Fundamental issues related to the prediction of coupled rotor/airframe vibration

TWENTYFIFTH EUROPEAN ROTORCRAFT FORUM

Paper n° G9

FUNDAMENTAL ISSUES RELATED TO THE PREDICTION

OF COUPLED ROTOR/ AIRFRAME VIBRATION

BY

Hyeonsoo Yeo, Inderjit Chopra

UNIVERSITY OF MARYLAND, COLLEGE PARK, USA

SEPTEMBER 14-16, 1999

ROME

ITALY

ASSOCIAZIONE INDUSTRIE PER L'AEROSPAZIO, I SISTEMI E LA DIFESA

ASSOCIAZIONE ITALIANA DI AERONAUTICA E ASTRONAUTICA

(

FUNDAMENTAL ISSUES RELATED TO THE PREDICTION

OF COUPLED ROTOR/ AIRFRAME VIBRATION

Hyeonsoo Yeo*

Inderjit Choprat

Alfred Gessow Rotorcraft Center

Department of Aerospace Engineering

University of Maryland, College Park, MD 20742, U.S.A.

Abstract

A comprehensive vibration analysis of a coupled rotor /fuselage system is carried out using detailed 3-D finite element models of the AH-lG airframe from the DAivlVIBS program. Predicted vibration results are compared with Operational Load Survey flight test data of

the AH-lG helicopter. Modeling of difficult

components(secondary structures, doors/panels, etc) is essential in predicting airframe natural

frequencies. Calculated 2/rev vertical vibration

levels at pilot seat show good correlation with the flight test data both in magnitude and phase, but 4/rev vibration levels show fair correlation

only in magnitude. Lateral vibration results

shmv more disagreement than vertical vibration results. Accurate prediction of airframe natural frequencies up to about 38Hz(7 /rev) appears essential to predict vibration in airframe. Second order nonlinear terms have an important effect on the prediction of vibration at high speed and high frequency. Third order kinetic energy terms generally have small inf!uence(about 7% change) on the prediction of vibration.

Introduction

As helicopter crew and passenger comfort has gained increased emphasis, vibration requirements

have become more stringent [1]. Even though

there has been enormous progress with vibration

suppression technology [2], cost and weight penalty

has been excessive in part because of inadequate

vibration prediction capability. To minimize

• Research Associate, Member AHS t Alfred Gessow Professor and Director,

Fellow ALJ\A, Fellow AHS

Presented at the 25th European Rotorcraft Forum, Rome, Italy, September 14-16, 1999.

the additional cost and weight penalty, accurate vibration prediction is necessary at the early design stage.

Even though considerable progress has been made to improve the mathematical analysis of rotors during recent years, reliable and accurate vibration prediction is still a challenging problem. In a recent validation study using the Lynx helicopter flight loads, it was found that most comprehensive analysis codes exhibit significant errors of as much as 60 percent from the measured vibratory loads [3]. Various analytical technologies were applied to evaluate their effects on vibration

predictions. Of all the technologies, free wake

models have been shown to have a dominant influence on vibration predictions at both low and

high speed conditions( [3] - [5]).

Airframe dynamics is also important in the prediction of helicopter vibration. NASA-Langley carried out a Design Analysis Methods for VIBrationS(DAMVIBS) program to establish

the technology for accurate and reliable

vibration prediction capability during the

design of a rotorcraft

[6].

Four major helicopter

manufacturers(Bell, Boeing, former McDonnell Douglas, and Sikorsky) actively participated

in this program. Systematic modeling and

analysis techniques were investigated for the

four technology areas: airframe finite element

modeling, modeling refinements for difficult

components(secondary structures, doors/panels, engine, fuel, transmission, cowlings, fairings, etc), coupled rotor-airframe vibration analysis,

and airframe structural optimization. All

participating companies developed

state-of-the-art fi.nite element models for the airframe, conducted ground vibration tests, and carried out comparisons of their predictions with test data. During this program, they improved the finite element modeling capability of both metal

and composite airframes. In conventional finite element modeling of an airframe, only the primary load carrying structures were represented in terms of their mass and stiffness characteristics, and the secondary structures were represented only

as lumped masses. Comparison of predicted

frequencies with measured values showed that agreement is less satisfactory above about 20

Hz with conventional modeling. The study

identified the critical role of difficult components

for vibration prediction. It was shown that

a detailed finite analysis of the airframe that included the effects of difficult components could predict frequencies with a deviation of less than 5% of measured values for modes with frequency up to 35Hz [7].

Under the DAMVIBS program, the four helicopter companies also applied their own methods to calculate the vibrations of the AH-1G helicopter, and correlated the predictions with an Operational Load Survey(OLS) flight test data. Most of the analyses were unable to predict vibration accurately for all flight conditions. These studies pointed out that the coupled rotor-fuselage vibration analysis should be improved in order to be useful for the design and development of a rotor-airframe system.

During the last two decades, coupled rotor-fuselage vibration analyses have been developed by many researchers using a variety of assumptions and solution methods (see reviews by Reichert [2], Loewy [8] and Kvaternik, et al. [9]). Simplified investigations such as those reported in Refs. [10] - [13] have made significant contributions to the understanding of the basic characteristics of rotorcraft vibration but are not sufficient

for accurate predictions. Most analyses also

incorporated highly idealized aerodynamics. For example, in Ref. (14] a coupled rotor/flexible fuselage model was developed for vibration reduction studies using 3-D fuselage. However, this analysis incorporated idealized aerodynamics such as uniform inflow and quasisteady aerodynamics so that vibration was substantially underpredicted. Helicopter vibration is due to the higher harmonic airloading of the rotor, thus nonuniform induced velocities caused by blade vortices can be a key factor in the prediction of vibration.

Recently, the present authors carried out a comprehensive vibration analysis of a coupled

rotor/ fuselage system incorporating refined

aerodynamic models such as free wake and unsteady aerodynamics (15]. Predicted vibration results were compared with Operational Load Survey flight test data of the AH-lG helicopter.

ivfodeling requirements for the vibration analysis of complex helicopter structures and rotor-fuselage coupling effects were identified. The importance of refined aerodynamic modeling was also addressed. The non-linear equations of motion of a coupled

rotor/ airframe are quite involved. Often, an

ordering scheme is used to systematically neglect higher order terms in the equations. Normally.

third order terms(<3 _{terms) are neglected in rotor}

aeromechanic analyses, so that equations are manageable and retain enough accuracy. Many aero elastic analyses of a rotor blades are focused to solve the aeromechanical stability that includes the calculation of blade steady periodic response and stability of linearized perturbation motion. These phenomena involve low frequency and retention of second order terms appears adequate. There are a few exceptions where higher order terms are included. For example, Crespo da Silva and Hodges [16], (17] derived equations of motion of a rotor blade retaining terms up to order of E3 and investigated equilibrium and stability of a uniform cantilevered rotor blade in hover. They emphasized the importance of higher order terms in the prediction of the behavior of blades with low torsional stiffness and at high thrust level. For aeromechanical stability, the rigid body modes appear adequate and the flexibility of fuselage is not considered. Since vibration analysis involves coupled rotor/fuselage equations, the modeling of both blades as well as airframe becomes important Since high frequency modes are involved in the vibration analysis, it may be possible that higher order terms may become important.

In this paper, the effect of higher order terms( especially third order) on the prediction of vibration is investigated. Since there are too many third order terms involved in the equations of motion, only higher order kinetic energy terms are investigated. Parametric studies are also conducted to examine the influence of several key factors on the prediction of vibration of a rotorcraft.

Vibration Analysis

The baseline rotor analysis is taken from

UMARC(University of Maryland Advanced

Rotorcraft Code). The blade is assumed to be an elastic beam undergoing flap bending, lag bending, elastic twist, and axial deformation. The analysis for a two-bladed teetering rotor is formulated and incorporated into UMARC. The elastic rotor coupled equations include six hub degrees of motion. The rotor vibratory loads are

(

transmitted to the fuselage through the hub and the effects of fuselage motion are included in the determination of blade loads.

The derivation of the coupled rotor /fuselage equations of motion are based on Hamilton's

variational principle generalized for a

nonconservative system.

i

to

biT

=

(

6U - 6T - 6W) dt

=

0

t,

( 1)

5U is the variation of the elastic strain energy,

liT is the variation of the kinetic energy, and oW

is the work done by nonconservative forces which are of aerodynamic origin. The contributions to these energy expressions from the rotor blades and fuselage may be summed as

where the subscripts b and F refer to the blade and

fuselage respectively and N, is the total number of

rotor blades. For example, the variation of the kinetic energy for the bth blade is expressed as

The equation of motion for the teetering degree of freedom of a two-bladed rotor is obtained from the equilibrium of the flap moment about the teeter hinge.

The 3-dimensional N ASTRAN finite element models of the AH-1G helicopter are used in the

coupled rotor/fuselage vibration analysis. The

airframe modal data( eigenvalues, eigenvectors, and generalized masses) are generated using NASTRAN and are used as an input to the coupled rotor/fuselage vibration analysis program. The couplings between rotor and fuselage are included in a consistent manner into UMARC.

Blade response equations, teetering motion equation, and fuselage response equations are solved simultaneously. To reduce computational time, the finite element equations are transformed into the normal mode space. Because the fuselage is in the fixed frame, the analysis is carried out in

the fixed frame by transforming the rotor equations using a multiblade coordinate transformation. The nonlinear1 periodic, coupled rotor/fuselage

equations are solved using a finite element method

in time. ·

Fuselage Models

First, the elastic line airframe structural

modeling capability was incorporated into

Ulv!ARC. The fuselage is discretized as an elastic beam using the same 15 degree-of-freedom beam element as that used for the rotor blade. Elastic line model of the AH-lG helicopter is shown in

Figure l. 39 beam elements are used in modeling

main fuselage1 tailboom1 wing1 and main rotor

shaft. Second, the 3-D NASTRAN finite element model of the AH-lG helicopter airframe developed in the mid 1970s, shown in Figure 2, is included

into UMARC. It consists of structural elements

such as scalar springs1 rods, bars, triangular and

quadrilateral membranes. The total number of elements is 2965. The main rotor pylon is modeled as an elastic line using bar elements. The main rotor pylon(Figure 5) provides the structural tie bet\veen the main rotor and the fuselage. It is attached to the fuselage through the elastomeric mounts and a lift link. The lift link is the primary vertical load path and is very stiff in the vertical direction. The elastomeric mounts are designed to produce low pylon rocking frequencies to isolate the main rotor in-plane vibratory loads from the fuselage and to balance the main rotor torque. This model was used for the coupled rotor/fuselage vibration analysis to correlate with Operational Load Survey flight test data in the DAMVIBS program. Third, a modified 3-D finite element model of the AH-lG helicopter including effects of difficult components is included into UMARC. The earlier 3-D finite element model was modified by Bell Helicopter to achieve better correlation of natural frequencies with test data. These updates included replacement of the original elastic line tailboom with a built-up rod and shear panel tailboom and inclusion of fastened panels, doors. and secondary structure in the forward fuselage. However, this model could not be used directly for the validation study because the overall weight of the test vehicle was different from that of

a OLS flight test vehicle. So, the NASTRAN

model was modified to convert it to the OLS test configuration by updating the weight of fuel, ammunition, etc. The final refined 3-D airframe model is shown in Figure 3. The total number of finite elements used in this study is 4373. A comparison of NASTRAN and test natural

I '

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"

"

"

14 IS

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Fig. 1 Elastic line fuselage Model

Fig. 2 3-dimensional

Model

finite element fuselage

Fig. 3 t;;efip,ed r,-~iTensional finite element

use age lV o e 30 i

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~ 51 a. 0 1 0 10% error / / 20% error / / .. /

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/ / / / 90 __ .-·

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2nd fuselage vertical bendtng

.

~ :..--,

'""-' t:. ~-· 1st fuselage lateral bending

,'-0

• • MIA pylon p1tch

5 10 15 20

Measured frquency (Hz)

(a) Elastic line fuselage

10% error 20% error / . / 25 / / / / 30 ,' / / _q.··

2nd fuselage vertiCal bendmg ~

0 / -"" •• • •

t

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1st tuselag!l vertiCal bending,,::,.. _,., / , • ~: 3rd fuselage lateral beno1ng,

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MIA pylon PitCh

5 10 15 20 25 30

Measured frequency (Hz)

(b) 3-D fuselage

/ Jrd fus.olage lateral bending, /

10% error .e:J / / 20% error / / /

.

/ / / / / /

--' , --' / / C)..2~d luii&lage lateral bendmg 1st fuselage vertical bonding ' / ...- A ·'

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,,

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, ~-:"'• 1st fuselage lateral bendmg MIR pylon pitch

5 10 15

io

25 30

Measured frequency (Hz}

(c) Refined 3MD fuselage

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Fig. 5 Main rotor pylon of AH-1G helicopter

frequencies is presented in Figures 4. The diagonal line represents perfect match between predictions and test data. Percentage error bandwidths are included to indicate trends in correlation. The elastic line model shows fair correlation up to 20 Hz. But, fuselage torsion and third fuselage lateral bending modes cannot be found within the frequency range up to 30 Hz. 3-D fuselage model shows fair agreement with test data except for the second and third fuselage lateral bending modes. With the modeling of difficult components, the natural frequency correlation at the higher frequencies is improved from 20% error to less than 10% error for up to 30 Hz. In particular, the improvement of fuselage lateral bending frequencies is noticeable.

Results and Discussion

The two-bladed teetering rotor of the

AH-1 G helicopter and its N ASTRAN airframe model

are used to calculate vibratory hub loads and

vibration levels at the pilot seat. Coupled

rotor/fuselage equations are solved in straight and level flight conditions. Estimated vibration results are compared with OLS flight test data of the AH-1G helicopter. Detailed blade properties and

test results are in Ref. [18]. For the calculation of inflow and blade loads, a pseudo-implicit free wake model [19] and a time-domain unsteady aerodynamics [20] are incorporated. The effects of compressibility(Prandtl-Glauert correction) and reversed flow are also included in the aerodynamic

model. For normal mode analysis, thirty

airframe modes( which covers frequencies up to 40 Hz(7.4/rev)) are used. Eight time elements with fifth order shape functions are used along the azimuth to calculate the coupled periodic response.

Effect of fuselage modeling

Vertical vibration levels at the pilot seat are presented in Figure 6 with airspeeds ranging

from 67 knots to 142 knots. There is a good

agreement of the magnitude of vibration level between predictions and test values and only slight differences exist between 3-D fuselage and

refined 3-D fuselage results. Rotor /fuselage

coupling reduces 2/rev vertical vibration by more than 50% and has a small effect on 4/rev vibration. Estimation of vertical vibration with

the elastic line model has a negligible effect on

2/rev vibration, but underpredicts 4/rev vibration. Lateral vibration levels at the pilot seat are

shown in Figure 7. Since there was a more

scatter in the prediction of fuselage lateral bending frequencies from measured values among fuselage models, significant improvement in the calculated lateral vibration levels was expected with modeling refinements. Estimation of 2/rev lateral vibration with the elastic line model shows large deviation from test results. However, the elastic line model shows fair correlation of 4/rev vibration. Lateral vibration levels with refined 3-D fuselage model are larger than those with regular 3-D fuselage model. Refined model improves somewhat correlation of 2/rev vibration but1 correlation becomes worse for

4/rev vibration. Both models overpredict 4/rev vibration.

Correlation of phase

For a systematic validation study of predicted vibration, both magnitude and phase of the hub loads should be compared. Hub vibratory loads, however, were not measured in the OLS flight test. Hence predicted vibration vectors(magnitude and phase) are correlated with measured vibration vectors.

Figures 8 and 9 show the effect of fuselage modeling on the phase of vibration at 101 knots. 2/rev vertical vibration results, shown in Fig. 8(a),

0.4 i

0.3~

:§

'

I

_§ 1! ₀₂₁ 0 0 u I u <: 0.1!' I 0.0 ' so 0.4 I

I

:§ 0.3l c

_I

.Q 0 I ~

0.2l

0 u 01

~

u <:

J

_so 0 0

-=--0 80 0 100 120 Airspeed (knots)

(a) 2/rev vertical vibration

-~----80 100 120

Airspeed {knots)

(b} 4/rev vertical vibration

0

140

140

Fig. 6 Vertical vibration level at pilot seat

show that detailed structural modeling helps to improve the correlation of phase angle even though

the magnitude is unchanged. The effect of

difficult component modeling on the phase of this component is negligible. 2/rev lateral vibration estimated using elastic line model shows a significant difference in both magnitude and phase from those using the detailed models(Fig. 9(a)). Significant difference of phase between an elastic line model and detailed airframe models is also

observed in the 4/rev vibration. 4/rev lateral

vibration results, shown in Fig. 9(b), show that

difficult component modeling has an influence on

the phase of this vibration component, and changes the phase angle by 9 degrees.

Predicted and measured vibration

vectors(magnitude and phase) are correlated

using refined 3-D fuselage model for three

different speeds(67, 101, and 142 knots) which respectively represent low speed, moderate speed,

160 160 0.4 § 03l c

02~

.Q

"

~ 0

01~

u u <: 0.0

!

60 0.4

I

0.3~ §

_I

c _I .Q 0.2J 1! 0

'

0 u u <: 01l 0.0 . 60

Relined 3·0 fuselage, No fuselage feedback

Elastic line fuselage

3·0 fuselage

Refined 3-D fuselage

0 Flight test

a'o 100 120

Airspeed (knots)

(a) 2/rev lateral vibration

~D 0 Q 0

so 100 120

Airspeed (knots)

(b) 4/rev lateral vibration

140

Fig. 7 Lateral vibration level at pilot seat

and high speed flight conditions. The 2/rev

vertical vibration result, shown in Figure lO(a), shows good correlation for both magnitude and phase at low and moderate speeds. At high speed, there is significant phase difference( about 20 degrees) between predicted and measured values. The 4/rev vertical vibration result, shown in Figure lO(b), shows considerable deviations for all speeds. For low frequency vibration, vertical hub loads seem to be modeled accurately up to moderate speed and fuselage model appears adequate in the vertical direction. The difference of 2/rev vibration at high speed is probably due to hub loads since the fuselage model is

not expected to change with speed. For high

frequency vibration, both hub loads and fuselage model may have errors. There is a large deviation in predicted and measured phase angles.

Figure ll(a) shows 2/rev lateral vibration result. Even though there is more disagreement

160

1 10··

I

5 10"2_..!

Cii 010°j

~

-5 10'

J

-1 10'

j

- ·- ·-- · Elastic line fuselage - - - 3-0 fuselage Refined 3-D fuselage - - Flight test -1.5 10"1

+!---~---~

-1.5 10"1 _{-1 10 -5 10"}2 _{o 1oo 5 10"}2 _{1 10"}1 Cosine (g)

(a) 2jrev component

1 10"1 , -5 10"2

~J

... _____ ,__

"'

c75 -5 10"2 -1.5 10"' +--,--~--,---~---' -1.5 w·l _, 1o·l -5 io·2 o _10° s _10·2 Cosine (g)

(a} 2/rev component

1 10"1 - , - - - , I I 5 10"2 ~ I I o; 0 10°~ - I "' I

a

-510'!

'

·1 10"1 -j

I

-1.5 10"1 +---~---~-,---.--:---1 _{·1.5 10"}1 +'---..,---~---~---'

-1.5 10"1 _{-1 10"}1 _{-5 10"2 o}

1oo

_{s 1'o·2 1 10·}1 _{-1.5 10"}1 _-11o·1 _-s1o·2 _o

1o

0 _{5 10"2}

Cosine (g) Cosine (g)

Fig. 8

(b) 4jrev component

Effect of airframe modeling on vertical vibration at pilot seat at 101 knots

between measured and calculated results than corresponding 2/rev results in the vertical direction, the trends appear quite consistent. Estimation shows the same phase angle at 67 and 101 knots as observed in the flight test data. At 142 knots, the test data shows a phase shift but the prediction does not show such a change. The measured phase difference at high speed differed about 19 degrees from those at low and moderate speeds. In Figure ll(b), 4/rev lateral vibration results show good correlation at low speed and the difference between predictions and test data increases with speed.

Contribution of airframe modes

The contribution of different airframe natural modes to vibration at the pilot seat is investigated next for a better understanding of airframe dynamics and its role in the prediction of

Fig. 9

{b) 4/rev component

Effect of airframe modeling on lateral vibration at pilot seat at 101 knots

helicopter vibration. Figures 12 through 19

show contributions of different airframe modes in vertical and lateral vibration at the pilot seat at 101 knots. Both a refined 3-D fuselage model and old 3-D fuselage model are used for the calculation

of vibration. The contribution of each mode

is presented in terms of magnitude of vibration nondimensionalized by the total vibration at the prescribed frequency.

Figure 12 shows that only four low frequency modes (M/R pylon pitch and roll, and 1st and 2nd fuselage vertical bending) have a dominant effect on the prediction of 2/rev vertical vibrations. The contribution of the M/R pylon pitch mode is due to the longitudinal hub force excitation. The contribution of fuselage vertical bending modes (primarily first and second modes) shows that the effect of vertical hub force on the pilot seat

vibration is about 20% of total level. The

110'' ,---~

I

5 10'2 .J 0 10'

~

s

·510..? j

i

-110'

~

I ·1.5 10'1

l

Relined 3·0 fuselage Fllght :est j/ I I I I I 101 kno1s ·2 10''..! ·2.5 10''

-i~--~--~--~--1

_,~

2

-kn_'_"~--~-__j

·2.5 10'1 _{·2 10'' ·1.5 10'' ·1 10'' ·5 10'}2 _{0 10°} Cosine (g)

(a) 2/rev vertical vibration

110"' ,---~

I

' 5 w·2 J 0 10°

~

I

i

·5

w•~

iii -I 10"'1 -1.5 10·'

I

·2 10'' J I 67 knots 101 kno1s -2 5 10"' +!~-,--:----~--~---,--,--_j ·2.510'' ·2 10'' ·1.5 10'' ·1 10'' -5 10'2 010° 510'2 1 10'1 Cosine (g}

(b) 4/rev vertical vibration

Fig. 10 Vertical vibration at pilot seat

40%), which is related to the lateral motion on

2/rev vertical vibration shows that the

cou~ling

between modes may have an important influence on vibratory response. For this roll mode, the vertical deflection at the pilot seat has almost same magnitude as the lateral deflection at this position. Thus, lateral hub force produces large vertical vibration as well as lateral vibration. Figure 13 shows 2/rev vertical vibration using an old 3-D N ASTRAN modeL Again, the main rotor pylon roll mode has the most dominant effect on the prediction of 2/rev vertical vibration. However, its contribution to total vibration is reduced to about 30% compared to 40% in the refined 3-D fuselage

modeL The contribution of pylon pitch mode

remains same at about 18%. The contributions of 3rd fuselage vertical bending and main rotor

mast fore-and-aft (F

I

A) bending modes increase to

twice compared to those of the refined 3-D fuselage modeL

For the 4/rev vertical pilot seat vibration

110'' , -1 5 w•

4

0 10°J 142 knots

Y

101 knots 101 knots 57 knots '<----Eo 142 knots 67 knots ::§ -sw·• 0

J5

·1 10'' -1.5 10'' ! ·2 10'' J

I

Refined 3-D fuselage Flight test ·2.5 10'' "1~-~---:--~--~---,--~---,---< ·2.5 10'' ·2 10'1 _{·1.5 10·' ., 1o·• -5 1o·• o 10° s 10·2} 1 10., Cosine (g)

(a) 2/rev lateral vibration

1 1 0 " ' 1 1 -5 10'2

J

I

0 10° i I ::§ ·5 10·2

i

0 ' ~ -110'1 ~ -1.5 10'1

~

·2 10''

J

101 knots 1 101 knots -ft 142 knols

i

142knots~-~ 67 knots

!

-2.5 10"' :+-:-.---:-"-:-.--~---:---~---! -2.5 10'' ·2 10'1 _{·1.5 10'}1 _{·1 10'' ·5 10'}2 _{0 10° 5 10'}2 _{1 10''} Cosine (g)

(b) 4/rev lateral vibration

Fig. 11 Lateral vibration at pilot seat

prediction using a refined airframe model, shown in Figure 14, several modes have similar contributions and most of them are high frequency modes. Since there is more error in the prediction of high frequency modes) prediction of vibration at this frequency is likely to be less accurate. The coupling between modes can be clearly seen in

this 4/rev vibration too. The contribution of

M /R mast lateral bending and 3rd fuselage lateral bending modes to the 4/rev vertical vibration shows that lateral hub force produces vertical vibration and its contribution is also important. The modes whose frequencies are above 33 Hz (6/rev) have a negligible effect on the vertical vibration. Figure 15 shows 4/rev vertical vibration at pilot seat at 101 knots using an old 3-D

NASTRAN model. Main rotor mast F

1

A bending

and 3rd fuselage vertical bending modes have

a dominant effect on this vibration. Unlike

the refined 3-D fuselage model which shows the important effect of 3rd fuselage lateral bending

/ I \ 17.91"•1

'1

.:()H! (7 34/ie~J (?.21/rev) _I (7.08/rev)

i

(S.S91rev) 7 (ii.60frev) (52~/rev) (5.94/rev) 6 (5.64/rev) (5.S6frev)

I

I I

3r.i hJsetage ta:erol tencl'.ng(~-92/rev) J

3rd :Uselage v!rncal ter.c::;r,; (U!Irev)S ;

-M'Rmastta:eraltliD:j:ng(O~rev)

t.

(~.58/rev) r

MIRmastFIA~endrq(4.Sllrev) j

41!lsk!d(t12/r<N)

4

~

2n.::! Wtage torsion (3.69/rav)

1

3rdsltid(3.6Jrev) 2ndsl00(3.2Sire-.-) "

2r>d fuselage ven:ical b!!rOng (3.t91rl!V)

r-Zr.d Wlage lateral bcncing (2.99/rov) 3

T

lSI ilJr.etage IOfS100(2.i4/mv)

j

1!tslod(2.5irev) I I

2F,

I Sl fuselage verocal tend~ (1 .SOlrev) 1st fuselage tal! rei bo..OO:ng (I Wrov)

1

.~1/R pyiJn rtl (0.71/rel')

!.1/R pylcn pit.'l (O.S&'ro~)

'1---30Hl

IOHl

OL-~--~--~~~~~~~

Fig. 12

0 20 40 60 ao

Percent contributions to total vibration (%) Contribution of refined 3-D airframe natural modes to 2/ rev vertical pilot seat vibration at 101 knots

100

mode on the prediction of 4/rev vertical vibration, an old 3-D fuselage model does not show such

a coupling effect. The modeling of difficult

components appears to produce the coupling between modes.

As shown in Figure 16, the M/R pylon roll mode of a refined airframe model has a dominant effect on the prediction of 2/rev lateral vibration

at the pilot seat. High frequency modes such

as M/R mast lateral bending and 3rd fuselage lateral bending modes also have an important influence on the low frequency vibration. For an

old airframe model(Fig. 17), M(R pylon roll, 2nd

fuselage lateral bending, and M/R mast lateral bending modes have almost same contribution on the prediction of 2/rev lateral vibration.

For the 4/rev lateral pilot seat vibration prediction, shown in Fig. 18, high frequency modes

S' ! i7 67/r~v) {6.~/rev) Yi 16 SJirev) i {6.20/rev)

j'

{6.03/rev) 6 {5 91/rl!'l)

r

{5 20/r€v) I I Jrd ~>e!ag.3 ver:!cal bend~{4 St/rev) 5

-(C'I!rev)

I

WR:nas1F/Aberd:1g{t59/ml-:\.._ (4A71rev) ----?-- :.., M.'Rmastlaler.!l bemir.g (4.42/nw) ~ 4:h?;id(U.:trev) 4

3rd ~selage la!er.!l berdng (3.7€/rl."f) 3rdskid(3.671rev)

2rd fuselage lc!S10n(l55/rev) 2nd snct (3.3Vrev)

2M 1\:sel:lge ve!'Dcal bel'610:J {3. 1$/rev) r--2rd !uselage laltlr.il berx!ilg(3.0tlrov) 3

i

tst!uselilgeto<Sl(f)(2}51rev) r'

lstsl00(2.71rev)

I

,jr'

---=··~~

I

ts: f'J~3gll ver:cal OOrd:ng {1.~6/rev)~ lSI fu~latetal ben6ng(1.3Jie'J)Ii

M!Rpylon roil 10.51/rev)

WFi ~ pil::h {O.SO!rev)

1-i ! I I 0~~~~~~~--~~--~--~ Fig. 13 0 20 40 60 80

Percent contributions to total vibration {%) Contribution of old 3-D airframe natural modes to 2/rev vertical pilot seat vibration at 101 knots

100

have a larger influence and the contribution of the low frequency modes is small. The lateral vibration due to longitudinal and vertical hub forces is small. For the prediction of the lateral vibration, airframe modes whose frequencies are up to 38 Hz(7 /rev) should be included. 4/rev lateral vibration results using an old 3-D airframe model.

shown in Figure 191 show a dramatic difference

from those using a refined 3-D fuselage model. M/R mast lateral bending mode of a 3-D fuselage has a significant contribution (about 50%) and 3rd fuselage lateral bending mode has an important contribution (about 10%) on the prediction of 4/rev lateral vibration. Compared to a refined 3-D fuselage model, the contribution of M/R mast lateral bending mode increases by more than 2.5 times and the contribution of 3rd fuselage lateral bending mode reduces by less than half. This shows that the airframe modeling appears to cause

(797/r~v) 8;

I

(7.34/rov) '-. - - - _ : : " : : , : : . " (7.21/rev)

J

(7.0&'tEv) (6.9!'/rov) 71 (6.60/rev)

I

(6.24/rev)

I

(5.9llrev) s...l,-(5.64/rev) I •• _____________ _::::O~H' (5.50/rcv) ~-zro tu:.etage tateraJ bending (4.92/revL -1_

3rdfusel~ver..ealbend:lg(~.1.3i~P ._ .\!,•?. mastlatelal ~ending (4.74/mv) ~ (4.58/re'l) -M>Rmas!F/Abel'ldro:!(4.511rev) ~ 41tl$kid(~.12fre<)

,,

J

2r.:JfurelagaiO!Sion(3.691rev) f -3/'dsl<id (3.5/rev) 2rdsl<>:!(3.2a/mv} !...

2nd~selagJvel'tiC<Iibm:ftng(3.191rev)-2rd fuselage l.ltl!r.ll berdr,. (2.WIM) 3 1st fus.etage loMn (2.74i'mv)

tslskd(2.5Jrev)

1S:Ius.!la9Jve<1lC31~(1.50irev)

I

1 sl f:Jsela9! talllral bencf.ng (1 .Wrev)

M'Rp'j1onmi(0.71/rev) Miil pylon ?Lith (O.S&'rev)

1 i i.,_

""

r

0·~----~---~~~--~ Fig. 14 0 20 40 60 80 100

Percent contributions to total vibration(%)

Contribution of refined 3-D airframe natural modes to 4/rev vertical pilot seat vibration at 101 knots

the differences of both magnitude and phase of the 4/rev lateral vibration prediction between two airframe models (Figure 9(b)). For the accurate prediction of vibration using 3-D fuselage model, airframe modes whose frequencies are up to 40 Hz (7.4/rev) should be included. This is similar to the conclusion with the refined 3-D fuselage.

Effect of Aerodynamic Coefficient

The section lift, drag, and pitching moment

coefficients used in the present analysis are expressed as

G=~+~a (6)

cd =do

+ d,lal +

d2a2 (7)

Cm

=

fo

+

fta

=

Cm.o

+

fta (8)

where co, Ct, do, db d2, Cm"", and

h

are airfoil section coefficients. The effects of these coefficients

(767/II!V) (7.431rev) 81 (6.99/rev) ?l' [6a31rev) ~ {6.20Jrev) ₁ : {6.03/revl ..; {5.91/fllV) fi

r-!

{5.30/rev) 1 3r.l

tusetageve~

OOnd:rq[4.9alrnv) 5

J~

(VI/rev)

lAiR mast FIA!Jen<tng

(4.59/rev>-:a.!----(4.47/rev) :, M.'Rm<!stl:ueralttndog(4AVrev) ;:-4ttl~d(4.34Jrvvj

I

3nltusclag.alilter.!IOOndiii91J-71>'rev)4l...,__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~_" 3n:tskid1J.S71~l I 2nd turelage 101!>00 (3.55/rev) 2ndsl:>d(3.321rev) i 2ndlilse!agevern:al00nding(3.19/lll'l)

r-21ld tuselago \J.teral ber.:t.ng (3.01/rev)

31

1st tuselago tors>on (2.7$/m'l)

'""""·"··>

I

2-~· ---="~"

I

1 Sl fuselage VEmcal OOndo'lgl\.4&1rev)L

,.,._

,

..

~.,,~,,·~·~ f.'/Ril'Jlon~l(o.sui¥'1)

Lr

lMl ~ p.ldl (0.5&lll'l)

0~~~~~~~~~~

Fig. 15 0 20 40 60 80

Percent contributions to total vibration{%)

Contribution of old 3-D airframe natural modes to 4/rev vertical pilot seat vibration at 101 knots

100

on the prediction of vibration at pilot seat are investigated. The baseline and modified values

of these coefficients are given in Table 1. The

Table 1 Aerodynamic coefficient variation

Baseline value Modified values

~ 0.0 0.05 0.1 Ct 6.16 5.7 6.28 do 0.0068 0.0 0.01 d, 0.0 0.1 0.3 Cmac 0.0 -0.01

O.Dl

h

0.0 -0.1 0.1

zero angle pitching moment coefficient, Cmac•

has the most dominant effect on the prediction of vibration among aerodynamic coefficients and the effects are shown in Figures 20 and 21. Negative pitching moment coefficient increases

(

(734/rev) [7.21/rev)~ (7.0Sirev) j (6,$9/rev)

?I

(6.5Wrev)

r

(5.2~/rev) (5.94/r&v) 6-'i (5.64/rev) IS.s&1ev) I

'

31d lus.e!ag~ i<lteral renO:ng(UZirev)

J

3rdl.Jse~ver.x:alber.{!;ng(4s:Jirev~

t""-1J.IR IN II lateral tler6ng ~~.7~/rev)~f

[458/rev)

/Nil mastFIA!;er.cl .... (4.51/rev)

<ttl sl\ld(4.t2frev) :-30Hz

41

201-'.x 2ndfuse>agerors>on(3.691nw) ~---­ :!rd~(3.S/rmr)

L

2nd skjj (3.28/rev) 2nd!usetaseveftieal!)e~(3.19/rev) j

zm ~lag; u:e~a~ th."'lC!~ (2.99/rev) 3

T

lstlJseiag.:JIOIS~ {2.74/rev)

r

"'"'"·""''

I

2~---~"~"'

Is: lus.tSasevertcal b-!r.ding (1.Sil!rev) _ lsttuseJasel.lteralti!rdng(LwrevJ

'r

1~

i !.'!Rpt.cnrol(0.71/rev) ~

i

-M'Flpy!Mpileh(O.Satr.!v) Fig. 16 0'~~----~--~~~~~~--0 20 40 60 80 100

Percent contributions to total vibration (%) Contribution of refined 3-D airframe natural modes to 2/rev lateral pilot seat vibration at 101 knots

the magnitude of 2/rev vertical vibration level and slightly improves the correlation with test

data. However, the difference of phase with

test data increases. Negative pitching moment coefficient reduces 4/rev vertical and 2/rev lateral vibration and has a small effect on the phase of these vibrations. Pitching moment coefficient has a large influence on the phase of 4/rev lateral vibration (Figure 21(b)). Modified pitching moment coefficients change the phase by 15 degrees and negative pitching moment coefficient improves the correlation of phase with test data.

Effect of second order nonlinearities

Figures 22 and 23 represent 2/rev and 4/rev vertical vibration levels at the pilot seat respectively. First, second order structural and aerodynamic terms are neglected. Second, only

(757/II!VI (7 ~3/tev) (6.~/tev) ₇

t

(£.83/rev) 1 (6201tev) (6.03/rev) J.. 15.91/lev) 6

r

(5.30/rev) ' I lrdh.lselagev<r11ealtlo!r.or,g(4.S!ll~) Si (4.71/teV) ~~ MIRmasiF!Abi!I'C.ng(J.$9/r~N) • (4.47/lev) j Wfl mastLaterol bo!ndog(4.4ZII!Vr*r--4~s>ld(tW~) ! 4J

3rd ilJs.ejage la~e~ bendog (3.7&1rev!*

L

3rdski:l(3.671rev) F

-Znd tuse~e torsion (3.S5/rev)

2n<l.wd(3.321~ 2odivselagHtH'ic:alberQr,;(3.1~tel') I Zndfu~!ater.Jberd:lg(3.0tlre'i)J+; -1stfusela~tofSICII(2.7Sirt"l)

r

""W)2.Uwl 1 21~---~"~~ ! lslfusela<}eveltealbel'ld:lg(U\iltev)l I st fuselaca taterol W;lng (Uirev)

.

J

'I WRpybnrtll(0.61/reY)

,!L_

i.I/RP)1onpi!Cl(0.5&tev) 0'~~~--~~--~----~--~ Fig. 17 0 20 40 60 so

Percent contributions to total vibralion (%) Contribution of old 3-D airframe natural modes to 2/rev lateral pilot seat vibration at 101 knots

100

second order aerodynamic terms are included. Third, only second order structural terms are included. And fourth, all second order nonlinear terms are included. Second order nonlinear terms have more influence on the prediction of vibration at high speed and high frequency. Second order nonlinear terms increase the magnitude of 2/rev vertical vibration by 5% and change the phase by 4 degrees at 142 knots. Nonlinear terms have a significant effect on both magnitude and phase of 4/rev vertical vibration. Second order nonlinear terms decrease the magnitude of 4/rev vertical vibration by about 60% and change the phase by almost 70 degrees. Figures 24 and 25 represent 2/rev and 4/rev lateral vibration levels at the pilot seat respectively. Especially nonlinear terms have an important influence on both magnitude and

phase of 4/rev lateral vibration. Second order

(7.971rev) Bl

'

(734/rl'l') [7.~1/nw) (7.081rl'l') J (6.99/r~) 7 I {6.601rllV) r' {6.24/r!iV) {5Mrevj6-1 -(5.Wrev) F"'---~-"-' {5.55/rev) ::rc

fuse~gelateral tlfflj~(4.92il'l!vl_

J

3:d~~ageverticalbef'<fii19(4.S3/I'l!vP

i:.,.---.'.lr'Rmasttateralbencir.g(4.74/r11V)

c--(4.5SIIl!v) .., MIRmastF/Allendf9(4.51hl'l') ! ~::h skJd (4.121rev) j 4i ZO!-'..: 2nc!lu5e!ag!!t01"1>0t1(3.69/1!1"1) i -3rdsl6d(3.611e'l) 2nd skid (3.2SirlN) 1st fuSolbgutO/WI\(2.74/rev) 2nd fuselage vem::alllero:f1119 (3.19/rev)f 2rd fuselage Lllr!ral bendirq RS9/rev) 3

'""""~"''

,I

""'

~---IS: fu~ago veoteal bendirlg {!.SO/rev)

ls\ fu5elage la:eralbef'l:flllQ (1 Wn;v)

i

M1Rpyonro~(0,711m't) ~

1--MIR P')1on p<te.'l (0.581rev) ~

0~~~--~~~~_w__w

Fig. 18

0 20 40 60 80

Percent contributions to total vibration{%)

Contribution of refined 3-D airframe natural modes to 4/rev lateral pilot seat vibration at 101 knots

100

lateral vibration by 65% and change the phase 35 degrees at 142 knots.

Effect of third order kinetic energy terms Kinetic energy of a blade retaining terms up to

third order(O(c3_{)) is derived and then higher order}

terms are selectively included in the equations of motion to examine their effect separately. Table 2 shows third order kinetic energy terms investigated. When each term is added in the blade equation, its effect on the fuselage is also included. Figure 26 shows the effect of third order terms on the magnitude of vibration at the pilot

seat. Since higher order terms have negligible

influence on the 2/rev vibration level, only 4/rev

vibration results at 142 knots(!' = 0.32) are shown.

Magnitude change of acceleration in the vertical

"l

i767irev) (7.43/rev) (699/rov) 7

-t

(QB3Jmv) i

i

! (5.20/te-<) 1503/rev) ~ {591/revj 6

i

'

30 "'"" '""" 0000:,:·9::,5

j

(4.7\ll"l'l')~i MIRmastFIAbeocling(~.59i!"l'l') !"'"' {4.47/rev)

r

WRrna$t\ater.~lbendr>g(4A2/revr->₁ r---4lhsi0<:1(4.:Wrev)

I

3rdfu~oseWer.!IOOnc:los(3.7&rev~,..,=---"-''

3rdsr.od(3.67/rev)

1

Wllirola()l!toflion(l5Sinr<) 1 2nd ~ {3.32-'rov) 2!ldfuSE~go:JverDc:albotidr>g(3.191rov) i 2ndiu~Ll:.)ralbD~(l01!1W)3--I lstlu~tago);)rs;on(2.75i1!1"1) lstsW(2.7/rr~) i

i

2'~,---"--H~

IS! lurolage vertieal be~g (l.~&mv)l

•Sl ~sef<lge lateral ber.ong {I :!J~~

WRpyion!Oil(061/it"i)

~

IJIR pylon p<tt.'l (0 55/ro~) I I

o~~_L~~~--~--~---Fig. 19

o 20 40 50 80

Percent contributions to total vibration(%)

Contribution of old 3-D airframe natural modes to 4/rev lateral pilot seat vibration at 101 knots

100

direction is calculated as follows

llzF,c ••• ,

1 - ZF(s.,.,.,, 1

ll

X

100 (9)

ZF(8o:u.din~)

Baseline vibration level is calculated using terms

up to second order. Generally, third order

kinetic term has a very small effect on the

prediction of vibration(less than 1%). Linear

lag-torsion coupling term(Case 3) changes 4/rev vertical vibration by 4.3% and linear flap-torsion coupling terms(Case 14 and 15) change 4/rev lateral vibration by 7.6% and 5.9% respectively. Figure 27 represents the effect of third order terms on the phase of 4/rev vibration at the pilot seat.

Third order linear flap-torsion coupling term( Case

14) changes phase angle of 4/rev vertical vibration by 7 degrees and third order linear lag-torsion coupling term(Case 3) changes phase angle of 4/rev vertical vibration by 5.3 degrees. The other

(

(

1 10"1 _-,---~ c,..,c = 0.01 c""'" =-0.01 Baseline (c = 0) 1 10"1 : -- 010° -9 Flight test m~c - 010° -9

~-J

(!) \ '', ~ -510"2_~ \ \\ , c=c = 0.01 j ' \ "' c • -0.01 _, 10"1 _J ~ _{-1 10''} s";~eline _{(c = 0)} ' ' (!) c75 -5 10'2

I

~---Ls w·'

-'-~---~--~---~---~---'

-1.s w·'

+---~---~

-1.5 w·' -1 1o·' -s1o·2 o 1o0 _{s ,·a-}2 _{1 10·' -1.510"1 -1 10·' -5 10·2 o 10° 5 10"2}

Cosine (g) Cosine (g)

(a) 2/rev vertical vibration (a) 2/rev lateral vibration

1 10'1 , - - - , 1 10' 1 ~, -5

10'~!

5 10'1 :§ 0 10'1 _ _ _ _ - - : " " ' :§ 0 10'i g? i ~ i U5 -510"2 ~ U5 -5 10"2 ~ -1 10''

J

-1 10''

j

.I

'

-1.5 10''

+ . - - - '

·1.510''

-:-1----,--~--~-..,---_.;

-1.s to·' -1 w·' -s 10·2 _{o 10° s w·}2 _{, 10·' -1.s10·' ., to·' -s w·< o 10° s 10·2} Cosine (g) Cosine (g)

(b) 4/rev vertical vibration

Fig. 20 Effect of zero angle pitching moment coefficient (cmac) on vertical vibration at pilot seat at 101 knots

terms have a negligible effect on the phase of 4/rev vertical vibration. For the 4/rev lateral vibration, all higher order terms have a small effect.

Figure 28 shows the consolidated effect of third order kinetic energy terms on the 4/rev vibration. Third order kinetic energy terms decrease the magnitude of 4/rev vertical vibration by about 7% and change the phase by 10 degrees. Their effects on the lateral vibration are smaller than those on the vertical vibration because the effects of Case 14 and Case 15 are canceled out.

Conclusions

From the validation and parametric studies, the following conclusions are obtained.

L Modeling

important

of difficult components is

for the accurate prediction

(b) 4/rev lateral vibration

Fig. 21 Effect of zero angle pitching moment

coefficient (cmaJ on lateral vibration at

pilot seat at 101 knots

of airframe natural frequencies, especially

for high frequency modes.

2. The correlation between calculated 2/rev vertical vibration at the pilot seat and measured data shows good agreement in both magnitude and phase, except at high speed where the phase discrepancy is as large as 20 degrees.

3. Estimated 4/rev vertical vibration at the pilot seat shows good correlation \vith test data only in magnitude. At 142 knots, there is a phase deviation of 115 degrees.

4. The correlation of 2/rev lateral vibration

at the pilot seat is generally fair(less than 0.02g difference), while calculated 4/rev lateral vibration is overpredicted at all speeds(maximum 0.03g).

Linear sturct. and aero. 2nd order nonlinear aero. 2nd order nonlinear struct. 2nd order nonlinear struct. and aero. Flight test (a) 67 knots 1 10'1 - , - - - , 5

w'~

0 10° j

I

Ol-510'2_-l

-;

I

.f: ·1 10'1 _j

w

I -1.5 10''

i

-2w•i

I ·2.510'1 +!--~--~--~-~--~---,----i

-2.510'1 _{-21o·l -1.5'10.}1 _{-1 1o·}1 _-51o·2 _{o 10° 5 1'o·}2 _{1 10'}1

0 10°

: :::1

§-s1o·2

"

.f: -1 10'1 (f) -1.5 10'1 Cosine (g) (b) 101 knots

,;,·r

' '

-JI'

:

,/

·!!'

;:)..

~ -2.5 10'1 _{+---~-~-,---.--,---,----~-~--,---i} -2.5 10·1 _{-21o·, -1.5 10·}1 _{-1 1o·t -51o·}2 _{o 10° 51o·}2 _{1 10·}1

Cosine (g)

(c) 142 knots

Fig. 22 Effect of second order nonlinearity on 2/rev vertical vibration

1 10'1 - , - - - , I 5 10'2 J I I 0 10° ~ Ol-5 10'2

J

- I C> I c__110-1j U5 ! -1.5 10'1 ..j I

Linear sturct. and aero. 2nd order nonlinear aero. 2nd order nonlinear struct. 2nd order nonlinear struct. and aero. Flight test

-2.5 10'1 -+'-,---~---,----~-~---,--~---,---~ -2.510'1 _-210'1 _{-1.510·' -1 1o·, -51o·}2 _{o 1oo 51o·}2

1 1o·1 Cosine (g) (a) 67 knots 1 10'1 , -0 10°

i

Ol-5 10·2

J

~ -110'~

-1.510'1

i

-2W'~

-2.510''

+1---,---~--~---~-...;

-2.s10·' -210·1 _-1.s'10·1 _-1

1o·

1 _-51o·2 _{o 10° s}

_1o·

2

Cosine (g) (b) 101 knots 1 10'1 , -5 10'1 0 1001 i 'Oi -5 10'2 I

~

-110'

~

-1.510'1 ~ -2W'1 -2.5 10'1 +!--~---~---i -2.5 10·1 _{-2 10'}1 _{-1.5 10'}1 _{-1 10'}1 _{-5 10·}2

o 1o

0 ₅

,·o-

2 _{1 10·}1 Fig. 23 Cosine (g) (c) 142 knots

Effect of second order nonlinearity on 4/rev vertical vibration

(

(

- 010° -9 Q)

J5

-5 10'2 _, 10'' J I

Linear sturct. and aero. 2nd order nonlinear aero. 2nd order nonlinear struct. 2nd order nonlinear struct. and aero.

Flight test -1.5 1 o·' -'----~--~c-::--~--,---,---1.5 10·' -1 10·' -s1a·2 _{o 10° s 10·2} 1 10'' Cosine (g) (a) 67 knots 1 10'1 _~---, Q) c s to·<

en

-5 10'2 -1.s w·l -1-: ---~---~---~---~---o -1.5 w·' -1 10·' .s1o·2 o

1ao

s ,·a-2

Cosine (g) (b) 101 knots 10'1 . - - - , 5 10"2

J

- 0 10°J

₉ ! Gl I c ! i:ij -5 10'2 ~ I -1 10"1

J

-1.5 10'1 +---~-:--~--,---,---i -1.5 w·' -1 w·' -5 10'2 _{0 1o}0 _{5 10'2 to·'} Cosine (g) (c) 142 knots

Fig. 24 Effect of second order nonlinearity on 2/rev lateral vibration

Linear sturct. and aero. 2nd order nonlinear aero. 2nd order nonlinear struct. 2nd order nonlinear struct. and aero. Flight test -1.5 10'1 -'---~---1.510'1 -1 10'' -5 10'2 0 10° 5 10'2 1 10'' Cosine (g) (a) 67 knots 1 10'1 . . , . --1.s to·l L---~---­ ·1.5 10'' -s w·< o 10° Cosine (g) (b) 101 knots w·l ---;---~ : ,1( /~ Oi 0100~

-

. Ql I c I U5 -5 10"2 ~

I

-1 10"1 _J

'

/ / / ·1.5 10"1

+i--:--~--,----..,----,.---:---·1.5 10"1 _{-1 10·}1 _{-s 10"}2 o _10° s _10·2 ₁ ,a·· Cosine (g) (c) 142 knots

Fig. 25 Effect of second order nonlinearity on 4/rev lateral vibration

10 ' ii s.J '

I

61 I

·~

I

i 2j

I

II

ol

.. II ..

• II

Caset Case3 Cases Case7 Cases Casett Caset3 Casets Case2 Case4 Cases Cases Caseto Case12 Case14

(a) 4/rev vertical vibration

l

10 '

!

"

_8~

0 ~

"'

I

o; u

61

u ~ 0

"'

•1

0> c ~ .c u I

"'

20 -o 3

I

·c 0> ~ ₀

:2 _{Caset Case3 CaseS Case? Case9 Case11 Ca.set3 easelS}

Case2 Case4 Case6 CaseS CasetO Caset2 Case14

Fig. 26

(b) 4/rev lateral vibration

Effect of third order kinetic energy terms on the magnitude of 4/rev vibration

5. The contribution of airframe modes to vibration between 3-D fuselage and refined 3-D fuselage models shows a significant difference on the prediction of 4/rev lateral

vibration. The modeling of difficult

components appears to produce the coupling

between modes. Accurate prediction of

airframe natural frequencies up to about 38Hz(7frev) appears essential to predict airframe vibration accurately.

6. Second order nonlinear terms have important effect on the prediction of vibration especially at high speed and high frequency. Second order nonlinear terms decrease the magnitude of 4/rev vertical vibration by about 60% and the magnitude of 4/rev lateral vibration by 65%.

"'

1 o

I

"'

2.

ai

"

.2

'"

:;;

6~

o; u u

"'

i 0

•i

"'

"'

"

I ~ 2j .c u _I

"'

'

00 _' ~ 0

!

""

0..

Caset Case3 CaseS Case7 Case9 Casett Caset3 easelS Case2 Case4 Case6 Case8 Case10 Case12 Case14

(a) 4/rev vertical vibration

a, 10

"'

2. c

"I

.2

'"

:;;

61

o; u u

"'

0 I 4J

"'

21

"'

"

"'

.c u

"'

00

"'

.c 0

!

0..

,

..

,-,

easel Case3 Cases Case7 Case9 Casett Caset3 Case15

Case2 Case4 Case6 CaseS Case10 Case12 Case14

(b} 4/rev lateral vibration

Fig. 27 Effect of third order kinetic energy terms on the phase of 4/rev vibration

7. Third order kinetic energy terms generally have a small influence on the prediction of vibration. Third order kinetic energy terms decrease the magnitude of 4/rev vertical vibration by about 7% and change the phase by 10 degrees.

Acknowledgments

This work was supported by the National Rotorcraft Technology Center under Grant No. NCC 2944; Technical monitor Dr. Yung Yu.

\

(

1 10"1 - , - - - . 1 1 0 " ' , - - - , Base line

/~-....,_~

3rd order kinetic energy

Flight test -5 10"2 5 10"2 03 ~ 010° c: (J) I -5 10"2

J

Base line Flight test

3rd order kinetic energy

-110"1 _{+----~----~---~---1}

_, 1o·, -s1o·2 o _10° s 1o·2 -1 10"

1

+---r----..----~---<

-1 10"1 _{-5 10"2 0}

1oo

5 10"2 1 10"1

Cosine (g) Cosine (g)

(a) 4/rev vertical vibration {b) 4/rev lateral vibration

Fig. 28 Effect of third order kinetic energy terms on 4/rev vibration

Table 2 third order kinetic energy terms investigated

Case1 axial-flap, linear Tu - -w sin (3p cos (3p

T w = -u sin (3p cos (3p

Case2 flap lag, nonlinear Tw'- e9(V v)v' sin Bo

Tv= e₉(v1

_W

1 _{sinBo-v'w' sin8o)}

Tv'= eg(v- v)w' sinBo

Case3 lag-torsion, linear Tv' = 2(k;,, - k;,,)¢cos (3p sin Bo cos Bo

T;, = -2(k;,, - k;,, )v' cos (3p sin 80 cos 80

Case4 lag-torsion, nonlinear Tv= -2e₉v'¢sinBo cos(3p

T;, = 2egvv' sin 80 cos (3p

Case5 lag-torsion, linear Tv= 2e₉¢cosBosin(3p

T;, = -2e₉vcos80sin(3p

Case6 lag, nonlinear Tv = -~e.v''(cos Bo

+

86 cos 8o

+

8o sin 8o)

Case7 lag, linear Tv= -e₉(2v'v'B0sin8o- v'' cos8o- v'v' cos80 )

CaseS flap, nonlinear Tw = -~e₉w''(8fi sin8o-8o cos8o)

Case9 lag-torsion, nonlinear Tv'= ~e

9

x¢2 cosBacos2_(3p

- 2

T;, = e₉x¢v'cos80cos (3p

Case 10 flap-torsion, nonlinear T _{w' - -2egX} 1 q,2 · _{Sin oCOS} 8 2{3 _p

T ' ' . 8 2_{3

J;

=

e₉x<pw sm acos P

Case 11 flap-lag, linear Tv' - e₉w sin {3p cos /3p cos Bo

T w = e. v' sin {3p cos {3p cos 8o

Case 12 lag, nonlinear Tv' - e₉(v v)v' cos8o

Case 13 flap, nonlinear Tw = e₉(u/ sin80

+

w'w' sin8o

+

2w'ulBo cos8o)

Case 14 flap-torsion, linear Tw' = 2(k;, cos2_8o

₊

_k;,,sin2

80 ) cos{3p¢

' 2 •

T;, = -2(k;,, cos2₈

0

+

k;,,sin 80 ) cos{3pw'

Case 15 flap-torsion, linear Tw = 2e₉¢sin{3psin8o

References

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