solution for the Coupled Rotor /Fuselage Response 
The procedure used for calculating the equilibrium state and the vibratory loads 
on the helicopter is based on a harmonic balance technique. In Ref.[20], different 
approaches to rotor-body coupling are discussed. In this paper, the "fully coupled 
equations approach" is used. Furthermore, in this study the trim state of the heli­
c:opter and the response solution are obtained in a single pass by simultaneously 
satisfying the trim equilibrium and the vibratory response of the helicopter for all 
the rotor and fuselage degrees of freedom. This is an extension of the harmonic 
balance technique which was initially developed for the aeromechanical stability 
problems, such as air resonance, in Refs. [21 J and [22]. A brief description of the 
method is provided below. The equations of motion for the coupled rotor-flexible 
fuselage system· can be symbolically writen as: 
(14) 
( 15) 
(16) 
( 17) 
The vector fb represents the flap,lag and torsional blade equations. The vector fr 
represents the fuselage rigid body motion equations. The vector fe represents the 
fuselage elastic deformation equations. Finally, f,1. represents the inflow equation. 
The trim solution is the vector qt , representing the quantities 
A, 80, 81c, 815 and a:R • The response solution represented by q , consists of the 
following : 
(18) 
The vector qb, for the case of the offset hinged spring restrained blade, contains the 
blade degrees of freedom /Jk, (k, and <Pk , and the generalized coordinates 
c;f1, c;f2, c;f3 , c;I 1, ~12 and c;t1 , for the case of elastic blade model. The vector qr 
consists of the five fuselage degrees of freedom R:'vfx, RMy, 'RMz, Bx, By- The 
fuse,lage yaw degree of freedom is not considered in the present analysis. The vec­
tor qe represents the generalized displacements ~i of the fuselage elastic modes. In 
forward flight a periodic solution in the form of Fourier series is assumed: 
91 - 76.10 
NHb 
qb = qbO + I qbnc cos nt/lk + qbns sin nr/Jk ( 19) 
n=l 
NHr 
qf = qfO + Iqfnc cos nNbr/1 + qfns sin nNbr/1 (20) 
n=l 
NHe 
qe = qeO + Lq enc cos nNbi/1 + qens sin nNbr/1 (21) 
n=l · · 
where NHb, NHf, NHe represent the number of harmonics for the blade, fuselage 
rigid body and fuselage elastic mode response, respectively. 
It is known that only those components of the loads that are integer multiples 
of the rotor passage frequency n x Nb will be transmitted to the fuselage through 
the rotor hub. Hence the response of the fuselage rigid body and elastic degrees 
of freedom contains only integer multiples of Nb per rev harmonics. The vibratory 
response of the fuselage rigid body degrees of freedom was evaluated about. an 
equilibrium state. Hence the constant pitch and roll attitudes of the fuselage are 
not included in the response expressions and they appear only in the trim vector 
qt . The substitution of equations Eqs. (19) - (21) in equations Eqs. (14) - (17) 
corn bined with the harmonic balance technique yields a system of non linear 
coupled algebraic equations. Solution of the nonlinear algebraic system is obtained 
using a Newton algorithm. 
5. Vibration Reduction Using Open Loop Pitch Control 
In vibration reduction schemes using higher harmonic blade pitch control, the 
nonlinear relationship between the Multiple Frequency Pitch Control (MFPC) in­
puts and the vibration outputs is usually represented by a linear quasi- steady 
matrix relationship, in frequency domain. The input control vector {8} consists of 
sine and cosine components of the harmonic pitch inputs and the output vector 
{Z} consists of sine and cosine components of the steady state vibratory loads and 
accelerations. In representing a linear relation between the pitch control vector and 
the vibration vector, two models can be formulated; they are: (a) a local model and 
(b) a global model [23,24]. 
The local model is essentially a linearization about the current step, while the 
global model represents a linearization about the baseline uncontrolled vibration 
level. The local model can be represented as 
{Z(i)} = {Z(i - l )} + [T] ( {8(i)} - {8(i - l )} ) (22) 
This equation can also be written as 
91 - 76.11 
{Z(i)} - {Z(i - I)} = [T] ( {8(i)} - {8(i - 1) } ) 
or 
{~Z(i)} = [T]{~8(i)} (23) 
In this local model, the transfer matrix [TJ relates the change in vibratory response 
{Z(i)} - {Z(i - 1) } due to an increment in the higher harmonic control input 
{8(i)} - {8(i - 1)} evaluated at the current control step. In the local model, the 
transfer matrix [TJ is treated as a function of the control input vector {8} therefore 
at every step i, the [T] matrix is re-evaluated. In the present study on vibration 
reduction using open loop MFPC, the index i in Eqns. (22) or (23) represents the 
computational iteration step, for updating the higher harmonic control pitch input. 
During each iteration step, the coupled rotor/fuselage nonlinear equations are 
solved with the control input evaluated in the previous iteration to obtain the vi­
bration vector {Z(i - l )} . If the vibration reduction is not satisfacory (i.e., if the 
fuselage vibration levels are greater than 0.005g ), the transfer matrix [TJ is up­
dated about the current control input and using a suitable control scheme ( two 
control schemes are described in the next two sections), the new control input is 
evaluated and .this process is repeated until the criterion for the reduction in 
fuselage acceleration levels is met. In the computational program, every iteration 
is initiated externally. Therefore, the analyst can also terminate the iteration proc­
ess at any stage, even if the convergence criterion is not satisfied. 
In the global model, the relation between the vibratory response and the har­
monic pitch input is represented by 
{Z(i)} = {Zo} + [T] {8(i)} (24) 
where {Z0} represents the baseline uncontrolled vibration level. The transfer matrix 
[TJ in the global model, is treated as a constant and it is evaluated about the 
baseline case with zero higher harmonic control input. In the present study, a local 
model is used for vibration reduction. 
5.1 Vibration minimization with pitch input {8E} 
The multi- frequency higher harmonic pitch control input for the k-th blade, as 
given by Eq. (4), consisting of a combination of two pitch inputs with frequencies 
wHH = 3/rev and 4/rev respectively, can be written as 
91 - 76.12 
(25) 
where the superscript 3 and 4 refer to the amplitudes of the pitch components 
corresponding to the harmonic frequencies wHH = 3/rev and 4/rev, respectively. 
The pitch input represented by Eq. (25) can be written as a control vector {8E} , 
having 12 elements as defined below. 
{8E} = {ebc ebs 8tc 8ts e~c e~s I etc ets etc 8ts etc e~s "} T (26) 
A local model is obtained by using Eq. (23) to relate the pitch control vector {8E} 
to twelve components of the vibratory response of the coupled rotor/ fuselage sys­
tem which are selected in an appropriate manner. From the solution of the coupled 
rotor/fuselage response problem, the components of hub shears and hub moments 
as well as the linear acceleration components at a particular location in the fuselage 
( such as C.G ) are available in three orthogonal directions. One of the objectives 
of this study is the simultaneous minimization of the sine and cosine components 
of the 4/rev hub shears together with the 4/rev fuselage acceleration at the C.G. 
For this particular case, the vibration vector {Z} is defined as 
(27) 
where the vector of hub shears ZF and the vector of C.G. accelerations ZA are de~ 
fined as: 
(28) 
T 
ZA = {Acx4c Acx4s Acy4c Acy4s Acz4c Acz4s} (29) 
The local model relating the control vector {8E} and the vibration vector {Z} can 
be written as 
{Z(i)} = {Z(i - I)} + [T EJ ( {8E(i)} - {8E(i - I)}) (30) 
This equation can also be written as 
{Z(i)} - {Z(i - I)} = [T EJ ( {8(i)} - {8(i - I)}) 
91 - 76.13 
The subscript E implies that the l2xl2 transfer matrix [T J corresponds to the 
pitch control vector {8E} . When this approach is implemented in a computational 
scheme for vibration reduction, the matrix [T J is evaluated numerically by a finite 
difference procedure, at every iteration step i. In every iteration, each column of the 
[T J matrix is evaluated by using_ a small increment of 0.005 rad for the corre­
sponding component of the current control input vector {8d and calculating the 
resulting changes in the 4/rev components of the hub shears and the fuselage ac­
celerations. The four columns of the [T J matrix corresponding to the control in­
puts 8bc, 8bs, 8tc and ets are obtained from the vibratory response, of the 
coupled rotor/fuselage system, to a single frequency harmonic input; i.e., 3/rev for 
8bc , 8bs and 4/rev for 8tc , ets . The remaining 8 columns of the transfer matrix 
[T J corresponding to the eight control inputs 
8bc , 8b , 8lc , 8ls , e~c , e~ , eic and eis are obtained from the response of the 
rotor/fuselage system to a pitch control input having two frequencies. This is evi­
dent from examining the pitch control input in Eq. (25) corresponding to these 
eight elements. For example, the pitch input eHHk corresponding to 8bc is 
8bc cos 31h cos 1/Jk which can be written as a sum of harmonics as 
0.5 8bc[ cos 41/!k + cos 21/!k] . The column of the [T EJ matrix corresponding to the 
element 8bc is evaluated by providing a small increment of 0.005 rad to 8tc and 
calculating the resulting changes in the 4/rev loads, from the nonlinear coupled 
rotor/fuselage equations. Since the input consists two frequencies, namely 2/rev 
and 4/rev, it is evident that the column of [T J corresponding to the element 8bc 
is obtained from the response of the nonlinear system to an input having two har­
monic frequencies. 
At each iteration step, the multi-frequency pith control input {8E} required 
for simultaneous reduction of hub shears and fuselage accelerations, is obtained by 
setting Z(i) = 0 in Eq. (30) and solving for {8E(i)} using the following relation 
(31) 
When the iteration is started, i = 1, corresponds to the baseline case without the 
control input {8E} i.e., {8E(O)} = 0 and {Z(O)} represents the baseline uncontrolled 
vibration vector. Note that in this procedure the control vector {8E} can be for­
mulated only when the transfer matrix [T J (which is a 12x12 matrix) 
has an inverse. 
5.2 Vibration minimization with pitch input {8R} 
An alternative and more straightforward approach to this vibration minimiza­
tion problem can be obtained by using directly the pitch control input provided in 
the rotating frame. The multi-frequency pitch control input eHHk in the rotating 
frame can be represented as a sum various harmonics. In this study, the pitch in­
put is assumed to consist of four frequencies, namely, 2,3,4 and 5/rev; which can 
be mathematically represented as 
91 - 76.14 
(32) 
where superscripts 2,3,4 and 5 refer to the frequencies 2,3,4 and 5/rev, respectively. 
For this case, the control input vector {BR} consists of 8 elements and is given 
by 
(33) 
The local control model, relating the 8 elements of the control vector {BR} to the 
12 elements of the vibration vector {Z} can be written as 
{Z(i)} = {Z(i - 1) } + [T RJ ( {BR(i)} - {BR(i - I)}) (34) 
In Eq. (34) the transfer matrix [T RJ is a rectangular matrix of size I2x8. In this 
case, the procedure adopted for calculating the vector {BR} required for simultane­
ous reduction of hub loads and fuselage accelerations, is based on a minimum 
variance control algorithm employing the deterministic controller approach 
[23,24]. The minimum variance control algorithm is based on minimizing a per­
formance index, which in our case; can be simply stated as 
J = {Z(i)} T [Wz ] {Z(i)} (35) 
where [W2 ] is a weighting matrix. Imposing the condition 
(36) 
and using the local linear model, given in Eq. (34), the control input vector {BR} can 
be obtained from 
(37) 
In the present study, the weighting matrix [Wz J is taken as a unit matrix. Thus, 
the performance index essentially represents the sum of the squares ot the 4/rev 
sine and cosine components of the hub shears and fuselage accelerations. When the 
91 - 76.15 
index i in Eq. (34) has a value i = I, it corresponds to the baseline case with zero 
values for the pitch control input {8R} . This deterministic controller approach is 
a simplified version of the control methodology described in Refs. 23 and 24. 
In every iteration, each column of the [T RJ matrix is evaluated by using a small 
increment of 0.005 rad for the corresponding component of the current control in­
put vector {8R} and calculating the resulting changes in the 4/rev components of 
the hub shears and the fuselage accelerations. The eight columns of the [T RJ ma­
trix corresponding to the eight control inputs 8t , 8§ , 8t , 8~ , e~ , ei , 8t and 8l 
are obtained from the responses, of the coupled rotor/fuselage system, to a single 
frequency harmonic input applied one at ·a time. This approach differs somev,'hat 
from that used in the evaluation of some of the columns of the [T EJ matrix, de­
scribed in the previous section. Recall that 8 columns of the [T J matrix were ob­
tained using control inputs which consisted of a combination of two harmonic 
frequencies. 
5.3 Relation between the transfer matrices [T RJ and [T EJ 
Since there is a linear relationship between the MFPC inputs, represented by the 
vectors {8E} and {8R} , a relationship between the [T EJ and [T RJ matrices corre­
sponding to these vectors can also be obtained. The linear transformation between 
{8R} and {8E} can be written, using Eq. (8) 
(38) 
where the vector {8E} is I2xl, (Eq. 26), and the vector {8R} is 8xl, (Eq. 33), thus 
the transformation matrix [PER] is 8xI2, Eq. ( 13). Combining Eqs. (34) and (38) 
one has · 
{Z(i)} = {Z(i - 1)} + [T RJ [PER] ( {8£(i)} - {8£(i - 1)}) (39) 
Comparing Eqs. (30) and (39) implies 
(40) 
In this relation, the transfer matrix [T J whose size is I2xl2, is formulated by 
multiplying two matrices [T RJ and [PER] whose rank is less than 12. Thus, the 
determinant of the matrix [T J must be zero and hence the matrix [T J is singular. 
However, in the vibration reduction calculations using the control vector {8E} , the 
transfer matrix [T J was evaluated following the procedure outlined in Section 5.1 
and the use of Eq. (40) was bypassed. Simultaneous reduction of hub shears and 
fuselage accelerations based upon the pitch control vector {8E} implies the existence 
of the inverse of [T J . However if the matrix [T J is singular, as implied by Eq. 
(40) its inverse will not exist and the method of vibration reduction based on the 
control vector {8E} and transfer matrix [T EJ , and described in Section 5.1, is 
doomed _to fail. The vibration reduction analyses performed with the pitch control 
vector {8E} did not pose any numerical problem and the scheme was successful in 
minimizing the hub loads and fuselage accelerations simultaneously [17 ,18]. This 
91 - 76.16 
poses an apparent paradox. To fully understand the situation, a term by term 
comparison was made between the transfer matrix [T J evaluated using the pro­
cedure outlined in Sec. 5.1 and the matrix [T J based on Eq. (40). It was foun.d 
that while the columns corresponding to the control inputs 8bc , 8bs , 8'bc and 8ts 
are identical, there were differences in the remaining eight columns corresponding 
to the other eight control inputs 8t~ , 8ts , 8lc , 8l5 , 8~c , 8~5 , eic and eis . The 
reason for this difference can be understood, if one recognizes the fact that the 
eight columns corresponding to the eight control inputs 
8tc , 8b , 8ic , 8is , e~c , e~s , eic and eis were obtained from the responses, of 
the coupled rotor/fuselage nonlinear dynamic system, to a pitch control input hav­
ing two harmonic frequencies. Since the equations of motion of the coupled 
rotor/fuselage system are nonlinear, the principle of superposition ( of two inputs 
having different frequencies ) does not hold. On the other hand the calculation of 
the [T EJ matrix based on Eq. (40) is equivalent to evaluating the columns of the 
[T EJ matrix by superposing the responses corresponding to each harmonic fre­
quency; since the columns of [T RJ are obtained using a single frequency harmonic 
input, as explained in Sec. 5.2. Therefore the transfer matrix [T EJ formulated using 
the procedure in Sec. 5.1 is different from the [T EJ matrix obtained from Eq. (40). 
It will be demonstrated in the results section that for the coupled rotor/fuselage 
example treated in this study, the MFPC vector {8E} obtained using the procedure 
described in Sec. 5.i and the control vector {8R} represent almost identical pitch 
variation in the rotating frame and both inputs produce simultaneous reduction in 
hub shears and fuselage accelerations. 
-6. Results -and Discussion 
The results presented in this paper deal with four separate items. The first part 
considers the problem of simultaneous reduction of hub shears and fuselage accel­
erations using open loop MFPC in the rotating reference frame, for coupled 
rotor/flexible fuselage systems. In the second part, the results of a sensitivity anal­
ysis is provided showing the behaviour of the 4/rev hub loads with respect to the 
frequency of the pitch control input in the rotating frame. A comparative study of 
the results obtained for the vibration reduction using the two control vectors {8E} 
and {8R} is presented, in the third section. In the final part, the results showing 
influence of fuselage flexibility on the simultaneous reduction of hub loads and 
fuselage accelerations is considered. 
6.1 Vibration reduction using Multiple Frequency Pitch Control 
The results presented in this portion of the paper are aimed at developing a 
physical undestanding for the open-loop MFPC scheme when applied to the vi­
bration reduction problem of coupled rotor/flexible fuselage systems. 
Two separate cases are considered. For the first case, the four bladed rotor is 
idealized as a offset hinged spring restrained blade model with coupled flap-lag­
torsional dynamics for each blade. The data used for this case is presented in Table 
I. For the second case, each blade of the four bladed rotor is modeled as a fully 
elastic hingeless blade, represented by six uncoupled rotating modes, corresponding 
to the first three flap, first two lag and a torsional mode. The data used for this 
caes is provided in Table 2. 
91 - 76.17 
For both cases the fuselage has five rigid body degrees of freedom combined 
with six elastic degrees of freedom. The fuselage elastic degrees of freedom consist 
of two flexible modes for bending in the vertical plane, bending in the horizontal 
plane and torsion along the X 1 axis, respectively. 
All the results were calculated for an advance ratio ofµ = 0.3, unless otherwise 
stated. In these vibration studies; the worst possible case with a fundamental 
bending natural frequency in the vertical plane equal to 4/rev, was considered. 
Very high stiffnesses ( fundamental natural frequency = 3000/rev ) in horizontal 
bending and in torsion, were assumed. This essentially implies that the fuselage 
flexible degrees of freedom in lateral bending and torsion are suppressed. 
Figure 2 illustrates the MFPC pitch angle variation together with its harmonic 
contents, in the rotating frame, for the first three iterations. The MFPC control 
signal corresponds to the control vector {8E} with the frequencies wHH ;:: 3/rev and 
4/rev. The first iteration case (d otted line) corresponds to the control input calcu­
lated about the baseline uncontrolled vibration level. Examination of the harmonic 
contents of this signal reveals that the 2/rev content has a maximum amplitude of 
I deg.; with 0.I5deg., 0.003deg., and 0.0007deg., respectively in the 3,4 and 5/rev 
harmonic components. It can be seen from Fig. 2 that the difference between the 
signals corresponding to the three itrations, is very small. This implies that the 
control pitch input has converged to a value which provides a very good reduction 
in vibratory loads and accelerations, even after the first iteration. 
The peak to peak values of the hub loads and fuselage C.G. accelerations for 
each iteration as well as the baseline values are shown in Table 3. For all practical 
purposes vibrations are completely suppressed after first iteration except for the 
hub rolling moment (MHX) for which a slight increase can be noticed in the second 
iteration. 
Figure 3 illustrates the MFPC pitch angle variation corresponding to the control 
vector {8E} obtained with two different combination of frequencies namely, (I) 
wHH = 3 and 4/rev and (2) wHH = 3 and 5/rev. It can be seen from Fig. 3 \hat 
the pitch control vectors {8E} corresponding to these two cases produce almost 
identical pitch angle variation in the rotating frame. When considering the har­
monic content associated with the 2, 3, and 4/rev components for these two control 
signals, it is evident that they have practically the same amplitudes. However, the 
amplitute of the 5/rev content is substantially greater in the case of 3 and 4/rev 
combination than that corresponding to the 3 and 5/rev MFPC combination. The 
amplitude corresponding to the 6/rev harmonic in the 3 and 5/rev MFPC signal is 
approximately 0.007 deg. Since the 3 and 4/rev combination MF.PC signal does not 
contain a 6/rev harmonic component in the rotating frame, its amplitude is zero. 
Figure 4 shows the baseline vibratory hub loads and the fuselage accelerations 
together with the controlled levels obtained with the two combinations of MFPC 
inputs. It can be seen from this figure that when a MFPC signal consisting of a 
superposition of 3 and 4/rev or 3 and 5/rev inputs is provided, the vibratory hub 
loads and the C.G. accelerations are reduced simultaneously. 
Figure 5 shows the pitch angle variation for the MFPC ip the rotating frame for 
three iteration steps; for the case where the blades are represented by a fully elastic 
blade model. The first iteration case (dotted line) corresponds to the control input 
calculated about the uncontrolled baseline vibration levels. A fourier analysis of 
91 - 76.18 
the final MFPC signal (solid line) obtained after the 2nd iteration indicates that the 
2/rev harmonic content is again dominant, with 30% content in 3/rev and a 10% 
to 20% content in 4/rev and 5/rev. 
The peak to peak values of the hub loads and the fuselage C.G. accelerations for 
each iteration as well as the baseline values are shown in Table 4. The effectiveness 
of the MFPC scheme in reducing t·he vertical acceleration at the fuselage C.G. is 
evident from this Table which indicates a reduction of 99% after the second iter­
ation. Furthermore, it is also evident from Table 4 that the hub shears and mo­
ments are also substantially reduced. 
6.2 Sensitivity of hub loads to hieher harmonic pitch input 
The results presented in the previous section indicate that for simultaneous re­
duction of hub shears and fuselage accelerations,the MFPC signal in the rotating 
frame appears to be composed of a predominantly 2/rev variation with very small 
contributions from other harmonics. The authors were intrigued by this behavior 
which seems to emphasize the importance of the 2/rev component. Therefore a 
careful sensitivity study was carried out so as to gain a better understanding of the 
effect of introducing a single harmonic pitch control input, in the rotating frame, 
on the vibratory 4/rev hub shears in a four bladed rotor. In this sensitivity study, 
the offset hinged spring restrained blade model was used. The data for the heli­
copter configuration are given in Table 1. During this sensitivity study, the 
fuselage was assumed to be rigid. This was achieved by setting the natural fre­
quencies of the fuselage elastic modes to 3,000/rev. 
When a single frequency pitch input, with frequency p/rev in the rotating frame, 
is introduced , the signal can be represented by: 
(41) 
where AP is the amplitude a_nd c/>P is the phase angle of the signal. The superscript 
p representing the frequency of the harmonic input can be 2,3,4 or 5/rev. Without 
loss of generality one can limit such a study on the cosine and sine components of 
the vertical hub shears F Hz4c and F Hz4s· 
Figure 6 shows the peak to peak vertical hub shear when a pitch input of the 
type given by Eq. (41) is applied. The amplitude AP was fixed at 0.0005 rad and 
the phase angle was varied between O to 360 degrees, in 60 degree increments. 
Four different frequencies: 2,3,4 and 5/rev were considered and the baseline case 
is also shown. It is evident from Fig. 6, that for a 4/rev input with an amplitude 
of 0.0005rad the peak to peak vertical hub shear is never reduced below its baseline 
value. However when the pitch inputs have frequencies of 2/rev, 3/rev and 5/rev, 
respectively, a reduction in vibratory vertical hub shear is achieved. 
In order to gain a better understanding of the curves plotted in Fig. 6, a more 
comprehensive parametric study was performed. In this detailed study plots of the 
peak to peak vertical hub shear as a function of the phase angle c/>P, with the am­
plitude AP as a parameter, were obtained. In order to choose the proper range of 
values for the amplitudes, a semi-theoretical anlysis was conducted first, so as to 
be able to identify the amplitude which produces a complete cancellation of the 
peak to peak vertical hub shear. This amplitude denoted by AP min, is called the 
91 - 76.19 
"critical higher harmonic amplitude". A derivation of this amplitude is presented 
in Appendix A. 
Figures 7 and 8 contain two series of plots each. Each series of plots depic\s the 
effect of a single frequency pitch input, having five different amplitudes ( 1/2 A 
min, Amin, 3/2 Amin, 2 Amin and 5/2 A min.), on the peak to peak vertical hub 
shears. Where A min is based on the approximate value calculated from Eq. (A3), 
given in Appendix A. For each of these amplitudes the phase angle was varied 
from O to 360 degrees, in 60 degree increments, and the coupled rotor/fuselage 
computer code was used to obtain the response. In Figs. 7 and 8 only the peak to 
peak vertical hub shear is plated, but similar plots can be obtained for all vibratory 
loads and fuselage accelerations. A comparison between the critical amplitudes A 
min for the frequencies 2/rev, 3/rev, 4/rev and 5/rev indicates that the peak to peak 
vertical hub shear is most sensitive to 4/rev input (Amin = 0.00016 rad) and least 
sensitive to 2/rev input (Amin = 0.0049 rad). 
In order to verify this important behavior, which has not been emphasized in the 
literature before, an independent analytical sensitivity study of the vertical hub 
shear of a four bladed rotor system was conducted. In this simplified model, each 
blade was represented by a centrally hinged spring restrained blade model having 
only flap degree of freedom. A multiple frequency pitch input was introduced in 
the rotating frame. The response was obtained from the linear flap equation, using 
a quasi-steady aerodynamic model with time varying pitch, and neglecting reverse 
flow effects. A concise description of this study can be found in Appendix B. The 
conclusions of this analytical study confirmed and reinforced the trends shown in 
Figs. 7 and 8. 
It is important to emphasize that amplitudes of the order of 0.00016 rad 
(0.00917 degrees) as in the case of 4/rev input are impossible to achieve in practice. 
For the 4/rev pitch input case, it is evident from Fig. 8, that when the amplitude 
exceeds 2 A min, vibration reduction cannot be achieved. Therefore amplitudes 
greater than 0.0183 degrees will result in an increase of the peak to peak vertical 
hub shear above the baseline value for any value of the phase angle </>4 • 
The sensitivity of the 4/rev hub shears, to the various single frequency pitch in­
puts, is the basis for understanding the large 2/rev content in the MFPC signals 
obtained in Refs. 17-19, and shown in Figs. 2,3 and 5. Because the 2/rev pitch in­
put is least effective in reducing the vibratory loads, a larger amount of 2/rev input 
is required to produce an amount of vibration reduction comparable to that asso­
ciated with 3/rev, 4/rev and 5/rev components, which despite their small ampli­
tudes, are much more effective in reducing vibration levels. Therefore, while the 
MFPC signal in the rotating frame appears to consist of a predominantly 2/rev 
signal, in reality much of the vibration reduction is achieved by 3/rev, 4/rev and 
5/rev components which are not clearly visible because they are overshadowed by 
the 2/rev component, due to their relatively small magnitude. For this reason the 
physical explanation given in Refs. 17-19, attempting to rationalize the importance 
of the 2/rev component is incorrect and misleading. Fortunately we persevered and 
found the correct explanation, which provides useful insight into the reduction of 
vibration levels using high frequency pitch control in the rotating frame. 
6.3 Comparison of vibration reduction schemes with control vectors {8E} and {8R} 
91 - 76.20 
The problem of simultaneous reduction of hub shears and fuselage accelerations 
was solved in Section 5.2 following the minimum variance controller approach. In 
this section, the MFPC signal represented by the control vector {8R} is introduced 
to the coupled rotor/flexible fuselage system and the vibration reduction achieved 
is compared with that obtained when using the the control vector {8E} . For con­
venience, this study is based on the offset hinged spring restrained blade model. 
The fuselage is assumed to be very stiff in lateral bending and torsion; however the 
fundamental frequency in vertical bending is assumed to be 4/rev. The calculations 
are performed for an advance ratio µ = 0.3 
Figure 9 depicts MFPC variation in the rotating frame corresponding to the nvo 
control pitch input vectors {8E} and {8R} , respectively; as well as their harmonic 
components. It is interesting to note that for the example considered, for simul­
taneous reduction of hub loads and fuselage accelerations, both control vectors 
{8E} and {8R} provide almost identical pitch variation in the rotating frame. This 
result indicates that for the example problem the control vectors converged to the 
same MFPC signal in the rotating frame while achieving the desired reduction in 
the vibratory response. 
Figure 10 shows the baseline peak-to-peak vibratory hub shears, hub moments 
and the accelerations at the C.G. of the fuselage together with the controlled vi­
bratory levels. Both approaches, namely vibration control with either the control 
vector {8E} or {8R}, provided a very good reduction in the hub loads and C.G. 
accelerations. The vibratory hub moments obtained with minimum variance con­
troller, Eq. (37), which in Fig. 10 is denoted optimal control is almost equal to that 
obtained with the control vector having a frequency combination of wHH = 3/rev 
and 4/rev. Note that these results were obtained with global controller which is 
equivalent to the local controller with one iteration. 
6.4 Influence of fuselage flexibilitv on the vibration reduction scheme 
Recall that the vibration reduction studies were conducted with a flexible 
fuselage which allowed only vertical bending. Very high stiffnesses in lateral 
bending and torsion were used to effectively suppress these two elastic degrees of 
freedom. In this section, the fuselage stiffnesses in lateral bending and in torsion 
are reduced so as to study the effect of fuselage flexibility on the vibration re­
duction using open loop pitch control. For this case the fundamental frequency in 
vertical bending of the fuselage is w 8 v1 = 4.0/rev; in horizontal bending 
w 8 tt 1 = 4.0/rev and the fundamental torsional frequency is wT1 = 3.5/rev . This 
represents an unfavorable fuselage frequency placement which can be easily excited 
by a four bladed rotor system. 
The approach used for simultaneous reduction of hub loads and the fuselage 
accelerations is based on the minimum variance control algorithm, utilizing the 
control vector {8R} . This approach was selected because it has a sound math-
ematical basis. . 
Figure I I depicts the MFPC input in the rotating frame together with its har­
monic content. Again the control signal is predominantly 2/rev with additional 
harmonic content consisting of: about 5<% in 3/rev, 2% in 4/rev and 11 % iQ 5/rev. 
The effect of this control signal on the peak-to-peak hub loads and fuselage ac­
celerations is shown in Fig. 12. While all the inplane hub shears show a substantial 
91 - 76.21 
reduction, the reduction in vertical hub shear is only marginal. The roll moment 
has increased by a factor of three from its baseline value. The acceleration levels 
at the C.G. showed a reduction in the Y-direction (ACY); but the acceleration in 
vertical direction (ACZ) has increased. This result indicates that a simultaneous 
reduction in all the components h_ub loads and the C.G. accelerations was not 
possible. The reason for the increase in some components of the vibratory loads 
can be explained by analyzing the harmonic contents of the vibratory hub loads 
and C.G. accelerations. 
The harmonic components of the hub loads and accelerations is given in Table 
5. It can be seen from this Table that the 4/rev content of the hub shears in all di­
rections is reduced;however the 8/rev content of the hub shears exhibits a substan­
tial increase. Recall that in this case the performance index consists only of the 
4/rev contents of the hub shears and C.G. accelerations. Since the 8/rev contents 
of the loads is essentially uncontrolled, these components are influenced by the in­
troduction of a higher harmonic MFPC control pitch input in the rotating frame. 
7. Concluding Remarks 
This paper describes an attempt to develop a multi frequency pitch control 
(MFPC) technique, which can produce simultaneous reduction of hub shears and 
fuselage accelerations in a coupled rotor/flexible fuselage helicopter model. Two 
types of control vectors were used in minimizing the vibratory hub loads and 
fuselage accelerations. The influence of fuselage flexibility on the effectiveness of 
the MFPC control pitch input was also studied. The most interesting conclusions 
obtained are summarized below. 
(I) When the fuselage flexibility .was limited to vertical bending only , ( and the 
lateral bending and torsion were essentially suppressed), the MFPC can reduce si­
multaneously the ht;b shears and fuselage C.G. accelerations. 
(2) The shape of the MFPC signal in the rotating frame depends on the partic­
ular model used to represent the blade flexibility. When the offset hinged spring 
restrained blade model was used, the MFPC signal has substantial 2/rev content 
with 17% content in 3/rev and a 4% content in 4/rev. For the fully elastic blade 
model, the 2/rev content in MFPC signal was reduced, however it still was the 
largest component with 30% content in 3/rev and about 10% to 20% content in 
4/rev and 5/rev. 
(3) A careful sensitivity analysis conducted revealed that the introduction of a 
single frequency pitch input with a frequency 2,3,4 or 5/rev in the rotating frame 
is capable of reducing the vibratory hub shears. For the four bladed rotor system, 
the vertical hub shear reduction required a pitch angle whose amplitude is highest 
for 2/rev pitch input and lowest f9r 4/rev pitch input in the rotating frame. When 
considering the the sensitivity of th~ hub shears to single frequency pitch input in 
the rotating frame, the 4/rev vertical hub shear is least sensitive to 2/rev pitch in­
put; highly sensitive to 4/rev pitch input;and moderately sensitive to 3/rev and 
5/rev pitch input. 
A high sensitivity with respect to a particular harmonic (such as 4/rev for a four 
bladed rotor system) implies the need for a precise control of the pitch inputs. For 
harmonics which have a strong influence on the hub shears, the requirement of a 
91 - 76.22 
very small amplitude may be physically unrealizable in an actual vibration re­
duction device. 
(4) The MFPC control signal obtained using two different control vectors, 
provided identical pitch angle variation in the rotating frame for simultaneous re­
duction of hub loads and C.G accelerations. This confirms that the.error found in 
our previous studies, Refs. 17-19, has been corrected. 
(5) Simultaneous reduction of all the components of hub shears and fuselage 
accelerations for a completely flexible fuselage, containing a broad frequency spec­
trum requires additional study. 
APPENDIX A 
Using the linear assumption, represented by Eq. (24) and expanding Eq. (4 1 ), 
the vertical vibratory hub shears can be written as: 
FHz4c = FoHz4c + T cc A sin</>+ T cs A cos q> 
FHz4s = FoHz4s + T SC A sin <P + T SS A cos q> (A.I) 
where FoHz4c, FoHz4s represent the baseline vetical hub shear components and the 
superscript p, in Eqs. (A.I) was deleted for the sake of convenience. For a linear 
system with time-invariant coefficients the [T] matrix has the following properties 
T CS = - T SC = Tb (A.2) 
which have been also noted in Ref. 23. For this case it can be shown that the 
minimum peak to peak vertical hub shear will be zero when the amplitude of the 
pitch input is equal to 
(A.3) 
and the minimum peak to peak vertical hub shear will be equal to the baseline 
value when the amplitude of the input is equal to 2 A min. If the computer code 
is used to calculate the coefficients T cc, T cs, T sc and T ss by a finite difference ap­
proach, such as the one employed for calculating the elements of the T matrix in 
Section 5, the property depicted by Eqs. (A.2) will be only approximately valid. 
This is due to the nonlinearity in the equations as well as the periodic coefficients 
associated with forward flight [23]. It is therefore possible to obtain approximate 
values for Ta, Tb in Eqs. (A.2) by averaging the coefficients 
Ta = T(T cc + T ss) 
91 - 76.23 
(A.4) 
These approximate values for Ta and Tb are used in Eq. (A.3) to obtain approxi­
mate values for A min. 
APPENDIX B 
The equation of motion for a centrally hinged spring restrained rigid blade 
undergoing flapping motion, in forward flight can be written as 
(B.l) 
In Eq. (B. I), the terms associated with noncirculatory lift are not considered, re­
verse flow effe~ts have been neglected and the inflow is assumed to be uniform. 
Equation (B. I) is obtained after simplifying the more general rotor blade equations 
derived in Ref. 25. The blade root shear in the vertical direction, also from Ref. 
25, can be written as 
z = -~a- -I + -µ 2 . µ 
F + µ sm t/1 - - 2 ) 
cos 2t/l 8 -
2Nb {( 3 2 2 
(+ +  µ sin t/1 )i - (+ +  ~ sin t/1 )iJ + 
b(+ 2 
+  µ sin v,)e - ( ~ cos ,J,+ ~ sin 2if,}} (B.2) 
The blade root shear is nondimensionalized with respect to pn R 2(QR)2 . 
Assuming that e and f3 consist of five harmonics (including a constant term) and 
substituting these harmonic expansions for f3 and e in Eq. (B.2) yields the blade 
root shear in the vertical direction as a function of harmonics of flap response and 
pitch input. 
91 - 76.24 
For a four bladed rotor, the hub vertical shear can be obtained by summing up 
Eq. (B.2) for four blades. The resulting 4/rev vertical hub shear, ::: terms of higher 
harmonic blade pitch input and higher harmonic flap response, c2.n be written as 
2 . 
2a a µ 2 -Tµ 3 3 3 
F Hz4 = [ cos 41/1 { -Tc 8s + Tbµ8c + 
(- 1 + -µ 2) e4 4 µ5 5 5 
5 - 2b8c - :::...ec - -bµ8
3 2 2 2 5 -
2 
-µ4f Jic - µ/J 3S + -43{
 
 J 4C - -µ/ J 5S }] (B.3) 
4 
From Eq. (B.3), it is evident that the 4/rev hub shear is least affected by the 
2/rev pitch input and the 2/rev blade response because these terms are multiplied 
by µ 2 , and therefore for advance ratios ofµ< 0.4 these terms will be significantly 
smaller than the other terms. On the other hand, the 3/rev and 5/rev pitch inputs 
are multiplied by µ ; and the 4/rev pitch inputs are multiplied by the term 
1/3 + µ 2/2 . The relative orders of magnitude of the coefficients of the harmonics 
of the pitch input and flap response clearly indicate that the 4/rev vertical hub 
shear is most sensitive to 4/rev harmonics, moderately sensitive to 3/rev and 5/rev 
harmonics and least sensitive to 2/rev harmonics of the pitch input in the rotating 
frame. 
Aknowledgements 
This research was funded by NASA Ames Research Center under grant 
NAG2-477, the useful comments of the grant monitor Dr. S. Jacklin are gratefully 
acknowledged. 
91 - 76.25 
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3. Ham, N., "Helicopter Individual-Blade-Control and Its Applications," 
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4. Richer, P., Eisbrecher, H.D. and Kloppel, V., " Design and Flight Tests 
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III.6.3.l-III.6.3.9 
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Control of Helicopter Vibration," 39th AHS Forum, St. Louis, Missouri, 
May 1983. 
7. Chopra, I., and J.L. McCloud, "A Numerical Simulation Study of 
Open-Loop, Closed-Loop and Adaptive Multicyclic Control Systems," 
AHS Journal, Vol. 28, No. l, January 1983, pp. 63-77. 
8. Robinson, L., and Friedmann, P.P., "Analy~ic Simulation of Higher 
Harmonic Control Using a New Aeroelastic Model," Proc. 30th 
AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics and 
Materials Conference, Mobile, Alabama, April 1989. AIAA Paper No. 
89.1321. 
9. Robinson, L., and Friedmann, P.P.," A Study of Fundamental Issues in 
Higher Harmonic Control Using Aeroelastic Simulation," AHS Journal, 
Vol. 36, No. 2, April 1991, pp. 32-43. 
I 0. Nguyen, K. and Chopra, I., "Application of Higher Harmonic Control 
(HHC) to Rotors Operating at High Speed and Maneuvering flight," 
Proceedings of the 45th Annual Forum of the American Helicopter 
Society, Boston, MA, May 1989, pp 81-96. 
91 - 76.26 
11. Molusis, J.A., Hammond, C.E., and Cline, J.H., "A Unified Approach to 
the Optimal Design of Adaptive and Gain Scheduled Controllers to 
Achieve Minimum Helicopter Vibration," AHS Journal , Vol.28, No.2, 
April 1983, pp. 9-18. . 
12. Lehmann, G., "The Effect of Higher Harmonic Control (HHC) on a 
Four-Bladed Hingeless Model Rotor," Vertica , Vol.9, No.3, 1985, pp. 
273-284. 
13. Shaw, J., Albion, A., Hanker, E.J., and Teal, R., "Higher Harmonic 
Control: Wind Tunnel Demonstration of Fully Effective Vibratory Hub 
Force Suppression," AHS Journal, Vol.34, No.I, January 1989, pp. 14-25. 
14. Wood, E.R., Powers, J.H., Cline, J.H., and Hammomd, C.E.,. "On 
Developing and Flight Testing a Higher Harmonic Control System," AHS 
Journal, Vol.30, No.I, January 1985, pp. 3-20. 
15. Miao, W., and Frye, H.M., "Flight Demonstration of Higher Harmonic 
Control (HHC) on S-76," 42nd AHS Forum, Washington, D.C., June 
1986. 
16. Polychroniadis, M., and Achache, M., "Higher Harmonic Control: Flight 
Tests of an Experimental System on SA 349 Research Gazelle," 42nd AHS 
Forum, Washington, D.C.,_ June 1986. 
17. Papavassiliou ,I., Venkatesan, C. and Friedmann, P.P., "A Study of 
Coupled Rotor-Fuselage Vibration with Higher Harmonic Control Using 
a Symbolic Computing Facility," Proceedings of the 16th European 
Rotorcraft forum, Glasgow U.K., Sept. 18-21, 1990, pp. 111.7.3.l-IIl.7.3.23 
18. Papavassiliou, I., " Nonlinear Coupled Rotor/Fuselage Vibration Analysis 
and Higher Harmonic Control Studies for Vibration Reduction in 
Helicopters," Ph.D. Dissertation, Mechanical, Aerospace and Nuclear 
Engineering Department, University of California, Los Angeles, January 
1991. 
19. I. Papavassiliou, P. P. Friedmann and C. Venkatesan, "Coupled 
Rotor-Flexible Fuselage Vibration Reduction Using Open· Looµ Higher 
Harmonic Control," AIAA Paper No. 91-1217, Proceedings of the 32nd 
AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics and 
Materials Conference, Baltimore, Maryland, April 8-10, 1991, pp. 
2011-2035. 
20. Stephens, W.B. and Peters, D.A., "Rotor-Body Coupling Revisited," AHS 
Journal , Vol. 32, No. 1, January 1987, pp. 68-72. 
91 - 76.27 
21. Takahashi, M.D. and Friedmann, P.P., "Active Control of Helicopter 
Helicopter Air Resonance in Hover and Fonvard Flight," AIAA Paper 
88-2407-CP, Procedings AIAA/ASME/ASCE/AHS 29th Structures, 
Structural Dynamics and Material Conference, Williamsburg VA, April 
1988, pp 1521-1532. 
22. Takahashi, M.D., "Active Control of Helicopter Aeromechanical and 
Aeroelastic Instabilities," Ph.D. Dissertation, Mechanical Aerospace and 
Nuclear Engineering Department, University of California, Los Angeles, 
June 1988. 
23. Johnson, W., "Self Tuning Regulators for Multicyclic Control of 
Helicopter Vibration," NASA TP 1996, 1982 
24. Davis, M.E., "Refinement and Evaluation of Helicopter Real-Time 
Self-Adaptive Active Vibration Controller Algorithms," NASA CR 3821, 
August 1984. 
25. Venkatesan C., and Friedmann, P., "Aeroelasic Effects in Multi-Rotor 
Vehicles with Applications to a Heavy-Lift System, Part 1 : Formu-lat..i on 
of Equations of Motion," NASA-CR 3822, August I 984. 
91 - 76.28 
TABLE I 
Data for coupled rotor/fuselage model used for offset hinged spring restrained 
blade configuration 
Rotor Data 
µ = 0.3 a = 5.7 
e = 0.0 Cdo = .01 
y = 5.0 Nb= 4 
CJ = .05 (3T = 0 rad 
WFI = 1.15 ex= .0 
Wu = 0.57 Cy = .0 
WT! = 4.5 cz = .0 
Fuselage Data 
lF = 1. Cw= 0.005 
XMH/IF = .196 XMc/lF = .196 
ZMH,1F = .233 ZMc/lF = 0.0 
XM.JlF = ; 196 ZMA,1F = 0.0 
TABLE 2 
Data for the coupled rotor/fuselage model used for the elastic blade configuration 
Rotor Data 
µ = variable a= 2n 
e = 0.0 Cdo = .01 
y = 5.5 Nb= 4 
CJ = .07 (3T = 0 rad 
WF = 1.123,3.41,7.65 ex= .0 
WL = 0.735,4.485 Cy = .0 
WT= 3.17 cz = .0 
Fuselage Data 
IF = 2. 
XMH/IF = .3 
ZMH,1F = .15 
XM.JlF = .3 
91 - 76.29 
TABLE 3 
Iteration history of peak to peak hub loads and C.G. accelerations for 3 and 4/rev 
MFPC combination using offset hinged blade model (local MFPC). 
Baseline ITER. #1 ITER. #2 ITER.#3 
Hub Forces ( Nt ) 
FHX 317.2 27.5 1.36 1.22 
FHY 301.8 27.5 1.08 0.86 
FHZ 1508.8 61.1 6.67 2.03 
Hub Moments ( Nt.m ) 
MHX 536.5 150.5 96.99 99.90 
MHY 506.5 31.6 1.26 0.96 
C.G. Accelerations ( g) 
ACX 3.9*1E-3 3.4*1E-4 l.6*1E-5 1. 5*1E-5 
ACY 3.7*1E-3 3.4*1E-4 l.2*1E-5 l.2*1E-5 
ACZ 1.463 0.0171 0.0051 0.0038 
91 - 76.30 
TABLE 4 
Iteration history of peak to peak hub loads and C.G. accelerations for 3 and 4/rev 
MFPC combination for the elastic blade model(local MFPC). 
Baseline ITER. #1 ITER. #2 ITER.#3 
Hub Forces ( Nt ) 
FHX 448.8 244.3 99.2 45.3 
FHY 1258.8 362.1 147.8 98.3 
FHZ 3615.0 318.2 147.5 61.0 
Hub Moments ( Nt.m ) 
MHX 3184.4 4108.0 781.6 1091.1 
MHY 2881.0 3444.0 612.6 417.1 
C.G. Accelerations ( g) 
ACX 5.9*1E-3 4.l*lE-4 6.7*1E-4 l.2*1E-4 
ACY l.6*1E-2 3.6*1E-3 l.3*1E-4 l.6*1E-4 
ACZ 7.731 0.485 0.24 0.0039 
91 - 76.31 
TABLE 5 
Hub shears, hub moments C.G accelerations for the MFPC scheme with the offset 
hinged blade model and fully flexible fuselage. 
Baseline with MFPC 
Hub Forces ( Nt) 
4p amplitude 303.23 44.48 
FHX 8p amplitude 0.23 19.04 
peak-to-peak 603.06 103.59 
4p amplitude 497.45 197.36 
FHY 8p amplitude 0.16 18.32 
peak-to-peak 989.18 400.34 
4p amplitude 52.81 24.95 
FHZ 8p amplitude 0.31 25.72 
peak-to-peak 104.97 86.97 
Hub Moments ( Nt*m ) 
4p amplitude 443.16 1700.97 
MHX 8p amplitude 0.06 14.36 
peak-to-peak 877. so 3370.39 
4p amplitude 411.62 67.23 
MHY 8p amplitude 0.07 14.08 
peak-to-peak 818.40 141.74 
C.G. A.ccelera tions ( g ) 
4p amplitude 0.37*1E-2 0.54*1E-3 
ACX 8p amplitude 0.28*1E-S 0.23*1E-3 
peak-to-peak 7.35*1E-3 l.22*1E-3 
4p amplitude 0.60137 0.14579 
ACY 8p amplitude 1.06*1E-S 0.62*1E-3 
peak-to-peak 1.19134 0.29107 
4p amplitude 0.14265 0.15892 
ACZ 8p amplitude · 2.24*1E-5 0.45*1E-3 
peak-to-peak 0.23701 0.31658 
91 - 76.32 
I 
OEFORME 
BLADE 
.... .... ~ 
, I 
E.A ' ', ' 
' 
. 'I'  I 
J..."./  I' I 
I }',, 
UNDEFORMEO BLADE ',~, ', 
X 
R,.., 
~Iyt, ch 
Hub Center 
E - Hinge orrset Point 
A - Point Ht on lr:-th Blade 
B - Blede Center or Mess 
C - ruselege Center or Mess 
Figure I: Coupled rotor/flexible fuselage model 
91 - 76.33 
-· 1s t iteration advance ratio=0.3 
-- 2nd iteration 
1.50 
- - 3rd iteration 
1.00 
Cl 
Cl) a. MFPC input 
·"C 
-- 0.50 
:::J 
c. 
C: 0.00 
.c: 
tJ 
~ -0.50 
u 
c.. 
~ -1.00 
-1.50----.--------------------
0 50 100 150 200 250 300 350 400 
azimuth (deg.) 
,.so 4 
2/rev content 
gi 4/rev content 
3 /\ 
c, 1.00 ."...C... . 
a, 
.."..C... . o 2 
0 
- 0.50 
::, 
c. X 
C: 
:;; 0.00 'i 0 
2. .: 
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> 2. 
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........ >
N -1.00 a,  
~-3 
~ 
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2.00 1.50 
....... 
g' OI 5/rev content 
1.50 a, 
."~ ...C... . 1.00 
0 1.00 0 
0 
- 0.50 
X 0.50 X 
:ki  0.00 
]-0.50 ..c: 
2, -0.50 
Q.. 
> -1.00 Q.. 
a
,...
,  > 
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in 
-2.001-+--...----,---,---....---,---,.---.----, -1.50+---,---,---,----,---,---,---,--
0 50 100 150 200 250 300 350 400 0 50 10 0 150 200 250 300 350 400 
azimuth (deg.) azimuth (deg.) 
b. Harmonic contents of MFPC input 
Figure 2: MFPC pitch angle variation in the rotating frame with 3 and 4/rev 
combination for the offset hinged blade configuration. Three iter­
ations of the local MFPC model are shown. (a) MFPC pitch input 
(b) Harmonic contents of MFPC input 
91 - 76.34 
-·· 3,4 MF"PC advance ratio=0.3 
-- 3,5 MF"PC 
1.50 
- 1.00 
Cl 
Q) 
'"C 
........ a. MFPC input 
0.50 
:i 
c. 
.E 0.00 
J::. 
t.l 
~-a.so 
u 
0.. 
i-1.00 
-1.50,---,---,---,---r----.--...----,.----, 
0 50 100 150 200 250 300 350 400 
azimuth (deg.) 
4 
I.SO 
2/rev content -C'l 
a, 3 .;,;~,: •h. .. h h 4/rev content 
, l ; ~ 
c:n "C I : i I \ I \ 
1.00 - 2 \ i\ :, :, i 
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.!: 0 I , I , 1 ,' I I 
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Q. I I I I , I ,, 
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C'l 5/rev content 
- 2.00 a, 
/\ /\ /\ 3/rev content -"C 1.00 
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"C / \ / \ I I 
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1 X (\ (\ 
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I \ I I I I c. o.oo 
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a
-..
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"' 
-2.00,+---r---,---,---.---....---r--....---, - 8 . " . " , 6/rev content 
0 50 100 1 50 200 250 300 350 400 C'l /\ ,, :~ ,, :~ :~ 
-a, 
azimuth (deg.) 6 
"C : \ : ! : \ : ! I ~ : ! 
I l I I I I I I : I I I 
0 4 : ~ : t : ! : t I : : ! 
0 1 I l I I I I I t I I 
I : I I I I I I : I I 1 
2 I I:  t I I I I I
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, : : : 
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b. Harmonic contents of MFPC input .c -2 : : : : : : :  : : : 
~ : : : : : ~ : ! : ~ : ~ 
c.._4 : !: ~: ~: ~: ~: ~ 
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l,C) 
-8-t----,----,----.---,---,----,-----,.---, 
0 50 10 0 150 200 250 300 350 400 
azimuth (deg.) 
Figure 3: MFPC pitch angle variation in the rotating frame with 3 and 
4/rev, 3 and 5/rev combinations for the offset hinged blade con­
figuration and global MFPC model. (a) MFPC pitch input (b) 
Harmonic contents of MFPC input 
91 - 76.35 
1 st vert. fus. bend. freq. = 4/rev 
20 
D baseline 
..0,-.. 
:::J • MFPC with 
.c:O 15 
0 (3/rev,4/rev) 
_x.,,... 
o~+z ,o Ill]]) MFPC with 
(3/rev,5/rev) 
..o.. '.-,'  
.x m 5 
amU. _  
D..~ 
0-+---JI.-Jll. ....- --.--'-.....- ------JL...IIIIIIIIIWIJ--
-soo FHX FHY FHZ 
E 
.!500 
z 
-400 
.x.? 
o c300 
·• ID 
o.E 200 
oo 
: E 100 
o..o 
D•. . .c::::J  0 -i----.---
MHX MHY 
81 
C 
O 1.00 
.Y. =CJ  
0 L. 
Cl CD 
o.~ 0.50 
Ou 
-o 
.Y. • 
De, 
:. (.) 0.00-+---.......- -..--------L-....I JIL-
ACX ACY ACZ 
Figure 4: -Hub shears, hub moments and fuselage C.G accelerations without 
MFPC and with 3 and 4/rev, 3 and 5jrev MFPC combinations for 
the offset hinged blade configuration and global MFPC model · 
91 - 76.36 
advance ratio=0.3 
1.50 
- ••• • 1s t iteration 
. - • 2nd iteration· 
1.00 
O> 
Q) - 3rd iteration ... .. ..   .. : .. "'O . .  
.-......... 0.50 ..
: ..  
:::J 
a. 
C 0.00 
..c ·•..  ... -(.) ... .. .   
 
·a. -0.50 ... . ..  
u 
a.. . 
~ -1.00 -.~ 
-1.50 
0 50 10 0 150 200 250 300 350 400 
azimuth (deg.) 
Figure 5: MFPC pitch angle variation in the rotating frame with 3,4/rev 
combination for the elastic blade model configuration. Three iter­
ations of the local MFPC model are shown. 
91 - 76.37 
Vertical Hub Shears 
Rigid Fuselage 
600 ,-------- HHC Amplitude 0.0005 rad 
,, ~,, 
......... , , 
,' ' ' 
, ' ' 
soo , , 
, ' ' 
2 , ' 
, , 
' ' 
, ' ' 
' ' 
Cl) ' , ,....---~ 
1.... 400 ','  , ,' ·· · 2/rev 
C ', , 
Q) ', 
~..c ,...... .. .. ' ,~ -· 3/r ev 
C en 300 ------' 
.c --- 4/rev 
(D 
c.. ::s 
..c ....... 5/rev 
I 
.E- 200 -base 
I C 
~-~ ,• 
~~ . 7.(:_. - . -- . -- .., 
100 .. ··•·· .......... ·,,, .. :.:. ... :.: .. .-.: ... :: ... ~. .: .. ::: .:·. . 
Cl..> 
0-+-----.---.....--------------
0 60 120 180 240 300 360 
phase angle ( degrees ) 
Figure 6: Influence of single frequency higher harmonic pitch inputs, in the 
rotating frame, on the vertical hub shear. 
91 - 76.38 
Vertical Hub Shears 
Rigid Fuselage 
600 2/rev input 
- (Amin=0.0046 rad) 
z 
-soo 
I..l.l  
C / 
ai ' 
.l: 
111 400 / ........ ········· ······· ... ,:.'. \ 
.c. 
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> 
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100 , ..'C   '--::-- ... -."!". ./. 
ai . / 
Q. 1/2 Amin 
·, I 
0-t----...----....---~----,-----'': .,_,------, Amin 
3/rev input -- 3/2 Amin j 
600 
- (Amin=0.001812 rad) • I 
2 Amin I 
• I 
z 
-soo .-··-··-. --5/2 Amin I 
I..l.l  / ' 
C - baseline I 
ai ./ '· 
.l: 
111 400 / /·...................... ················· ... ' ' 
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] 300 
.: 
ai 
> -.. -·<·:>:,/ --·- -,'~,,::-.> 
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•• • I • • ' •• 
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I 
0 
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'.. , . ' 
100 . . . -: :-- :-. ., /.· 
.: 
C 
ai I 
Q. 
' I 
0-+----.,-------,----.-----,-----,-----, 
0. 60. 120. 180. 240. 300. 360. 
phase angle (degrees) 
Figure 7: Influence of amplitude of 2/rev and 3/rev signals in the rotating 
frame, on the vertical hub shear. 
91 - 76.39 
Vertical Hub Shears 
Rigid Fuselage . 
600 4/rev input 
- {Amin=0.0001 6 rad) 
z 
-soo ,,,,,,-··-·· ......... 
. .. .... 
I..l.l  / 
C 
ID 
.r:. / ' 
111 400 
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' ', ··. ····...... ........ .. ,'/ I .  / 
.£ 
Jc 100 
C 
ID 
c. .. 1/2 Amini 
-· Amin I 
5/rev input -- 3/2 Amin 
600 
- (Amin=0.00059 rad) ..... 2 Amin 
z 
-soo ,,,,,.,. .. -·- ......... --5/2 Amin 
... ..... 
Ill / 
C 
ID / 
.r:. ' baseline · 
111 400 / .... ········" ··········· ... ' -
..Q 
:, / .... ··· ------ ···.... ' 
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/ .. ···········~·,.,,,,"'"'- .............. ~·,········· ....'  ' 
-~ 300 
.:: / , -·-·-., 
ID 
> ......... ···,,,,,." ./ / ..... '',,,········· ... ' ·-.. -··· 
.:,r. . / .. .. .. .. ' . 
g 200 .. .... ,, ,,' / . . ' ··· ... 
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I . ,, '' ················· 
.£ , .. 
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100 \; ... . 
C 
ID 
Q. \ I 
0...-----.-------.----,-----r----.....---"' '---, 
0 60 120 1 BO 240 300 360 
phase angle (degrees) 
Figure 8: Influence of amplitude of 4/rev and 5/rev signals in the rotating 
frame, on the vertical hub shear. 
91 - 76.40 
advance ratio=0.3 
·-· pitch input 8 E 
1.50 
- - - pitch input fJ R 
;'' 
1.00 JJ  ' \ 
O'> ,-, i \ 
Q) J ' i \ 
"C I ' ' 'I  \ 
.-........ 0.50 J/  \ i \\  a. MFP'c input 
I \ i \ 
:, 
c. I'  \ \ .i \ 
I \ 
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.r:: IJ  ' \ i'  \\  
t, J \ I \ 
~ -0.50 I \ i \ 
J \ I \ 
J \ I \ 
u J \ I \ 
c.. 
~ -1.00 / '
/ · -I ' 
\ 
-1.so-1---...-----,------.----------------
0 50 100 150 200 250 300 350 400 
azimuth (deg.) 
1.50 4 
2/rev content ~ /\ /\ f\ f\ 4/rev content 
,....._ 1.00 .,-,, ,.,,-,\ .,.1 J ~ 1 \ 1 J ,. 
Cl \ / \ f \ I \ 
-a, ., ' J \ 
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2.oo 1.50 
....... 
g' 1.50 /\ /\, /\ 3/rev content C> 5/rev content 
a, 
I \ I \ lj \\ -0 1.00 :°..., 
~ I I I I 
O 1.00 I \ I , I \ - :f  ,:I..  
I 
I I I I i \ 0 I 
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X 0.50 / ', / \ f;  \ X ,,1,  
:5 I I I I 1 :5 :, .r 
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  I: 
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I I / I / I .: h if 
]-0.50 I I / I / I .s:: :\ :, 
I \ I \ .!: -a.so lt 
·c. II  ' ~ I J: 
I ' , ! l 
> -1.00 I I I I I 'c. 
.a..,  I I / I / \ > 
,,.-., ,.so I \ I \ / \ ~ -1.00 
\/ \/ ~.,., ......... 
oil 
-2.00+--,---,---,--...,..---,---.---.---~ -1.so+--..---,.---,---,---,.--.....-----~ 
0 50 10 0 15 0 200 250 JOO 350 400 0 50 100 150 200 250 JOO 350 400 
azimuth (deg.) azimuth {deg.) 
b. Harmonic contents of MFPC input 
Figure 9: MFPC pitch angle variation in the rotating frame with pitch in­
puts {8R} and {8E} for the offset hinged blade configuration and 
global MFPC model. (a) MFPC pitch input (b) Harmonic contents 
of MFPC input 
91-:- 76.41 
1 st vert. fus. bend. freq. = 4/rev 
20 
.a- D baseline 
.:.,c g 15 • MFPC with 
(3/rev,4/rev) 
.Y...-
=C •­ [IIID MFPC with 
a.Z 10 (3/rev,5/rev) 
-0 ...... D optimal control 
" 
.Y. 111 5 
0U 
ID i.. 
o...2 
o-L--1..,__ ___ 1_1-,_.___..Lillllllld!rJc:J~ 
FHX FHY FHZ 
'"'SOO 
E 
!.SOO 
z 
-400 
.!C.? 
o c.300 
QI ID 
o.e 206 
oo 
: e·,oo 
o.c 
• :::J 0 
a...c -~--MHX 
"'"'1.so 
-Cl 
1111 
C 
O 1.00 
+= 
..l.( C 
C i.. 
CD CD 
o.~ a.so 
Ou 
-o 
..Y. • 
:C_ cu, 
o .oo,-1-------------_J.--1....a:C11=:L..., 
ACX ACY ACZ 
Figure 10: Hub shears, hub moments and fuselage C.G accelerations with­
out MFPC and with various vibration reduction schemes for the 
offset hinged blade configuration. 
91 - 76.42 
Pitch input {8R} Flexible fuselage in 
bending and torsion 
3 
- 2 
Ol 
a, 
"C a. MFPC input 
-::, 
c.. 
C: 0 
..c: 
u 
:c::. .- 1 
u 
Cl.. 
~-2 
-3-+---,---..---,---.....--...-----------. 
0 50 100 150 200 250 300 350 400 
azimuth (deg.) 
3 
- 0.60 4/rev content 
2/rev content 
C) 
"ii 2 
- ! 0.40 
Cl) 
"C 
0 
:i - 0.20 
0. X 
C 
; 0 :i 
fa.ea 
l: 
c._, .r. 
.>.. 
l: -0.20 
Cl)  c. 
'N -2 > 
~ -0.40 
'.. ., 
-0.60-1--.---.....----------~ 
1.50 3 
-"ii 
3/rev content - 5/rev content 
en 
~ 1.00 ! 2 
0 0 
- a.so 
X X 
:i :i 
Ea.co f o 
.r. .r. 
cJ:.  -0.50 lc:.  '-1 
> > 
~ -1.00 
',, ., 
~-2 
'If ) 
-1.501-t---r---i---.---r--....------ -3,-1--,-----,.---.---.--...------
o 50 100 15 0 200 250 300 350 400 0 50 100 150 200 250 300 350 400 
azimuth (deg.) azimuth (deg.) • 
b. Harmonic contents of MFPC input 
Figure 11: MFPC pitch angle variation in the rotating frame for a fully 
flexible fuselage. (a) MFPC pitch input (b) Harmonic contents 
of MFPC input 
91 - 76.43 
Flexible fuselage in bending and torsion 
,a 
.c ...... D baseline 
.:cJoo  B • optimal control 
.:::J...-
Ca ,•- 6 
c.Z 
o'--' 4 
+-c,, 
.:::J. a, 
Ca, u....  2 
Q.~ 
- 0-+-----..-
FHX FHY FHZ 
040 
.a...  
ACZ 
Figure 12: Hub shears, hub moments and fuselage C.G accelerations with­
out MFPC and with minimum variance control input for a fully 
flexible fuselage. 
91 - 76.44